Oxford Studies in Metaphysics Volume 11 9780198828198, 9780198828204, 0198828195

Table of contents :
Cover
OXFORD STUDIES IN METAPHYSICS: Volume 11
Copyright
PREFACE
CONTENTS
THE SANDERS PRIZE IN METAPHYSICS
PART I: RELATIONALISM AND SUBSTANTIVALISM
1: A New Approach to the Relational–Substantival Debate
1. INTRODUCTION
2. SPATIOTEMPORAL STRUCTURE AND THE MATCHING PRINCIPLE
3. A DISAGREEMENT ABOUT GROUND
3.1. Relationalism in terms of ground
3.2. Substantivalism in terms of ground
3.3. Further clarifications
3.4. Something old, something new
4. AN ARGUMENT FOR SUBSTANTIVALISM
5. A CHALLENGE FOR RELATIONALISM
6. CONCLUSION
REFERENCES
2: Relative Locations
1. LEIBNIZ’S SHIFT ARGUMENT
2. RELATIONISM
2.1. Mathematical relationism
2.2. The prospects for alternative versions of relationism
3. MULTIPLE LOCATION
3.1. Locations in ordinary language
3.2. The combined theory of part and location
3.3. The theory of part and location
3.4. Combining the theory with shift invariance
4. CONCLUSION
5. APPENDIX A: FIELDS
5.1. Demathematizing fields in orthodox Newtonian physics
5.2. Fields and the theory of multiple location
6. APPENDIX B: THE FIRST-ORDER THEORY OF PART AND LOCATION
ACKNOWLEDGMENTS
REFERENCES
PART II: TIME AND CHANGE
3: A Passage Theory of Time
1. A SEARCH FOR PASSAGE
2. FRAGMENTATION ACROSS TIME
3. REGIMENTING PASSAGE
4. CONCLUDING REMARKS
REFERENCES
4: Fragmenting the Wave Function
1. STRATEGIES FOR B-THEORETIC ENDURANTISTS: RELATIVIZING, OUTSOURCING, AND DEFUSING
1.1. Relativizing accounts
1.2. Outsourcing accounts
1.3. Defusing accounts
1.4. Fragmentalism
2. SPOOKY ISOLATION AND SPOOKY COINCIDENCE
2.1. Spooky isolation
2.2. Spooky coincidence
3. SPOOKY ACTION AT A DISTANCE
4. CONCLUSION
BIBLIOGRAPHY
PART III: RECOMBINATION, RELATIONS, AND SUPERVENIENCE
5: Possible Patterns
1. TWO COMBINATORIAL IDEAS
2. PATTERNS OF PROPERTIES
3. PATTERNS OF RELATIONS
4. PLURALITIES OF WORLDS
5. UNRESTRICTED PATTERNS
APPENDIX A: SET STRUCTURES
APPENDIX B: PLURAL STRUCTURES
REFERENCES
6: Plural Slot Theory
1. INTRODUCTION
2. SLOT THEORY AND ITS PROBLEMS
3. STATING SLOT THEORY MORE PRECISELY
4. AN INITIAL FORMULATION OF POCKET THEORY
5. A PROBLEM AND A REVISED FORMULATION
6. CONCLUDING REMARKS
ACKNOWLEDGMENTS
REFERENCES
7: Local Qualities
1. HUMEAN SUPERVENIENCE
2. ANTI-HUMEAN INSPIRATION: LOCALITY WITHOUT INSULATION
3. NON-PIECEMEAL HUMEANISM
4. CONSEQUENCES: NON-LOCAL BASING
ACKNOWLEDGMENTS
REFERENCES
PART IV: VAGUENESS
8: Vague Naturalness as Ersatz Metaphysical Vagueness
1. THE PLAUSIBILITY OF (V1)–(V3)
1.1. (V1)
1.2. (V2)
1.3. (V3)
2. THE TENSION BETWEEN (V1)–(V3)
2.1. Preliminaries
2.2. The precise naturalness strategy
2.3. The reference magnetic strategy
3. VAGUE NATURALNESS AS ERSATZ METAPHYSICAL VAGUENESS
4. FUNDAMENTAL INDETERMINACY
5. THE REFERENCE MAGNETIC STRATEGY, REDUX
6. NO RADICAL INDETERMINACY
7. NATURALNESS IS NATURAL
8. CONCLUSION
9. APPENDIX: SOPHISTICATED REFERENCE MAGNETISM
REFERENCES
9: Against ‘Against “Against Vague Existence”’
1. INTRODUCTION: VAGUE EXISTENCE AND SEMANTIC INDECISION
2. SUPER-VAGUENESS AND MULTIPLE PRECISIFICATIONS: A CHALLENGE
3. IS SUPER-VAGUENESS VAGUENESS?
4. A SUPER-META-LANGUAGE FOR SUPER-VAGUE EXISTENCE?
5. SUPER-VAGUE EXISTENCE AND WORLDLY INDETERMINACY
6. CONCLUSION
REFERENCES
Author Index

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OXFORD STUDIES IN METAPHYSICS

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OXFORD STUDIES IN METAPHYSICS Editorial Advisory Board Elizabeth Barnes (University of Virginia) Ross Cameron (University of Virginia) David Chalmers (New York University and Australasian National University) Andrew Cortens (Boise State University) Tamar Szabó Gendler (Yale University) Sally Haslanger (MIT) John Hawthorne (University of Southern California) Mark Heller (Syracuse University) Hud Hudson (Western Washington University) Kathrin Koslicki (University of Alberta) Kris McDaniel (Syracuse University) Trenton Merricks (University of Virginia) Kevin Mulligan (Université de Genève) Laurie Paul ( Yale University) Jonathan Schaffer (Rutgers University) Theodore Sider (Rutgers University) Jason Turner (University of Arizona) Timothy Williamson (Oxford University)

Managing Editor Christopher Hauser (Rutgers University)

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OXFORD STUDIES IN METAPHYSICS Volume 11

Edited by Karen Bennett and Dean W. Zimmerman

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Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © the several contributors 2018 The moral rights of the authors have been asserted First Edition published in 2018 Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2018940688 ISBN 978–0–19–882819–8 (hbk.) 978–0–19–882820–4 (pbk.) Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work.

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PREFACE Oxford Studies in Metaphysics is dedicated to the timely publication of new work in metaphysics, broadly construed. The subject is taken to include not only perennially central topics (e.g. modality, ontology, and mereology) but also metaphysical questions that emerge within other subfields (e.g. philosophy of mind, philosophy of science, and philosophy of religion). Each volume also contains an essay by the winner of the Sanders Prize in Metaphysics, an biennial award described within. K. B. & D. W. Z. New Brunswick, NJ

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CONTENTS The Sanders Prize in Metaphysics

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PART I. RELATIONALISM AND SUBSTANTIVALISM 1 A New Approach to the Relational–Substantival Debate

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Jill North 2 Relative Locations

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Andrew Bacon PART II. TIME AND CHANGE 3 A Passage Theory of Time

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Martin A. Lipman 4 Fragmenting the Wave Function

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Jonathan Simon PART III. RECOMBINATION, RELATIONS, AND SUPERVENIENCE 5 Possible Patterns

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Jeffrey Sanford Russell and John Hawthorne 6 Plural Slot Theory

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T. Scott Dixon 7 Local Qualities

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Elizabeth Miller PART IV. VAGUENESS 8 Vague Naturalness as Ersatz Metaphysical Vagueness

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Rohan Sud 9 Against ‘Against “Against Vague Existence” ’

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Roberto Loss Author Index

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THE SANDERS PRIZE IN METAPHYSICS Sponsored by the Marc Sanders Foundation* and administered by the editorial board of Oxford Studies in Metaphysics, this essay competition is open to scholars who are within fifteen years of receiving a PhD and students who are currently enrolled in a graduate program. (Independent scholars should enquire of the editors to determine eligibility.) The award is $10,000, and the competition is now biennial. Winning essays will appear in Oxford Studies in Metaphysics, so submissions must not be under review elsewhere. Essays should generally be no longer than 10,000 words; longer essays may be considered, but authors must seek prior approval by providing the editor with an abstract and word count by 1 November. To be eligible for the next prize, submissions must be electronically submitted by 31 January 2020. Refereeing will be blind; authors should omit remarks and references that might disclose their identities. Receipt of submissions will be acknowledged by e-mail. The winner is determined by a committee of members of the editorial board of Oxford Studies in Metaphysics, and will be announced in early March. At the author’s request, the board will simultaneously consider entries in the prize competition as submissions for Oxford Studies in Metaphysics, independently of the prize. Previous winners of the Sanders Prize are:

Thomas Hofweber, “Inexpressible Properties and Propositions”, Vol. 2; Matthew McGrath, “Four-Dimensionalism and the Puzzles of Coincidence”, Vol. 3; Cody Gilmore, “Time Travel, Coinciding Objects, and Persistence”, Vol. 3; Stephan Leuenberger, “Ceteris Absentibus Physicalism”, Vol. 4; Jeffrey Sanford Russell, “The Structure of Gunk: Adventures in the Ontology of Space”, Vol. 4; Bradford Skow, “Extrinsic Temporal Metrics”, Vol. 5; Jason Turner, “Ontological Nihilism”, Vol. 6; Rachael Briggs and Graeme A. Forbes, “The Real Truth About the Unreal Future”, Vol. 7; Shamik Dasgupta, “Absolutism vs Comparativism about Quantities”, Vol. 8; Louis deRosset, “Analyticity and Ontology”, Vol 9;

* The Marc Sanders Foundation is a non-profit organization dedicated to the revival of systematic philosophy and traditional metaphysics. Information about the Foundation’s other initiatives may be found at .

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The Sanders Prize in Metaphysics Nicholas K. Jones, “Multiple Constitution”, Vol. 9; Nick Kroll, “Teleological Dispositions”, Vol. 10; Jon Litland, “Grounding Grounding”, Vol. 10; Andrew Bacon, “Relative Locations”, Vol. 11; T. Scott Dixon, “Plural Slot Theory”, Vol. 11.

Enquiries should be addressed to Dean Zimmerman at: [email protected].

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PART I RELATIONALISM AND SUBSTANTIVALISM

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1 A New Approach to the Relational–Substantival Debate Jill North 1. INTRODUCTION The traditional relational–substantival debate is about whether space—in modern terms, spacetime—exists. The substantivalist says that it does. The relationalist says that it doesn’t. According to the relationalist, all that exists, in the physical world, are material bodies related to one another spatiotemporally; there is no further thing in which these bodies are located. This is a debate with a long history. Yet there is still surprisingly little agreement not only on what is the right answer, but also on how to understand the very question at issue and the potential answers to it—and even on whether there is any genuine dispute here. For example, we can try to formulate the debate in a way that harkens back to the traditional Leibniz–Newton dispute, as the question of whether space exists as a substantial entity. But then what it means to call something a substantial entity is disputed, so that it may start to seem like the two sides are simply talking past each other. Some people have concluded that the debate is not substantive. Perhaps it is merely a verbal dispute about which things to call ‘space’ versus ‘matter’, with no objectively correct answer to be had (Rynasiewicz, 1996). Others have thought that the dispute has stagnated or become divorced from physics.¹ A review of the historical dispute and its central examples (Newton’s bucket and globes, Leibniz’s shifts, Kant’s glove, as well as the more recent ¹ Claims that the traditional debate is non-substantive, unclear, or removed from physics, either in certain contexts or in general, can be found in Stein (1970; 1977b); Malament (1976); Horwich (1978); Friedman (1983, 221–3); Earman (1989); DiSalle (1994); Leeds (1995); Rynasiewicz (1996; 2000); Dorato (2000; 2008); Belot and Earman (2001, sec. 10.7); Pooley (2013, sec. 6.1, 7); Curiel (2016); Slowik (2016). Earman (1989) advocates the need for a tertium quid.

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hole argument, all of which live on in today’s discussions) may reasonably suggest a stagnated debate. Each of these aims to show that the opposing side recognizes either too few or too many spatiotemporal facts for the physics; but there are various maneuvers, well hashed-out in the literature, allowing each side to escape the charge. Relatedly, given the variety of different understandings of the dispute, you might think that there is no overarching, well-posed question in the vicinity (Curiel, 2016). David Malament is not alone in wondering whether there is any clear-cut dispute between the two sides: “Both positions as they are usually characterized . . . are terribly obscure. After they are qualified so as to seem intelligible and not too implausible, it is hard to retain a firm grasp on what divides them” (1976, 317). Certainly all of this hints at “the fragile health of the substantival–relational debate” (Belot, 1999, 38). These are reasonable concerns when leveled at traditional conceptions of the dispute. Nonetheless, I believe that there is a debate that is substantive, not stagnant, and relevant to physics. The debate that I will present is not exactly the traditional one. But it is close enough in spirit that I think it is the best way of understanding that dispute, updated to take into account more recent developments in physics and philosophy. And once we frame the debate in this way, we unearth a novel argument for substantivalism, given current physics. At the same time, that conclusion could be overridden by future physics. A seemingly subtle shift yields surprising progress on a longstanding issue that many people feel has stagnated. In Section 2, I discuss an idea that will play a central role: structure in general, and spatiotemporal structure in particular. I will argue that, regardless of whether you are a relationalist or substantivalist, you should think that there are objective, determinate spatiotemporal facts about a world: you should be a realist about spatiotemporal structure in my sense. This follows from a general principle we rely on in physics. (The traditional debate was about the existence of space and time separately. I discuss the question of spacetime, or spatiotemporal structure, updating things to the terms of modern physics.) In Section 3, I will argue that, regardless of whether you are a relationalist or a substantivalist, you can be a realist about spatiotemporal structure. I do this by framing the debate in terms of fundamentality and ground, notions that have gotten lots of press recently in metaphysics. I show that this way of putting things captures traditional conceptions of the dispute, while allowing us to formulate the most plausible—if not entirely traditional—versions of the two main positions on it. (Although I put things in terms of ground, what’s most important is that we make use of some notion of relative fundamentality.) Finally (Sections 4 and 5), I put all the pieces together to show that there is a powerful argument for substantivalism, or at least a powerful challenge to relationalism, given much of current physics.

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At the end, I briefly discuss how the conclusion in favor of substantivalism may change with future developments in physics. Yet however the physics turns out, the question of relationalism versus substantivalism should be settled by means of the new type of argument offered here. Hence, if I am right, the substantivity of the debate is secured regardless of future developments in physics, while the conclusion in favor of one view or the other will ultimately be decided by the physics.

2 . S PA T I O T E M P O R A L S T R U C T U R E A N D THE MATCHING PRINCIPLE I’ll begin by arguing that both the relationalist and the substantivalist should posit enough, and not too many, spatiotemporal facts for the physics. As I will put it, they both should countenance the spatiotemporal structure that is needed for the physics. (In Section 3, I turn to whether they both can do this.) I argue that there is a certain methodological principle we are used to relying on in physics, even if it is not usually mentioned. This principle guides our inferences from the mathematical formulation of a theory to the nature of the world according to the theory. I show by example that we do generally, and successfully, rely on this principle. The conclusion about spatiotemporal structure will follow from it. Consider classical Newtonian mechanics. What does this theory tell us about the world? Newton thought it tells us that absolute space, a space that persists through time, exists. He argued that phenomena involving inertial (unaccelerated) and non-inertial (accelerated, in particular rotated) motion reveal this. (Think of his bucket experiment and the spinning globes example.) Although we nowadays agree that the phenomena indicate a real distinction between inertial and non-inertial motion, we think that Newton was wrong about what’s required to account for this distinction. In today’s terms, Newton was arguing for substantivalism about what is often called Aristotelian, or Newtonian, spacetime.² This spacetime has the structure to support Newton’s idea of absolute space, for it has structure that identifies spatial locations over time. But we now know (as Newton did not) that Galilean, or neo-Newtonian, spacetime also supports the distinction between accelerated and unaccelerated motion, without absolute space. Spelling this out, Aristotelian spacetime has all the structure of Galilean spacetime, but it also has absolute space, or an absolute standard of rest or ² Not to be confused with the spacetime that Earman (1989, sec. 2.6) calls ‘Aristotelian.’ I follow Geroch’s (1978) use of the ‘Aristotelian’ and ‘Galilean’ labels.

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preferred rest frame. To remind you of what this means, think of an observer on a platform and another observer on a train moving with constant velocity relative to the platform. Each observer feels that he or she is at rest and that the other is moving. Galilean spacetime says that neither one is “correct” or at rest in any absolute, observer-independent sense. Each is simply in motion relative to the other, and at rest in her own frame of reference. (Think of a reference frame as a coordinate system attached to an observer, representing her own point of view.) According to an Aristotelian spatiotemporal structure, there is an observer- or frame-independent fact, from among all the observers in constant relative motion, about which one is at rest in an absolute, frame-independent sense—namely, the one at rest in absolute space. For there is a frame-independent fact about whether a given spatial location is the same location over time, so that an object located there is at absolute rest. In other words, there is a preferred rest frame: the one that’s at rest with respect to absolute space. Intuitively, an Aristotelian spatiotemporal structure has more structure than a Galilean one. It has all the same structure, plus an additional absolute-space, or absolute-velocity, structure. It recognizes all the same spatiotemporal facts, but it also says that there are facts about how fast an object is moving with respect to absolute space. It turns out that these additional facts are not needed for, or recognized by, the physics here. Newton’s laws are the same in any inertial frame—they are invariant under changes in inertial frame—which means that they can be formulated without mentioning or presupposing a preferred frame. Since a preferred frame isn’t needed in the mathematical formulation of the laws, we infer that it doesn’t correspond to anything physical in the world. An absolute standard of rest isn’t part of the theory’s, or world’s, spatiotemporal structure. The physics does not recognize objective, frame-independent facts about what velocity an object has. Conclusion: Aristotelian spacetime has excess, superfluous structure, as far as Newton’s laws are concerned. It recognizes more spatiotemporal facts than the laws do. These laws do recognize facts about objects’ accelerations (as Newton argued). Think of Newton’s first law: an object travels with uniform velocity unless acted on by a net external force. This law assumes that there is a distinction between accelerated and unaccelerated motion, since it tells things to behave differently depending on whether they are accelerating or not. In terms of spatiotemporal geometry, the law assumes a distinction between straight and curved trajectories or paths through spacetime, with the straight ones corresponding to inertial motion, the curved ones to noninertial motion. And Galilean spacetime has the structure to support this distinction. It has an affine connection, or inertial structure, which provides a standard of straightness for these trajectories. We might put it like this: this

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spatiotemporal structure supports a notion or quantity of absolute acceleration but not of absolute velocity—“absolute” not in Newton’s sense, which assumes the existence of absolute space, but in the sense of being invariant or frameindependent.³ All of this suggests that a Galilean spatiotemporal structure is the right structure for Newton’s physics. This is the structure that’s required for, or presupposed by, the dynamical laws; the structure that recognizes the spatiotemporal facts that the laws do.⁴ Newton was wrong to think that a classical world must contain absolute space and a concomitant quantity of absolute velocity: the physics doesn’t require it. (If the laws were not invariant under changes in inertial frame, then we would infer that extra structure. Such laws would implicitly refer to a preferred frame.) Notice that we reached this conclusion about the structure needed for the laws independently of the relational–substantival debate, an idea that I will return to soon.⁵ First let me say a bit about “structure.” On my understanding (and as it is often used in physics and mathematical physics),⁶ structure has to do with the invariant features or quantities, which are the same in all allowable reference frames or coordinate systems. Inertial structure, for example, is part of a classical spatiotemporal structure: there is an absolute, frameindependent notion of accelerated versus unaccelerated motion. But there is no “absolute-velocity structure.” An object’s velocity depends on the inertial frame we use to describe it. Since Newton’s laws are invariant under changes in inertial frame, we infer that the choice of frame is an arbitrary choice in description, and that any quantity depending on that choice, like velocity, is merely frame-dependent, not out there in the world apart from that choice. Similarly, we think that a choice of origin is just an arbitrary choice in description, not corresponding to genuine structure in the world. Choose a coordinate system with a different origin, and the laws always remain the same. Since the laws are invariant under changes in origin—they “say the

³ I believe that this sense evades Rynasiewicz’s (2000) arguments against the clarity of any absolute/relative distinction. ⁴ Although the inference to a Galilean structure is now relatively standard (Earman (1970); Stein (1970); Huggett (1999, 194–5); Maudlin (2012, ch. 3)), there is room for debate. Saunders (2013) and Knox (2014), in different ways, argue that Newtonian physics requires a different structure. I continue as though the above inference is correct. It is in any case agreed that absolute space is not needed, and whatever structure is required, the example illustrates our reliance on the upcoming principle. ⁵ A similar point is made by Stein (1970, 271–2), although he goes on to say that, “the question whether . . . this structure of space-time also ‘really exists’, surely seems to be supererogatory” (277). In a way I agree, but I also think that there remains a substantive dispute. ⁶ More is in North (2009).

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same thing”⁷ regardless—we infer that this choice is merely a conventional or arbitrary choice in description. There is no preferred-location structure in the world, no coordinate-independent fact about whether a given point is “really” the origin. By contrast, the laws of Aristotle’s physics are not invariant in this way. According to them, there is a preferred-location structure in the world—a location toward which certain elements naturally fall and away from which others naturally rise—and preferred coordinate systems for describing this structure, namely those with an origin at that location. We likewise think that different choices of unit of measure are conventional or arbitrary choices in description. Change from feet to meters or some other unit for measuring distances, for instance, and the physics always remains the same. Since the physics says the same thing regardless, we infer that there is no “preferred-unit-of-measure structure” in the world. As I see it, structure corresponds to the intrinsic, genuine, objective features or quantities, which don’t depend on arbitrary or conventional choices in description. By contrast, frame-, coordinate-, or unit-dependent quantities depend to some extent on our arbitrary or conventional choices in description—arbitrary, since according to the physics any choice is equally legitimate. Such quantities aren’t wholly about the world as it is in itself, but are in part about our descriptions of the world, whereas structural features are agreed upon by all the allowable descriptions, and so correspond to genuine features of the world apart from any of those descriptions. No matter which description you use, after all, you get the same result. Spatiotemporal structure in particular concerns the intrinsic, genuine, objective spatiotemporal features of a world, which don’t depend on arbitrary or conventional choices—that two objects are separated by some amount under a Euclidean metric, say, or that a particle’s trajectory is straight according to a given inertial structure. Notice that this idea of structure is neutral between substantivalism and relationalism. Both of these views can recognize that there is a distinction between spatiotemporal facts that are more objective, and those that are frame-, observer-, unit-, or coordinate-relative. We are still working up to the general principle. Here’s an idea that we have reached so far, which will motivate the principle. As we can see from the inference to a Galilean structure for Newton’s laws, any physical theory will constrain, or help dictate, a world’s spatiotemporal structure. We infer the structure from the physics in this way. This is because any theory will require or presuppose a certain spatiotemporal structure. In particular, it will require the structure needed to support the laws, in that the laws cannot ⁷ Brading and Castellani (2007) discuss different ways of spelling out this idea.

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be stated or formulated without assuming it—they wouldn’t make sense without it.⁸ Two examples illustrate this. Recall Newton’s first law, which tells objects to behave differently depending on whether they are traveling inertially, with uniform velocity, or not. This law would not make sense if there weren’t a distinction between uniform and accelerated motion: it presupposes it. So the world must be such that there is this distinction. The world’s spatiotemporal structure should distinguish between inertial and noninertial trajectories. Assuming that the laws are about the objective nature of the world, there must be objective facts about whether objects are traveling inertially or not.⁹ Consider a different example that I’ll return to later. If the laws are not time reversal invariant—if they “look different” when we flip the direction of time, swapping past and future—then this suggests a structural, physical distinction in the world between the two temporal directions. Newton’s laws are symmetric in this sense: any behavior allowed by the theory can also happen backward in time. The film of any Newtonian process (a ball thrown in the air, billiard balls colliding) run backward also depicts a process that evolves with the laws. These laws don’t distinguish past versus future: they say the same thing regardless of the direction of time. By contrast, the second law of thermodynamics says that entropy increases to the future, not the past: gases expand, ice melts, not the reverse. A reverserunning film shows something disallowed by the law. Non-time reversal invariant laws like this mention or presuppose the distinction between past and future, telling things to behave differently depending on the direction of time. Such laws would not make sense if there weren’t a past–future distinction in the world, corresponding to an asymmetric temporal structure, or objective facts about past versus future: they presuppose it. (If you are worried about this conclusion in the case of the second law, stay tuned: I return to it later in this chapter.) Finally, the principle. The above examples are familiar instances of how we draw certain conclusions about the physical world from the laws that govern it. These examples all suggest that we rely on a certain methodological principle, which says to posit in the world the structure that’s presupposed by the laws. We generally posit physical structure in the world corresponding to the mathematical structure needed to formulate the laws—such as a Galilean spatiotemporal structure for Newton’s laws, an asymmetric temporal structure for non-time reversal invariant laws, or a ⁸ Consider Earman’s statement that “laws of motion cannot be written on thin air alone but require the support of various space-time structures” (1989, 46). ⁹ Compare Maudlin (2012, 9–12); Pooley (2013, sec. 3).

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preferred-location structure for Aristotle’s laws. We infer to the world whatever the laws presuppose, whatever there must be in the world for the laws to make sense and be true of it. There should be a match in structure between the laws and the world. Theories obeying what I will call the matching principle are “well-tuned,” to borrow a phrase that John Earman (1989, ch. 3) uses for a somewhat different idea.¹⁰ (I take it this is motivated by a kind of realism. I won’t argue for realism here.) As with any guiding methodological principle, this principle won’t yield conclusive inferences, yet it is still a reasonable guide. We cannot be certain that there is no absolute space in a Newtonian world, but it is reasonable to infer that there isn’t. Or take special relativity. The matching principle lies behind the thought that there is no preferred simultaneity frame. Since the laws are invariant under changes in Lorentz frame, we infer that there is no absolute, frame-independent simultaneity relation. We can’t be certain about this, and some people argue that we have other reasons to posit this structure (for presentism or for certain theories of quantum mechanics, for example). Still, we do generally, and reasonably, rely on this principle. We take it to be successful. As the case of special relativity shows, we need an extra reason to disobey it. To put it another way: all other things being equal, we should infer a match in structure between laws and world. Those who believe in a mismatch are saying that other things are not equal, and must argue as much.¹¹ It is sometimes said that the reason to posit a Galilean rather than Aristotelian structure in a Newtonian world is that the latter would yield in-principle undetectable physical facts.¹² Since Newton’s laws are invariant under changes in inertial frame, no experiment could ever detect which is the preferred frame. Choose any frame in which to run your experiment, and the laws always predict the same results. That’s right. But I think that there is a deeper reason for the inference to a Galilean structure, which is the match between the mathematical structure of the theory and the physical structure of the world. This match is part of our evidence that we have inferred the correct structure of the world. This is a more fundamental reason for the inference than the verificationist-sounding principle to avoid undetectable physical facts.

¹⁰ Earman suggests that there should be a match between the symmetries of the laws and of the spacetime, as a condition of adequacy on theories. ¹¹ Those who argue from quantum mechanics aren’t proposing a mismatch, but that the laws of quantum mechanics trump special relativity when it comes to inferring this structure. ¹² Mentioned, with varying support, in Earman (1989, ch. 3); Ismael and van Fraassen (2003); Roberts (2008); Dasgupta (2009); Maudlin (2012, ch. 3); Pooley (2013, secs. 3–4).

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I have argued that the matching principle is a core methodological principle we use to guide our inferences from a physical theory to the nature of the world according to that theory. Now we can see that this principle tells us to posit, or countenance, or somehow be able to talk about, spatiotemporal structure. For the laws generally talk about, they mention or presuppose, a particular spatiotemporal structure. We should countenance the particular spatiotemporal structure or facts required for the laws; ipso facto, we should countenance spatiotemporal structure or facts in general. In other words, the matching principle says that we should be realists about spatiotemporal structure, since the laws presuppose such a thing, and we should generally posit in the world the structure that’s presupposed by the laws. Importantly, this conclusion is independent of the relational–substantival debate. Regardless of your position on that debate, the matching principle tells you to believe that there are objective facts about the spatiotemporal structure of a world; to recognize the spatiotemporal facts that are recognized by the laws. You should believe that a Newtonian world has a Galilean spatiotemporal structure, for example (although this claim may be understood differently by the relationalist and substantivalist, as I discuss below). Who would reject the principle? The conventionalist, for one, like Reichenbach or Poincaré, who denies that there is an objective fact about the “right” spatiotemporal structure of a world: there are no objective spatiotemporal facts. Against such a view, the matching principle suggests that spatiotemporal structure is out there in the world. It is not conventional or arbitrarily chosen, as is an inertial frame or origin or unit of measure.¹³ This structure exists; it is part of reality. There is an objective, determinate fact about what spatiotemporal structure a world has, evidenced by its laws. The matching principle is not Quine’s criterion for ontological commitment. Quine says that we are ontologically committed to what the variables of our theories must range over in order for those theories to be true. This has to do with ontology, with what entities exist. The matching principle is about what structure we should posit. It says to align physical structure in the world with the mathematical structure required to formulate the laws. This has to do with what spatiotemporal facts we should recognize, which is not simply a matter of ontology. To see that these come apart, notice first that a given spatiotemporal structure, say a Galilean one, can be understood by different

¹³ We can agree with Reichenbach and Poincaré that those things are arbitrary, since the laws indicate that different choices are equally legitimate. Spatiotemporal structure is different. We cannot arbitrarily alter the metric, for instance, and keep the laws the same, not without major compensating changes elsewhere.

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people as involving different entities: by a certain substantivalist¹⁴ as involving points of spacetime and a relationalist as involving material bodies. (As Tim Maudlin (2015) puts it, to attribute “a mathematical structure to physical items” is to say that those items “have some physical features that make them amenable to precise mathematical description in some respects”. In particular, it is not yet to say what the items must be.) Second, two people might agree on what entities exist—say, points of spacetime—but disagree on the spatiotemporal structure, for instance on whether the points are arranged in a Galilean or Aristotelian way. This will become clearer as we proceed. Question: How should we formulate the laws? It seems as though different formulations can presuppose different structures. If so, then in order to adhere to the matching principle, we will first need to know how to formulate the laws, which is a big question. Trust me for now that we can make progress in advance of answering this question. I will return to it at the end of this chapter. Some have argued for a third view, neither substantivalist nor relationalist, called ‘structural spacetime realism.’¹⁵ Since that view emphasizes realism about spacetime structure, you might think that it is what I am advocating. I don’t have space to address the alternative in detail,¹⁶ but I will note that, despite superficial similarities, it is importantly different from my overall approach. First, I claim that both the relationalist and the substantivalist should (and can: below) be realists about spatiotemporal structure, whereas spacetime structural realism aims to be distinct from either of those views. Second, I understand the idea of spatiotemporal structure differently, to encompass any objective, intrinsic spatiotemporal fact about a world. In particular, countenancing spatiotemporal structure in my sense does not mean eschewing fundamental physical objects (alternatively, intrinsic properties) altogether, nor the possibility of our knowing about such things, as the structural spacetime realist often seems to do. That said, below we will see one way in which my account mirrors certain claims of the spacetime structural realist.

3 . A D I S A GR E E M E N T A B O U T G R O U N D In order to say that the relationalist and substantivalist both should countenance spatiotemporal structure, I must be able to say that they both can do this. ¹⁴ See Section 3.3. ¹⁵ Different versions are in Dorato (2000; 2008); Slowik (2005); Bain (2006); Esfeld and Lam (2008); Ladyman and Ross (2009). ¹⁶ See Greaves (2011).

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You might wonder: How can the relationalist believe in spatiotemporal structure? Isn’t this the very sort of thing the relationalist rejects? On the other hand, if the relationalist can believe in spatiotemporal structure, you might then wonder what could be left for the two views to disagree about. I’ll now suggest that the notion of ground gives the sense in which the relationalist as well as the substantivalist can countenance spatiotemporal structure, and that this yields a real disagreement that’s relevant to physics. The basic idea will be this. Both views can countenance, or believe in the existence of, spatiotemporal structure. (Whether each one is able to recognize the particular structure needed for the laws is a question that I will be sidestepping here, for reasons to come.) The views differ on what underlies this structure. Essentially, the substantivalist says that spatiotemporal structure is fundamental to the physical world, whereas the relationalist says that it arises from the relations between and properties of material bodies. Putting this in terms of ground. A grounding relation is an explanatory relation that captures the way in which one thing depends on or holds in virtue of another, without implying that the dependent thing doesn’t exist. Ground captures a “metaphysical because” in answer to questions about why something exists or some fact holds. (I use the general idea, without entering into debates over its metaphysics. I won’t take a stand on whether ground is properly a relation between facts or objects, but deliberately use both ways of talking. It is generally thought that the grounding relation is transitive and irreflexive, and that the grounds metaphysically necessitate the grounded. None of these assumptions have gone uncontested, but I assume them here.¹⁷) Using the notion of ground, the relationalist and substantivalist can each say that spatiotemporal structure exists, that there are objective spatiotemporal facts about a world. They disagree on what the spatiotemporal structure holds in virtue of; what metaphysically explains the spatiotemporal facts. The relationalist says that a world’s spatiotemporal structure is grounded in the features and behaviors of material bodies. All the spatiotemporal facts are grounded in the facts about material bodies. The substantivalist says that spatiotemporal structure isn’t grounded in anything else more fundamental to the physical world; in particular, it is not grounded in material bodies. There are fundamental spatiotemporal facts that are not grounded in facts about material bodies. Both views can countenance spatiotemporal structure or facts; they disagree on what, if anything, grounds this structure or those facts.

¹⁷ Different accounts are in Fine (2001); Schaffer (2009). Rosen (2010) defends the idea.

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I spell out the two views more in a moment. First, a few notes on the use of ground in this context. Jonathan Schaffer (2009, 363) and Shamik Dasgupta (2011) also suggest that we can understand this debate in terms of ground, but they put things a little differently. They say that the relationalist and substantivalist both believe that spacetime exists, while differing on what grounds the existence of spacetime. I say that both (can and should) believe that spatiotemporal structure exists, while differing on what grounds the existence of that structure. I prefer this way of putting things because, we’ll see, it allows us to flesh out the competing views in different ways, all the while maintaining a genuine dispute that the physics will weigh in on. It may seem unexciting to exchange a debate about the existence of spacetime for one about the fundamentality of spatiotemporal structure. There has been much discussion in metaphysics of late about doing a similar kind of exchange with other existence debates (as in Schaffer (2009)), so that this instance may feel like old hat. There have been some related thoughts about the spacetime debate in recent philosophy of physics as well. Thus Carl Hoefer (1998) frames the question in terms of fundamentality, as that of how “to understand the basic ontology of the physical world,” although he formulates aspects of the dispute more traditionally, saying for instance that substantivalism is committed to the existence of “a substantial, quasi-absolute entity.”¹⁸ Gordon Belot (1999; 2000; 2011) says that the relationalist, like the substantivalist, can be a realist in the sense of “attribut[ing] to reality a determinate spatial structure,” while disagreeing on “the nature of the existence of space” (2011, 1).¹⁹ This is close to my own way of putting things, although his account is not spelled out in the same way (it does not use notions like ground or my conception of spatiotemporal structure, and it focuses on certain traditional examples), nor does he draw the same conclusions. The more prevalent attitude in philosophy of physics, especially among those who complain about the substantivity of the dispute, is that the debate concerns the existence question. So although my proposed way of understanding of the dispute is not without precedent, even then there are differences, and it is anyway not the prevalent viewpoint. If you disagree with that assessment, though, it will soon be clear that novel avenues of argument open up once we are completely explicit about this shift. ¹⁸ Hoefer similarly argues that this is a substantive dispute, which is likely to remain so with future physics, and that general relativity supports substantivalism. Yet he puts various things differently from how I do, drawing these conclusions for different reasons. ¹⁹ Belot also says that his formulation, while unorthodox, yields a debate that is substantive, relevant to physics, and reminiscent of the traditional dispute.

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3.1. Relationalism in terms of ground The relationalist says that certain material bodies, and various of their properties and relations, are fundamental, and a world’s spatiotemporal structure holds in virtue of them. All spatiotemporal structure or facts are grounded in (facts about) material bodies. In saying that “certain material bodies are fundamental,” this means whichever material objects turn out to be most fundamental: certain particles, say. (I assume the fundamental relations can include spatiotemporal ones,²⁰ although the relationalist might want a different kind of relation to be fundamental, causal ones being a familiar candidate. I leave this open here. The upcoming argument takes aim at all these versions of relationalism equally.²¹) So, for example, the fact that a world has a Euclidean spatial structure is grounded in, holds in virtue of, the fact that its particles are, and can be, arranged in various ways, with various distance relations between them. (I return to this “can be” phrase soon.) The world has a Euclidean structure because (in the metaphysical sense) its particles are, and can be, arranged in those ways; this is what the spatial structure consists in. Similarly, the fact that a Newtonian world has a Galilean spatiotemporal structure is grounded in the fact that its particles do, and can, behave in various ways, with various spatiotemporal relations between them. The fact that a world has a particular spatiotemporal structure is made true by the facts about material bodies. A world has the spatiotemporal structure it does because material bodies (can) behave in certain ways. Three notes on this use of ground. First, a grounding explanation is importantly different from a causal explanation. In Kit Fine’s words, ground yields “a distinctive kind of metaphysical explanation,” in which the objects or facts are connected by “some constitutive form of determination” (2012, 37). Particle behaviors don’t cause a Euclidean spatial structure. This is rather what the spatial structure consists in or depends on, in a metaphysical sense. Compare this to more familiar cases, such as the grounding of facts about the macroscopic world in facts about subatomic particles, or the grounding of mental facts in non-mental facts, or moral facts in nonmoral facts. Ground captures this metaphysical “in virtue of ” explanation.²² As I understand it, when we say that “the fact that x grounds the fact that y,”

²⁰ Contra Nerlich (1994a, ch. 1). ²¹ I also assume that the objects and relations are equally fundamental, though there may be a view with only one fundamental “ontological category” in the sense of Paul (2013). ²² Loewer (2001) discusses the relevant sense of “in virtue of.”

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this just means that “the fact that y holds in virtue of the fact that x”; i.e. that the holding of the grounded fact consists in nothing more than the holding of the grounding fact. Second, ground aims to give a “looser” connection between the facts or objects involved than that given by a definition. An analogy: I am thinking of ground in such a way that it can articulate the view that the biological facts are nothing over and above the facts about these systems’ particles. (You may not hold such a view, but ground can specify what it amounts to.) The history of failed attempts in twentieth-century philosophy of science to spell out a “tighter” connection between the reduced and reducing facts by means of correspondence rules that define the biological quantities in terms of physical ones suggests that this won’t work. Yet there is still a way of capturing the sense in which the biological facts “are nothing but” the physical facts, which is to say that the biological facts are grounded in the physical ones. In an analogous way, the relationalist can say that the facts about spatiotemporal structure are “nothing but”—are grounded in—the facts about material bodies, even if she can’t explicitly define the spatiotemporal structure in terms of the relations between material bodies. A grounding relation can hold even in the absence of a definitional connection. (This is one reason the notion of ground can help the relationalist, since finding such explicit definitions is notoriously difficult. Of course, it is not easy to give an account of the grounding of spatiotemporal structure in material bodies either, but replacing the definitional requirement with the looser constraints of ground can ease some of the burden.) Third, there must be some account of how the facts that the relationalist takes to be fundamental manage to ground all the spatiotemporal facts needed for the physics. (For instance, there can’t be two worlds with the same fundamental relationalist facts but different spatiotemporal structures, since the fundamental facts necessitate the grounded facts.) Simply being a realist about spatiotemporal structure does not guarantee the ability to generate the particular structure required by the laws as the matching principle demands. You might be skeptical that the relationalist can do this. Much of the literature is taken up with this question of how, and whether, the relationalist’s more meager ontology can recognize all the spatiotemporal facts we want.²³ This is a big question, but I won’t try to answer it here. I won’t try to tell you exactly how the relationalist grounds all the spatiotemporal facts in facts

²³ A repeated complaint against the varieties of relationalism surveyed by Pooley (2013) is that the relationalist’s resources are too thin to yield predictions of the phenomena.

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about material bodies.²⁴ As we’ll see, I think there is an argument for substantivalism that goes through even if we grant the relationalist the ability to ground all the relevant facts in ones she takes to be fundamental. So for the purposes of that argument, I am going to grant the relationalist that ability. It is worth mentioning one thing that I do think will be required to ground that structure, which is some version of “modal relationalism.” I suspect that the relationalist will have to countenance facts not only about the actual features and behaviors of material bodies, but about their possible ones as well—facts about what spatiotemporal relations can hold, in some sense. This is because the actually instantiated relations won’t in general suffice to fix the full spatiotemporal structure required for the physics. (As long as the relationalist can embed the actual relations uniquely into a certain structure, it seems as though she can talk of the spatiotemporal structure of a world. The problem is that the actual relations may not uniquely fix the structure (up to isomorphism) needed for making predictions about material bodies.²⁵) In order to adhere to the matching principle, the relationalist will have to go modal. I refer you to Carolyn Brighouse (1999) and Belot (2011) for discussion of ways the relationalist might do this and what sort of modality may be involved.²⁶ (Modal relationalism arguably allows the view to countenance vacuum worlds, which seem possible according to both classical and relativistic physics. Such worlds contain no material bodies and yet can have a spatiotemporal structure. Now, it is open to the relationalist to deny that vacuum models correspond to physically possible worlds. Nonetheless, the modal relationalist should be able to allow for these possibilities. All the facts about spatiotemporal structure will still be grounded in facts about material bodies—in facts about how these bodies would behave, if there were any. Such a relationalist can arguably even countenance different spatiotemporal structures in different vacuum worlds, as general relativity seems to allow for. This is not to say exactly how the relationalist can do this, just as I haven’t

²⁴ From this perspective, those such as Manders (1982); Mundy (1983; 1992); Huggett (2006); Belot (2011) can be seen as giving accounts of how this grounding project might go. ²⁵ Examples are in Mundy (1986); Maudlin (1993, 193–4, 199–200); Nerlich (1994a); Belot (2000; 2011, ch. 2). Field (1984) argues that the modal view is necessary for the relationalist to solve the problem of quantity. An alternative is conventionalism (Earman, 1989, sec. 8.6). ²⁶ The view may sound newfangled, but even Leibniz, according to many, held it: Belot (2011, Appendix D). The liberalized relationalism of Teller (1991) is a precursor to more recent versions. See also Sklar (1974, III.B2); Horwich (1978); Mundy (1986). Objections are in Malament (1976); Field (1984); Earman (1989, sec. 6.12); Nerlich (1994a).

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said how the relationalist can ground any particular structure in material bodies. Yet once we grant the (modal) relationalist the ability to ground all the spatiotemporal facts in facts about material bodies, there needn’t be a special problem for vacuum worlds.) Keep in mind that the relationalist might not deny the fundamentality of any spatiotemporal fact or structure. Depending on the version of the view (see the beginning of this subsection), the fundamental facts may include ones such as that two particles are separated by some distance, or that one particle lies between two others.²⁷ What’s important is that the relationalist only allows certain kinds of spatiotemporal facts (if any) to be fundamental, namely those that essentially involve material bodies and their relations— facts that the substantivalist takes to be nonfundamental. The fact that a world has a given spatiotemporal structure is grounded in the facts about material bodies, even though these latter facts may include certain spatiotemporal ones. More exactly, there is no fundamental spatiotemporal fact or structure apart from the structure of, or facts about, material bodies. For ease of exposition, I put this as the claim that all spatiotemporal facts are grounded in facts about material bodies. All spatiotemporal structure is grounded in the relations between and properties of material bodies. So, using the notion of ground, the relationalist can say that there are facts about a world’s spatiotemporal structure, which are distinct from the facts about material bodies and their relations, but are also nothing over and above those facts about material bodies—just as one might say that there are real facts about macroscopic systems, which are distinct from the facts about their particles, but are also nothing over and above the facts about the particles. This is a non-standard (if not wholly unprecedented) way of formulating relationalism, which captures traditional thoughts about the view, for instance that spacetime doesn’t “really exist”: “spacetime” is nothing but various features of material bodies; certain material bodies are fundamental, and any spatiotemporal talk or fact is really about them. At the same time, this formulation allows the relationalist to say that spatiotemporal structure exists, that there are objective truths about what spatiotemporal structure a world has, as the matching principle says we should do. It’s just that these things all hold in virtue of what’s true about material bodies.

²⁷ Which of these depends on whether the relationalist thinks that fundamental relations can be quantitative.

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3.2. Substantivalism in terms of ground The substantivalist denies that all spatiotemporal facts hold in virtue of facts about material bodies. A world’s spatiotemporal structure is not grounded in features and behaviors of material bodies. The fact that a world has a given spatiotemporal structure is a fundamental fact about the physical world; in particular, it is not grounded in facts about material bodies. (Clarifications below.) The facts about a world’s spatiotemporal structure, in turn, ground the facts about the spatiotemporal relations between material bodies. (The former may only partially ground the latter, since the grounds may include occupation relations that material bodies bear to spacetime points or regions, depending on the version of the view—see Section 3.3.) For example, the fact that two particles are some distance apart is grounded in, made true by, the fact that they are separated by that amount according to the fundamental metric structure (where the metric will itself be understood in different ways by different substantivalists—see Section 3.3— but will in any case not be grounded in features of material bodies). The fact that a particle is traveling inertially in a Newtonian world is likewise grounded in facts about the fundamental spatiotemporal structure: the particle is following a straight trajectory because (in the metaphysical sense) its path is straight according to the world’s Galilean structure. (The substantivalist then recognizes nonfundamental spatiotemporal facts or structure of a sort, about the spatiotemporal relations between material bodies. More exactly, the view holds that there are fundamental spatiotemporal facts or structure not grounded in (facts about) material bodies. Notice that certain facts about material bodies, for instance about their fundamental intrinsic properties, will be fundamental. What’s not fundamental are the spatiotemporal facts about them.) By contrast, for the relationalist, a world’s spatiotemporal structure is Galilean because the particles behave in certain ways. On that view, the facts about material bodies metaphysically explain the fact that a world has the given structure. For the substantivalist, facts about the spatiotemporal relations between material bodies are nothing over and above facts about how these objects are arranged according to a given spatiotemporal structure. Facts about a world’s spatiotemporal structure, on the other hand, are not grounded in facts about material bodies, and in that way are “over and above” any facts about material bodies. This captures the traditional conception of the view as holding that spacetime exists “independently of ” material bodies: there is spatiotemporal structure that is not metaphysically due to material bodies. You may worry that this conception of substantivalism is already disconfirmed by our current best theory of spacetime. According to general relativity,

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the presence of matter affects the local spatiotemporal geometry, which in turn affects the behavior of matter; whereas on my conception of substantivalism, there is spatiotemporal structure that is independent of matter. This worry is evaded by noticing that the interdependence between spatiotemporal structure and material bodies in general relativity is of a different, causal or nomological, kind from that given by ground. Although the substantivalist says that there is spatiotemporal structure that is independent of material bodies in not being grounded in them—these facts about spatiotemporal structure are “metaphysically over and above” the facts about material bodies—she can still allow that the behavior of material bodies causes a certain spatiotemporal structure in accord with the physical laws. Compare: although the dualist says that mental events are not grounded in physical events—mental events are “metaphysically over and above” physical ones—she can still allow that physical events cause mental events in accord with the scientific laws. Substantivalism and relationalism, as I understand them, disagree about the fundamental nature of the physical world. They both countenance spatiotemporal structure or facts, but disagree on whether all such structure or facts hold in virtue of material bodies. Both views can recognize the fact that two particles are separated by some distance under a Euclidean metric, for instance, or that a world has a Euclidean metric structure. But they will disagree on whether the metric is itself fundamental or grounded in the behavior of material bodies. To borrow a phrase that Helen Beebee uses for a different debate, these views “have completely opposite conceptions of what provides the metaphysical basis for what” (2000, 580). The substantivalist sees a world’s spatiotemporal structure as the metaphysical basis for the spatiotemporal relations between material bodies. The relationalist sees material bodies and their relations as the metaphysical basis for a world’s spatiotemporal structure. If we ask, of a Newtonian world, “why (in the metaphysical sense) does it have a Galilean spatiotemporal structure?” the relationalist will answer: “because the particles (can) behave thus and so.” The substantivalist will have no answer (or if there is any answer, it won’t reference material bodies: see below). This is a substantive debate about what makes it the case that the spatiotemporal structure needed for the physics holds.

3.3. Further clarifications The substantivalist might not take a world’s spatiotemporal structure to be absolutely fundamental. Newton held that absolute space is a necessary consequence of God’s existence, so that the facts about the world’s spatial

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structure are not fundamental but grounded in facts about God. Yet Newton is still a substantivalist, on my understanding, since the facts about the spatial structure are more fundamental than the facts about bodies’ spatial relations.²⁸ To put it another way: the facts about the spatial structure are fundamental to the physical realm. Analogously, the relationalist will say that all spatiotemporal facts are grounded in facts about material bodies, regardless of her other metaphysical views, such as whether there is something yet-more-fundamental that lies outside the physical realm. The views still disagree over whether spatiotemporal structure apart from material bodies is fundamental to the physical world. For ease of presentation, I continue to put the dispute as the question of whether spatiotemporal structure is fundamental (to the physical world). What if there is no fundamental physical level? In that case, the views might still be distinguished by means of the relative fundamentality of the behaviors of material bodies and a world’s spatiotemporal structure, depending on the details. This may suggest that the debate should be framed in terms of relative fundamentality. Substantivalism would then be the view that the facts about a world’s spatiotemporal structure are more fundamental than the spatiotemporal facts about material bodies, and relationalism would be the view that the facts about material bodies are more fundamental than the facts about spatiotemporal structure. But I don’t want to put it this way. That way of putting things would imply that either relationalism or substantivalism is bound to be true, regardless of future physics, so long as the two kinds of facts are not equally fundamental. Yet intuitively, if nothing like either spatiotemporal structure or material bodies turns out to be fundamental to the physical world, then neither view has been vindicated. You could insist that substantivalism would still be correct so long as the facts about the world’s spatiotemporal structure are more fundamental than the spatiotemporal facts about material bodies, and contrariwise for relationalism. This strikes me as too far removed from the original views. More generally, I don’t think that one of these views must be correct regardless of future physics, and it will depend on the details of that future physics whether one or the other, or neither, is correct. There is another way to put the difference between the views, which I want to be careful with. The substantivalist says that there exists a fundamental physical space(time); the relationalist denies this. Similarly, the relationalist denies, whereas the substantivalist accepts, the existence of spacetime points (or regions) as fundamental physical objects. This way of putting things is familiar and in keeping with traditional conceptions

²⁸ Some argue that Newton wasn’t a substantivalist: Stein (1970); DiSalle (2002).

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of the dispute.²⁹ The problem is that it is not entirely clear what it means to say that a physical space—this “peculiar entity” (Belot and Earman, 2001, 227)—does, or doesn’t, exist; relatedly, whether spacetime points or regions exist as concrete entities. I suspect that this is an underlying reason for the unclarity of the debate in many people’s minds, especially in the philosophy of physics community. Some philosophers of physics have worried about taking spacetime points to be concrete physical entities in particular. As Malament says, in the context of discussing whether spacetime points are nominalist-friendly, “They certainly are not concrete physical objects in any straight-forward sense. They do not have a mass-energy content. . . . They do not suffer change. It is not even clear in what sense they exist in space and time” (1982, 532). Others have worried more generally that this kind of ontological dispute—a dispute that is just about what things exist—is nonsubstantive or merely verbal.³⁰ Howard Stein, in discussing the spacetime debate, says that, “For me, the word ‘ontological’ itself presents seriously problematic aspects”; in particular, “Quine’s usage [is] not a very useful one for philosophy of physics” (1977a, 375). As I see it, the debate is about the fundamentality of spatiotemporal structure, in particular about whether there is any spatiotemporal structure (fact) not grounded in the structure of (facts about) material bodies, where the substantivalist says that there is and the relationalist says that there isn’t. Within this framework, there is some flexibility as to how exactly to put the dispute. Neither the matching principle nor my conception of spatiotemporal structure says how we must construe the nature of spatiotemporal structure; and I have not taken a stand on whether ground is primarily a relation between objects or facts. As a result, although we can put the disagreement as being about whether there exists a fundamental physical spacetime or fundamental spacetime points, we do not have to. Anyone squeamish about putting things in ontological terms can still see the debate as being about the fundamentality of spatiotemporal structure, understanding this as being not about whether there exist certain objects (over and above material bodies), but about whether there are certain facts (over and above the facts about material bodies): the relationalist says that the fact that a world has a certain spatiotemporal structure holds in virtue of the fact that material bodies behave thus and so; the substantivalist denies this,

²⁹ See Field (1980, ch. 4); Mundy (1983); Earman (1989, 12); Brighouse (1994). ³⁰ This seems the spirit behind Stein (1970; 1977a); Curiel (2016); perhaps Belot (2011) and some others in note 1; in a different way Wallace (2012). There have been similar thoughts in metaphysics, for example in Hirsch (2011), but it’s not clear that this is exactly the same idea.

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seeing it as a fundamental fact about the physical world. This allows us to discuss the dispute, and to evaluate the evidence for either side, while remaining neutral on how the substantivalist wants to understand the instantiation of that structure or the ontology behind this fact. This dovetails with an idea in spacetime structural realism. Jonathan Bain (2006) argues that classical field theory (this includes general relativity), standardly given in terms of a tensor formalism, can be formulated in ways that do not presuppose a differential manifold of points. He describes three alternative formalisms one could use (twistor theory, Einstein algebras, and geometric algebra), none of which treat points as fundamental. My understanding leaves it open for the substantivalist to spell out the spatiotemporal structure in any of these ways, or even to refuse to choose among them, as Bain himself proposes. (Bain argues that we should be realists about spacetime structure and not any particular instantiation of it. He sees this as a third view, since according to him the substantivalist is committed to spacetime points, but it counts as substantivalist by my lights.) To be explicit, there are four different kinds of view that my conception of substantivalism is meant to encompass, each of which holds that there are spatiotemporal facts or structure not grounded in material bodies. First is what we might call Bainianism, on which one is a realist about spatiotemporal structure but not about any particular instantiation of it, i.e. not about any of the (non-material) objects that could be said to instantiate it. On this view, the different possible descriptions or formulations or instantiations of spatiotemporal structure do not really differ from one another: one is an anti-realist about those. Second is what we might call uncommitted substantivalism, on which one is a realist about a particular instantiation of spatiotemporal structure—there is a single best way of describing or formulating the spatiotemporal-structure facts, in terms of a certain kind of nonmaterial object—but one doesn’t know what that instantiation or best formulation is; hence we cannot state the view as propounding one or another such formulation. Third is what we might call committed substantivalism, on which one is a realist about a particular instantiation of spatiotemporal structure, one thinks that there is a best formulation of it, and one does claim to know what it is; e.g. it might be the one in terms of points (in which case the view approaches traditional substantivalism). Fourth is the “qualitativist” substantivalism of Dasgupta (2009; 2011), on which the fundamental spatiotemporal facts are purely qualitative, not mentioning any entities at all; spacetime is not an entity but a “purely qualitative structure.” One of the things I am claiming is that, when it comes to the relational–substantival debate, we needn’t choose among these versions of substantivalism. The argument in Section 4 will support each of them in the same way.

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3.4. Something old, something new There are too many different notions of “relational,” “substantival,” and related concepts in the literature to survey them all here and compare them to my own account.³¹ It should be clear that this is a non-standard conception of the dispute, which captures core ideas behind more familiar conceptions, both contemporary and traditional. For example, my understanding captures the thought that the substantivalist believes in “the independent existence and structure of space and time” (Sklar, 1974, 163)—that spacetime exists “independently of material things . . . and is properly described as having its own properties, over and above the properties of any material things that may occupy parts of it” (Hoefer, 1996, 5)—so that “space is something as real as matter and whose existence does not require matter, but which is not the same stuff as matter” (Huggett, 1999, 129). It encompasses the idea that for the substantivalist, “space-time points (and/or space-time regions) are entities that exist in their own right” (Field, 1980, 34); “[s]pace is an entity in its own right—a real live thing in our ontology” (Nerlich, 1994a, 3), a “genuine entity of a fundamental kind” (Pooley, 2013, 526). These ideas are captured by the claim that spatiotemporal structure is fundamental to the physical world. There is spatiotemporal structure that is not grounded in, and is in that way independent of, any material bodies. My conception also captures the thought that the relationalist “denies that space, or spacetime, is a basic entity, ontologically on a par with matter” (Brown and Pooley, 2002, 183, n.1), so that “the universe consists solely of objects and events exemplifying various properties and relations” (Horwich, 1978, 397); “all that exists is material bodies” (Arntzenius, 2012, 153). As a result, “all our talk of space and time can be reconstructed out of talk about spatial relations between objects” (Brighouse, 1999, 60), and we “regard the use physical theory makes of space-time and its geometrical structure merely as a convenient way of saying something about the spatio-temporal properties and relations of concrete physical objects” (Friedman, 1983, 216). These statements are captured by the claim that spatiotemporal structure apart from material bodies is nonfundamental; whereas certain material objects, and certain of their properties and relations, are fundamental. At the same time, this is a non-standard, non-traditional take on things, which allows us to sidestep many of the reasons people feel that the usual dispute has stagnated or become non-substantive. Most importantly, it leaves room for future physics to provide an answer, so that this dispute cannot be “merely verbal” or “purely metaphysical.” We think that there ³¹ See the many notions listed in Horwich (1978); Friedman (1983); Earman (1989).

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is a real difference between a world in which spatiotemporal structure is fundamental, and one in which it arises from some pre-spatiotemporal structure, for instance. Physicists treat these as genuinely different possibilities, governed by different theories. This is evidence of a genuine difference between the views as I see them. Against tradition, I claim that the relationalist as much as the substantivalist can recognize “absolute” or frame-independent facts about—quantities of, structures that support—objects’ motions.³² In particular, it needn’t be the case that “all motion is relative” for the relationalist, since there can be objective facts about objects’ motions even in a world devoid of other material bodies.³³ The traditional question about the relativity of motion, then, is not of primary concern.³⁴ In addition, we needn’t distinguish the two views by means of how they count possibilities, contrary to tradition as well as some recent accounts.³⁵ Further, against some other understandings of the dispute, this one allows for both sides to believe in, to be realists about, spatiotemporal structure.³⁶ (I have argued that they both should do this, in order to respect our usual inferences in physics.) I even leave it open for the relationalist to posit the same spatiotemporal structure to a world as the substantivalist, whereas some have taken the dispute to be over the relevant structure.³⁷ My conception also avoids having to draw some of the distinctions that people have been skeptical of. It does not require that we definitively distinguish between container and contained, substance and non-substance, absolute and relative, to name a few.³⁸ There are three distinctions presupposed by my understanding of the dispute, but they are not as unclear as those required by more traditional conceptions. First, there is the distinction between the fundamental and the nonfundamental. This is a distinction that we have a reasonably clear pre-theoretic grasp of, clear enough to be useful here even without spelling it out in more detail. Second, my conception requires that we can identify what structure counts as spatiotemporal. ³² Hoefer notes that traditional relationalism “is connected essentially to the denial of absolute motion” (1998, 460). ³³ Huggett and Hoefer (2009) note other relationalist views denying the relativity of motion. ³⁴ This aligns with a similar shift away from that question in recent literature, exemplified in Stein (1970; 1977b); Sklar (1974); Friedman (1983); Earman (1989); Belot (1999; 2000; 2011); DiSalle (2006). ³⁵ Huggett (1999, ch. 8) discusses the traditional arguments. More recent examples are in Earman and Norton (1987); Belot (2000). ³⁶ Statements intimating that the relationalist cannot believe in spatiotemporal structure are in Field (1984, 34); Nerlich (1994a); Pooley (2013, 542); Maudlin (2012, 66). ³⁷ Earman (1989) suggests this at points. ³⁸ Rynasiewicz (1996; 2000) worries about the clarity of all these (and other) distinctions.

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This is something that the physical laws give us a handle on, in ways discussed earlier, though I admit that there is more that could be said. Perhaps there is nothing else that makes some fact or structure spatiotemporal; perhaps there is.³⁹ Either way, I take the idea to be relatively familiar from physics. At least we have some clear cases of spatiotemporal structures, such as those discussed here. Third, my conception requires a distinction between material bodies and other things in the world. Although people have worried about the clarity of this distinction,⁴⁰ I think that it is clear enough for our purposes. At the least, I suggest that we understand the debate in this way, on the assumption that we will be able to locate such a distinction. For now I follow Earman, who says that, “It is a delicate and difficult task to separate the object fields into those that characterize the space-time structure and those that characterize its physical contents,” while also noting that “the vagaries of this general problem need not detain us here, since there are clear enough cases for our purposes” (1989, 155–6). For those wanting argument that the distinction can generally be made, I refer you to Carl Hoefer (1998) and also David Baker (2005). One will find, in contemporary discussions, the thought that the relationalist can believe in the existence of spacetime, understanding this as being (somehow) constructed out of material bodies and their features. So it may seem like even the traditional dispute (and contemporary versions of it) was never about the existence of spacetime but its fundamentality, and my own formulation may seem like just a new label for an old dispute. This however is something of an anachronism. Traditional participants, like Newton and Leibniz, weren’t focused on questions of fundamentality: they were not thinking explicitly in those terms. Neither, of course, were they thinking in spatiotemporal terms. At the same time, to the extent that we can understand what they were saying in these terms, this shows that my understanding is, as I claim, an updating of the traditional dispute, using more recent developments in physics (involving spacetime and its structures) and philosophy (fundamentality and ground).

4 . A N A R GU M E N T F O R S UB S T A N T I V A LI S M I now suggest that if we do understand the debate in this way, then there is a powerful argument for substantivalism, given much of current physics. ³⁹ Belot (2011) and Brighouse (2014) are two different accounts. ⁴⁰ See especially Rynasiewicz (1996).

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Above I argued that the relationalist should go partway⁴¹ toward adhering to the matching principle by countenancing spatiotemporal structure, and that she can do this by understanding all the facts about spatiotemporal structure as being grounded in facts about material bodies. I am now going to argue that really the relationalist can’t adhere to this principle, properly understood. The argument differs from the more familiar charge that the relationalist cannot countenance a particular spatiotemporal fact or structure. Recall that the matching principle says to posit in the world the structure presupposed by the laws; that is, to posit physical structure in the world corresponding to the mathematical structure needed to state the laws. Now here is something else about the principle I haven’t yet mentioned. It applies, in the first instance, to the fundamental laws. (By saying “in the first instance,” I mean to indicate that the principle applies at least to the fundamental laws, and that this is where we begin constructing our picture of the world from physics, in that we build a world “from the bottom up.” I leave it open whether an analogous idea holds for nonfundamental laws.) Given the fundamental laws, we should posit in the world the structure they presuppose. This is clear from our usual inferences about spatiotemporal structure. Assuming that Newton’s laws are fundamental, we infer a Galilean structure to the world. From different fundamental laws, we infer a different spatiotemporal structure—such as a Minkowskian structure for special relativity, a preferred-location spatial structure for Aristotle’s physics, or a variety of different spatiotemporal structures for general relativity. The matching principle also tells us to posit, in the fundamental level of the physical world, whatever those laws presuppose. The fundamental laws, after all, are about what’s fundamental. They don’t “care about” or “know about” or mention the nonfundamental. I take it this is part of what we mean when we say that they are fundamental. I also take it that this is a familiar thought. (Michael Townsen Hicks and Jonathan Schaffer (2017) call it orthodoxy.⁴²) For example, it lies behind our dislike of quantum laws that mention things like “measurement” or “the observer.” This isn’t to deny that fundamental laws have consequences for nonfundamental things. These laws yield predictions for nonfundamental phenomena when we plug in initial conditions and use various bridge principles. On their own, though,

⁴¹ Partway, since I haven’t shown that the relationalist can ground the particular structure needed. ⁴² They argue against the idea, concluding that fundamental laws can, and do, mention nonfundamental properties. I agree that an alternative formulation can be useful in practice, but I think that the best formulation won’t mention such things.

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fundamental laws only mention or presuppose or know about things at the fundamental level.⁴³ Another way to see this comes from the idea of “the structure presupposed by the laws.” The sense in which the laws presuppose or require some structure is akin to an idea familiar from mathematics. In mathematics, we can define different levels of structure by starting with a lowest level, such as a set of points, and then defining other objects that add more structure. These levels of structure form a hierarchy. The ones “higher up” assume or presuppose or constrain levels lower down, in that the higher-level objects cannot be defined until the lower-level ones have been assumed or defined. For example, think of adding differential structure to a topological space. This structure indicates, from among the continuous curves specified by the topology, which ones are smooth to varying degrees. In this way the differential structure assumes or presupposes a topology: it cannot be defined, it doesn’t make sense, absent a topology. Higher-level structure is not similarly constrained by levels lower down—as different metrics, or none at all, can be added to a differential manifold. In other words, a given level of structure only “knows”—requires, constrains, presupposes, assumes—things about that level and below.⁴⁴ Analogously for the structure required by the physical laws. This structure is presupposed by the laws in that it must be assumed in order for the laws to be formulated or make sense. The laws don’t similarly know about— require, constrain, presuppose, assume—higher-level structure. For fundamental laws, the result is that they only know about fundamental structure. Note that the fundamental laws may constrain things higher up in a different, metaphysical sense: given the fundamental laws and ontology, everything else may be “fixed” in some sense. This is a different sense of constraining from the mathematical notion, which concerns what is needed for something to make sense or be defined. The other sense is a metaphysical notion that requires additional metaphysical principles concerning the relation between different levels of reality. An example illustrates and motivates the primary reading of the matching principle. Recall the discussion of non-time reversal invariant laws. Earlier ⁴³ This is different from Sider’s (2011, ch. 7) purity principle. Purity is a very general principle about what the fundamental facts or truths can mention. (It says that they cannot mention nonfundamental concepts.) The above is specific to the physical laws and what they presuppose and therefore tell us about the physical world. ⁴⁴ In mathematics one also talks of a higher-level structure “inducing” a lower-level one (e.g. “the topology induced by the metric”). This makes it sound as though the higher-level structure is defined first and it then constrains the lower, but in fact it amounts to the above idea (e.g. once we have defined a metric, there must already be implicitly a topology).

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I said that if the laws are asymmetric in this way, then we infer an asymmetric temporal structure in the world. The idea is that such laws presuppose this structure, for they mention or presuppose a distinction between past and future, by telling things to behave differently depending on the direction of time. But there is more to the story. Take the second law of thermodynamics. This law is not time reversal invariant, so it may seem to indicate an asymmetric temporal structure. However, the second law of thermodynamics is not a fundamental law. It doesn’t mention a system’s particles or other fundamental constituents. It is formulated in terms of higher-level macroscopic quantities like entropy. Whether to infer an objective past–future distinction in the world then really depends on what fundamental theory accounts for the second law, and whether that theory’s laws are symmetric in time. (It is natural to think that if a past hypothesis account of thermodynamics is correct, then there is no asymmetric temporal structure; whereas if a non-time reversal invariant theory like GRW quantum mechanics is true (and able to account for thermodynamics) then there is.⁴⁵) The nonfundamental law on its own does not tell us about fundamental temporal structure: it is too far removed from the fundamental level to do that. Only a fundamental law can tell us about this. In other words, we posit fundamental structure in the world needed for the fundamental laws. We recognize as fundamental the facts that are recognized by the fundamental laws. The matching principle applies, in the first instance, to the fundamental laws and fundamental level of physical reality. The matching principle as discussed in Section 2 says that the world should “look like” or “fit” its laws. The primary reading of the principle says that the fundamental level of the world should look like or fit its fundamental laws. Now to the argument for substantivalism. First notice that the kinds of fundamental laws we are most familiar with are formulated to presuppose spatiotemporal facts apart from material bodies. These laws mention or presuppose a spatiotemporal structure in addition to material bodies and their features. Newton’s laws presuppose a Galilean spatiotemporal structure in addition to the existence of massive particles. These laws assume or require that the world has this structure, just as the laws of special relativity assume or require a Minkowskian structure. The laws of Aristotle’s physics mention a preferred-location spatial structure in addition to the elements that move toward their natural places. Similarly for the laws of general relativity, even though they allow for different spatiotemporal structures.

⁴⁵ Albert (2000) discusses these two accounts. See North (2008) on why these conclusions about temporal structure are natural.

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Think of the usual way of understanding the field equations, as saying how the distribution of matter and energy relates to the spatiotemporal geometry, which in turn affects the behavior of matter. These equations are formulated directly in terms of—they mention or talk about—a spatiotemporal structure apart from material bodies, coded up in the metric tensor, distinct from the stress-energy tensor. (See Hoefer (1996; 1998) for arguments that the metric is most naturally seen as characterizing a spatiotemporal structure that is not the structure of a material field. This is not uncontroversial, but is assumed in standard presentations.) The fundamental laws that we are familiar with make reference to material bodies, but they also presuppose or make reference to a spatiotemporal structure apart from those bodies.⁴⁶ Given that the fundamental laws are typically like this, a problem arises for the relationalist. The problem is not that the relationalist doesn’t recognize enough spatiotemporal facts for the physics, a concern lying at the root of classic arguments like Newton’s, as well as many contemporary ones (see notes 23 and 25). Grant the relationalist enough stuff to ground those facts and make the relevant predictions, and there is still a problem. According to the core of the view, all the facts about spatiotemporal structure are grounded in more fundamental facts about material bodies. The kinds of fundamental laws we are used to, though, presuppose or mention spatiotemporal facts apart from material bodies—facts that, for the relationalist, are nonfundamental. This violates the principle that the fundamental level of the physical world should contain whatever is needed for or presupposed by the fundamental laws. So the argument is this. First premise: the fundamental laws are about what’s fundamental to the physical world; they refer to or presuppose things about the fundamental physical level. Second premise: these laws are about, they presuppose or refer to, a spatiotemporal structure, or spatiotemporal facts, apart from material bodies. Third premise: for the relationalist, this kind of structure or fact exists at a nonfundamental level,

⁴⁶ There is a difference between the laws mentioning and presupposing something. That a law explicitly mentions something implies that the law presupposes it, but not vice versa. The laws of general relativity explicitly reference both material bodies and spatiotemporal structure. The usual Newtonian laws explicitly mention the former yet only presuppose the latter. (Hence a difference from Quine’s prescription (see pp. 11–12): Newton’s laws, as usually formulated, presuppose a Galilean spatiotemporal structure; they don’t explicitly mention or quantify over that structure, which the matching principle tells us to posit.) This difference does not matter here. We use the matching principle to infer structure in the world regardless of whether it is explicitly mentioned or presupposed. Either way, the laws require it.

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above that of material bodies. Fourth premise: the primary reading of the matching principle. Conclusion: relationalism is incorrect. Substantivalism posits the spatiotemporal structure or facts needed for the laws at the fundamental level. General relativity provides an example. This theory establishes a nomological connection between material bodies and a spatiotemporal structure apart from them. On their own, the laws do not say whether material bodies and spatiotemporal structure are at the same level of physical reality, nor which is more fundamental if not. Without some further principle, both relationalism and substantivalism seem satisfactory: both recognize facts about material bodies as well as a world’s spatiotemporal structure. Enter the matching principle. The substantivalist does, the relationalist does not, adhere to it. You may wonder why the spatiotemporal structure presupposed by the laws is apart from material bodies, as premise two claims. After all, the relationalist, in my view, can countenance this structure, but will say that it has to do with the (actual and perhaps possible) spatiotemporal relations between material bodies. In what way do the laws presuppose a spatiotemporal structure that is in addition to material bodies? The answer comes from the way that the fundamental laws are usually formulated. (I turn to potential reformulations in Section 5.) These laws are typically formulated to directly mention material bodies, with a term that directly refers to them—such as the mass term of Newton’s dynamics, or the mass density of some formulations of Newtonian gravitation, or the elements mentioned in Aristotle’s laws, or the stress-energy tensor of general relativity.⁴⁷ At the same time, these laws also presuppose that the world has a spatiotemporal structure apart from those bodies—apart in that it is presupposed by the laws in the mathematical sense given above, or else is directly mentioned by or coded up in a distinct term. Recall that the matching principle tells us to infer that a special relativistic world lacks an absolute simultaneity structure. The laws don’t require this mathematical structure, which suggests that the world doesn’t have the corresponding physical structure. To fail to adhere to the matching principle is to fail to heed this evidence from the laws about what the world is like. The relationalist fails to adhere to the primary reading of the principle in the same way. The fundamental laws are giving us evidence that spatiotemporal structure is fundamental to the physical world, which the relationalist fails to heed. The relationalist may respond that there are good reasons to

⁴⁷ In the context of this debate, both views take certain material objects to exist at the fundamental level. (Supersubstantivalism would then deny this.)

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disregard this apparent evidence from the laws. The burden is then on the relationalist to show this, just as the burden falls on the proponent of absolute simultaneity. You might think that there are two distinct notions, that of what’s physically fundamental versus metaphysically fundamental; that the matching principle governs the first whereas substantivalism and relationalism are views about the second; and conclude that the argument from the matching principle doesn’t make contact with those views. In particular, you might think it open for the relationalist to say that spatiotemporal structure is metaphysically nonfundamental, in accord with relationalism, yet physically fundamental, in accord with the matching principle—that a world’s spatiotemporal structure is less metaphysically fundamental than, but more physically fundamental than, the spatiotemporal relations between material bodies. I suppose that such a view is possible, but it seems implausible on its face. Imagine an analogous reductionist who says that macroscopic systems (boxes of gas) are metaphysically nonfundamental, grounded in more fundamental microscopic objects (their particles), yet physically fundamental. This is a puzzling view. Surely the thought that microscopic objects are metaphysically fundamental goes hand in hand with evidence from physics suggesting that they are physically fundamental. Relative physical and metaphysical fundamentality cannot plausibly go in opposite directions. More generally, I’m inclined to reject the idea that there are two distinct notions of fundamentality here. Suppose that what I have been calling “spatiotemporal structure” involves, at least in part, facts that must be stated using universal generalizations. On a standard axiomatic approach to geometry, for instance, a given spatiotemporal structure will be defined via a universal generalization over a domain of points. Suppose further that generalizations are not fundamental but grounded in their instances, in accord with a familiar way of thinking about grounding. Then it may seem as though the substantivalist doesn’t adhere to the matching principle either, simply because spatiotemporal structure, qua generalizations, cannot be fundamental. However, the substantivalist will avoid the worry, for one of the following reasons. First, one might for independent reasons think that generalizations are fundamental, a not-unprecedented (to my mind, not implausible) view, even among grounding proponents. Second, even if spatiotemporalstructure-qua-generalizations is not absolutely fundamental, it is very close to being fundamental, so that the fundamental structure of the world almost directly matches the structure for the fundamental laws. The only “gap” there is between spatiotemporal structure and the fundamental level is the one created by the gap between generalizations and their instances. This is an intuitively smaller gap than that between a world’s spatiotemporal

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structure and features of material bodies. The former is just a “gap in logical form”—the “size” of the separation between a generalization and the collection of particular claims that grounds it—whereas the latter is a larger, physical gap. The substantivalist then adheres to the matching principle more than the relationalist does. Finally, notice that even if the generalizations that axiomatize a given structure are not absolutely fundamental, the various facts about the points still can be, and these facts are included in my conception of spatiotemporal structure; in which case there are still fundamental spatiotemporal facts or structure apart from material bodies. (The worry would also seem to go too far. It would force us to say that no particular collection of fundamental facts is to be preferred to any other on the basis of the physical laws, simply because any structure required for those laws takes the form of a generalization, and no generalization is fundamental. But surely a matching-type argument can sometimes work—as when we want to say that Berkeleyan idealism posits a world that radically fails to match the structure indicated by the laws. It seems we might reject that view for the reason that the fundamental nature of the world does not match the structure for the laws—even though that structure is given by generalizations, and even if generalizations are not fundamental but grounded in their instances.) Notice that the argument for substantivalism is independent of one’s view on the metaphysics of laws. The question of what makes a statement a law is distinct from the injunction to posit, assuming that a certain statement is a law, the requisite structure in the world. Even the Humean, who denies that laws of nature are metaphysically fundamental, can agree to posit, in the fundamental physical level of the world, the structure presupposed by the fundamental physical laws. To put it another way, the content of the law claim, the proposition p of the statement “it is a law that p,” is what indicates structure in the world. It is irrelevant whether what makes it the case that p is a law is itself metaphysically fundamental. Whatever your account of laws of nature, you can, and should, adhere to the matching principle. Current physics therefore gives us reason to believe that substantivalism is correct. Nonetheless, it is open for future physics to turn the tide. If a quantum theory of gravity or some other future fundamental theory contains laws that only presuppose things about material bodies and their relations, which in turn give rise to the spatiotemporal structure presupposed by current theories, we can conclude that relationalism is correct. Future laws might even suggest a view that doesn’t look like either relationalism or substantivalism, presupposing facts about neither material bodies nor spatiotemporal structure but something else. (A causal set theory approach to quantum gravity, for example, might support relationalism, depending on the particulars, or it could be a case on which

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neither view is correct.⁴⁸) In this way the debate will remain relevant to, and continue to be informed by, future developments in physics.

5 . A C HA L L E N G E F O R R E L AT I O N A L I S M Finally, let me turn to the question raised at the end of Section 2. I have been assuming that the fundamental laws we currently have are formulated to presuppose a spatiotemporal structure apart from material bodies. This reveals one other way for the tide to turn: the relationalist could try to reformulate these laws to only presuppose things about material bodies. If such a reformulation is possible, then the argument will turn on how we should generally formulate the laws, which is a big question that I can’t fully answer here. Even so, the argument poses a significant challenge to any relationalist attempt to reformulate the laws. Consider an illustrative example: the relationalist reformulation of Newtonian mechanics initially suggested by Bas van Fraassen (1970, sec. 4.1) and filled out in one way by Nick Huggett (2006). According to their idea, we can reformulate Newtonian mechanics to include the statement that, “Newton’s Laws hold in some frames,” where these will be the inertial frames. (There is also a force law, and on Huggett’s account a law about the spatial geometry.) These laws then pick out a standard of inertia or straightness of trajectories—they recognize a quantity of, or facts about, acceleration—without assuming that spacetime exists. In my terms, they only presuppose spatiotemporal facts about material bodies. This is because, according to Huggett, the facts about inertial frames—indeed, all the spatiotemporal facts—themselves supervene on facts about the history of relations between material bodies. (Huggett rejects modal relationalism.) This is a genuinely relationalist formulation, on my construal, which respects the primary reading of the matching principle. The truth of the laws in certain frames effectively substitutes for an inertial structure, so that the laws themselves do not have to mention or presuppose this structure. The problem is that this is a worse formulation of the laws, for a couple of reasons. First, this formulation does not respect the idea that fundamental laws only mention fundamental things. These laws are given in terms of facts about inertial frames, which for Huggett are not fundamental but grounded in facts about the relations between material bodies.

⁴⁸ See Huggett and Wüthrich (2013) and the other papers in that journal issue on the emergence of spacetime in quantum gravity.

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Second, this formulation is given in terms of reference frames. Why is this worse? I take it that fundamental physical laws are best formulated in terms of things about the world itself, and reference frames don’t fit the bill. According to Newton’s laws, inertial frames are like units of measure or coordinate systems, in that a choice of frame is an arbitrary choice in description. Now, Huggett’s formulation does not mention any particular frame, nor does it directly mention inertial frames. Instead it says that there are frames you can choose such that Newton’s laws are true.⁴⁹ But the fact that a choice of inertial frame is arbitrary suggests that inertial frames in particular, and reference frames in general—these objects as a group or kind of thing—are merely descriptive or labeling devices we use, not inherent in physical systems themselves;⁵⁰ hence they should not, other things equal, be mentioned in the fundamental physical laws. I gather that this is what underlies the general feeling in foundational discussions that formulating the laws in geometric, coordinate-free terms is desirable. (Consider formulations of classical mechanics in terms of so-called generalized coordinates, which do not mention any particular coordinate system. Even this reference to coordinates is seen as ideally replaceable by geometric objects with no mention of coordinates.) An idea from Hartry Field bolsters the thought that such a formulation is worse in this way. Field draws a distinction between ‘intrinsic’ and ‘extrinsic’ explanations. The former “explain what is going on without appeal to extraneous” entities, things “extrinsic to the process to be explained ” (1980, 43). As a result, intrinsic explanations are better, more “illuminating” (1980, 43) or “satisfying” (1989, 18). He says, [E]xtrinsic explanations are often quite useful. But it seems to me that whenever one has an extrinsic explanation, one wants an intrinsic explanation that underlies it: one wants to be able to explain the behaviour of the physical system in terms of the intrinsic features of that system, without invoking extrinsic entities . . . whose properties are irrelevant to the behaviour of the system being explained. If one cannot do this, then it seems rather like magic that the extrinsic explanation works. (1989, 193; original italics)

The best explanations cite intrinsic features relevant to the system’s behavior. By analogy to Field’s idea, call formulations of the laws in terms of reference frames or coordinate systems or the like “extrinsic formulations.” ⁴⁹ See Dorr (2010) for argument that “existential quantification as such is a distinctive source of badness” (166; original italics). ⁵⁰ Compare Einstein on a coordinate system, which is “only a means of description and in itself has nothing to do with the objects to be described ” (2002, 203; original italics).

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Extrinsic formulations are then worse for the same reasons Field says that extrinsic explanations are worse: they reference things outside the system or world itself, whose properties aren’t directly relevant to the system’s behavior.⁵¹ This makes the success of the formulation seem like magic. All things equal, it is better to have an intrinsic formulation—or what I prefer to call a direct formulation, since extrinsic entities, like coordinate labels, can tell us about the system in question; only they do so in an indirect, and therefore less preferable, way. It’s analogous to characterizing the geometry of the Euclidean plane by saying that, “there are coordinate systems in which the distance formula takes the usual Pythagorean form,” rather than by giving the metric tensor (or, for that matter, Euclid’s axioms). That characterization gives the structure of the plane, but in a needlessly indirect way, by means of the kinds of coordinate systems we can lay down on top of it. Better to have a formulation of the laws that more directly reflects reality. (It is not uncommon for physics books to state the laws in terms of reference frames or coordinate systems. The claim is that this is not the best formulation.) Of course, direct formulations may seem preferable only if you are a realist to begin with—only if you think that it is the job of a physical theory to tell us what the world is like. An instrumentalist may be unbothered by indirect formulations and extrinsic explanations. (The instrumentalist should be used to the charge that the success of science seems like magic.) Since it is not my aim to argue for realism here, I leave it to the anti-realist to parry the objection that such formulations are worse. Let me note, though, that indirect formulations seem particularly problematic for fundamental laws, since the elements that feature in them, like reference frames or coordinate systems, don’t seem the sorts of things that can be truly fundamental or explanatory. There are other relationalist reformulations to consider in more detail than I have space to do here. However, the above strikes me as indicative of the kinds of problems that any such reformulation will face. In order for relationalism to be victorious, the proffered reformulation must be genuinely relationalist, presupposing facts only about material bodies; it should be direct; and it should respect the primary reading of the matching principle. A brief look at three more examples further suggests that a relationalist reformulation meeting these constraints will be hard to come by. (1) Julian Barbour’s relationalist mechanics (Barbour (1982; 2000; 2001); Barbour ⁵¹ Consider Field’s reason that a scientific explanation citing direct relations between physical objects and numbers is extrinsic and therefore worse: “[T]he role of the numbers is simply to serve as labels for some of the features of the physical system: there is no pretense that the properties of the numbers influence the physical system whose behaviour is being explained” (1989, 192–3). The role of reference frames in physics is similar.

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and Bertotti (1982)), which eschews any fundamental temporal structure, arguably presupposes a spatial structure above that of material bodies,⁵² in which case the theory is substantivalist, on my understanding. Setting that aside, the theory is not formulated directly.⁵³ (2) David Albert (1996) suggests that in classical mechanics, the Hamiltonian energy function gives rise to a three-dimensional spatial structure. Since the Hamiltonian is defined in terms of particle features, this may count as a relationalist theory, on my construal. (Albert is not arguing for relationalism.) Yet there is also a case to be made that the mathematical formulation presupposes a spatial structure apart from material bodies (in particular for the kinetic energy term), in which case it would either count as substantivalist, or fail to respect the primary reading of the matching principle. (3) Huggett mentions another law of his reformulation of Newtonian mechanics: “ ‘There is an embedding of the relational history into G ’, for some specific Riemannian geometry G” (2006, 53), where for him the privileged embedding supervenes on the history of relations between material bodies. Facts about the embedding geometry (spatial structure) are not fundamental but grounded in facts about material bodies. This makes the law relationalist. The problem is that it, too, explicitly mentions nonfundamental things, and is formulated indirectly, in terms of a structure into which the relations can be embedded. (A similar charge applies to Albert’s (2012) suggestion for a relationalist Newtonian mechanics that says: “The physically possible histories of inter-particle distances are those which can be embedded in a full substantivalist Newtonian space, or imagined as taking place in such a space, in such a way as to satisfy F ¼ ma.”) This does not prove that no relationalist reformulation can succeed, and more work must be done to fully evaluate the various proposals on offer in these terms.⁵⁴ But it does suggest that it won’t be easy to find a relationalist ⁵² See the presentation in Earman (1989, secs. 2.1, 5.2). Arntzenius (2012, sec. 5.11); Pooley (2013, sec. 6.2) suggest this for Barbour’s reformulation of general relativity in particular. ⁵³ The indirectness enters in recovering the topological temporal structure and the inertial structure: Arntzenius (2012, chs. 1, 5). ⁵⁴ A few more examples. On the dynamical approach of Brown (2005); Brown and Pooley (2006), a world’s spatiotemporal structure holds in virtue of the behavior of material bodies via the laws and their symmetries. This seems relationalist, on my conception (in particular if the laws are grounded in facts about material bodies). They presumably reject my idea that the laws presuppose a certain structure in order to be formulated. Another relationalist theory is that of Belot (1999; 2000), which seems indirectly formulated (cf. Brown and Pooley (2002, 192–3); it also presupposes a temporal structure apart from material bodies: Brown and Pooley (2002, 194)). Another is that of Albert (2017), on which there is no fundamental, pre-dynamical spatiotemporal structure: all spatiotemporal facts are grounded in facts about the behaviors of material bodies. Albert reformulates the laws in an indirect way.

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reformulation that has the features we want of fundamental laws. Current laws are generally formulated to presuppose a spatiotemporal structure apart from material bodies. The problem is that typical relationalist substitutes for that kind of structure—facts about things like reference frames or coordinate systems or embedding geometries—are not candidates for direct formulations of the laws. Future laws, however, may be different.

6 . C O N C LU S I O N Many people have thought that the arguments for relationalism or substantivalism will have to resort to considerations like simplicity, ontological parsimony, or explanatory power.⁵⁵ Some have said that the relationalist’s ontology is more parsimonious, and therefore favored by Occam’s razor.⁵⁶ Others have said that the substantivalist’s theory is simpler, and therefore favored by ordinary criteria of theory choice.⁵⁷ Some have argued that the relationalist’s theory is more explanatory. Others have claimed that the substantivalist’s is.⁵⁸ You might conclude that the debate is hopelessly vague, since the criteria of simplicity, parsimony, and explanatory power needed to adjudicate it are themselves vague; nor is it clear which to favor when these virtues compete.⁵⁹ I don’t object to relying on such considerations even so, but it is worth noting that the argument from the matching principle is different. The matching principle doesn’t say to refuse to posit unnecessary entities or to go with the simplest or most explanatory theory. It says to posit in the world the structure presupposed by the laws. The argument based on this principle escapes those particular worries about the status of the debate. The matching principle is a familiar and successful guiding principle. It applies, in the first instance, to the fundamental laws and fundamental level ⁵⁵ Dasgupta (2015) discusses the effects of these criteria on the spacetime debate for classical physics. ⁵⁶ Huggett (2006); Huggett and Hoefer (2009); Pooley (2013). ⁵⁷ Huggett (2006); Arntzenius (2012, ch. 5). ⁵⁸ Earman (1989) suggests that the relationalist’s theory will be worse; Brown and Pooley (2002) argue against this. Maudlin (1993, 196) says that the substantivalist’s theory is more explanatory in some ways; Nerlich (1994a; 1994b) argues that it is more explanatory in general. ⁵⁹ See Horwich (1978); Earman (1989, sec. 3.3); Huggett (2006, 70) for this kind of complaint. Sklar (1974, 231) notes a tradeoff between the substantivalist’s explanatory power and relationalist’s parsimony; Mundy (1983, 207) notes one between the relationalist’s parsimony and substantivalist’s simplicity. Belot (2011) suggests that parsimony in fact favors substantivalism (while arguing against using simplicity considerations to draw metaphysical conclusions).

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of physical reality. The substantivalist and relationalist, as I see them, disagree about the fundamental physical level, which is why the matching principle can distinguish between them. This is a substantive debate about the fundamental nature of the world according to physics; a debate about what makes it the case that the spatiotemporal structure required by the physics holds. The traditional debate centered on whether we need to posit an independently existing space in order to account for objects’ motions. The debate that I have presented is a natural descendant: a debate about whether we need to posit a spatiotemporal structure apart from material bodies to support the theory that best accounts for objects’ motions. This is a substantive debate, which we currently have reason to believe the substantivalist is winning.⁶⁰ Rutgers University

R E F E REN C E S Albert, David (1996). “Elementary Quantum Metaphysics.” In J. T. Cushing, A. Fine, and S. Goldstein, eds., Bohmian Mechanics and Quantum Theory: An Appraisal, pp. 277–84. Dordrecht: Kluwer Academic. Albert, David (2000). Time and Chance. Cambridge, MA: Harvard University Press. Albert, David (2012). “Philosophy of Physics.” In Encyclopaedia Britannica Online Academic Edition. Encyclopaedia Britannica Inc.: https://academic.eb.com/. Albert, David (2017). “On the Emergence of Space and Time.” Unpublished manuscript. Arntzenius, Frank (2012). Space, Time, and Stuff. Oxford: Oxford University Press. Bain, Jonathan (2006). “Spacetime Structuralism.” In Dennis Dieks, ed., The Ontology of Spacetime, Volume I, pp. 37–65. Amsterdam: Elsevier. Baker, David John (2005). “Spacetime Substantivalism and Einstein’s Cosmological Constant.” Philosophy of Science (Proceedings) 72(5): 1299–311. Barbour, Julian (1982). “Relational Concepts of Space and Time.” British Journal for the Philosophy of Science 33(3): 251–74. Barbour, Julian (2000). The End of Time. New York: Oxford University Press. ⁶⁰ For comments and discussion, I am grateful to Ori Belkind, Karen Bennett, Jim Binkoski, Carolyn Brighouse, Andrew Chignell, Ted Sider, and audiences at Rutgers University, the University of Illinois-Champaign, the University of Western Ontario, the University of Wisconsin-Milwaukee, Brown University, the Philosophy of Science Association, the University of North Carolina-Chapel Hill, Tel Aviv University, the Hebrew University of Jerusalem, the University of Massachusetts-Amherst, and the University of Oxford. Many thanks also to the anonymous reviewers for, and the editors of, this volume. This research was funded in large part by the National Science Foundation STS Scholars Award No. 1430435.

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Barbour, Julian (2001). The Discovery of Dynamics. New York: Oxford University Press. Barbour, Julian and Bruno Bertotti (1982). “Mach’s Principle and the Structure of Dynamical Theories.” Proceedings of the Royal Society A 382(1783): 295–306. Beebee, Helen (2000). “The Non-Governing Conception of Laws of Nature.” Philosophy and Phenomenological Research 61: 571–94. Belot, Gordon (1999). “Rehabilitating Relationalism.” International Studies in the Philosophy of Science 13(1): 35–52. Belot, Gordon (2000). “Geometry and Motion.” British Journal for the Philosophy of Science 51: 561–95. Belot, Gordon (2011). Geometric Possibility. Oxford: Oxford University Press. Belot, Gordon and John Earman (2001). “Pre-Socratic Quantum Gravity.” In Craig Callender and Nick Huggett, eds., Physics Meets Philosophy at the Planck Scale, pp. 213–55. Cambridge: Cambridge University Press. Brading, Katherine and Elena Castellani (2007). “Symmetries and Invariances in Classical Physics.” In Jeremy Butterfield and John Earman, eds., Handbook of the Philosophy of Science: Philosophy of Physics, Part B, pp. 1331–67. Amsterdam: Elsevier. Brighouse, Carolyn (1994). “Spacetime and Holes.” In David Hull, Mickey Forbes, and Richard Burian, eds., PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association, 1994, Volume 1: Contributed Papers, pp. 117–25. East Lansing, MI: Philosophy of Science Association. Brighouse, Carolyn (1999). “Incongruent Counterparts and Modal Relationism.” International Studies in the Philosophy of Science 13(1): 53–68. Brighouse, Carolyn (2014). “Geometric Possibility—An Argument from Dimension.” European Journal for Philosophy of Science 4(1): 31–54. Brown, Harvey R. (2005). Physical Relativity: Space-Time Structure from a Dynamical Perspective. Oxford: Oxford University Press. Brown, Harvey R. and Oliver Pooley (2002). “Relationalism Rehabilitated? I: Classical Mechanics.” British Journal for the Philosophy of Science 53: 183–204. Brown, Harvey R. and Oliver Pooley (2006). “Minkowski Space-Time: A Glorious Non-Entity.” In Dennis Dieks, ed., The Ontology of Spacetime, pp. 67–89. Amsterdam: Elsevier. Curiel, Erik (2016). “On the Existence of Spacetime Structure.” British Journal for the Philosophy of Science. doi: 10.1093/bjps/axw014. Dasgupta, Shamik (2009). “Individuals: An Essay in Revisionary Metaphysics.” Philosophical Studies 145: 35–67. Dasgupta, Shamik (2011). “The Bare Necessities.” Philosophical Perspectives 25: 115–60. Dasgupta, Shamik (2015). “Substantivalism vs Relationalism about Space in Classical Physics.” Philosophy Compass 10(9): 601–24. DiSalle, Robert (1994). “On Dynamics, Indiscernibility, and Spacetime Ontology.” British Journal for the Philosophy of Science 45(1): 265–87. DiSalle, Robert (2002). “Newton’s Philosophical Analysis of Space and Time.” In I. Bernard Cohen and George E. Smith, eds., The Cambridge Companion to Newton, pp. 33–56. Cambridge: Cambridge University Press. DiSalle, Robert (2006). Understanding Space-Time: The Philosophical Development of Physics from Newton to Einstein. Cambridge: Cambridge University Press.

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Dorato, Mauro (2000). “Substantivalism, Relationism, and Structural Space-time Realism.” Foundations of Physics 30(10): 1605–28. Dorato, Mauro (2008). “Is Structural Spacetime Realism Relationism in Disguise? The Supererogatory Nature of the Substantivalism/Relationism Debate.” In Dennis Dieks, ed., The Ontology of Spacetime, Volume II, pp. 17–37. Amsterdam: Elsevier. Dorr, Cian (2010). “Of Numbers and Electrons.” Proceedings of the Aristotelian Society CX: 133–81. Earman, John (1970). “Who’s Afraid of Absolute Space?” Australasian Journal of Philosophy 48(3): 287–319. Earman, John (1989). World Enough and Space-Time. Cambridge, MA: MIT Press. Earman, John and John Norton (1987). “What Price Spacetime Substantivalism? The Hole Story.” British Journal for the Philosophy of Science 38: 515–25. Einstein, Albert (2002). The Collected Papers of Albert Einstein, Volume 7, trans. Alfred Engel. Princeton, NJ: Princeton University Press. Esfeld, Michael and Vincent Lam (2008). “Moderate Structural Realism about Space-Time.” Synthese 160: 27–46. Field, Hartry (1980). Science Without Numbers. Oxford: Blackwell. Field, Hartry (1984). “Can We Dispense With Spacetime?” In P. Asquith and P. Kitcher, eds., PSA 1984: Proceedings of the 1984 Biennial Meeting of the Philosophy of Science Association, Volume 2, pp. 33–90. East Lansing, MI: Michigan State University Press. Field, Hartry (1989). Realism, Mathematics and Modality. Oxford: Blackwell. Fine, Kit (2001). “The Question of Realism.” Philosopher’s Imprint 1: 1–30. Fine, Kit (2012). “Guide to Ground.” In Fabrice Correia and Benjamin Schnieder, eds., Metaphysical Grounding: Understanding the Structure of Reality, pp. 37–80. Cambridge: Cambridge University Press. Friedman, Michael (1983). Foundations of Space-Time Theories. Princeton, NJ: Princeton University Press. Geroch, Robert (1981). General Relativity from A to B. Chicago, IL: University of Chicago Press. Greaves, Hilary (2011). “In Search of (Spacetime) Structuralism.” Philosophical Perspectives 25: 189–204. Hicks, Michael Townsen and Jonathan Schaffer (2017). “Derivative Properties in Fundamental Laws.” British Journal for the Philosophy of Science 68(2): 411–50. Hirsch, Eli (2011). Quantifier Variance and Realism: Essays in Metaontology. New York: Oxford University Press. Hoefer, Carl (1996). “The Metaphysics of Space-Time Substantivalism.” The Journal of Philosophy 93(1): 5–27. Hoefer, Carl (1998). “Absolute versus Relational Spacetime: For Better or Worse, the Debate Goes On.” British Journal for the Philosophy of Science 49: 451–67. Horwich, Paul (1978). “On the Existence of Time, Space and Space-Time.” Noûs 12(4): 397–419. Huggett, Nick (1999). Space from Zeno to Einstein: Classic Readings with a Contemporary Commentary. Cambridge, MA: MIT Press.

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Huggett, Nick (2006). “The Regularity Account of Relational Spacetime.” Mind 115: 41–73. Huggett, Nick and Carl Hoefer (2009). “Absolute and Relational Theories of Space and Motion.” Stanford Encyclopedia of Philosophy (Fall 2009 Edition): https:// stanford.library.sydney.edu.au/archives/fall2009/entries/spacetime-theories/. Huggett, Nick and Christian Wüthrich (2013). “The Emergence of Spacetime in Quantum Theories of Gravity.” Studies in History and Philosophy of Modern Physics 44: 273–5. Ismael, Jenann and Bas C. van Fraassen (2003). “Symmetry as a Guide to Superfluous Theoretical Structure.” In Katherine Brading and Elena Castellani, eds., Symmetries in Physics: Philosophical Reflections, pp. 371–92. Cambridge: Cambridge University Press. Knox, Eleanor (2014). “Newtonian Spacetime Structure in Light of the Equivalence Principle.” British Journal for the Philosophy of Science 65: 863–80. Ladyman, James and Don Ross (2009). Every Thing Must Go: Metaphysics Naturalized. Oxford: Oxford University Press. Leeds, Stephen (1995). “Holes and Determinism: Another Look.” Philosophy of Science 62: 425–37. Loewer, Barry (2001). “From Physics to Physicalism.” In Carl Gillet and Barry Loewer, eds., Physicalism and Its Discontents, pp. 37–56. Cambridge: Cambridge University Press. Malament, David (1976). “Review of Lawrence Sklar, Space, Time, and Spacetime.” Journal of Philosophy 73(11): 306–23. Malament, David (1982). “Review of Hartry Field, Science Without Numbers: A Defense of Nominalism.” Journal of Philosophy 79(9): 523–34. Manders, Kenneth L. (1982). “On the Space-Time Ontology of Physical Theories.” Philosophy of Science 49(4): 575–90. Maudlin, Tim (1993). “Buckets of Water and Waves of Space: Why Spacetime is Probably a Substance.” Philosophy of Science 60(2): 183–203. Maudlin, Tim (2012). Philosophy of Physics: Space and Time. Princeton, NJ: Princeton University Press. Maudlin, Tim (2015). “How Mathematics Meets the World”: https://fqxi.org/ data/essay-contest-files/Maudlin_How_Mathematics_Mee.pdf. Mundy, Brent (1983). “Relational Theories of Euclidean Space and Minkowski Space-Time.” Philosophy of Science 50: 205–26. Mundy, Brent (1986). “Embedding and Uniqueness in Relational Theories of Space.” Synthese 67(3): 383–90. Mundy, Brent (1992). “Space-Time and Isomorphism.” Philosophy of Science (Proceedings) 1: 515–27. Nerlich, Graham (1994a). The Shape of Space. 2nd edn. Cambridge: Cambridge University Press. Nerlich, Graham (1994b). What Spacetime Explains: Metaphysical Essays on Space and Time. Cambridge: Cambridge University Press. North, Jill (2008). “Two Views on Time Reversal.” Philosophy of Science 75(2): 201–23. North, Jill (2009). “The ‘Structure’ of Physics: A Case Study.” Journal of Philosophy 106(2): 57–88.

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Paul, L. A. (2013). “Categorical Priority and Categorical Collapse.” Aristotelian Society Supplementary Volume 87: 89–113. Pooley, Oliver (2013). “Substantivalist and Relationalist Approaches to Spacetime.” In Robert Batterman, ed., The Oxford Handbook of Philosophy of Physics, pp. 522–86. Oxford: Oxford University Press. Roberts, John T. (2008). “A Puzzle about Laws, Symmetries and Measurability.” British Journal for the Philosophy of Science 59: 143–68. Rosen, Gideon (2010). “Metaphysical Dependence: Grounding and Reduction.” In Bob Hale and Aviv Hoffmann, eds., Modality: Metaphysics, Logic, and Epistemology, pp. 109–35. Oxford: Oxford University Press. Rynasiewicz, Robert (1996). “Absolute Versus Relational Space-Time: An Outmoded Debate?” Journal of Philosophy 93(6): 279–306. Rynasiewicz, Robert (2000). “On the Distinction Between Absolute and Relative Motion.” Philosophy of Science 67: 70–93. Saunders, Simon (2013). “Rethinking Newton’s Principia.” Philosophy of Science 80(1): 22–48. Schaffer, Jonathan (2009). “On What Grounds What.” In David Chalmers, David Manley, and Ryan Wasserman, eds., Metametaphysics, pp. 347–83. Oxford: Oxford University Press. Sider, Theodore (2011). Writing the Book of the World. Oxford and New York: Oxford University Press. Sklar, Lawrence (1974). Space, Time, and Spacetime. Berkeley, CA: University of California Press. Slowik, Edward (2005). “Spacetime, Ontology, and Structural Realism.” International Studies in the Philosophy of Science 19(2): 147–66. Slowik, Edward (2016). The Deep Metaphysics of Space: An Alternative History and Ontology Beyond Substantivalism and Relationism. Switzerland: Springer. Stein, Howard (1970). “Newtonian Space-Time.” In Robert Palter, ed., The Annus Mirabalis of Sir Isaac Newton, 1666–1966, pp. 258–84. Cambridge, MA: MIT Press. (Originally published in Texas Quarterly (1967) 10: 174– 200.) Stein, Howard (1977a). “On Space-Time and Ontology: Extracts from a Letter to Adolf Grünbaum.” In John S. Earman, Clark N. Glymour, and John J. Stachel, eds., Foundations of Space-Time Theories: Minnesota Studies in the Philosophy of Science, Volume VIII, pp. 374–402. Minneapolis, MN: University of Minnesota Press. Stein, Howard (1977b). “Some Philosophical Prehistory of General Relativity.” In John S. Earman, Clark N. Glymour, and John J. Stachel, eds., Foundations of Space-Time Theories: Minnesota Studies in the Philosophy of Science, Volume VIII, pp. 3–49. Minneapolis, MN: University of Minnesota Press. Teller, Paul (1991). “Substance, Relations, and Arguments about the Nature of Space-Time.” The Philosophical Review 100(3): 363–97. van Fraassen, Bas C. (1970). An Introduction to the Philosophy of Time and Space. New York: Columbia University Press. Wallace, David (2012). The Emergent Multiverse: Quantum Theory According to the Everett Interpretation. Oxford: Oxford University Press.

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2 Relative Locations Andrew Bacon Substantivalism is the view that locations exist independently of the objects that they are locations of. Thus, for example, the moon’s present location would have existed even if it had not been located there and, reciprocally, there are regions that could have been the location of some material object, but aren’t (unoccupied regions). Substantivalism can have a number of different motivations. In ordinary English we frequently talk about, and quantify over, locations. For example, I can talk about where Emily went on vacation, or where I left my keys; I can in some cases talk about unoccupied locations, such as the place I would have been had I traveled halfway to the moon. The substantivalist has a ready answer for what we are doing when we quantify in this way: we are quantifying over regions of space-time. If this were the central motivation, however, it would suggest that the dispute is primarily about the existence of locations. This is not quite right: even those who oppose substantivalism— the relationists—are often happy to accept ordinary talk about locations provided that that talk can be recovered from a more acceptable fundamental ontology. (For these theorists, locations are a bit like shadows and holes: they are, metaphysically speaking, ‘second-rate’ entities in some sense. We can readily quantify over them, but their existence depends on other kinds of things: a location on the thing that occupies it,¹ much like a shadow on the thing that casts it, and a hole on the thing it penetrates.) Substantivalists arguably should not accept this reduction of our ordinary talk of locations to quantification over regions of space-time either. My pocket is not a region of space-time, and nor is Paris—for example, my pocket changes its shape as it goes through the washing machine, and a region of space-time cannot change its shape. But my pocket and Paris are

¹ Or perhaps, if the region is empty, the possible objects that could have occupied it. See Forbes (1993).

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the kinds of things I might be asking about when I ask where my keys are or when I ask where Emily went on vacation.² Since I find the above sorts of objections compelling, I shall place little weight in what follows on the arguments from ordinary language. In my view, the strongest motivations for substantivalism—the motivations that play a central role in this paper—are slightly more theoretical. Space-time points tend to appear in the formulation of many of our simplest physical theories and this gives us good reason to take their existence seriously. Unfortunately the most straightforward versions of substantivalism and relationism suffer from having a pair of undesirable, and arguably related, consequences. Substantivalism predicts the existence of physically indistinguishable worlds that differ solely concerning where things are located. For according to substantivalism there could be two worlds that differ from each other only in that each object’s location in one world has been displaced relative to the other by some fixed distance in some particular direction (see Leibniz and Clarke 2007). By refusing to take talk of locations and regions of space-time at face value the relationist does not face this problem in its most acute form. However, relationists are subject to a somewhat similar concern. The most straightforward versions of relationism take geometrical properties and relations to be a matter of fundamental relations holding between material objects and abstract objects, such as numbers to represent distances between objects, vectors to represent forces, and so on. However much like the substantivalist’s location relation, I shall argue, one can have physically indistinguishable worlds in which the relations between the material objects and the appropriate abstracta have undergone a similar kind of displacement. In this paper I shall in response to these worries be advocating for a novel version of substantivalism underwritten by a non-standard account of the relation between material objects and the regions they occupy. As I cautioned against earlier, the theory is not intended to model our ordinary way of speaking about locations. However, the theory does allow one to give a story about how the shapes of objects, the distances between them, and other such facts can emerge from the relations they stand in to a sufficiently structured space-time manifold. Here is an outline of the paper. In section 1, I review Leibniz’s shift argument against substantivalism and argue that it provides us with reasons to look for alternatives to the orthodox version of substantivalism. In section 2.1, I consider the analogous problem involving abstracta for ² Of course some people—the supersubstantivalists—do identify my pocket with a region; however, some fairly elaborate maneuvers are needed to make sense of the idea that ordinary things could have had different shapes.

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(versions of ) relationism, and consider the options for a relationist who wishes to give an account of geometrical properties without reference to abstract objects. To this effect I outline in section 2.2 a result that says that any first- or second-order language that can express, relative to some class of models, all possible configurations of a system of n particles either must have an infinite set of fundamental primitives or must admit a model with an infinite ontology (whether that be an ontology of space-time points, abstract objects, or something else). In section 3.1, I develop a substantivalist theory according to which an object is located at a region of space-time if and only if it is located at every transformation of that region. Thus unlike orthodox substantivalism, objects are multiply located in a fairly far-reaching way. Finally, I consider two further technical issues in the appendices: the treatment of space-time fields and the first-order theory governing the interaction of parthood and location.

1 . L E I B N I Z’ S SH I F T A R G U M E N T Substantivalism is subject to an old objection that traces all the way back to Leibniz (Leibniz and Clarke 2007). Pick some direction and distance, and imagine a world at which the location of every object has been translated uniformly in that direction by that distance, but in which every other property of the object has been kept the same. According to the substantivalist, worlds related by such operations are genuinely different because they disagree about facts concerning which particular regions of space-time each object is located at. Substantivalism thus falls afoul of the following seemingly attractive thesis: NO SHIFTS: There are no differences between shifted worlds. A pair of worlds are ‘shifts’ of one another if they are related by a translation of every object’s location of the sort described above. The crucial feature of shifts to highlight here is that they preserve all the laws of physics, and arguably all observable properties and relations between objects.³ Very similar arguments could be made by appealing to other spatial transformations operations: rotating about some axis or reflection about some plane.

³ One might object here on the grounds that one can have singular thoughts about particular regions of space-time, and thus knowledge, that isn’t preserved under transformations. (I can know that I’m located right here, at this particular region of space-time for example—I clearly wouldn’t know this in a world where I wasn’t located right here). I’m going to set this issue to one side for the time being, but see Maudlin (1993) for more discussion.

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And just as we can consider spatial transformations, we may also consider operations involving time: time reversal (a kind of temporal reflection), temporal shifts (moving every event forwards or backwards a fixed amount of time) and boosts (uniform shifts of velocity), and of course, arbitrary combinations of any of these operations. We shall call a combination of these operations a Galilean transformation.⁴ The principle NO SHIFTS has been given a number of distinct motivations over the years, some less convincing than others. While some have overtly theistic or verificationist premises, NO SHIFTS has remained central to the debate about the existence of space-time, even though attempts to motivate it from more general principles have changed. Most contemporary philosophers take the best justification for NO SHIFTS to be a defeasible one: theories that postulate undetectable structure that play no role in explaining the observable world are not to be preferred over theories that do not postulate this structure but are otherwise just as simple and explanatory (see, for example, Russell 2014 and Pooley 2013). As a silly example, one could imagine a theory which postulates the existence of an absolute origin from which every point can be assigned an absolute distance. Although a particular origin could well make a nominal appearance when the dynamical laws are formulated in terms of a particular coordinate system, a world in which an alternative point were the absolute origin would be identical to our own in all physical respects. This kind of additional structure is completely undetectable and is also unnecessary to the formulation of the dynamical laws. A less silly example is Newton’s original formulation of classical mechanics, in which there was a distinguished velocity called ‘absolute rest’: objects traveling at that velocity counted as being at absolute rest, and objects moving relative to it had absolute motion. The move from Newtonian space-time to Galilean space-time, in which this redundant structure is eliminated, is now universally accepted. The main problem for this line of reasoning is that unlike the case of the absolute origin and absolute rest, we don’t have a clear alternative theory in which the redundant structure is eliminated. Thus, while many philosophers take this version of Leibniz’s challenge to be a serious one, ultimately they reject it on the grounds that the most prominent alternative to

⁴ Note on terminology: sometimes reflections of various sorts are excluded from the definition of a Galilean transformations—I always intend them to be included in what follows. Under the operation of composition these transformations form a group called the Galilean group. Note, however, that I do not include enlargements in the space of transformations: these are not in fact symmetries of the laws of Newtonian physics, since the rate at which two particles of equal mass will accelerate towards one another depends on the distance they are separated.

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substantivalism—relationism—doesn’t share the theoretical virtues of the substantivalist theory (see, for example, Maudlin 2012: 65–6 and Pooley 2013).⁵ This strikes me as sufficient reason to explore alternative theories which don’t posit this sort of redundant invisible structure.⁶

2. RE LA TI ONISM The best argument for NO SHIFTS rests on general considerations of theory choice, and is thus successful only if there is a simple alternative to the standard version of substantivalism that respects it. It is often assumed that relationism is this alternative. Unfortunately relationism faces a somewhat similar set of issues.

2.1. Mathematical relationism According to relationism, the motions and geometrical properties of material objects are not grounded by the relations they stand in to a background space-time manifold. Thus the relationist is in need of some other kind of theory of these sorts of properties and relations. To keep things simple we shall restrict our attention to a relatively sparse world consisting of three point particles, arranged in a certain configuration, traveling with certain momenta as governed by the laws of Newtonian gravitation. Since it is clear that there is a difference between worlds in which the particles are arranged in, say, a regular triangular shape, and those in which they are colinear, or worlds in which the masses of the particles, and thus the forces between them, are greater or smaller, the relationist owes us an account of what these differences consist in. According to the most straightforward versions of relationism, geometrical properties like these consist in fundamental relations holding, in certain configurations, between the material objects, numbers, and other kinds of abstract objects. Thus, for example, distances can be represented by a fundamental three-place relation, Dxyz, taking two particles and a real ⁵ Although there have been several attempts to develop a ‘Machian’ theory of this sort, it’s unclear how successful they are. See Barbour and Bertotti (1977; 1982). See Pooley (2013) for an overview. ⁶ This general sort of project has been attempted by Dasgupta (2009) and Russell (2014), but both sorts of theories involve contentious metaphysical ideology such as a many–many grounding relation or a primitive factuality operator. The approach I will be recommending here is straightforwardly intelligible to anyone who has the concepts presupposed in the original relationism/substantivalism debate—specifically, anyone willing to theorize in terms of the location relation.

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number, satisfying the axioms of a metric space. (In mathematical contexts, it is standard to equivalently describe such a relation as a function d(x,y) mapping x and y to the distance between them.) Thus three particles arranged in a regular triangle are such that each pair of them is related to the same number. The gravitational force exerted by one particle on another can similarly be represented by a relation, Fxyz, taking two particles and a vector from some suitable normed three-dimensional vector space, and mass can be represented by a binary relation between particles and real numbers Mxy (equivalently a function f(x,y) mapping pairs of particles to vectors and a function m(x) mapping particles to real numbers).⁷ Given this kind of background theory, one can begin to attempt to reconstruct a relationistically acceptable version of Newtonian physics: for example, we might subject F to the restriction that if F holds between x, y, and v it holds between y, x, and –v according to Newton’s third law.⁸ Although there is room for much variation in the details let us call this general kind of approach, in which geometric and physical properties are grounded in relations between the physical and the platonic realms, mathematical relationism. (For a prominent theory of this sort, which freely employs relations to mathematical objects, see Barbour and Bertotti 1982.) The framework just outlined seems to be susceptible to an objection similar to the one afflicting substantivalism: it seems that one can describe two different configurations of these relations between objects and numbers that correspond to the same physical scenario. For consider any legal arrangement of our three-particle system: this will consist in the relations D,F, and M holding between our particles, numbers, and vectors in some configuration. In particular, each pair of particles, x and y, will be related by F to some vector v in an abstract vector space V, representing the force that x exerts on y. Now consider another configuration which agrees with the original regarding which things are related via M and D, but in which the vector related to each pair of particles by F has been uniformly switched, at every time, for the result of rotating that vector by 90 degrees about some fixed axis in the abstract vector space it belongs to. Notice that this operation ⁷ I have set aside the issue of time here. A proper treatment may involve introducing another argument place to these predicates for a time, or giving them a tense-logical treatment (the latter seems natural for those relationists, such as Arthur Prior, who reject time as well as space). ⁸ One might wonder if one could eliminate forces just by appealing to distances and masses (or perhaps masses in favor of distances and forces). This is far from clear: for example, a pair of equally massive particles orbiting one another in a circular motion will have constant masses, and be at a constant distance from one another, although the force acting between them is constantly changing, so that forces can’t be recovered from mass and distance alone.

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will preserve everything of importance for the modeling of forces—crucially, it will preserve the inner product which represents the lengths of the vectors and the angles between them. This operation bears many of the marks of Leibniz’s shift argument: the operation of rotating the vector that represents the force between any pair of particles appears to be a symmetry of our relationist theory. Although this is not on its own a sufficient reason to reject this theory, it does mean it lacks the advertised advantages over substantivalism. We must, however, be careful to distinguish the symmetry just described from a different operation. For example, one can imagine a possible world that is not legal by the standards of the Newtonian theory of gravitation, in which the force that the particle y exerts on x is not parallel, but orthogonal to the line passing between them—see the ‘Illegal world’ diagram below. (Note: each diagram is supposed to represent three different states all at the same time t.)

Legal world 1 x

v

y

Legal world 2 x

u

y

Illegal world u x

y

This is not the operation I am describing: if the force y exerts on x really is acting perpendicular to the line between them, then at later times x would accelerate in a direction perpendicular to that line, so that the distance between x and y ought to increase for a period of time (absent other forces).⁹ In the original example, however, I stipulated that the distance facts are the same at both possibilities at all times, so if the original case is legal, the distance between x and y will decrease for a period in the variant world. The ‘shifted’ world is therefore a world in which the physical force in fact points from x to y (since that is the direction that x will move in), but in which its so pointing consists in x and y standing, via F, to a different abstract vector (u instead of v, as in ‘Legal world 1’ and ‘Legal world 2’). Of course, one could insist that one of these configurations is not really a metaphysical possibility. Perhaps, in that configuration, only v but not u could possibly represent the force exerted by y on x. Similar moves can also be made in the substantival case: perhaps no two worlds that agree about the non-locational profile of the material objects can disagree about their locations in space-time. But such responses do little to assuage the feeling of arbitrariness—why, for example, is it necessary that when the particles

⁹ We are assuming here that the world is legal in at least the sense that particles accelerate in the direction of the forces acting on them, even though it is illegal in the sense that the force exerted by a particle does not point towards the particle.

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are arranged this particular way, they are always located at this region of space-time? Note also that the vector representing the force y exerts on x can differ between times, even when those times agree about the mass and distance facts. For consider a system of two equally massive point particles, x and y, orbiting one another in a circular motion. The distance between the particles, and their masses, remain fixed over time. However, the vector representing the force between x and y is rotating at a constant rate.¹⁰ Thus, in fact, the states depicted in ‘Legal world 1’ and ‘Legal world 2’ could represent the state of the same world at two different times. So the thought that it’s impossible for v or impossible for u to represent the force acting between x and y when the distance and mass facts are in such a state seems to rule out certain kinds of physically possible scenarios.¹¹ It may strike the reader that the above problem is specific to the particular form of mathematical relationism I have outlined above, which relied on the use of vectors. However, the problem outlined above is but an instance of a more general problem for relationism outlined by Hartry Field (1980). For example, Field notes that we can describe the world equally well using different units. Thus a configuration of M, D, and F in which M relates each particle to its mass in kilograms, and the configuration in which M relates each particle to its mass in pounds but is otherwise the same, correspond to the same physical scenario. The ‘shift’ that we are performing in this case is that of multiplication by a factor of a scalar quantity, rather than a rotation of a vector quantity.¹² There is something very suspect about a fundamental metaphysics in which one ¹⁰ This is part of the reason why this version of relationism can avoid Newton’s bucket. ¹¹ One could try to make this argument more rigorous by appeal to the principle that whatever sometimes happens could happen (see Dorr and Goodman Forthcoming), for in the two-particle world described, x and y sometimes bear F to v and sometimes to u, but always bear M and D to fixed numbers. (Although I do not myself subscribe to this principle, there is something very compelling about the intuition in this case.) ¹² In the case of scale dependence there’s a fairly straightforward fix: instead of having a primitive binary relation Mxy relating each object to a number, have a ternary relation Mxyz relating two physical objects to a number that simply tells us what the ratio of the mass of x and y is. Whatever units we choose, the ratio of two masses will remain the same. However, this doesn’t really speak to the underlying problem that when abstract numbers appear in physics there are usually other abstract objects that would do the job just as well; usually the choice of mathematical objects to use is made based on convenience. A slightly contrived version of this problem applies even to ratios of physical quantities—there are many abstract objects that could do the job of ratios equally well. Ratios are real numbers, and set theoretically we can construct these entities in several equally natural ways: we can represent them by certain sets of rational numbers, called Dedekind cuts, or by certain kinds of converging sequences of rational numbers called Cauchy sequences. Even if real numbers are sui generis entities, and are not identical to either of these constructions out of rational numbers, that just means that we have one more isomorphic mathematical structure to choose between.

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has to choose between kilograms and pounds, quite aside from its the fact that it generates invisible distinctions. The issue to do with units above appears to be part of a broader set of problems afflicting theories that postulate relations to mathematical objects in order to account for the fundamental properties of physical objects. In general there is a lot of arbitrariness in our choices when we represent the world using abstracta: not just in the choice of units, but sometimes in more far-reaching ways. For example, Newtonian mechanics is simple enough to be formulated in terms of affine spaces—in which pairs of space-time points are related to objects in a mathematical vector space—but it could also be formulated in the language of differential geometry in which case each space-time point is instead related to an equivalence class of real-valued smooth functions on a neighborhood of that point. Similar points extend to most uses of abstracta by physicists, whether it concerns the choice of origin and orientation of a coordinate system, the choice of units, or sometimes the very choice of mathematical formalism itself.¹³ Thus some philosophers, such as Field, have maintained on this basis that ‘relations between physical things and numbers are conventional relations that are derivative from more basic relations that hold among physical things alone’ (1984: 46). Field has in mind relations of magnitude in particular and, although he is presumably at least partly being motivated by his nominalism, also mentions considerations such as those about units mentioned above. I am not a nominalist, but I think the idea is compelling nonetheless. Even if one rejects nominalism, it would be puzzling if the fundamental laws of motion, for example, depended for their truth on the existence of abstract objects. One could dramatize this intuition by imagining that all numbers were to disappear tomorrow: insofar as we can entertain such a hypothesis, it doesn’t seem likely that there would be any serious consequences for the non-mathematical universe—it’s not like the earth would stop orbiting the sun, or that planes would start falling out of the sky. Gravity, for example, seems to be a physical force whose existence does not depend on the numbers we use to represent it: surely concrete things could have moved about in the way that physics demands even if there hadn’t been any numbers.

2.2. The prospects for alternative versions of relationism It is best, I think, for a relationist to maintain that material objects have their topological, metrical, and geometrical structure intrinsically: structure that is ¹³ For example, some physicists prefer to theorize in the language of category theory, which often leads to certain kinds of more familiar abstracta being replaced with categories.

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internal to the domain of material objects and is not merely inherited by their relation to a platonic realm of numbers. There are a couple of ways we could go about this project, depending on whether we take there to be an infinite number of metaphysically primitive geometrical properties or a finite number: (1) GEOMETRIC PRIMITIVISM: Take each geometrical property and relation to be metaphysically primitive. On this picture all geometrical properties are equally fundamental. This theory will have an infinite number of primitives: for each possible value of α there is a primitive relation between particles of being α meters apart; for each possible shape, there is a primitive fundamental property of having that shape; and so on. (2) GEOMETRIC REDUCTIONISM: Attempt to fix the geometrical structure by a smaller finite set of primitives. Perhaps to the notion of an object being an open sphere (Tarski 1983), or to the notion of two pairs of particles being congruent to another, and a particle being between two others (Hilbert 1899; Tarski 1959).¹⁴ If one thought that our fundamental properties and relations are governed by simple laws, the first option should strike us a deeply unsatisfactory. There are, for example, general geometrical laws relating collections of particles with certain shapes, and moreover physical laws governing how those shapes should evolve as the particles are attracted to and repel one another. In the first kind of theory we have no hope of writing these sorts of laws down: to achieve quantification over distances (which we need to do in order to talk about rates of change, for example) would require one to employ large infinitary conjunctions and disjunctions.¹⁵ The latter sort of approach to geometry has been developed extensively by Hilbert and Tarski, and has most prominently been championed in the philosophical literature by Field (1980) (see also Casati and Varzi 1999, Maudlin 2010, and Arntzenius and Dorr 2012 for some similar approaches to geometrical and related structure). To illustrate let us focus on the work of Hilbert (1899) and Tarski (1959). The usual way to represent distances between points would be to introduce a metric—a function from pairs of space-time points to real numbers—telling us how far apart the points are.

¹⁴ If we are interested in properties relating to the smoothness of the manifold, the notion of a converging sequence of points (see Arntzenius and Dorr 2012). Or if we are interested in topological properties, the notion of two things touching (see Casati and Varzi 1999) or a closed line (see Maudlin 2010). ¹⁵ For a comprehensive discussion of related issues, see Skow (2007); see also the further discussion in Kleinschmidt (2015).

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The mark of the Hilbert–Tarski program, by contrast, is that it employs a small number of geometrical properties and relations whose relata are concreta. These geometrical properties can be understood in both a substantivalist and a relationist setting by interpreting the primitives as applying to either regions of space-time or material objects respectively. In Tarski’s geometry of solids, for example, we have the following primitives: (1) A binary relation x
y whose intended interpretation says that x is a mereological part of y. (2) A unary predicate, Sx, stating that x is an open sphere.¹⁶ Hilbert’s axiomatization, also later refined by Tarski, instead only invokes a pair of relations whose arguments are point-like objects: (1) A three-place betweenness relation, Bxyz, whose intended interpretation states that x lies on the straight line segment between y and z. (2) A four-place spatial congruence relation, Cxyzw, whose intended interpretation states that the distance between x and y is the same as that between z and w. (For short: xy is congruent to zw.) It should be noted that in this setting the quantifiers are understood as ranging over point-like objects (space-time points or point-like material objects); if one wanted to make that restriction explicit one could introduce another primitive applying to mereological atoms, or one could simply stipulatively understand congruence so that it applies to no complex objects and define atomicity as standing in congruence relations to some things. The general approach is not limited to quantities representing distances either. Field has shown that the latter sort of theory can be extended to other quantities. If I want to talk about the numerical value of a field at a given point—for example, the mass density field or the gravitational potential—I can employ a similar trick. One can introduce a congruence relation stating that the difference in gravitational potential, for example, at x and y is the same as the difference of the gravitational potential at z and w, and a betweenness relation saying the potential at x is between the potential at y and z respectively (see Field 1980).¹⁷ ¹⁶ This is what modern mathematicians would call an open ball; the word ‘sphere’ is now typically reserved for the two dimensional surface of a ball; however, this is not what Tarski meant by a sphere. ¹⁷ In order for this to work there must be a rich enough variety of field comparisons for us to be able reconstruct the field values (up to a scale) from the comparisons. This is guaranteed if we make the assumption, typical in physics, that physical fields are always continuous, so that a field is either constant everywhere or inhabits an open interval of the space of possible field values.

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One might hope that a relationist could employ a small set of geometrical primitives, such as those above, and attempt to recover geometric structure that way. Unfortunately, as Hartry Field has noted in (Field 1984), some of the above theories are simply not available to the relationist. In particular the Hilbert-style theory, employing congruence and betweenness, fixes the relevant geometric structure only if there is a sufficiently large number of geometric objects hanging around. This can be illustrated with a simple example: consider a situation in which the distance between x and y is two times the distance between z and w. While this may on the surface look as though we have a relation between four points and the number two, in the Hilbertian setting reference to the number two can be eliminated. This relation can be stated instead as follows: There is some point u between x and y such that both xu and uy are congruent to zw. If we wanted to say the distance between x and y was three times the distance between zw we’d say there was a u₁ and u₂ such that xu1 , u1 u2 , and u2 y were each congruent to zw. One can see, without much trouble, how to paraphrase away talk of arbitrary rational ratios of distances in this fashion. Note that once you have pinned down the rational distances between points, all remaining distances between points are fixed. So, given the existence of enough point-like objects, betweenness and congruence facts are enough to pin down all metric structure: no two worlds can agree about the betweenness and congruence facts and disagree about the geometrical facts. The above statement captures the notion of a pair of particles being twice as far apart as another pair perfectly well in a substantivalist setting, since according to that theory, for any two space-time points there is another space-time point between them. However, Field notes that this is not so for point-sized material objects: if I take the closest point to the earth on the edge of the moon, p, and take the closest point to the moon on the edge of the earth, q (suppose for a moment that both have definite boundaries) then, at least by the relationist’s lights, there are no entities between p and q, not even a space-time point. Thus, even if I were twice as far away from the earth as the moon is, I wouldn’t count as such by the lights of Hilbert’s analysis given a relationist ontology. Indeed, it is easy to see that there are distinct arrangements of me, the moon, and the earth, that agree about all betweenness and congruence facts but are nonetheless very different geometrically. The problem Field has identified here seems to be much more general. For example, Tarski defines the topological notion of two spheres touching

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(i.e. touching but not overlapping) in terms of the notion of a sphere as follows: x touches y if and only if, x is disjoint from y, and any two spheres containing x and disjoint from y are such that one is part of the other. Neither this definition nor any other will work in a relationist setting. Imagine two worlds containing two perfectly spherical balls, x and y, in an otherwise empty space. Suppose also that in the first they are touching, and in the second they are not. Both worlds agree with one another concerning which things are spheres and which things are parts of what, but they disagree about which things are touching. Thus, given a relationist ontology, the touching facts are not fixed by the sphere and parthood facts (in particular, because the only sphere that contains x is x itself, according to the relationist ontology, x and y count as ‘touching’ in both worlds by Tarski’s definition). It follows that in order to pin down the geometrical structure the relationist needs to introduce twice the distance and touching as new primitives. It is natural to wonder whether this can be done by adding only finitely many primitives; thus evading the undesirable aspects of PRIMITIVISM. To address this question let us focus on a simple example world consisting of only three point particles, arranged in some shape. There are an uncountable infinity of arrangements like this that differ regarding the ratios of the distances between the three particles. In this setting the primitivist strategy involves introducing, by brute force, an uncountable infinity of primitive ternary relations, so that for each possible arrangement a of the particles x,y,z there is a fundamental relation Ra that holds between x,y,z (in any order) iff those particles are in that arrangement. It is natural to wonder whether we can do better. Can we specify a theory of three point particles with finitely many primitives without expanding our ontology to include space-time points or numbers? In order to answer this question we need to state it a bit more precisely. An arrangement of three particles can be represented by the three particles x,y,z and the ratios of the distances between any pair of them. Formally, the set of possible arrangements of three particles, A, is the set of metric spaces (M, d) such that M ¼ fx, y, zg quotiented out by scale.¹⁸ Someone hoping to write down a theory capturing the geometry of three point particles must choose a language L—given by specifying some set of non-logical primitives—and present a theory which can be either specified axiomatically, or by a class of

¹⁸ That is to say, an arrangement is an equivalence class of such metrics where ðM , d Þ and ðM 0 , d 0 Þ are equivalent iff d ¼ α:d 0 for some α 2 ℝ .

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intended models, C. For example, in the Hilbert-style theory the primitives of our language are the congruence and betweenness relations, three constants denoting the three particles, and a location relation for stating the location of each particle. The class of intended models consists of threedimensional Euclidean spaces with the three particles x,y, and z located at three points of that space. Each model of the theory ought to correspond to some arrangement of the particles: there ought to be a surjective function Arr : C ! A. In the Hilbert-style theory Arr is easy to specify—each model easily determines an arrangement because the underlying metric of the Euclidean space tells us what the distances between the three distinguished points are. The Hilbert-style theory has a nice feature. For any two models M and M 0 in C of the theory in which the arrangement of x,y, and z differ (i.e. Arr(M) 6¼ Arr(M 0), one can find a sentence in the language of congruence and betweenness, ϕ, such that M  ϕ and M 0 ⊭ ϕ.¹⁹ In this way any two arrangements can be distinguished by some sentence of the language. The primitivist relationist theory also satisfies this constraint. The language of this theory has a primitive relation Ra xyz for each possible arrangement a 2 A. The models of this theory have a minimal relationist ontology: the domain of each model contains only the three particles x, y, z. Moreover, for any arrangement a 2 A there’s some model in which the objects are arranged that way: a model in which Ra applies to x,y, and z in any order, but in which Rb doesn’t apply, in any order, for any b distinct from a. Each model is therefore associated with a unique arrangement ArrðM Þ ¼ a where a is the arrangement such that the extension of Ra is non-empty in M. As with the Hilbert theory, whenever M and M 0 correspond to different arrangements— i.e. when ArrðM Þ 6¼ Arr M 0 —there is a sentence that is true in M but not M 0 , namely Ra xyz where a ¼ ArrðM Þ. The principle we have appealed to in each case is the following principle. Suppose that L is a theory with a class of intended models C, that includes models representing each possible arrangement of the three particles (that is to say, the function Arr associating each model with an arrangement is surjective). Then if the primitives of L express a physically complete set of

¹⁹ Proof: let the distances between x and y in a model M be denoted dM ðx, yÞ. If the arrangement of particles in M and M 0 differ then for some a, b, c, d 2 fx, y, zg the ratio dM ða, bÞ=dM ðc, d Þ and dM 0 ða, bÞ=dM 0 ðc, d Þ differ (if all these ratios agreed, they would be in the same arrangement). That means there are a pair of rational numbers, q and q 0 such that q < dM ða, bÞ=dM ðc, d Þ < q 0 but doesn’t hold when M is substituted for M 0 . As we indicated earlier, it is quite easy to express facts about rational distance ratios using a single sentence.

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fundamental relations, any two models representing different arrangements of the particles ought to be distinguishable by some sentence of L: DISTINGUISHABILITY: If M and M 0 correspond to different arrangements of the particles (i.e. Arr(M) 6¼ Arr(M 0 )) then there is some closed sentence ϕ 2 L such that M  ϕ and M 0 ⊭ ϕ. This constraint is quite important and is effectively a way of saying, in model theoretic terms, that the arrangement of the particles supervenes on the facts expressible in L. Without it one could not formulate an adequate physical theory: the forces particles exert on each other, for example, depend on the arrangement of those particles. In particular, particles arranged differently can behave differently. If the fundamental primitives do not distinguish between two possible arrangements of the particles then the behavior of the particles will not be determined by kinds of facts we are taking to be fundamental. Neither the arrangements nor the motions of the particles will supervene on the distribution of the fundamental properties and relations; particles could be arranged differently in two worlds even when the two worlds agree about all the fundamental facts as stated in our fundamental language, L. We can now prove the following limitative theorem. Theorem 2. Let L be a first or second order language, C a collection of models of L, and Arr : C ! A a surjective association of arrangements to models. Suppose that our class of models also satisfies the distinguishability constraint. Then one of the following is true: (1) L has infinitely many non-logical primitives. (2) C contains an infinite model.

This theorem is not particularly deep from a mathematical perspective.²⁰ However, it does clarify the situation: the two most salient responses to the theorem are (i) to adopt infinitely many primitives, as with PRIMITIVISM, or (ii) to adopt an infinite ontology, as with the substantivalist theory or the theory that postulates mathematical objects to account for the relations between the three particles. Note, however, that our theorem is quite general—we have allowed, for example, that which ‘extra’ objects exist can

²⁰ If L has a finite signature, then for each finite cardinality n, there are only finitely many different isomorphism classes of models of L of that cardinality (for an k place k relation there are only 2n possible relations over a domain of cardinality n, so there are nk1 þ ... þ nkm l :n possible models in a language with l constants and predicates of arity 2 k1 , . . . , km ). It follows that if each model in C is finite then there are at most countably many isomorphism classes of models of M. But since there are uncountably many arrangements, Arr must map two models in the same isomorphism class to the same arrangement, contradicting DISTINGUISHABILITY.

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depend on the arrangement of the particles. Perhaps if the particles are nicely and symmetrically arranged one can characterize that arrangement with a finite set of primitives without postulating extra objects. What the theorem shows is that whatever your theory, if it has finitely many nonlogical primitives, there will always be some arrangements for which one needs infinitely many objects present to distinguish those arrangement from other such arrangements. Note that one could also question the constraints placed on the kind of language employed in our theorem: we allowed ourselves first-order and second-order resources, but one might think it possible to do better with modal resources. A particularly natural strategy would be to combine modal resources with second-order resources. With a modest finitary principle of recombination for possible particles, guaranteeing that the outer domain of quantification is infinite, one could force the domain of the second-order quantifiers to range over an infinite collection of intensions even whilst keeping the inner domain of the first order quantifiers finite at each world. This could in principle allow one to simulate the kinds of things one would normally do with space-time or mathematical objects, whilst technically keeping our first-order ontology finite.²¹ Even if one grants that secondorder quantification does not in itself carry a commitment to abstracta (as I am inclined to think myself ), such proposals are not without their own difficulties, although it is beyond the scope of this paper to survey them in full (but see the discussion in section 2 (on using properties to simulate spacetime) and (on modal approaches) in Field 1984, section 9). (See also Mundy 1987 and Eddon 2013 for property theoretic account of quantities, and Belot 2011 for a fairly sophisticated example of a modal treatment of geometry.)²²

²¹ Here is an example of this kind of strategy, where we employ modal operators and higher-order quantification into sentence position: one could adopt a primitive Aðx, y, PÞ taking two terms and a sentence, roughly meaning P is an arrangement of x and y: in a possible worlds style model, for some distance d , P is the set of worlds at which the distance between x and y is d . From this one can define what it means for P to describe the arrangement of any finite collection of particles, and helping oneself to propositional quantification and a necessity operator, one can recover the congruence and betweenness relations. Thanks to Jeremy Goodman for discussion here. ²² According to the Mundy–Eddon view, there are infinitely many properties corresponding to each quantity (e.g. one property for each possible mass) and primitive higherorder relations between these properties that determine their quantitative structure. It is natural to view the Mundy–Eddon view as falling under the ‘infinite ontology’ branch of our dilemma, as particles are inheriting their quantitative structure by standing in relations to an infinite collection of mass properties. However, one could also formalize it in third-order logic with primitive third-order relations over second-order entities, in which case the theory is not within the remit of our theorem, and the first-order ontology could well be finite (even though the higher-order domains are infinite).

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One could respond to these arguments not by introducing relations to space-time points or numbers, but by introducing some other infinite physical structure that, unlike space-time, is invariant under Leibnizian symmetries. Although not relationism as traditionally conceived, one might hope to find an alternative to space-time from which distances and other quantities can be recovered without relinquishing NO SHIFTS. There are, of course, lots of ways of going about this strategy, and the project is too broad to say anything too conclusive about its prospects. A particularly natural way to go about this is to take a group of displacements of the universe of material objects as a primitive physical entity in its own right. According to this theory, particles do not get locations in this space, but rather pairs of particles get assigned ‘locations’—i.e. vectors in our space of parallel displacements representing the displacement between them.²³ Crucially, this ‘location’ remains the same under Leibniz shifts.²⁴ Note, however, that these projects all involve accepting a form of substantivalism, even if not the familiar sort. Rather than evaluating the prospects for theories like this—a worthwhile project for another time—I want to focus on the possibility of carrying out this sort of idea within the confines of orthodox substantivalism.

3 . M U L T I P LE L O C A T I O N These sorts of considerations strike me as a good prima facie reason to be a substantivalist: by appealing to relations between material objects and spacetime points we can recover enough structure to represent the distances and other quantitative properties and relations, without placing an undue significance on any particular mathematical representation of these facts. ²³ Note that the order of the pair matters: if v is the displacement between p and q, v is the displacement between q and p. ²⁴ For simplicity we might focus on a physical structure that is invariant under translations in space—the group of parallel displacements—however, mathematically it is straightforward to extend this idea to rotations using the notion of an angular displacement, and boosts and other continuous symmetries using other similar notions. A parallel displacement is just a vector quantity that points from one point of space-time to another, telling us the displacement between the two points; this quantity is invariant under translations in the sense that if one pair of particles is a translation of another the displacement vector between the first pair is the same as the second pair. The group of parallel displacements of Galilean space-time is itself a four-dimensional manifold, just like Galilean space-time, but it has more structure: it is a normed vector space, and thus unlike Galilean space, has a special point that is distinguished from the others (the 0 vector). The manifold is rich enough that the mathematical structure can be recovered in a Hilbert–Tarski–Field-style setting.

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A relationist, on the other hand, must either take the role of abstracta in physics more seriously than seems wise or attempt to emulate substantivalism using certain kinds of higher-order and modal resources. For someone following this line of reasoning, it would not be unreasonable to take these considerations to also reflect negatively on the NO SHIFTS principle. After all, substantivalism is often taken to entail that there are differences between shifted worlds. Note, however, that the role substantivalism plays in fixing the geometrical structure of material objects is quite different from the role that regions of space-time play in the shift argument, which make some specific assumptions about the location relation. To address the issue of geometric structure one postulates a manifold with its own intrinsic geometric structure. Geometric relations between material objects then exist only in a derivative sense: in virtue of material objects standing in some relation—call this the ‘location relation’—to space-time regions that have the geometric properties in question. Nothing we have said so far, however, rules out the possibility that the location-like relations material objects bear to space-time regions are invariant under Leibnizian transformations. Note that as I have introduced it above, the location relation is defined implicitly by its role as that relation between objects and space-time from which the geometrical properties of objects can be recovered. As such, one shouldn’t assume that it corresponds to the pretheoretic of notion of location we are used to employing when we are not doing fundamental metaphysics. The ordinary notion of location usually relates us to places—like Paris, the moon, and so on—and not regions of space-time.²⁵ With this caveat in mind, we can ask what it would mean for the world to remain unchanged by a uniform shift of the locations of each material object. Ignoring time for a moment, and restricting ourselves to a three-dimensional universe it would require the following: SHIFT INVARIANCE: If x is located at y and y 0 is a Euclidean transform of y (a combination of shifts, rotations, and reflections) then x is located at y0 . Later we will see that SHIFT INVARIANCE can be formulated very simply without appeal to Euclidean transforms (see section 3.4). SHIFT INVARIANCE arguably follows from a certain conception of the location relation. As mentioned above, we are introducing the location relation by the job it plays, regardless of the distance from our pretheoretic notion. Assuming that an object’s geometric properties are entirely determined by its shape, the job to be satisfied can be summarized by the idea

²⁵ More importantly, objects are usually thought to be located at no more than one place on this conception.

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that the shape of an object is given by the shape of its location.²⁶ Thus the location relation must satisfy the constraint that if it holds between a material object O and a region R, then O has the shape of R. The simplest relation satisfying this constraint is the relation ‘O has the same shape as R’, and on this interpretation SHIFT INVARIANCE is true.²⁷ (Of course, there are also one–one relations that satisfy the job, relating each object to a unique region: but there are infinitely many of them that are equally good, and if the job description is our only criteria for choosing, to pick one of them seems arbitrary.) Of course SHIFT INVARIANCE requires material objects to be multiply located in a fairly radical way. The crucial point is that if we are at a world in which objects are multiply located in this sort of way, then uniformly subjecting every object to a Euclidean transformation will leave each object with the locations it had before the transformation. Thus we do get to maintain the principle NO SHIFTS. This idea generalizes to other kinds of transformations. For example, if you thought that embiggenings don’t generate genuine differences then perhaps a world in which the displacement between some particular point, p, and the location of each particle had been doubled (an embiggening around p) also results in a state of affairs no different from the one you started out with. (In this case, however, it is less clear that the transformation is a symmetry of the underlying physics.²⁸) In the special theory of relativity, the relevant transformations are Lorentz transformations. And in the context of general relativity, the transformations that seem natural are diffeomorphisms. In this latter setting the metric structure of the manifold seems like a contingent and changeable feature of the world, and thus transformations which preserve the differential structure but not the metric properties seem like the natural transformations to use. (Although, perhaps surprisingly, this means that properties like shape are not independent features of ²⁶ More generally, we can read off geometrical relations between more than one object from the shape of their fusion. ²⁷ Note that although one can identify the location relation with the relation of having the same shape, that needn’t be the order of reduction: in particular, it’s consistent to assume, as we have been, that space-time has its geometric structure intrinsically and that the shape of a material is given by its location(s). ²⁸ In a world in which the distances between two point particles of mass m has been doubled, they will accelerate towards each other at different rates, because the forces between them are inversely proportional to the square of the distance between them. (Note that talk of ‘doubling the distance between two points’ must be treated with some care. One can of course change between units in a way that doesn’t change the physics, but if we keep the scale the same and move each particle so that the distance between pairs of particles is doubled we won’t in general keep the physics the same, unless we also change other properties like their masses to compensate.)

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an objects location in GR, and only emerge in the presence of a gravitational field). Thus although classical physics, with its Galilean symmetry group, has been our focus the fundamental idea can be generalized to other theories in natural ways. SHIFT INVARIANCE is not enough: for all we’ve said, a single fusion of particles could be located at a table-shaped region of space-time and a chairshaped region of space-time, so long as it is located at every region of spacetime that has that table or chair shape. To rule this out we want to require that any two locations of an object are Euclidean transforms of one another: SHIFT EQUIVALENCE: If x is located at y and x is located at y 0 then there’s a Euclidean transformation taking y to y 0 . SHIFT EQUIVALENCE is required if we are to carry out the project described as GEOMETRIC REDUCTIONISM: if the shape of a material object, for example, is determined by the shapes of its locations, but SHIFT EQUIVALENCE failed, objects simply wouldn’t have well-defined shapes.²⁹ If we want to satisfy NO SHIFTS and GEOMETRIC REDUCTIONISM at once, we had therefore better accept SHIFT INVARIANCE and SHIFT EQUIVALENCE. (Note, on the other hand, that we could in principle attempt to take geometric properties as primitive properties of material objects and regions, and attempt to reduce the location relation to them: an object is located at a region if and only if they both have the same shape. This would, of course, guarantee SHIFT INVARIANCE and SHIFT EQUIVALENCE, but would require us to be a geometric primitivist.³⁰) Finally, it is natural to require that the locations of mereological simples be themselves mereological simples (space-time points): SIMPLE LOCATIONS: If x is mereologically simple then its locations are too. This rules out extended simples. With these three principles in place we are in a position to see, at least in outline, that the present view is in as good a position as the relationist is. For the relationist the arrangement of a collection of particles is given by the distances between each pair of particles. SIMPLE LOCATIONS tells us that the locations of a mereological simple are themselves mereologically simple; they are space-time points. Given a natural principle governing the interaction of parthood and location, to be discussed in the section 3.3, it follows that the location of the fusion of two mereological atoms is the fusion of two ²⁹ Note that if we weakened Euclidean transformations to diffeomorphisms, then objects won’t have well-defined shapes. This is perhaps not unexpected given the consequences of general relativity for the naïve notion of shape. ³⁰ Thanks to Jeff Russell and Shieva Kleinschmidt for discussion here.

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space-time points. (For convenience we shall call the fusion of two mereological atoms a ‘diatom’.) By SHIFT EQUIVALENCE we know that all the locations of a diatom must be congruent to one another, and thus all consist of a fusion of two space-time points that are the same distance from one another. Thus the distance between two mereological atoms is always uniquely determined from the locations of the diatom they fuse, so that all distance facts between point particles (and thus the geometric properties of their fusions) can be recovered from their relation to the space-time manifold.

3.1. Locations in ordinary language Now one might wonder how the radical proliferance of locations posited by this kind of view is consistent with our experiences. After all, if I am looking at the General Sherman I see a solitary tree. If I were looking at a bilocated tree—a single tree with two exact locations—one expects to see two tree-like shapes belonging to the same tree. This is at least how philosophers typically describe paradigm cases involving multiple-location. By this reasoning, if the tree were multiply located at each of its Euclidean transforms we should expect to see a tree smeared out over all of space (if that is even possible to visualize), but this is emphatically not what we see. One crucial difference between the bilocated tree and the present case is that in the former the observer is not herself bilocated. Indeed, in the former case the distance between the observer and the tree is not obviously well-defined: there is the distance between the observer’s location and the tree’s first location, and the distance between the observer’s location and the tree’s second location. In such a case SHIFT EQUIVALENCE could fail, since the locations of the fusion of the observer and the tree might not be congruent to one another if the observer is not symmetrically positioned between the two locations of the tree (this result can be demonstrated more rigorously in the theory of part and location to be developed in section 3.2). As we have just seen, however, in the present setting in which we have SHIFT EQUIVALENCE, the distance between the observer’s eyes and the tree is completely well-defined: it simply falls out of a property that the locations of the eye–tree fusion share—roughly, being a disconnected eye–tree-shaped region whose connected parts are separated by a certain distance. Of course it would be a tall order to give a complete theory of perception in more fundamental terms, but surely whatever the correct theory is, the position of the tree in our visual field will depend only on the relative distances between our eyes, the trees, and other background objects, and will not depend on which particular regions of space-time the tree is located at. (One way to convince oneself that there is no conflict with experience, perhaps, is to

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appeal to the fact discussed in section 2.2. The pattern of locations on this picture determines all the facts about relative distances that the relationist needs. Thus the present view has the means to account for perceived relative distances if the relationist does.) Similar puzzles can be warded off in analogous ways. One might think that it is possible to uniquely specify the General Sherman’s location simply by gesturing towards a particular location. I could point at the General Sherman and say something like ‘the General Sherman is located at that region of space-time over there, and not anywhere else’. But on the present view I haven’t really succeeded in singling out a unique region of space-time; since my hand is multiply located the gesture I made is simultaneously related to every region of space-time with that shape. All of this admittedly sounds a little wild at first. To put it in perspective, it might be worth recalling our opening remarks in which we distinguished between two kinds of motivations for substantivalism. The reasons I have been discussing so far have been rather theoretical—the simplest way to formulate a physical theory of distances and other physical quantities without giving particular mathematical objects undue physical significance appeals to space-time points. However, one might have much more direct motivations for being a substantivalist: perhaps you think that regions of space-time play a more explicit role in our lives than I have been acknowledging. Perhaps when I wonder where my keys are I am implicitly wondering which region my keys occupy, for example, or when I learn where Jones went on holiday I learn something about a particular region. On this picture facts about particular regions of space-time can be revealed simply by observing the objects that occupy those locations, and the theory that best explains our observations is the orthodox theory in which every object has no more than one location. I suspect that locations in the everyday sense exist and are indeed the subject of our ordinary talk of ‘locations’. But I also suspect that locations in this sense aren’t fundamental entities. Perhaps they are places: ‘my pocket’ and ‘Paris’ both refer to places, and seem like reasonable answers to the question of where my keys are and where Jones went on holiday, respectively. Countries and pockets, like other material objects, will be multiply located in the more fundamental sense. Or perhaps places in the colloquial sense are material objects, but are ontologically ‘lightweight’ objects like holes and shadows—certain kinds of non-fundamental entities whose existence supervenes on the properties of other more fundamental objects. (Holes and shadows, then, are also multiply located in the fundamental sense.) Relationists are often perfectly happy to engage in talk of places and locations in the way understood above. In this way the kind of substantivalist I have been describing can also accept this sort of non-fundamental

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talk of locations. But like the relationist, I deny that this way of talking is a perspicuous description of the fundamental metaphysics. Indeed, even the ordinary substantivalist should think twice before attempting to identify ordinary locations, like my pocket and Paris, with regions of space-time, since the former have much more interesting modal profiles than the latter.³¹ To a substantivalist motivated by the more direct sort of reasons mentioned above, the view that people are multiply located in this radical manner might seem particularly bad. But the fact is that the methodology of taking our ordinary use of language at face value is a notoriously bad way to do fundamental metaphysics. One can see the present view as what one gets from taking a relationist picture and then expanding the ontology with a richly structured physical entity that plays the role in a theory of quantities that a mathematical object would otherwise have played. Counterintuitive results arise when one attempts to identify that physical entity with the nonfundamental way of talking about places and locations; but I think this kind of identification is ill-advised in the first place.

3.2. The combined theory of part and location According to orthodox substantivalism there is a fairly simple way to determine the location of a complex object. If you know the locations of its parts—in particular, if you know the locations of all its atomic parts—then the location of the whole is just the fusion of those parts. In short: the locations of the atoms determine the location of the whole. This is consistent with the broader Humean thesis that the properties of and relations between point-sized objects determine the properties of and relations between all objects. The fact alluded to, however, does not come for free—it is delivered by a plausible theory governing the interaction of parthood and location. One can concoct formal models in which the location of a whole is not determined by its atomic parts and which thus violate this theory. A pair of point particles, both located at space-time points, has a fusion that is located at the fusion of those two points. However, it is simple enough to construct formal examples where the fusion is located elsewhere: on one not so far-fetched model the fusion might be located at a larger extended region containing the original two points—on this picture ordinary objects like tables and chairs can be located at extended regions (with non-zero volume) even if they are finite

³¹ Of course some people do make these identifications—see footnote 2—but they are not completely pain free (see e.g. Sider 2001: §4.8).

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fusions of zero-volume point particles.³² But there are models that deny any connection between the locations of the atomic parts and the fusion: it’s logically consistent that the location of the fusion of two particles be any region whatsoever, whether containing the locations of the individual particles or not. However, insofar as these kinds of situations are deemed pathological they should be ruled out by our theory of part and location. The situation for the multiple location theory is not so simple. It seems clear that whatever kind of theory one adopts connecting parthood and location it cannot be true that the locations of a complex object are determined by the locations of its atomic parts. To see this, note that by SHIFT INVARIANCE, SHIFT EQUIVALENCE, and SIMPLE LOCATIONS, a mereological atom has an exact location at each space-time point and is located only at space-time points. It follows that any two mereological atoms have exactly the same locations. Yet a complex object can have much more interesting locations. Consider two different objects each composed of three atomic parts: one could have locations that are all equilateral triangles, whereas the other might have locations that are all the shape of some particular irregular triangle. These two composite objects have different locations, yet as we have seen their atomic parts have exactly the same locations. In general, knowing the locations of the particles is not enough to determine the locations of the things they compose. Indeed it might at first seem that on this picture one must completely relinquish the principle that the location of a complex object is determined from the locations of its smaller parts. As it turns out things aren’t this bad: the location of a whole is determined by the locations of its diatomic parts (objects with exactly two proper parts)—but in order to see this we must develop the theory of parthood and location a little more.³³ To that end let us begin with the pure theory of location and part: the theory one gets by looking only at the relation between the parthood and location relations, ignoring any geometrical constraints such as SHIFT EQUIVALENCE and SHIFT INVARIANCE. Since, to my knowledge, no one has given a thorough analysis of the relation between locations and parts among multiply located objects we shall need to develop a little bit of theory. ³² The fusion of the locations of the particles composing a table is an extremely disconnected object—the gaps between the particles is significantly greater than the sizes of the particles—so this intuition holds even if we do not assume that particles are point-sized. On this view the table itself is a solid, connected object merely having a location that contains the fusion of locations of the particles (see Fine 2003). ³³ In addition to the above, there are also quite general problems surrounding the interaction of mereology and location when multiple location is permitted (see Kleinschmidt 2011). The following is an attempt to formulate a simple logic of mereology and location that is not subject to these sorts of worries.

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A natural constraint to impose on our theory is that it should have what I’ll call the classical theory of part and location as a special case: it should entail that theory given the assumption that every object has a unique location. The classical theory effectively says that the location relation determines an isomorphism between the parts of a thing and the parts of its location (this theory is sometimes called ‘mereological harmony’; see Uzquiano 2011 and Saucedo 2011). For simplicity I shall assume classical extensional mereology. I shall write x þ y to denote the fusion of x and y and x ny to denote the relative complement of y from x, which exists whenever x is not a part of y (formally, it is the fusion of x’s parts disjoint from y). (1) The domain of objects that have locations is closed under fusions and parthood. (2) The location relation, L, is functional (on the domain of objects that have locations). We will write l ðxÞ, to denote the unique y such that Lxy. (3) l preserves atoms: if a is mereologically atomic then l ðaÞ is mereologically atomic. (4) l preserves relative complements l ðx nyÞ ¼ l ðxÞnl ðyÞ. (5) l preserves fusions: for any x and y, l ðx þ yÞ ¼ l ðxÞ þ l ðyÞ. (More generally, the location of the fusion of some things is the fusion of the locations of those things.)³⁴ Regarding the theory of parthood as it applies to both material objects and regions, the classical theory consists of classical extensional mereology. On the intended interpretation both regions and material objects individually and jointly form a complete Boolean algebra with the bottom element removed. The classical theory rules out many of the interesting possibilities that have traditionally preoccupied philosophers interested in location. (5), for example, says that an object is located at the sum of its parts locations; thus ruling out the pathological example we opened with. (1) ensures that fusions and parts of located objects have locations, (2) rules out the possibility of multiple location, (3) rules out extended simples and (4) effectively rules out colocation and partial colocation (when mereologically disjoint objects are located at overlapping regions), and (3), (4), and (5) together rule out unextended complexes.

³⁴ The parenthetical part of (5) is stated using plural quantification. If a first order axiomatization of this theory is sought, one should replace this axiom with a schema, saying: if x fuses the ϕs then x’s location is the fusion of the locations of the ϕs. Note that without the parenthetical these principles do not entail that l preserves arbitrary fusions.

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These principles jointly entail that the location function preserves other important mereological properties: for example that l preserves parthood and disjointness, and that no two things can have the same location.³⁵ Although many of the possibilities suggested above are worthy of philosophical attention, our purpose at present is to explore the consequences of multiple location, and the best way to do this is to screen off these other kinds of non-standard behavior. Thus the theory we will consider is the minimal generalization of the classical theory that allows for multiple location. If l is a function that satisfies these conditions we shall call it a location function. The intended models of the classical theory of location and part are therefore models in which the location relation is a location function. (More formally the intended models of the classical theory of location and part consist of a tuple ðD,
, o, l Þ where ðD,
Þ is a standard model of classical extensional mereology, o is an element of D whose improper parts represent the objects that have locations, and l is a location function whose domain consists of the improper parts of o and whose range is disjoint from o.) In order to develop the multiple location theory further, we need to start by specifying the intended models of that theory. By way of motivating our choice of model, let us start with a simple example. Consider a simple Lego construction consisting of three Lego bricks, B1–B3, that have been put together in a pyramid shape, with two at the bottom and one brick on top holding them together. Now suppose further that this Lego pyramid is bilocated. By examining our intuitions about this simple case we shall attempt to tease out some general principles connecting the location and parthood relations. To keep our intuitions clean we shall make a few simplifying assumptions. (i) The three bricks themselves might be composed of smaller parts— indeed assuming they are not extended simples they must—but for simplicity we shall pretend that these three bricks and their fusions are the only parts of the pyramid. (ii) We shall assume that the two locations of the Lego pyramid are both congruent to one another (in accordance with SHIFT EQUIVALENCE). This geometric constraint is not required by the pure theory

³⁵ (5) entails that l preserves parthood: if x is a part of y then x þ y ¼ y and thus l ðxÞ þ l ðyÞ ¼ l ðyÞ by (5), which means that l ðxÞ is a part of l ðyÞ. Without this principle one could be located in one city even when one’s arms, legs, torso, and head are located in another. By contrast (4) ensures that l preserves disjointness: if x is disjoint for y then xny ¼ x and so l ðxÞnl ðyÞ ¼ l ðxÞ which means that l ðxÞ is disjoint from l ðyÞ. It also ensures that no two things can have the same location, for if x is distinct from y then either x ny or ynx exists, but if l ðxÞ ¼ l ðyÞ then neither l ðxÞnl ðyÞ nor l ðyÞnl ðxÞ would exist.

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of part and location—the different locations of the pyramid could in principle have been any other shape or size (although it would then become unclear what grounds we have for calling it a Lego pyramid, if some of its locations did not have that shape). (iii) We shall assume that the bricks B1– B3 that compose the pyramid also have two locations each, and that these two locations are congruent. (iv) We shall also assume the three bricks are arranged symmetrically at each location—one could in principle have had B1 be the ‘top’ brick relative to one location and B2 on top at the other. Again this is completely consistent with the pure theory of part and location we are exploring in this section. (v) Lastly we shall assume for simplicity that the two locations of the whole pyramid do not overlap. To make things vivid we’ll suppose one of the pyramid’s locations is entirely contained within a box, and the other within a jar.³⁶ Our theory ultimately won’t require this, and of course any constraint like this will have to be dropped if we are to accommodate SHIFT INVARIANCE. The interaction between parthood and location here is relatively clear: there are seven (i.e. 23  1) parts of the pyramid: B1, B2, B3, B1 þ B2, B2þ B3, B1 þ B3, and B1 þ B2 þ B3. Each of these parts, including the pyramid itself, has two locations—a box location and a jar location. Insofar as we are generalizing the classical theory, then the box location of B1 þ B2 should just be the fusion of the box location of B1 and the box location of B2, and similarly for the other three composite parts of the pyramid. Since we have only finitely many objects this in effect gets us the classical theory of location and part when we restrict quantification over locations to locations within the box. None of the deviant behavior ruled out by (1)–(5) ought to arise when restricting ourselves to box locations (ignoring, for the moment, that the bricks are extended simples): thus there ought to be a location function—a function satisfying (1)–(5)— mapping parts of the pyramid to regions within the box. Parallel reasoning suggests that there should be a location function mapping the pyramid and its parts to regions inside the jar. These location functions, of course, are also defined on material objects disjoint from the pyramid; however, assuming there is no other multiple-location going on, we can assume that the two location functions agree about the locations of every material object disjoint from the pyramid. An arbitrary object is thus located at a region iff one (or both) of these two location functions maps the object to ³⁶ Note if the two locations of the pyramid did overlap, that would be completely consistent with our ban on colocation. That ban rules out two objects being entirely located at the same region, or entirely located at overlapping regions: but this is a case where only one object is entirely located at two overlapping regions, and so is not subject to this ban.

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that region: formally this means that the location relation is just the union of these two location functions (recalling that both functions and relations are sets of ordered pairs). This example is hopefully instructive enough as to make the following model of a location relation seem particularly natural: A proper location relation, L, is a union of location functions, where each location function is a mapping of material objects to regions of space-time satisfying (1)–(5). Thus any location relation, L, can be decomposed into a set of location functions roughly telling us how objects and their parts are located relative to certain locations. The intended models therefore consist of all tuples ðD,
, o, LÞ where ðD,
Þ is a standard model of classical extensional mereology, o is an element of D whose improper parts represent the objects that have locations, and L is a proper location relation whose domain consists of the improper parts of o and whose range is disjoint from o. It should be stressed that the decomposition of L into location functions is not unique: one can have two sets of location functions that have the same union. Here is a very simple example of this phenomenon. Suppose that two point particles a and b are both triply located at each of three space-time points, x, y, z, arranged in an equilateral triangle. Suppose further that the fusion a þ b is multiply located at the three locations x þ y, y þ z, x þ z respectively (note that this stipulation is consistent with the constraint that L be a union of location functions). This situation can be represented by three location functions as follows: Here each box represents how a and b are located in space relative to each location function.

b

a

a

b

b

a

Note that it can also be represented by the following three location functions:

a b

b a

a

b

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The unions of these two sets of location functions are identical, and locate a and b in exactly the way we described above.³⁷ It should also be obvious that the union of all six location functions gives us yet another representation of the same relation. However, this latter representation is special in the sense that it is the largest set of location functions whose union is L. Thus although there is no unique decomposition of a location relation into location functions, there is always a unique maximal decomposition: the set of location functions that are subsets of the location relation. When we talk about the decomposition of a location relation we shall always mean the maximal decomposition. This observation suggests that while location functions are a useful way to specify the structure of the location relation they don’t have any independent reality; indeed in the next sections we shall look at ways of characterizing the location relation directly without appealing to location functions. It is worth comparing our theory with prima facie similar responses to the shift argument that place more importance on the role of location functions. For example, Jeff Russell (2014) has recently defended the view that material objects have unique locations but it is a completely indeterminate or nonfactual matter which exact locations they have. Each ‘precisification’ of the location relation, on this view, is a location function, telling us where each object is located according to that precisification. It is interesting to note that if one were to apply this model to the above example there would be a genuine difference between a world where it is indeterminate whether the locations of a, b and a þ b are given by the first, second, or third location functions, and another world where it is indeterminate whether the locations of a, b and a þ b are given by the fourth, fifth, or sixth (so that the admissible precisifications of ‘located at’ are given by the first three locations functions in the first world, and the second three in the second world). On the picture where they’re multiply located, rather than indeterminately located, there would be no difference between these two possibilities because they determine the same location relations. Ultimately these distinctions get washed out in Russell’s theory: he accepts a principle analogous to SHIFT INVARIANCE that effectively forces us to choose the maximal set of precisifications. But this demonstrates that our theory of multiple location is at least conceptually very different; it does not make the kinds of invisible distinctions that would have to be acknowledged if we started with a set of admissible location functions instead of the single location relation that is their union.

³⁷ We also constructed it so that it satisfied SHIFT EQUIVALENCE.

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3.3. The theory of part and location Now that we have specified the intended models of the theory, let us turn to axiomatizing it. A major choice-point is whether to pursue a first- or secondorder axiomatization. In what follows I shall present both; however, there are several substantial technical questions concerning the first-order theories that remain open. I outline these in further detail in Appendix B. These mainly concern completeness (is every sentence true in all intended models provable in our system), and the status of representation theorems (can every relation that satisfies the axioms of the theory be expressed as a union of location functions). The plural language augments first-order logic with plural variables, written xx, yy, zz, and so on, a plural quantifier 8xx and a singular–plural relation xxx meaning that ‘x is one of the xx’. In both the first- and secondorder cases, the non-logical vocabulary consists of a binary relation, L, a unary predicate O, and a binary function symbol þ . L represents the location relation, O applies to material objects, and þ is a fusion operation. We assume, for simplicity, that every object is either a material object or a region of space-time so that regions of space-time can be defined as :Ox. We have chosen to take the binary fusion operator as primitive for convenience, but note that many of the more familiar mereological relations can easily be defined from it. Parthood is defined by x
y ¼ df x þ y ¼ y, overlap by x  y ¼ df 9 zðz
x ^ z
yÞ, and atomhood AtðxÞ ¼ df 8yðy
x ! x ¼ yÞ. We write xx  yy as short for 8zðz  xx ! z  yyÞ. Finally, if x and y overlap, it is also convenient to have a term x u y—the product of x and y—that denotes the fusion of things that are parts of both x and y, and a term FusðxxÞ which, whenever there is at least one xx, denotes the fusion of the xx. We may now start laying out the axioms of this theory. We shall start with the plural version, which results from adding to a standard axiomatization of plural logic, the following (self-explanatory) principles: The axioms of classical mereology (in terms of þ ).³⁸

³⁸ From the first three axioms and our definition of parthood one can prove the reflexivity, transitivity, and anti-symmetry of the parthood relation. (For example, to show anti-symmetry suppose that x
y and y
x. Thus by definition x þ y ¼ y and y þ x ¼ x. But by the second axiom x þ y ¼ y þ x so x ¼ y. The other principles are similarly straightforward.) Thus every principle of classical mereology is provable given our definition of parthood (the fusion principle needs no modification). Conversely, the below principles can be proven in classical mereology if
we add the linking principle that  x þ y ¼ z iff z is the fusion of x and y: x þ y ¼ z $ 8u u  z $ ðu  x∨u  y .

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x þx¼x x þy¼yþx x þ ðy þ zÞ ¼ ðx þ yÞ þ z 8xx 9 x8yðx  y $ 9 zðz  xx ^ y  zÞÞ All and only material objects are located somewhere. Ox $ 9 yLxy Locations aren’t material objects. Lxy ! :Oy Atoms are located at space-time points. AtðxÞ ^ Lxy ! AtðyÞ: In the classical theory of part and location, we had an axiom that said that the location relation preserved fusions: l ðx þ yÞ ¼ l ðxÞ þ l ðyÞ. Our approach to axiomatizing the theory of part and location will be to produce a suitable analog of this axiom. However, one must be careful: it is not true, for example, that the locations of x þ y are all possible fusions of the locations of x and the locations of y. In the case of the Lego pyramid, for example, the fusion of B1’s box location and B2’s jar location is neither wholly located in the box or the jar, and is thus not a location of B1 þ B2. Here is something that is true, however. If the pyramid (the fusion B1 þ B2 þ B3) is located at a region R, then there is a way of cutting R up into three disjoint pieces R1, R2, and R3 such that (i) B1 is located at R1, B2 at R2, and B3 at R3, and (ii) B1 þ B2 is located at R1 þ R2, B1 þ B3 at R1 þ R3, B2 þ B3 at R2 þ R3 . . . and (iii) B1 þ B2 þ B3 is located at R1 þ R2 þ R3. This idea can be generalized in the following way. Suppose that an arbitrary object x is located at a region y. Then for any way of cutting x up into smaller disjoint pieces, ðxi Þi2I (indexed by some set I ), there is a way of cutting up y into an equal number of little pieces ðyi Þi2I such that whenever J ⊂I , Fusðfxj j j 2 J gÞ is located at Fusðfyj j j 2 J gÞ. Thus by letting J range over singletons, we can see that each part of x, xi say, is located at the corresponding part of y, yi . Moreover, each fusion of these parts of x (say, xi þ xj þ xk ) are located at the fusion of the corresponding parts of y, (that is, yi þ yj þ yk ). (And similarly for infinite collections of these parts.) This brings us to the final and most important axiom of our theory, which effectively encodes this thought in plural logic. A partition

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of an object, x, are some things xx which are pairwise disjoint ð8xyðxxx ^ y  xxÞ ! ðx  y ! x ¼ yÞÞ but fuse to x (FusðxxÞ ¼ x). A partition xx divides x if and only if x is a fusion of some of the xx. Finally write matðxÞ for the material part of x (i.e. x u Fusðhx : OxiÞÞ and regðxÞ for the space-time part of x ði.e. x u Fusðhx : :OxiÞÞ.³⁹ The final axiom says: ARBITRARY PARTITIONING: Suppose that x is located at y and that the xx are a partition of the material objects that divides x. Then there are some things, the zz, that are a partition and that divides x þ y, such that: (a) If z and w are one of the zz, and matðzÞ ¼ matðwÞ or regðzÞ ¼ regðwÞ then z ¼ w.

 (b) If the ww  zz then mat FusðwwÞ is located at reg FusðwwÞ . (c) If z  zz and matðzÞ is atomic, regðzÞ is atomic. Our formalization of the intuitive thought, which uses indexing sets, is a little less direct, since we are restricted to a plural language. The zz effectively encode a pairing between the given partition of x and some partition of y, and a little reflection should reveal that ARBITRARY PARTITIONING correctly captures the thought explained above. The above plural axiomatization has an important property: it accurately captures the intended models of the pure theory of part with multiple location. Any full model of the axioms is one in which L can be represented as a union of location functions.⁴⁰ In order to state a first-order version of this theory we must first replace the mereological composition axiom above with a schema:
 9 zϕ ! 9 x8y x  y $ 9 zðϕ ^ y  z

³⁹ Here hx : ϕxi is a plural expression for the ϕs. ⁴⁰ A model of a plural language is full if the plural quantifiers range over every subset of the domain. We show that L is the union of the location functions that are subsets of L (here we use L to denote the interpretation of the location function in a model). In particular this means showing that if Lxy then some location function that is a subset of L maps x to y (we show this under the assumption of atomism; a more intricate argument is needed without the assumption). Suppose that x is located at y and consider the partition that consists of all the material atoms. Arbitrary partitioning guarantees that there is a bijection f between these atoms and the atoms of a region of space-time containing y, whose graph is determined by material and space-time parts of the elements of the partition zz. This bijection will be such that (i) the fusion of any set of those atoms, X , is located at the fusion of the image of X under f , f ðX Þ. If we let l ðaÞ be the fusion of the image of as atoms under f then l is a location function that is a subset of L. Moreover, since zz divides x þ y, l maps x to y as required.

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We must also find a suitable first-order replacement for ARBITRARY PARTITIONING. The most straightforward substitute is the following finitary version: PARTITIONING: If x is located at y, and x1 , . . . , xn partition x, then there are y1 , . . . , yn such that each xi is located at yi , each xi þ xj is located at yi þ yj , each xi þ xj þ xk is located at yi þ yj þ yk (and so on). Unlike ARBITRARY PARTITIONING, PARTITIONING is a schema, with a different instance for each choice of n. That this schema really is expressible in firstorder logic is shown in Appendix B. Here the logical issues are not as clean as with the second-order case. In any model of the theory in which the extension of O is finite, the extension of L is a union of location functions. However, there are unintended models when the extension of O is infinite. Two natural questions then present themselves: are there any first-order axiomatizations that rule out unintended models, and are there any first-order axiomatizations that are sound and complete for the class of intended models (even if such axiomatizations admit unintended models)? Both these questions are explored further in Appendix B.

3.4. Combining the theory with shift invariance We can eliminate reference to the mathematical notion of a Euclidean transformation in our statements of SHIFT INVARIANCE and SHIFT EQUIVALENCE, so that our entire theory can be stated in the internal language of congruence, betweenness, location, and part. In fact the statement of both principles takes a particularly simple form, and turns out to be a restriction of both principles to diatoms. For example, ‘diatomic shift invariance’ says that if a diatom x is located at p þ q (a fusion of two space-time points) then it is located at every Euclidean transform of p þ q. Note, however, that p0 þ q 0 is a Euclidean transform of p þ q iff p and q are congruent to p0 and q 0 , which is something we can state in our fundamental vocabulary consisting of congruence and betweenness relations. Diatomic shift equivalence similarly becomes the claim that if x is located at p þ q and p0 þ q 0 then p and q are congruent to p0 and q 0 . Putting both together we get:
 CONGRUENT DIATOMS: Lxðp þ qÞ ! Lxðp0 þ q 0 Þ $ Cpqp0 q 0 where p, q, p0 and q 0 range over space-time points. To see that this principle is adequate, we need to know that if the diatoms are located at all and only the Euclidean shifts of their locations then every

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object is located at all and only the Euclidean shifts of their locations. In other words, we need to know that CONGRUENT DIATOMS entails SHIFT INVARIANCE and SHIFT EQUIVALENCE. In fact this follows from a fact we mentioned earlier and which we are now in a position to prove: the locations of a whole are determined by the locations of its diatomic parts (even though, as we saw earlier, they are not determined by the locations of their atomic parts). Theorem 3.1 (The Locations of the Diatoms Determines the Locations of Everything) Let L be a proper location relation (a union of location functions), between some domain of material objects and Euclidean space E 3 . Suppose that for every diatom there is a pair of space-time points such that the diatom’s locations are given by the collection of all pairs of space-time points congruent to that pair. Then for any material object, there is some region such that the object’s locations are given by the collection of all regions congruent to that region.

Proof. Suppose that L is a union of a set of locations functions, F , and that d is some metric that coheres with the congruence and betweenness structure on E 3 . We want to show that if Lxy and Lxz then y is congruent to z: there is some Euclidean transformation that maps y to z. It is a standard result that y and z are Euclidean transformations of one another if and only if there is a distance preserving bijection (an ‘isometry’) between the points in y and the points in z. From the fact that Lxy and Lxz it follows that there are location functions f , ɡ 2 F such that f ðxÞ ¼ y and ɡ ðxÞ ¼ z. I claim that the mapping ι ¼ g  f 1 restricted to points is an isometry between points in y to points in z. One can see by inspection that it maps y to z. Suppose that p and q are two points in y and that ιðpÞ ¼ p0 and ιðqÞ ¼ q 0 . Now let a ¼ f 1 ðpÞ and b ¼ f 1 ðqÞ. It follows that ɡ ðaÞ ¼ p0 and ɡ ðbÞ ¼ q 0 . Note also that a and b must be mereological atoms because they are mapped to atoms in E 3 by a location function. Thus a þ b is a diatom, and it is located at p þ q by f (note that f ða þ bÞ ¼ f ðaÞ þ f ðbÞ ¼ p þ q) and located at p0 þ q 0 by ɡ (since ɡ ða þ bÞ ¼ ɡ ðaÞ þ ɡ ðbÞ ¼ p0 þ q 0 ). By C ONGRUENT D IATOMS it follows that p þ q is congruent to p0 þ q 0 , which means that
 0 0 d ðp, qÞ ¼ d ðp , q Þ ¼ d ðιðpÞ, f ιðqÞ as required of an isometry. A small modification to the argument must be made to extend to the four-dimensional case: regions are transforms of each other in the relevant sense iff there is an isometry that also preserves simultaneity and nonsimultaneity of each pair of points.

4 . C O N C LU S I O N We have considered two sorts of theories of the geometry of material objects: theories in which they inherit their structure by their relation to a

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space-time manifold which has its geometric structure internally, and theories in which they inherit their structure by their relation to a platonic realm of abstracta. We have seen that the most straightforward versions of both these views give rise to certain kinds of objectionable invisible structure. That does not mean we should reject these approaches, for they may have other theoretical virtues that could not be achieved otherwise. However, I have outlined a theory that can provide an adequate account of the geometry of physical objects that does without these invisible differences, and is otherwise quite simple. To be sure, the view does not describe (nor does it purport to describe) our untutored ideas about the nature of spacetime as it relates to our ordinary talk of locations. But it strikes me that this talk is just a convenient shorthand for describing facts about the relative locations of different objects (much like the physicists use of coordinate systems) and should not be taken seriously as a guide to how things are fundamentally. 5. APPENDIX A: FIELDS Fields, as they are ordinarily conceived, are properties of space-time: a field determines a field value for each point in space-time, which specifies the strength (and possibly also direction) of the field at that point. On this view the strength of the field values at a particle’s location determines the behavior of that particle. This picture, however, is fundamentally at odds with the present view, since each particle has many locations, and so no unique field value can be associated with a particle. This obstacle is not fatal to the present approach: Newtonian mechanics can be formulated as an ‘action at a distance’ theory—a theory that works directly with the forces acting on each particle by other particles without invoking fields to mediate these forces. However, this approach is not in the spirit of (if not incompatible with) the idea that forces and other quantities usually represented by mathematical objects are fundamentally reducible to more basic relations between concrete entities like space-time points. For example, the action at a distance theory assigns a number to each ordered pair of particles telling us the force one exerts on the other. A problem similar to that posed for relationism then arises: unless there are a continuum of particles between each pair of particles, relations encoding the forces between the particles alone will be insufficient to specify all the possible kinds of forces a particle can exert on another. If one could consider the forces exerted at a continuum of points between the two particles (as one might in some sort of field theory) we could employ the methods discussed in section 2.2 to

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eliminate numbers. At any rate, the use of space-time fields are so pervasive in modern physics it would be remiss not to treat them. In what follows I’m going to focus on strategies, in the spirit of those proposed by Field (1980), for demathematizing scalar and vector fields by reducing them to certain kinds of congruence and betweenness facts. This strategy comes in two parts, the first of which is to note that most vector fields of physical interest can be equivalently described by a scalar field, so that our problem can in effect be reduced to the problem of giving an account of scalar fields. The second is to see that a scalar field can be represented, up to an affine transformation, by a pair of congruence and betweenness relations. If a vector field is ‘conservative’—a property most physical fields share— then it can be represented as the gradient of a scalar field.⁴¹ To get a feel for this result it helps to visualize the special case of a scalar field on twodimensional space: one can picture this as a kind of hilly terrain imposed over the surface where the height of the field above the surface represents the magnitude of the scalar field. The slope of the terrain at a point—the vector pointing downhill with a magnitude proportional to the steepness—will be a vector field of the requisite kind. This idea easily generalizes to higher dimensions. When a vector field is generated by a scalar field in this way, we call the scalar field a ‘potential’ for the vector field. It should be noted that many different scalar fields generate the same vector field—one can uniformly raise or lower the height of a terrain without changing the direction or magnitude of the slope at each point. For this reason people normally don’t consider the potential scalar field to be fundamental. When the laws of physics depend only on the values of a vector field, and thus do not discriminate between different scalar fields that generate that vector field, it is extremely natural to think that the only physically real distinctions are those that give rise to different assignments of vectors to points. It thus seems prima facie inadvisable to attempt to reduce vector fields to scalar fields. However, our strategy is to reduce both scalar and vector fields to congruence and betweenness relations, so, for example, the potential betweenness relation would say

⁴¹ A vector field is conservative if the ‘work-done’ to get from one point to another does not depend on the path you take. Most fields that have a chance of representing real physical fields are conservative so our restriction won’t cost us much in the way of generality. Note, though, that an arbitrary vector field on an n-dimensional manifold can be decomposed into n scalar fields representing its n-components (although this decomposition is highly non-unique). So, if needed, a more general reduction procedure is available.

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‘the potential at x is between the potential at y and z’. Interestingly, in this setting the correspondence between vector fields and the potential congruence and betweenness facts becomes one-to-one. In particular, one can change the height of a potential without either changing the vector field it generates or the congruence and betweenness facts it generates. Thus the strategy of reducing conservative vector fields to facts about potential congruence and betweenness at space-time points seems like a promising place to start. However, once we adopt SHIFT INVARIANCE we quickly encounter a completely independent difficulty that has nothing to do with the project of demathematizing fields. To illustrate the problem, consider a single particle with external forces acting on it. We would normally represent the gravitational potential by a distribution of quantitative properties over space-time points and determine the particle’s motion by the distribution of these properties in a neighborhood of the particle’s location. If the particle is multiply located at every space-time point this method breaks down: the gradient of the potential is different at each point of space, so we can’t tell anything about the force acting on a particle just by looking at its locations. It is natural to think that this means that fields have to be multiply located as well: if a point p has a certain gravitational potential, every Galilean transformation of p must have that potential as well. But this is not sufficient either, for it would entail that every field value that’s had anywhere is had by every space-time point at once (since every spacetime point is a Galilean transform of every other space-time point). The location of the field has to be somehow correlated with the locations of the material objects for this to work.⁴² The following is an exploration of one way of meeting this challenge.

5.1. Demathematizing fields in orthodox Newtonian physics We shall start by getting a clearer handle on the representation of forces in the ordinary substantivalist setting, in which each particle has a single ⁴² Let me note in passing that one way to do this would be to expand our ontology of material objects to include fields as well, consisting of point-sized parts. Field values can then be construed not as properties of space-time points, but as properties of the pointlike parts of the field. Thus the field values are multiply located in a way that correlates with locations of the field itself, and any other material objects we might want to include. This strategy is not without contention, however. Since each material object has a unique ‘location’ within a given field, this maneuver potentially reinstates the problems we originally had with space-time. For example, Arntzenius (2012) points out (in the context of a discussion of relationism) that you can shift the field values at each point of a field to some other point on the field in a uniform fashion, in a way completely analogous to a Leibniz shift on space-time.

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location. Let us begin by considering a very simplified example consisting of two point particles, a and b, constrained to one dimension with no external forces acting on them. Both a and b generate a force field defined over space: the value of the field generated by a at the location of b determines how quickly and in what direction b will accelerate. Such forces can be equivalently represented by a scalar potential (the potential energy): in this case it will be the values of the potential in a neighborhood of a and of b that will determine their respective motions.⁴³ Now we can imagine what would happen if we were to uniformly translate the positions of the two particles in some direction by some fixed amount without also shifting the fields. The result would not be legal: each force field has a source—a point from which the field emanates—and these sources would not match the locations of the particles. In particular the force acting on a and b respectively would depend on their distances from the sources, and not on the distance from each other as it ought to. In order to preserve the laws we must translate a, b, and the gravitational field as well. Similarly, when the two particles move in a lawlike manner, we must also provide laws telling us how the field values change, so that the field sources ‘keep up’ with the particles. A metaphysically distinct (although mathematically equivalent) way to represent our two particle example is to imagine the system having a single location in a higher-dimensional space called configuration space, where each point in configuration space represents all the particles’ locations in ordinary space. Recall that both particles in our example reside in a one-dimensional space, and there are only two of them, so in this case configuration space is two-dimensional. (More generally, when we are considering all three dimensions, and there are N particles, configuration space will have 3N dimensions.) The gravitational forces can similarly be represented by a vector field on this two-dimensional space, with the first component of the vector at a point representing the force a exerts on b and the second component representing the force b exerts on a when a and b are located according to that point in configuration space. As before, this vector field can be equivalently represented by a scalar potential. A uniform shift of the locations of a and b in configuration space corresponds to moving the location of the system diagonally in configuration space (north-east or south-west, as it

⁴³ Sometimes it is natural to consider a single vector field defined over space—the gravitational field—telling us the force per unit mass that would be exerted at each point of space-time due to every other particle. I have chosen to illustrate things with two distinct fields, rather than merging them into one, because it makes for a more straightforward comparison with views considered later.

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were). Unlike in the previous example, we do not also need to shift the field, since the value of the gravitational field on configuration space is constant along all diagonal lines of this sort. The forces between two particles depends only on the relative distances between the particles, so the field values on configuration space must be constant along all paths in configuration space that leave the relative distances between particles fixed.⁴⁴ (Although we are presently focusing on Newtonian mechanics in the ordinary setting, one can already see how this second way of thinking might be more friendly to the multi-locational view I am endorsing.) These two examples suggest two possible ontologies one might adopt. According to the first ontology one has two particles whose fundamental properties include their positions and masses, and two vector force fields (or equivalently, scalar potentials) whose fundamental properties include its strength at each point in ordinary space. To specify the dynamical evolution of this system we need two kinds of laws. One of the laws will tell us how the positions of the particles change over time—these laws tell us how positions change in terms of the strength of the field at (or at a neighborhood) of those positions. Another kind of law is needed to specify how the fields change their field values over time—these laws tell us how the field values change in terms of the positions of the particles. Conceptually this seems a little unsatisfactory, since there is an obvious circularity, and it’s a mathematically non-trivial fact that the circularity is not vicious. This picture also seems to be unnecessarily complex: we have to provide both laws telling us how the positions of the particles evolve (depending on the strength of the field) and laws telling us how the gravitational field evolves (depending on the positions of the particles). This might seem especially unsatisfactory if one thought that the existence of fields were only postulated to explain the motions of particles; on the present picture one also needs to explain the motions of the fields, and to do so one invokes the motions of the particles. According to the second ontology we have particles as before, except now the thing playing the role of the two force fields (or potentials) is a single field over configuration space. In this setting the force field on configuration space remains constant over time, and the only thing that changes is the system’s position in configuration space. Thus in this setting we only need laws telling us how the position of the particles change in configuration space in terms of a constant field.

⁴⁴ In more detail, the value of the force  at ðx, yÞ—the point where a is at x and b at y—is ðGma mb =ðx  yÞ2 , Gmb ma =ðx  yÞ2 . It is the same value at ðx þ c, y þ cÞ since 2 2 ððx þ cÞ  ðy þ cÞÞ ¼ ðx  yÞ .

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Let us develop the second idea a little more. In the general setting with N particles and three spatial dimensions, the field can be represented by a scalar potential field U on a 3N dimensional configuration space, and the system gets assigned a position in that space. To get an intuitive handle on U , suppose that a’s position is described by the ith argument of U , and imagine that somehow the positions of every particle except for a have been frozen in place in positions p1 . . . , pi1 , pi þ 1 . . . pN . We can then imagine the forces that would be exerted on a if a was located at each point p: these forces can be represented by the potential at each point in space. This is a function of p, Ui , that is given exactly by our N place function with N  1 of its degrees of freedom removed by filling them with the parameters p1 . . . , pi1 , pi þ 1 . . .pN ðthat is, Ui ðpÞ ¼ U ðp1 . . . , pi1 , p, pi þ 1 . . . pN Þ).⁴⁵ We shall simplify this idea in three stages: we shall firstly show how to do without configuration space substantivalism, then we’ll show how to rid our primitives of haecceitistic involvement of particular particles, and finally we’ll show how to understand these fields without fundamentally involving numbers in our description. The first thing to note about this picture is that it seems to commit us to configuration space substantivalism: the view that configuration space is just as real and concrete as ordinary space, or perhaps that it should replace ordinary space. This is not in itself a problem, and some have suggested configuration space substantivalism for independent reasons (see Albert 1996). However, it is natural to figure out to what extent this picture commits us to configuration space substantivalism, and how far we can get without reifying configuration space. It turns out that it isn’t that hard to do without it provided we are ordinary substantivalists about space-time. To get a quantity defined over ordinary space that will do the job, all one needs is a quantity that requires many arguments to determine its value, N spatial points to be precise, rather than a single point of configuration space. Formally this is a real function U ðp1 , . . . , pN Þ taking N space-time points and returning a scalar quantity. For conceptual clarity, it’s important not to confuse an N -place function with a 1-place function taking a single argument with N components to a real number (i.e. a scalar field on configuration space)—even though they are mathematically isomorphic, the latter formalism assumes a richer ontology of points of configuration space. The second thing to note is that the potential field U is not a perspicuous way to represent reality since its values depend covertly, and in a non-perspicuous ⁴⁵ Note, of course, that to determine the motion of a, one only needs to know the field values in a neighborhood of a. U specifies those values everywhere, which is important, for example, when we are looking at the Lagrangian formulations of Newtonian mechanics.

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way, on the properties of three particular particles.⁴⁶ For example, in a system with three particles, a, b, and c, the potential is completely described by all the facts of the form U ðp1 , p2 , p3 Þ ¼ α, stating that when a is at p1 , b at p2 , and c at p3 , the potential is α. Clearly the value of U ðp1 , p2 , p3 Þ depends on the masses of a, b, and c: as we vary the masses of a, b, and c through modal space this value changes, whereas if we were to vary the masses of three distinct particles the value wouldn’t change. Thus two qualitatively identical worlds could differ over the value of U ðp1 , p2 , p3 Þ if a, b, and c existed and exerted forces in one world, and didn’t exist (and thus didn’t exert forces) in the other.⁴⁷ It is somewhat strange to think that the fundamental laws governing our example above are essentially specific to the motions and forces generated by the particular particles a, b, and c, and not about how three particles in general would interact with one another. It should also strike us as unsatisfactory that in order to say what the potential is when three different particles with the same masses are located at p1  p3 one must introduce a new fundamental primitive. These issues can easily be resolved if we adopt a more general relation as our fundamental primitive, that takes both particles and space-time points as arguments, and tells you what the potential is when those particles are located at those positions. In the three-particle case, for example, we’d employ a six-place function U ða, p1 , b, p2 , c, p3 Þ ¼ α which tells us what the potential is when a is at p1 , b at p2 , and c is at p3 . Thus the three-place function we were working with above is just the result of substituting three particular particles as arguments into our more fundamental function. ⁴⁶ A related point is made by Bradley Monton (2006) about configuration space realism in the context of quantum mechanics. ⁴⁷ In a world where a, b, and c do not exist, U ðp1 , p2 , p3 Þ is 0, as we have introduced it, since a, b, and c do not exert any forces. One might try to come up with a different way of interpreting U so that its values somehow depend on whichever three particles happen to exist (assuming there are only three), but the hopes for that project seem dim. Consider the set of worlds, X , in which only the particles a, b, and c exist in different possible arrangements. Now imagine two sets of worlds, X ½a=d  and X ½a=d 0 , whose worlds are qualitatively identical to the worlds in X in which particles d and d 0 respectively have been substituted for a in each world in such a way that they play the some qualitative role as a (and assume b and c remain in their qualitative roles). We would presumably want the first argument of U to track the mass of d throughout X ½a=d  and d 0 throughout X ½a=d 0 . By completely symmetrical reasoning, if we were to substitute b in each world in X ½a=d  with d 0 we’d expect the second argument of U to track the mass of d 0 throughout throughout the resulting space of worlds, X ½a=d ½b=d 0 . Similarly, if we were to substitute b with d in each world in X ½a=d 0 , we’d expect the second argument of U to track the mass of d in X ½a=d 0 ½b=d . However, this is impossible because X ½a=d 0 ½b=d  ¼ X ½a=d ½b=d 0  (since for every world in X there’s a qualitatively identical world in X in which a and b have switched roles).

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The third and final modification we need to make is to eliminate reference to numbers in our formulation of the potential. Following Field (1980) we achieve this by means of congruence and betweenness relations. For example, assuming configuration space substantivalism we could adopt a four-place congruence relation CU ðx, y, z, wÞ and three-place betweenness relation BU ðx, y, zÞ on configuration space. The former tells us when the difference in potential between two points in configuration space is the same as between two other points, and the latter tells us when its value at a point is between the values at two other points respectively. As we noted above, we can reformulate this theory in a way that doesn’t assume configuration space substantivalism, by introducing relations over ordinary points of space with  Þ and betweenmore arguments. Thus congruence becomes a CU ðx , y , z , w ness a BU ðx , y , z Þ independent variables standing for space-time points, rather than a single variable standing for a point in configuration space. The large number of arguments should not come as a surprise: if we are not assuming configuration space substantivalism, even our potential function U has to have a large number of argument places.⁴⁸ Finally, we noted that gravitational potential is most perspicuously represented by including the particles as arguments as well; U ða1 , p1 , . . . , aN , pN Þ. This function can be captured using betweenness and congruence relations in the usual way, although now we double the number of arguments.⁴⁹ General laws—sentences in which all the arguments of primitive predicates are occupied by bound variables—stated in this language will state qualitative facts.

5.2. Fields and the theory of multiple location The framework we have just described is at least a gesture at what I take to be the most promising assortment of fundamental relations that can recover the fields of Newtonian physics without postulating fundamental relations between the physical and the abstract. What is particularly interesting, however, is that this formalism extends without any modification to the setting of interest to us, in which every object is located at every Galilean transform of its locations.

⁴⁸ The large number of arguments is a feature that was introduced when we rejected configuration space realism; it is not the use of congruence and betweenness relations themselves that are responsible for this. ⁴⁹ If you’re keeping track, congruence now has 8N arguments and betweenness 6N , where N is the number of particles.

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To get an intuitive feel for this, note that the fundamental relations we have outlined above determine (via a potential) the forces acting at each point in configuration space. For Newtonian systems the potential on configuration space has the following form: X X U¼ U ðjx  x jÞ ¼ Gmi mj =jxi  xj j ij i j i n. As m and n increase the number of conjunctions in the first order definitions of ‘located’, ‘partition’, and ‘subpartition’ increase, and as k increases the number of alterations of the quantifiers increases as well. The case k ¼ 0 is given by our original notion of location for sequences. We can thus strengthen PARTITIONING to: k-PARTITIONING: If x is located at y, then for some y 0 disjoint from y, x, x c is k-strongly located at y, y0 . Once again, if these principles are not provable from our initial system, it is natural to ask questions analogous to the two questions above about a strengthened system containing these principles. University of Southern California A C K N OW L E D G M E N T S Many thanks to Jim van Cleve, John Hawthorne, and Jeff Russell for invaluable discussion and to an audience at Toronto who saw an earlier version of this paper. Thanks also to the judges Cian Dorr, Brad Skow, and Jessica Wilson and the editor Karen Bennett for many helpful suggestions.

R E F E REN C E S Albert, David. 1996. “Elementary Quantum Metaphysics.” In J. T. Cushing, Arthur Fine, and Sheldon Goldstein, eds., Bohmian Mechanics and Quantum Theory: An Appraisal, pp. 277–84. Dordrecht: Kluwer Academic. Arntzenius, Frank. 2012. Space, Time, and Stuff. Oxford: Oxford University Press. Arntzenius, Frank and Cian Dorr. 2012. “Calculus as Geometry.” In Frank Arntzenius, Space, Time, and Stuff, pp. 213–68. Oxford: Oxford University Press. Barbour, Julian B. and Bruno Bertotti. 1977. “Gravity and Inertia in a Machian Framework.” Nuovo Cimento 38: 1–27. Barbour, Julian B. and Bruno Bertotti. 1982. “Mach’s Principle and the Structure of Dynamical Theories.” Proceedings of the Royal Society A 382(1783): 295–306. Belot, Gordon. 2011. Geometric Possibility. Oxford: Oxford University Press. Casati, Roberto and Achille C. Varzi. 1999. Parts and Places: The Structures of Spatial Representation. Cambridge, MA: MIT Press. Dasgupta, Shamik. 2009. “Individuals: An Essay in Revisionary Metaphysics.” Philosophical Studies 145(1): 35–67.

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Dorr, Cian and Jeremy Goodman. Forthcoming. “Diamonds Are Forever.” Noûs. Eddon, M. 2013. “Fundamental Properties of Fundamental Properties.” In Karen Bennett and Dean Zimmerman, eds., Oxford Studies in Metaphysics, Volume 8, pp. 78–104. Oxford: Oxford University Press. Field, Hartry. 1980. Science Without Numbers. Princeton, NJ: Princeton University Press. Field, Hartry. 1984. “Can We Dispense with Space-Time?” In P. Asquith and P. Kitcher, eds., PSA: Proceedings of the 1984 Biennial Meeting of the Philosophy of Science Association, Volume 2, pp. 33–90. East Lansing, MI: Michigan State University Press. Fine, Kit. 2003. “The Non-Identity of a Material Thing and Its Matter.” Mind 112 (446): 195–234. Forbes, Graeme. 1993. “Time, Events, and Modality.” In Robin Le Poidevin and Murray MacBeath, eds., The Philosophy of Time, pp. 80–95. Oxford: Oxford University Press. Hilbert, David. 1899. The Foundations of Geometry. Open Court Publishing. Kleinschmidt, Shieva. 2011. “Multilocation and Mereology.” Philosophical Perspectives 25(1): 253–76. Kleinschmidt, Shieva. 2015. “Shaping up Location: Against the Humean Argument for the Extrinsicality of Shape.” Philosophical Studies 172(8): 1973–83. Leibniz, Gottfried Wilhelm and Samuel Clarke. 2007. “The Leibniz–Clarke Correspondence.” In Elizabeth S. Radcliffe, Richard McCarty, Fritz Allhoff, and Anand Jayprakash Vaidya, eds., Late Modern Philosophy: Essential Readings with Commentary, pp. 123–32. Malden, MA: Blackwell. Maudlin, Tim. 1993. “Buckets of Water and Waves of Space: Why Spacetime Is Probably a Substance.” Philosophy of Science 60(2): 183–203. Maudlin, Tim. 2010. “Time, Topology and Physical Geometry.” Aristotelian Society Supplementary Volume 84(1): 63–78. Maudlin, Tim. 2012. Philosophy of Physics. Princeton, NJ: Princeton University Press. Monton, Bradley. 2006. “Quantum Mechanics and 3NDimensional Space.” Philosophy of Science 73(5): 778–89. Mundy, Brent. 1987. “The Metaphysics of Quantity.” Philosophical Studies 51(1): 29–54. Pooley, Oliver. 2013. “Substantivalist and Relationalist Approaches to Spacetime.” In Robert Batterman, ed., The Oxford Handbook of Philosophy of Physics, pp. 522–86. Oxford: Oxford University Press. Russell, Jeffrey Sanford. 2014. “On Where Things Could Be.” Philosophy of Science 81(1): 60–80. Saucedo, Raul. 2011. “Parthood and Location.” In Karen Bennett and Dean Zimmerman, eds., Oxford Studies in Metaphysics, Volume 6, pp. 225–84. Oxford: Oxford University Press. Sider, Theodore. 2001. Four Dimensionalism: An Ontology of Persistence and Time. Oxford: Oxford University Press. Skow, Bradford. 2007. “Are Shapes Intrinsic?” Philosophical Studies 133(1): 111–30. Tarski, Alfred. 1959. “What Is Elementary Geometry.” Symposium on the Axiomatic Method, with Special Reference to Geometry and Physics, University of

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California, pp. 16–29. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1. 1.477.3340&rep=rep1&type=pdf. Tarski, Alfred. 1983. Logic, Semantics, Metamathematics: Papers from 1923 to 1938. Indianapolis, IN: Hackett. Uzquiano, Gabriel. 2011. “Mereological Harmony.” In Karen Bennett and Dean Zimmerman, eds., Oxford Studies in Metaphysics, Volume 6, pp. 199–224. Oxford: Oxford University Press.

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PART II TIME AND CHANGE

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3 A Passage Theory of Time Martin A. Lipman

Time passes. The fact that you read these words passes into the fact that you read these words instead, which passes into the fact that you now read these words, and so it goes. This paper explores a view of time that takes passage to be the most basic temporal notion, instead of the usual A-theoretic and B-theoretic notions. When we describe the facts of one time as passing into the facts of a next time, we describe the world from an atemporal point of view. There are the facts that are constitutive of each moment of time and there is the passing of the one collection of facts into the other. But, when we adopt an atemporal view on which all times are on par, and yet admit a genuine passage of time, it seems that we adopt an incoherent view. Only the passage of a fact into a contrary fact can make for a genuine change of the objects involved in those facts. So if the facts of distinct times equally obtain, and those facts make for genuine change across time, then it seems that contrary facts equally obtain. Many conclude that passage and change are therefore incoherent notions that we should dispense with in favor of more kosher substitutes. This paper explores an alternative approach, namely one according to which passage indeed involves contrary facts and yet really obtains. We can make sense of this if the world is metaphysically fragmented. The proposed theory will build on the fragmentalist view that was introduced by Fine in his ‘Tense and Reality’ (2005). Unlike Fine’s A-theoretic fragmentalism though, the proposed view will be a fragmentalist view based on a primitive notion of passage. The essay consists of the following three sections. Section 1 argues that the standard A-theory, standard B-theory and Finean fragmentalism do not capture the passage of time. Section 2 spells out the version of fragmentalism that will be the basis of the proposed passage theory of time. Section 3 proposes the passage theory itself. The main objective of this paper is simple: to describe a conception of time that is of intrinsic interest. I will be fairly quick when I discuss worries to the more standard theories in the understanding that these worries are

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not meant to refute these theories but only help introduce and shape another contender.

1 . A S E A R C H F O R P A S S AG E There is a widespread view that for time to pass is for certain tensed facts to obtain. Let an A-theory be any theory formulated in terms of tensed notions.¹ The tensed descriptions of the world are typically stated with the help of a past tense operator ‘P’ (‘it was the case that . . . ’), a future tense operator ‘F’ (‘it will be the case that . . . ’), and the principle that any unembedded sentence ‘A’ is read as stating how things are right now.² So a sentence ‘P(A)’ says that it was the case that A or, equivalently, that it is now the case that it was the case that A. Using a tense-based language of this kind, one might think that time passes when certain things were the case and will be the case that aren’t the case right now.³ But the tensed descriptions do not capture the passage of time at all. Given that any sentence A states what obtains right now, any fact whatsoever is a current fact; in particular, that something will obtain and that something has obtained are themselves current facts. At the heart of the A-theory lies the mentioned principle that any sentence states what currently obtains, that any sentence is merely descriptive of the current state of the world. This implies that any sentence only ever specifies the contents of a single momentary stage in time, namely the current one. To state that something obtains is just to describe more of the current stage in history, and not the passing from that stage of history to the next. Price made the objection vivid: ‘[W]hat did God need to create, in order to create the whole of reality, as our exclusive presentist describes it? Not a long series of worldstages, but just a single moment, complete with its internal representation of a past and future’ (Price 2011: 279; cf. Fine 2005: §7). ¹ Though a tensed-based conception of time is often combined with a presentist view (according to which only current objects exist), it naturally features in other conceptions of time as well. Even on a growing block view, one will want to say that the block will include more; and even on a moving spotlight view, one will want to say that a different time will be qualitatively privileged (cf. Sider 2001: 22). ² This basic framework is due to Prior (1967). The principle that A is equivalent to NOW(A) doesn’t hold for sentences embedded under tense operators. Kamp (1971) showed that prefixing embedded sentences under a now operator can affect the truthvalue of the sentences they are embedded in. ³ Thus, for example, Prior: ‘ “It was the case that p, but is not now the case that p” – this formula continues to express what is common to the flow of a literal river on the one hand (where it was the case that such-and-such drops were at a certain place, and this is the case no longer) and the flow of time on the other’ (Prior 1962/2003: 19).

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The closest that a standard A-theory comes to capturing the passage of time is in the constant rewriting of its description of the world. It states that the world is (now) this way. And then we wait. And then it states that the world is (now) this way. But the crucial bit is in the waiting, this is where time passes, and the passing itself isn’t captured in any of the descriptions that the theory offers us. To emulate passage is not to capture it (cf. Park 1971 and Savitt 2002), just as we do not capture the nature of redness by writing our theory in red ink. Nor do the tensed descriptions offer an animated picture of the world simply because they include bits that fix what is to come, and what came before. What an A-theory really offers us, across time, are the still snapshots of that which passes away if and when time passes and not a picture of that very passing itself. A passing picture isn’t a picture of passage. One might reply that the A-theory really proposes a reductive account of passage. The fact that new things will be the case, one might say, captures everything that is worth capturing about our ordinary concept of passage; the theory teaches us what passage must at bottom consist in. This reply does not help the dialectic forward, however. There is a difference between a satisfactory reductive account and an incomplete account of a target phenomenon. Whether the A-theory’s reductive account of passage is satisfactory or simply misses the mark by failing to capture certain central aspects of passage depends on what passage consists in, and that is precisely at issue in the worry raised above. To the extent that the above worry sways us, we have reason to think that there is more to passage than is offered by the A-theory. If the tensed descriptions of a standard A-theory do not capture the passage of time, what sort of view does? The culprit seems to be the principle that any statement is a statement about the way things are right now, which makes any fact a current fact. It is this principle that confines us to describing the world from the current perspective in time, rendering any statement a statement of just more momentary content. So perhaps we should free ourselves from this reading of free-standing sentences and describe the world from an atemporal point of view instead. Surprisingly, this might make one turn to B-theories in search for passage, since a B-theory assumes an atemporal perspective on reality and does not take reality to be confined to what is currently the case. But the B-theory assumes an atemporal perspective in a particular way, namely by thinking of time as a dimension similar to space. On a standard B-theory, there is assumed to be a series of times in our ontology, ordered by an earlier-than relation, at which objects are said to be ‘located’ and ‘at which’ objects are said to have their properties. There are different accounts of the way in which objects have their properties ‘at times’ (Lowe 1988: 73). One might

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think that a tree is straight ‘at t’ when (1) the tree has a temporal part that is straight and located at t (Lewis 1986: 202–4), or (2) the tree bears a straightat relation to t (Mellor 1981: ch. 7), or (3) the tree instantiates in a t-relative way the straightness property ( Johnston 1987: 128), or (4) the tree is involved in the type event [the tree is straight] which is tokened at t (Haslanger 2003: §9.3).⁴ These views have in common that cross-temporal relations only hold between facts (or events) that are not incompatible, and between objects that are not involved in incompatible facts. Only compatible facts make up the various regions of the block universe. It seems however that B-theories leave no room for a passage of time precisely to the extent that cross-temporal relations only hold between facts (or events) that are not incompatible and between objects that are not involved in incompatible facts.⁵ Passage should make for change (McTaggart 1908: 459), and there is change only if a fact passes into a contrary fact. The following conditions all seem necessary conditions for change and yet are in direct conflict with the mentioned accounts of persistence (cf. Haslanger 2003): Identity condition: If an object persists through change, the object before the change is one and the same as the one existing after the change. Proper subject condition: The object undergoing the change is the very thing that is the different ways before and after the change. Contrary ways condition: The way an object is before the change conflicts with the way the object comes to be through the change. These conditions are all underwritten by a simple picture of change: an object a changes across time if a’s being a certain way passes into a’s no longer being that way. There is change when we pass from the presence of a fact to the absence of that fact, when we pass from it being the case that A to it no longer being the case that A (or vice versa, when we pass from not A to A). We cannot draw how things are across time in the way we can draw how objects are across space because things change across time, and change implies involvement in contrary facts. Again, B-theorists may reply that they aim to offer a reductive account of change. A perdurantist, for example, might insist that an object’s having different temporal parts with incompatible properties captures everything that is worth capturing about change, and that the theory teaches us what change must at bottom consist in. But again this reply does not help the ⁴ This is by no means an exhaustive list of options; see also e.g. Ehring (1997) and MacBride (2001). It also does not represent the only way of articulating perdurance and endurance views; see e.g. Hofweber and Velleman (2011). ⁵ For a detailed ‘no-change’ objection to perdurantism, see Mellor (1998: 89). For a detailed ‘no-change’ objection to relationalism, see Rodriguez-Pereyra (2003: 191–2).

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dialectic forward. Whether the B-theoretic account of change is satisfactory or just incomplete depends on what change consists in. The no-change objections provide reasons to think that there is more to change than is offered by the B-theory. The B-theory leaves no room for passage because it adopts an atemporal point of view in the wrong way, namely by spatializing the way objects are present in time. This suggests that we need to adopt an atemporal point of view whilst thinking of the contents of times in a way that is closer to the way they are thought of within the A-theory. We should not revise our conception of the objects that are involved in change, nor revise the intrinsic properties they have across time, nor revise the way objects have these properties—more generally, we should not attempt to render the facts across time compatible. So we need a view that adopts an atemporal point of view and yet leaves the conflicting facts across time in place. Fine has introduced precisely such a view, which he called fragmentalism (see Fine 2005: §5).⁶ To adopt this view, we step back from our temporally embedded perspective, and take the tensed contents of any time to be all equally part of reality. We admit the tensed facts of past times as they were back then (so we admit the fact that Aristotle is (now) sitting), and we also admit the tensed facts of future times as they will be in due time (so we admit the fact that a human is (now) walking on Mars). The tensed contents of all times are deemed equally real. This means that conflicting facts are part of reality. There is however a primitive notion of coherence that holds between some but not all facts, thus forming maximally ‘coherent’ collections of facts, the so-called fragments of reality. Any fragment is internally coherent as only non-conflicting facts ‘cohere’ and so the overall incoherent collection of facts is taken to consist of multiple internally coherent sub-collections of facts. Does this tense-based fragmentalism succeed in capturing the passage of time? It seems not. At most, Fine’s fragmentalism provides the facts of which we want to say that some pass into others. It still doesn’t provide the passing itself. Fine himself is aware of this: [C]learly, something more than the equitable distribution of presentness is required to account for the passage of time. But at least, on the current view, there is no obvious impediment to accounting for the passage of time in terms of a successive now. We have assembled all of the relevant NOWs, so to speak, even if there remains ⁶ The A-theory’s failure to capture the passage of time also motivates another nonstandard A-theory, which Fine calls external relativism. I will not discuss this view here or compare it to the view that will be proposed in this paper. For a discussion of the relation between fragmentalism and relativism, and why fragmentalism is superior, see Fine (2005: §11).

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some question as to why the relationship between them should be taken to constitute a genuine form of succession. (Fine 2005: 288)

Fragmentalism is only a necessary part of a theory that captures the passage of time. It’s not the full story. So, then, what is the full story? Tallant, in a discussion of Fine’s fragmentalism, sees various obstacles to the addition of a relation that could constitute passage. First, he notes: [S]uch a relation would have to be in neither of the fragments that it relates—it must bridge the gap between them. The first (obvious) problem is that it is entirely unclear what sort of relation is suited to relating distinct fragments of reality. (Tallant 2013: 12–13)

Tallant is right: passage cannot be a further way things are ‘at’ a moment in time. The relevant relational fact could not simply be just-more momentary content again; after all, we are not interested in further additions to the momentary states of the world, we are interested in the passage of one momentary state into another. Tallant continues: The second problem is that, even if we can locate a relation to relate the distinct fragments, it remains unclear how this relation is to suffice for passage. [ . . . ] Only particular relations can generate temporal order. If that is right, and the relation between distinct fragments of reality is temporal, then presumably said relation will have to be the tenseless ‘earlier than’ and ‘later than’ relation, that is the fundament of the B-theory. (Tallant 2013: 13)

I agree that we need a relation that doesn’t just hold ‘at’ a time but is nevertheless a temporal relation. But Tallant is too quick in thinking that it can in that case only be the earlier-than relation that is added to the fragmentalist’s view. In fact, we can be sure that it is of no help to add the earlier-than relation, as the temporal order that this introduces is already captured by the tensed contents of the fragments as Fine is thinking of them. Let frag₁, frag₂, . . . refer to the fragments, understood as certain sets of tensed facts, and let [A] refer to the fact that A.⁷ We can then define an earlier-than notion, symbolized with ‘
’, as follows (cf. Meyer 2013: 61): frag1
frag2 iff , if ½A 2 frag1 , then ½PA 2 frag2 8 Since the earlier-than relation can be defined in this way from the tensed contents of the fragments, it isn’t temporal order that is lacking. We have no ⁷ Fine makes clear that this talk of facts as things, and of reality as a thing composed of facts, is mere loose talk, and not the idiom that reflects the fragmentalist’s conception of the world (Fine 2005: 268). The difference between the strict and loose talk doesn’t matter for the current point. ⁸ We can also define the earlier-than relation using the future operator: frag₁
frag₂ iff, if [A] 2 frag₂, then [FA] 2 frag₁. Of course, these two definitions only generate a

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problem in ordering the fragments in such a way that they follow the trajectory of actual history, and yet the very passing of time seems absent. Mere temporal order doesn’t make for passage. This of course aggravates the question of what on earth the fragmentalist could add to the ‘equitable distribution of presentness’ in order to capture the passage of time. There are at least two possibilities. The first is that we find something—distinct from passage itself and distinct from the A-theoretic and B-theoretic notions—that constitutes the passing from one fragment into the next. I do not see what this further notion could be, however. A second possibility is that we started in the wrong place by assuming that we could explain the passage of time on the basis of other temporal notions. Perhaps there is something to passage that is basic, and cannot be captured in any other terms. The very fact that many of us can recognize that passage seems lacking from the standard A-theory, the standard B-theory, and Fine’s A-theoretic fragmentalism, suggests that at least many of us possess a concept of passage that is not exhausted by any of the theoretical primitives currently at play in these theories. If we indeed possess such a concept, we are free to employ it within in our theories of the world, and regiment it directly.⁹ In that case we no longer search for a reductive explanation of what the passage of time consists in but instead investigate what the world must be like if we assume that there is a real passage of time. Our starting point is now that the contents of one time pass into the contents of another time in the very sense in which this seems absent from the theories discussed above. Let me summarize the dialectic. The A-theory seems not to capture the passage of time because it describes only current momentary facts, suggesting that we need an atemporal view of the world. The B-theory seems not to capture passage because it does not allow conflicting facts across time, suggesting that we need an atemporal view that admits such conflicting facts. Fine’s tense-based fragmentalism is precisely such a view, and yet it still fails to capture passage, suggesting that passage does not admit of a reductive explanation in terms of tense even when we adopt an atemporal satisfactory earlier-than relation if the tense-operators behave in the right way. Meyer (2013: §4.3) points out that a very weak tense logic suffices for this. Where H is the ‘always has been the case’ operator and G is the ‘always going to be the case’ operator, all we need are the axioms: H(A!B)!(HA!HB) and G(A!B)!(GA!HB), the tense-logical analogues of K’s distribution axioms, and the two rules: if ‘ A then ‘ HA, and if ‘ A then ‘ GA, the tense analogues of necessitation. ⁹ The intelligibility of the concept of passage is the subject of much debate, most of which I cannot address here. Influential discussions of passage, or aspects of passage (such as its directionality), are found in Smart (1949), Williams (1951), Price (2011), Earman (1974), Maudlin (2007: ch. 4), Savitt (2002), and Norton (2010).

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perspective that admits the conflicting facts that appear to obtain across time. Taking these considerations together, it seems we must make room for conflicting facts in our conception of the world and relate those facts through a non-reductive notion of passage. I will proceed in two steps. First I will propose an understanding of fragmentalism (that differs from Fine’s in certain ways) and discuss it in some detail; and then I will discuss how we can situate passage within the fragmentalist conception of the world.

2. F R A G M E N T A T I O N A C R O S S T I M E Fine’s characterization of fragmentalism is firmly based in a certain conception of reality and its role in metaphysics (see Fine 2001: §§8–10; and 2005: §2). Instead of discussing Fine’s (metametaphysical) framework, I will set out my preferred understanding of fragmentalism from scratch.¹⁰ On this view, fragmentalism does not rely on a primitive notion of reality. Consider a tree that starts out growing straight up but then gradually grows into a crooked and bent tree. If we consider this from an atemporal perspective and say that the tree is straight and bent, we quickly want to add that the tree is these ways only at different times. We add this temporal qualification because we take this to explain how the tree can be both straight and bent when considered from an atemporal perspective. But why does this temporal qualification explain this? The temporal qualification explains how incompatible facts can obtain because, I submit, the temporal separation implies a kind of metaphysical separation of the incompatible facts, a lack of co-reality. The fact that the tree is straight and the fact that the tree is bent do not obtain together in the sense that they do not constitute a unified chunk of world. The first fact only obtains insofar as the second fact doesn’t and, vice versa, the second fact only obtains insofar as the first doesn’t. Both facts obtain, they just obtain separately from each other. The point of talk of facts obtaining ‘at different times’ is not just to relate facts to some entities, times. The point of such talk is that, by relating incompatible to distinct times, we convey that the relevant facts do not co-obtain in a certain sense. We can abstract this failure of co-obtainment from the relativization that we use to convey it. Pursuing this line of thought further, we can make the involved understanding of co-obtainment explicit. It’s normally assumed that for A and B to co-obtain is just for A and B each to obtain. Call this co-obtainment in

¹⁰ I discuss Fine’s conception of fragmentalism and compare it to the conception of fragmentalism proposed below, in Lipman (2015).

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the thin sense. Fragmentalism assumes that there is also a thick sense of co-obtainment according to which A and B can each obtain without coobtaining. Given this distinction between a thin and thick sense of co-obtainment, it seems that we normally slide between them, reflecting an implicit assumption that the world is a metaphysically unified place. This slide goes as follows: we assume that if something is straight and bent, it must thereby be straight and bent together, and that, because something cannot be straight and bent together, it cannot be straight and bent, period. We cannot conceive of the co-obtainment in the thick sense of the tree’s being straight and the tree’s being bent, that is, we cannot conceive of a unified bit of world with a tree in it that is both straight and bent. But, from the fact that this is inconceivable, we draw a conclusion concerning co-obtainment in the thin sense: we conclude that it’s impossible that the tree is straight and bent. We make an implicit assumption that, necessarily, any two facts that co-obtain in the thin sense, co-obtain in the thick sense. This is a substantive metaphysical assumption that is not at all obvious when we consider the way things are across time. It’s implausible to think that Aristotle’s sitting and my sitting form a unified bit of world within which we are both sitting just because the facts both obtain across time. Fragmentalism denies this unity assumption, and hence denies that there is a legitimate inference from the impossibility of being straight and bent together to the impossibility of being straight and bent. The thin sense of co-obtainment is captured by ordinary conjunction. It’s the case that A ^ B just when A and B are each the case. Ordinary conjunction is simply silent about whether A and B also form a single unified chunk of world. This can be distinguished from the thick notion of co-obtainment that I will express with a sentential connective ‘  ’, which does imply metaphysical unity: it is the case that A  B just when there is a single unified bit of world which is such that A and B. I propose that we read ‘A  B’ as ‘A insofar as B’, so that for example ‘the sun shines  the tree is leafless’ is read as ‘the sun shines insofar as the tree is leafless’. This is to some extent a theoretical regimentation of the ordinary language phrase ‘insofar as’.¹¹ Since we are introducing a new concept, our options are either to use a notion whose ordinary language sense comes close, or to introduce a new phrase, say ‘shqand’ and read ‘A  B’ as ‘A shqand B’. I prefer the former option, using a well-known phrase that comes close, in a regimented way. For ease of expression, I will also sometimes talk of facts co-obtaining, ¹¹ In ordinary language, the phrase ‘insofar as’ has various readings. I’m using the phrase here to offer an informal reading of the introduced notion, and it’s not assumed that any of the ordinary language meanings coincides neatly with the way it is used here; see also Lipman (2016: 49).

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in the understanding that this talk about reified facts can be translated into the official idiom: ‘the fact that A and the fact that B co-obtain’ can always be translated into ‘A insofar as B’. ‘Co-obtainment’ always refers to co-obtainment in the thick sense from now on. Given the sharp distinction between two conjunctive notions ‘ ^ ’ and ‘  ’ we can now make a distinction between two kinds of conflicting facts: ‘A’ and ‘B’ state contrary facts iff they cannot both obtain, i.e. iff, necessarily, :(A ^ B). ‘A’ and ‘B’ state incompatible facts iff they cannot co-obtain, i.e. iff, necessarily, :(A  B). Call sentences such that the one is the negation of the other (‘A’ and ‘ :A’) contradicting sentences. It’s natural to hold that contradicting sentences only express contrary facts, i.e. that, necessarily, :(A ^ :A). The conflict in what contradicting sentences state arises arguably from the meaning of negation and not from the particular predicates involved, and the kind of conflict that arises from negation is naturally taken to be contrarity. This stands in contrast to sentences such as ‘the tree is bent’ and ‘the tree is straight’, where any conflict we see between such sentences doesn’t arise from their logical structure but from the meaning of the predicates involved. According to fragmentalism, there are pairs of facts which are incompatible but not contrary; for example, we might think that it’s possible that the tree is straight and bent but impossible that the tree is straight insofar as it is bent. If contradicting sentences always state contrary facts, there is a constraining connection between, on the one hand, the logical structure we attribute to sentences and, on the other hand, the kind of conflict we think there is in what is stated by the relevant sentences. Take the pair of sentences ‘Aristotle is alive’ and ‘Aristotle is dead’. Is it possible that Aristotle is alive and dead? Well, one might think that to be dead is just not to be alive, i.e. that ‘Aristotle is dead’ expresses that Aristotle is not alive.¹² If we believe that ‘Aristotle is dead’ expresses that Aristotle is not alive, we thereby believe that ‘Aristotle is alive’

¹² One might think this, or one might not. Fragmentalists are not beholden to the grammatical structure of the language they use to describe the world. The fact that ‘is dead’ or ‘is non-alive’ are unary predicates in no way determines that it’s possible that Aristotle is both alive and dead, or alive and non-alive. Whether predicates are unary doesn’t determine anything. The point here is logical structure: a sentence with surface structure ‘Ga’ might sometimes be understood as expressing that :Fa, and when that is the case, the fragmentalist is no longer free to hold that Fa and Ga, and vice versa, when she believes that Fa and Ga, she is no longer free to hold that the sentence ‘Ga’ expresses that :Fa.

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and ‘Aristotle is dead’ express contrary facts, i.e. that it is impossible that Aristotle is alive and dead. Vice versa, if we believe that it is possible that Aristotle is alive and dead, this means that we cannot subsequently understand ‘Aristotle is dead’ as expressing that Aristotle is not alive. Our judgments concerning logical form and the involved kind of conflict constrain each other.¹³ What about compatible facts, do they necessarily co-obtain? I see no reason why they should. If two compatible facts seem to obtain at distinct moments in time only, such as Aristotle’s sitting and my sitting, then they do not co-obtain, regardless of their compatibility. We can distinguish the possible case in which compatible facts co-obtain from the possible case in which they each obtain but fail to co-obtain. This renders the co-obtaining of compatible facts a contingent and substantive matter. A complete description of the world doesn’t just need to capture everything that obtains, it needs to capture what co-obtains with what and what things obtain yet fail to co-obtain. One may wonder how, exactly, fragmentalism is meant to be a coherent view. To have a better sense of this, and of the formal properties of co-obtainment more generally, we need to have a closer look at the inferential role of co-obtainment. It’s helpful to resort to some simple model theory for this purpose.¹⁴ Let the set of sentences S consist of atomic sentences p, q, r, . . . and be such that, if A and B are sentences, so are :A, A ^ B, and A  B (nothing else is in S ). A model M is a pair where T is a set of points and v is a function that assigns either 1 or 0 to each of the atomic sentences relative to each point t 2T.

¹³ To decide tricky cases, it can be a good heuristic to think in terms of the existence of properties (although this is not an official commitment of the framework—just a heuristic). Do ‘Aristotle is alive’ and ‘Aristotle is dead’ express contrary or merely incompatible facts? Here it might help to ask whether there are the properties of being alive and being dead, or merely the property of being alive that some things have and other things lack. In the case of ‘the rose is red’ and ‘the rose is blue’, for example, it seems intuitive that there are two properties, and that ‘is red’ is not adequately understood as ‘is neither blue nor green nor orange . . . ’ but that there is a positively qualitative way that things are when they are red. When it sounds right to say that there are two distinct properties that the relevant sentences attribute to an object, this is some reason to think that they state merely incompatible facts. ¹⁴ The logic proposed is inspired by the ‘discussive logic’ of Jaśkowski (1948/1969)— with the important difference that Jaśkowski’s discussive logic is paraconsistent, whereas the logic below isn’t. For discussions of other closely related logics, see Rescher and Brandom (1980), Lewis (1982), Priest (2008), and, in particular, Restall (1997). For an accessible introduction to non-adjunctive logic, see Varzi (1997) and Priest (2007: §4.2). The logic presented here is discussed in a little more detail in Lipman (2016).

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The valuation v for the atomic sentences relative to points is extended to a valuation for all the sentences via the following recursive clauses (where t ranges over points in T ): • vt ðA  BÞ ¼ 1 iff vt ðAÞ ¼ 1 and vt ðBÞ ¼ 1 • vt ðA ^ BÞ ¼ 1 iff vt ðAÞ ¼ 1 and vt ðBÞ ¼ 1 • vt ð :AÞ ¼ 1 iff vt ðAÞ 6¼ 1 Note that conjunction and co-obtainment have the same clauses here. Truth in a model, written M ⊩ A is defined via the following recursive clauses (where p is an arbitrary atomic sentence): • M ⊩ p iff 9tðvt ðpÞ ¼ 1Þ • M ⊩ A  B iff 9tðvt ðA  BÞ ¼ 1Þ • M ⊩ A ^ B iff M ⊩ A and M ⊩ B • M ⊩ :A iff M ⊮ A Note that conjunction and co-obtainment have different clauses here. We define validity and logical truth in the standard way (where Σ is a set of sentences): An argument from Σ to A is valid, written Σ  A, iff, for every model M, if M ⊩ Σ then M ⊩ A.¹⁵ A formula A is logically true, written  A, iff, for every model M, M ⊩ A. Note that the points in T are used to represent the fragmentation across facts and can for heuristic purposes be thought of as moments of time. One can also think of them more abstractly as representing unified bits of world.¹⁶ That, within a model, A  B is true if and only if there is a point at which A and B are true, reflects our metaphorical paraphrase of A  B as saying that there is a single unified bit of world which is such that A and B. But, just as we should not confuse metaphorical paraphrases with the notion paraphrased (the view is not that we quantify over ‘unified bits of worlds’), we should not confuse the structure of the models with the structure of what they are models of: the set-theoretic machinery is merely a heuristic tool to draw out whatever logical structure the co-obtainment notion needs to have

¹⁵ By M ⊩ Σ we mean that M ⊩ B for all B 2 Σ. ¹⁶ Alternatively, the points can be interpreted as the possible worlds known from standard modal logics, so that a single fragmented world (here represented by a single model) corresponds to a set of possible worlds in a frame of modal logic; see Restall (1997).

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in order for it to capture the metaphysical picture that we are after. It’s not part of our metaphysical view that sentences are true ‘relative to’ or ‘at’ points, and the points do not correspond to anything in the fragmentalist’s ontology. Certain facts obtain insofar as other facts do; that is how the fragmentalist understands things. The v-clauses say what is true and false at each of the points in the model theory. There are v-clauses for conjunction, negation, and co-obtainment because the logic needs to handle the embedding of logically complex sentences within co-obtainment sentences. For example, A  (B  C ) is true in the model only if A and B  C are true at a point and this requires that B  C has a truth-value at a point. The same applies to A  :B and A  (B ^ C ). We can now clarify the proposed version of fragmentalism. Note first of all that a sentence is true at a point if and only if its negation isn’t true there. This means that we never have a point where both a sentence and its negation are true. That is:  :ðA  :AÞ It cannot be the case that something obtains insofar as it doesn’t obtain. Similarly, any sentence in our language is true in a model if and only if its negation isn’t true in the model. This means that the law of noncontradiction holds:  :ðA ^ :AÞ It cannot be the case that something both obtains and doesn’t obtain. To illustrate the consequences of this, consider a model where we have t₁ at which p is true but q isn’t, and t₂ at which q is true but p isn’t. As there are points at which p and q are true, they are true in the model. This means that :q isn’t true in the model. But given that :q is true at t₁, p  :q is true in the model. So in this model, :q is false, yet true insofar as p is true (i.e. :q is false but p  :q is true). The worldly fragmentation gives rise to negative sentences being true insofar as certain other things are true, even though they are false simpliciter. This understanding of negative facts reflects a natural understanding of local absences versus global absences. Compare the way existence-at-alocation and existence simpliciter interact: an object exists when there is a location at which it exists but it doesn’t fail to exist when there is a location at which it doesn’t exist. An object doesn’t exist only when there is no location at which it exists. Whereas local existence suffices for global existence, local non-existence doesn’t suffice for global non-existence. There may be local non-existence without global non-existence. Similarly,

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in a fragmented world we can think of atomic sentences as stating the positive contents of the world. When an atomic fact obtains insofar as other facts obtain (or ‘within a fragment’), this suffices for it to obtain simpliciter (or ‘in the world as such’), but when the fact is absent insofar as other facts obtain (or absent ‘in a fragment’), this does not suffice for the fact to be absent simpliciter (or absent ‘from the world at large’). This is the view we arrive at when we think of negation as behaving classically, both within a fragment as well as in the world at large, and there is no reason why the fragmentalist should adopt a non-standard understanding of negation, and hence no reason why the fragmentalist requires a paraconsistent logic. It can easily be seen from the semantics that co-obtainment is commutative and associative: AB  BA A  ðB  C Þ  ðA  BÞ  C Co-obtainment is not an asymmetric affair in any way, and whenever A co-obtains with the co-obtainment of two other facts, this just means that all three facts co-obtain with each other. Co-obtainment is however not idempotent: AA  = A17 A = A  A18 It may be that A co-obtains with itself, and yet doesn’t obtain. The reason for this is the earlier noted emergence of negative facts that do not obtain, yet do co-obtain with other facts. As a limit case of this, there are negative facts that co-obtain with themselves but which do not obtain as such. Vice versa, there are facts that obtain without co-obtaining with themselves. The failure of this arises from descriptions of multiple fragments. The conjunctive fact that the tree is bent and straight will fail to co-obtain with itself; there is no single unified bit of world that is characterized by the conjunction.

¹⁷ To see why we have A  A  = A, consider a model where some atomic sentence p is false at one point t₁ but true at a different point t₂. In this model, :p  :p is true given that there is a point where each is true (viz. t₁). And yet :p is not true simpliciter, given that p is true at t₂. = A  A, consider a model in which we have t₁ at which p is ¹⁸ To see why we have A  true but q isn’t and t₂ at which q is true but p isn’t. Here p ^ q is true, but (p ^ q)  (p ^ q) isn’t true, as there is no single point at which p ^ q is true.

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We may furthermore note the failure of adjunctive and simplifying rules for co-obtainment: A, B  = A  B 19 AB  = A20 The fact that A obtains and B obtains doesn’t mean that A obtains insofar as B obtains. This is the central feature of the co-obtainment notion we discussed above. Simplification fails, again, because a negative fact may obtain insofar as another fact obtains and yet fail to obtain. Co-obtainment is also non-transitive: A  B, B  C

 = A  C 21

This failure of transitivity allows fragments to overlap, without the fragments collapsing into one. An object may be red insofar as it is straight, and it may be red insofar as it is bent, but in no way should this imply that the object is thereby straight insofar as it is bent. This should suffice in building some formal grasp of the introduced sense of co-obtainment. The semantics teaches us how to use the notion of co-obtainment even if our understanding of it is admittedly still thin. Beyond the metaphorical paraphrases and the offered logical constraints on its use, our understanding of co-obtainment can only become richer through its application in concrete cases. In the case at hand, that of fragmentation across time, the fragmentalist language affords us with an atemporal view of the world that doesn’t force us to take the facts that constitute the world at various times all to co-obtain, or be compatible. We can describe the way that reality is at a single time as the co-obtainment of a large collection of facts, each one of which co-obtains with every other, and not all of which co-obtain with the facts that constitute the world at a different time. So the overall conception of the world, thus far, is reflected in a description of the following form:

¹⁹ For the failure of adjunction, consider a model where we have t₁ at which p is true and t₂ at which q is true. Here p is true and q is true (and hence p ^ q is true) because they are atomic sentences and there are points at which they are true. But p  q isn’t true, given that there is no point at which p and q are both true. ²⁰ For the failure of simplification, consider a model where we have t₁ at which p is true and t₂ at which q and :p are true. Here q  :p is true in the model, but :p is not true in the model, given that p is true at t₁. ²¹ Consider a model where we have a point t₁ at which p and q are true but r isn’t, and a point t₂ at which q and r are true but p isn’t. In such a model, p  q and q  r are true, but p  r isn’t.

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. . . ^ (a brachiosaurus walks the earth  Aristotle does not exist  . . . ) ^ . . . ^ (no brachiosaurus walks the earth  Aristotle is alive  . . . ) ^ . . . ^ (no brachiosaur walks the earth  Aristotle is dead  Napoleon sits on his horse  . . . ) ^ . . . ²² The long co-obtainments that feature within these descriptions state what obtains ‘at various times’ as facts that mutually co-obtain. The fragmentalist will believe that, for example, the changing tree is straight and bent. This only makes sense if we can resist thinking of the world as one continuous fabric of facts. Some of the facts in the overall collection are only real insofar as some of the other facts in the overall collection aren’t real. Focus for a moment on the way things are around you while you are reading this. Insofar as things are those ways, Napoleon is entirely non-existent, as unreal as a unicorn. However, we are able to abstract from our current perspective in time. Now it remains the case that Napoleon is entirely non-existent insofar as things are the ways they are around you while you are reading this, but this does not mean that Napoleon cannot exist insofar as things are some other way.²³ The current view of the world shouldn’t be read as featuring tensed descriptions of facts. Contra Finean fragmentalism, the fragmentalist framework that will be the basis for the passage theory of time doesn’t feature tense at all, not even the present tense. The descriptions should all be understood as tenseless descriptions. The reason for this is simple. If the claim that ‘Aristotle is alive’ were understood as saying that Aristotle is now alive, then, in treating all times on a par, the fragmentalist would be claiming that it is now the case that Aristotle is alive. But it’s a straightforward historical fact that Aristotle isn’t now alive.²⁴ In entertaining the fragmentalist view,

²² As co-obtainment has been symbolized using a binary connective, it should be clear that I have left out some unnecessary bracketing. It might well turn out that co-obtainment is more aptly conceived of as a multigrade or even infinitary connective. ²³ One might wonder here: what keeps us from adopting an even more expansive perspective that also includes how the world is across modal space? I agree with Fine when he writes that ‘there is not the same wide metaphysical gulf between the present and other times as there is between the actual world and other possible worlds. What goes on in the present and at other times is somehow part of the same all-encompassing reality in a way in which what goes on in the actual world and in other possible worlds is not’ (Fine 2005: 285). The ways things are across time stand in various explanatory connections that one does not find across possible worlds. How things could have been doesn’t explain how things are in the way in which how things were explains how things are now. Also, there is nothing like passage in the modal case; we are not taken through modal space in the way we are taken through time. ²⁴ Fine avoids denying the historical fact by resorting to a non-factive reality operator ℜ : that it is the case that ℜ (Aristotle is now alive) doesn’t imply that Aristotle is now

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I’m simply not concerned with the way things are now. The fragmentalist’s predications must be tenseless predications if we are to adopt a truly neutral standpoint and treat the contents of all times on a par. We can think of ‘Aristotle is alive’ as expressing that Aristotle instantiates a certain property, not as expressing that he instantiates the property now, nor that he always or eternally instantiates the property, just that he has the property. We need to think of predications in a temporally naïve way. The predication of properties and relations to objects is stripped from any temporal meaning, not implying anything about where in time the object has the property. 3 . R E G I M E N T I N G P A S S A GE Nothing thus far represents the passing of time. Let us now appeal directly to the notion of passage that seems to be lacking in other theories. The result will of course not be a theory of passage; the result will be a theory of time based on passage. We use many metaphors when we describe the passage of time: time is often compared to ‘a river’ that ‘flows’ and ‘carries’ us into the future. The resort to such metaphors is taken by many to show that passage is a confused or obscure notion (see e.g. Smart 1949; Williams 1951). We will take passage to be a basic phenomenon, any description of which in different terms is bound to be metaphorical precisely because it is a basic and, indeed, elusive temporal phenomenon.²⁵ When we adopt a non-reductive theory of passage, we embrace these metaphors as providing much-needed elucidation. As time passes, there is indeed a sense in which we are driven ‘forwards’ and there is indeed a sense in which the reality of one stage ‘flows’ into the next stage. Though the metaphors help convey aspects of the passage of time, they are also risky. When we describe the passage of time as a ‘flow of time’ or a ‘moving now’, this can suggest that passage itself changes or flows. But the passage of time has to be carefully distinguished from something that itself changes or moves. As Maudlin explains: Except in a metaphorical sense, time does not move or flow. Rivers flow and locomotives move. But rivers only flow and locomotives only move because time

alive (see Fine 2005: 297–8). But the fact that ℜ (Aristotle is now alive) worries me as much as the fact that Aristotle is now alive. ²⁵ The non-analyzability of our notion of passage was recognized by Broad: ‘I do not suppose that so simple and fundamental a notion as that of absolute becoming can be analysed’ (Broad 1938/1976: 281).

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passes. The flow of the Mississippi and the motion of a train consist in more than just the collections of instantaneous states that have different relative positions of the waters of the Mississippi to the banks, or different relative positions of the train to the tracks it runs on. The Mississippi flows from north to south, and the locomotive goes from, say, New York to Chicago. The direction of the flow or motion is dependent on the direction of the passage of time. Given the essential role of the passage of time in understanding the notion of flow or motion or change, it is easy to see why one might be tempted to the metaphor that time itself flows. (Maudlin 2007: 110)

Notions such as ‘flow’, ‘change’, and ‘movement’ are ultimately parasitic on the passage of time; the passage of time is a precondition for any change to occur and should not be confused with it. When we embrace the notion of passage, we can try to understand it better through regimentation instead of through reduction. To express passage within our metaphysics, I will use a sentential passage-operator ‘ ↪ ’ (‘ . . . passes into . . . ’). We add this notion to our metaphysical vocabulary and turn to models that precisify the added notion of passage. We can use a simple adaptation of the models we saw in section 2. We add sentences of the form A ↪ B to the language, and take a model M now to be a triple , where T is a set of points, v is a function that assigns 1 or 0 to the atomic sentences relative to each point in T, and O is a set of ordered pairs of points taken from T, representing an order relation on T that is irreflexive, antisymmetric, transitive, and connected. The valuation v for the atomic sentences relative to points in T is first extended to a valuation for all the sentences via the following recursive clauses (where t ranges over points in T ): • • • •

vt ðA ↪ BÞ ¼ 0 vt ðA  BÞ ¼ 1 iff vt ðAÞ ¼ 1 and vt ðBÞ ¼ 1 vt ðA ^ BÞ ¼ 1 iff vt ðAÞ ¼ 1 and vt ðBÞ ¼ 1 vt ð :AÞ ¼ 1 iff vt ðAÞ 6¼ 1

Note that any passage sentence is false at the points in T. This reflects the intuition that passage is not itself part of that which passes. We will discuss this below. The valuation v is further extended to an evaluation of the sentences relative to each ordered pair of points in the relation, i.e. relative to each 2 O : • v ðpÞ ¼ 0 • v ðA ↪ BÞ ¼ 1 iff vt1 ðAÞ ¼ 1 and vt2 ðBÞ ¼ 1 and ðA ¼ B or A ¼ :B or :A ¼ BÞ • v ðA  BÞ ¼ 1 iff v ðAÞ ¼ 1 and v ðBÞ ¼ 1

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• v ðA ^ BÞ ¼ 1 iff v ðAÞ ¼ 1 and v ðBÞ ¼ 1 • v ð :AÞ ¼ 1 iff v ðAÞ 6¼ 1 Note that the only true passage sentences are those that feature either the same sentence on both sides, or a sentence and its negation. This captures the thought that a case of passage consists in a fact’s recurrent obtaining, in its ceasing to obtain, or its coming to obtain. Note also that atomic sentences are false at the points in O; contrary to passage facts, they do not obtain insofar as matters pass. Again, we will discuss the motivation for these clauses below. The clauses for the points in T and O together fix the truth of each sentence in a given model. This is defined via the following recursive clauses (where x ranges over T [ O, that is, over both the points in T and the ordered pairs of points in O): • M ⊩ p iff 9xðvx ðpÞ ¼ 1Þ • M ⊩ A ↪ B iff 9xðvx ðA ↪ BÞ ¼ 1Þ • M ⊩ A  B iff 9xðvx ðA  BÞ ¼ 1Þ • M ⊩ A ^ B iff M ⊩ A and M ⊩ B • M ⊩ :A iff M ⊮ A Validity and logical truth are defined as before. To illustrate the model-theoretic machinery, consider the following timeseries (with columns representing points in T, and the rows stating sentences that are true relative to those points according to function v):

t1

t2

t3

t4

A B C

A B C

A B C

A B C

A model like this determines the following sort of truths. There is the passage of the individual facts, for example, in the case of A we have: A ↪ :A (t₁ to t₂), :A ↪ A (t₂ to t₃), A ↪ A (t₁ to t₃), and :A ↪ :A (t₂ to t₄). These passage facts do not co-obtain; for example, we have it that :((A ↪ :A)  ( :A ↪ A)). Other passage facts do co-obtain however, forming bundles of passings as it were. We have for example: (A ↪ :A)  (B ↪ B)  ( :C ↪C ), A’s passing into :A co-obtains with B ’s passing into B which both co-obtain with :C ’s passing into C (t₁ to t₂). Co-obtaining with these is also the passage of logically complex facts, in particular we have:

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(A  B  :C ) ↪ :(A  B  :C ): the fact that A, B, :C all co-obtain passes into the fact that they no longer all co-obtain (t₁ to t₂). The passings of these large co-obtainment facts constitute the passings of large unified chunks of world (i.e. of moments of time). In the remainder of this section, I will motivate the various clauses of the model-theoretic machinery and the formal properties that they fix. One may first of all worry that the model theory seems B-theoretic. It clearly helps to think of the points in T as moments of time, and to think of the ordering relation O as an earlier than relation on times. In my view, this shows at best that the B-theoretic conception lends itself to an elucidation of the formal features of passage. In no way does this show that passage is really B-theoretic. The situation here is similar to that of standard tense logic, where the model theory also avails itself of points and an ordering relation that can be glossed as the earlier than relation (see e.g. Burgess 2002). How we like to think of the models doesn’t determine how we think of the tense operators themselves, and the same applies here. Indeed, the model theory serves to regiment the formal properties of passage regardless of how we think of the points or the ordering relation; for example, we could just as well think of the points as numbers and the ordering relation as the larger than relation. As we will see, passage will also turn out to be formally different from the ordering relation that is used in the model theory (for example, whereas O is irreflexive and asymmetric, passage is neither irreflexive nor asymmetric). Let us now have a closer look at some of the proposed features of the passage of time, and the rationale behind them. As noted, the only true passage sentences are those that feature either the same sentence on both sides, or a sentence and its negation. The fact that I sit doesn’t pass into the fact that it rains, even if it rains at some later time. The fact that I sit either passes into the fact that I sit, or into the fact that I do not sit. This reflects the idea that, as Maudlin notes, ‘the passage of time underwrites claims about one state “coming out of ” or “being produced from” another’ (Maudlin 2007: 110). The fact that it rains doesn’t ‘come out of ’ the fact that I sit, at least not in the sense of ‘coming out of ’ that is sensibly said to be constitutive of time. One might want to object that the clause is too restrictive in focusing on contrary facts, leaving out the passage from one fact into a fact that is merely incompatible with the first, such as from my sitting into my standing. Though this is a fair worry, introducing such passage-facts raises tricky model-theoretic questions; in particular, it requires that we somehow regiment the incompatibility between facts (in such a way that the models ‘see’ the incompatibility of the facts). Furthermore, our current language already allows us to capture the notion of qualitative change in

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terms of the loss of one property co-obtaining with the gain of a property incompatible with it: Qualitative change: a changes from being F to being G iff (Fa ↪ :Fa)  ( :Ga ↪ Ga) and, necessarily, :(Fa  Ga). Put informally, an object changes qualitatively when it loses a property insofar as it gains a property that is incompatible with it; for example, the tree changes from being straight to being bent if and only if (Straight (tree) ↪ :Straight(tree))  ( :Bent(tree) ↪ Bent(tree)).²⁶ If we introduce the direct passage from the tree’s being straight and the tree’s being bent, we introduce this passage as something over and above the fact that the loss of straightness co-obtains with the gain of bentness. One might still worry that, if we should have either of these kinds of passage—between contrary facts or between incompatible facts—it is the passage from one fact to an incompatible fact because such passage is more fundamental and grounds the gain and loss of properties. For example, one might think that the tree’s being straight passes into its not being straight because the tree’s being straight passes into its being bent.²⁷ But I’m not convinced that the passage between incompatible facts is indeed the more fundamental kind of change, for the simple reason that it seems perfectly possible that there are cases of change where there is not obviously any change to the instantiation of a new property. For example, if we assume that to be dead is not to be alive, then it seems that Aristotle changes when there is passage from his being alive to his not being alive, even though there is only the loss of a property here, and not the change from one property to another. Thus, whereas change from one to another property can be understood in terms of the loss of one property and the gain of another, cases of a mere loss or a mere gain of properties cannot be understood in terms of change from one to an incompatible property. The proposed account can straightforwardly define these other kinds of change:

²⁶ This account of change results in an endurance account in the sense that one and the same object is involved in facts that obtain across time. Note that the offered account requires no revisions in our understanding of the objects that are involved in change, nor revisions in the intrinsic properties they have across time, nor revisions in the way objects have these properties. The offered account is thus conservative with regard to both our overall ontology and ideology (with the exception of introducing passage and co-obtainment of course). Moreover, we have made no reference to times at all within our official idiom, and we do not need to in order to avoid incoherence. The account is neutral on the question whether we should admit times to our ontology; if times serve a purpose other than relativizing incoherences away, we can add times to our ontology. ²⁷ Thanks to Dean Zimmerman for pressing this worry.

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Ceasing to be a certain way: a ceases to be F iff Fa ↪ :Fa. For example: the tree ceases to be straight iff Straight(a) ↪ :Straight(a). Coming to be a certain way: a becomes F iff :Fa ↪ Fa. For example: the tree becomes straight iff :Straight(a) ↪ Straight(a). The current proposal is thus more general. There might of course be other good reasons to complicate the simple story offered here and the passage between incompatible facts does seem plausible. But there is a danger that we would make things more complicated only to introduce facts that do not obviously add anything to what is already captured in the simpler picture given here. The passage theory describes a world in which things genuinely change, or so I want to claim. The crucial difference with B-theoretic accounts is that the passage theory can appeal directly to a cross-temporal phenomenon, the passage of time, which involves genuinely conflicting matters. There is no need to admit that a is F and that a is not F in order to say that a changes from being F to not F because we can identify the change with the passage from a’s being F to a’s not being F. The reason for this is that passage is not taken to be a factive notion: A ↪B  = A

= B A↪B  Say that a rose’s being red passes into its not being red. If passage were factive, this would imply that the rose is red and not red, and that cannot be the case. Across time we can accept (1) that the rose is red and (2) that the rose is not red insofar as it is blue. Although it’s not the case that the rose is not red (given that it is red), the passage from the rose’s being red to its not being red is genuinely the case—and that is all we need for genuine change. As mentioned, the model theory sharply distinguishes between truth at a point in T and truth at an ordered pair in O. At points in T, atomic sentences are true and passage sentences aren’t, whereas at points in O, passage sentences are true and atomic sentences aren’t. This reflects the intuition that momentary states are not constituted by their own transition, and that the passage of states are not themselves constituted by the momentary states that pass. The points in T represent the contents of times, or what happens ‘in’ time. The points in O represent passings as taking us from one time to another, and the very passing of momentary contents is not itself part of the momentary contents that pass. In more metaphysical terms: the passing of time itself is not ‘in’ time, but constitutes it.

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Since the relation O is transitive, O includes pairs of points in T that are not adjacent in the temporal order. The contents of one time do not just pass into the contents of the next time (if there is one), they also pass into the contents of any time that comes after it. Is this right? Does the way things are now pass into the way things will be in some time from now? If time passes to a subsequent moment, and that second moment passes into a third moment, there is a sense in which the first moment thereby passes into the third moment—this just is what it is for the first moment to pass into that third moment, leaving little room to deny the overarching passage. We can think of the passage from the current facts to facts of later moments as being like a determinable, realized in the passing between the facts of intervening times. Just as we can recognize ways in which something is colored (a red way, a blue way, etc.), we can recognize ways in which certain passage facts obtain. Say the following are successively true: A, :A, A. We might then say that it being the case that A ↪ :A and :A ↪ A is how it is the case that A ↪ A.²⁸ Note that it is only when there are no contrary facts in between, that passage from one time to another constitutes true stasis across time. There is also a more theoretical pressure to think of passage as transitive. Time is plausibly thought to be continuous, that is, passage is naturally thought to be a passage through a continuous series and not naturally thought to consist of staccato jumps between discrete units of time. If we were to deny the overarching passage, and replace the order relation in the model theory with its transitive reduction (i.e. a non-transitive order relation), so that one moment only passes into the very next moment (cf. von Wright 1965), we would then be at a loss to account for the passage of time if time is continuous, i.e. such that between any two moments of time there is another, so that there is no ‘next’ time. Put differently: if time is continuous, which seems plausible, and if the way things are now passes into the way things are now, which is our starting assumption, then we thereby have good reasons to accept that there is overarching passage.

²⁸ Alternatively, we could invoke some suitable notion of metaphysical grounding or realization and say that A ↪ A because or in virtue of the fact that A ↪ :A and :A ↪A. This may however raise issues depending on our commitments concerning grounding. That is, the approach may imply infinite chains of grounding facts in the case where time is continuous. It may also be problematic when we have a time series in which we successively have: A, A, A. Are we going to say that A ↪ A because A ↪ A (and A ↪ A)? This would be an objectionable case of self-grounding. Now this may in turn be avoided if we hold that the fundamental kind of passage is only ever that of maximal co-obtainment facts; but this may run contrary to the intuition that the passing of maximal facts is built up from the passing of facts that are ‘part’ of those co-obtainment facts.

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Besides its transitivity, it might be surprising that passage is not asymmetric: A↪B  = :ðB ↪ AÞ It’s allowed that both A ↪ B and B ↪ A. We should all agree that passage is an essentially directional affair, that passage is understood to take reality from somewhere to somewhere. But it is a mistake to think that the formal property of asymmetry (i.e. the property that A ↪ B  :(B ↪ A)) is sufficient or even necessary to capture the temporal directionality. Asymmetry would tell us that when we pass from A to B, there is no passage from B back to A. But that is not directionality, that is non-recurrence across time, telling us that if A passes away, it cannot pass back into reality again.²⁹ Such a logical ban on non-recurrence seems implausible; it seems clear that certain facts can recur over time. If I sit, then stand up, and then sit down again, my sitting passes into my not sitting, and my not sitting passes into my sitting. Which is not to say that there may also be specific kinds of states that may not recur in this way; for example, the Second Law of Thermodynamics tells us that, when a closed system’s state of entropy passes away, we never pass back to that very system having that very state of entropy again. There may thus be various asymmetries running along the passage of time. But that should not make us think of passage itself as being asymmetric. Although passage is not asymmetric, it is also not symmetric:

= B ↪A A ↪B  But, again, even though passage is non-symmetric, this is not what provides or captures the directionality of passage; it only reflects the possibility that we pass one way and never back, as in the mentioned case of states of entropy. It is tempting to think that asymmetry and non-symmetry somehow capture the directionality of passage but they really only capture what points we are allowed to pass through. So then how do we capture the directionality of passage? As far as I can see, the directionality of passage is a primitive aspect of the notion of passage, something we only really express by saying that we pass from one to another thing, that passage drives us forwards. I do not see how this aspect of passage can be captured by anything other than the notion of passage ²⁹ It is thus also a mistake to think that, simply because the earlier than relation is asymmetric, it thereby captures something of the directional aspect of passage. That the earlier than relation is asymmetric merely means that there is a certain order to the contents of time.

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itself. Much has been written about the direction of time. There are various types of physical asymmetries running along the direction of time, such as the increase of entropy, the expansion of the universe, and the causation of events. Many have explored reductive views of the direction of time (see Mellor 1998: ch. 12; Zeh 2007; and the papers in Savitt 1995), an example of which is the view that the direction of time runs from a first state to a second state when the entropy of the second state is larger than that of the first. The proposed view of the direction of time clearly stands in contrast to such reductive views. The directionality of time does not emerge from anything that is the case at the various times, but is part and parcel of passage itself. Not only is it hard to see how any directional passage can truly be seen to emerge from the proposed reductive bases, the non-reductive view has other advantages. The reductive views make it hard to see what the content is of various important physical laws. As Maudlin (2007: 129) points out, if the direction of time is reduced to the increase in entropy for example, this seems to drain the content out of the Second Law of Thermodynamics, which surely doesn’t just state that entropy increases as entropy increases. The non-reductive view of the passage of time enables a straightforward reading of physical laws that concern asymmetries over time. On the one hand there is the increase in entropy (or any other physical asymmetry) and on the other hand there is the passage of time. So the relevant physical law relates one thing to another and is therefore substantive. To sum up: passage is either that of one fact into itself or that of one fact into a contrary fact, it is non-factive, transitive, and neither asymmetric nor symmetric. Of course, this merely regiments the formal features of passage and much more needs to be done to show that this regimentation fits the metaphysical work that the notion of passage is meant to do for us, if we accept it as basic. One next step is to explore what the truth conditions are of tensed sentences and sentences that involve dates, in light of the proposed metaphysical view of the world. Another step is to consider the relation of the view to the special theory of relativity. We have assumed a single foliation of facts constituting one definite order in which things come to pass, and we know that things cannot be that simple.³⁰ Also, given the various choice points we encountered, we can explore variants of the view proposed here. Some of the choices made above are not set in stone.³¹ ³⁰ See e.g. Gödel (1949/1990). Fine argues that fragmentalism is superior to the A-theory precisely because it is compatible with the special relativity theory; see Fine (2005: §10). ³¹ Just as there are many systems of tense logic, so we can naturally expect various systems of passage logic. We could for example explore different logics by changing the

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One may have noticed how we retraced some of the steps of McTaggart’s argument for the unreality of time. McTaggart’s argument targets both the A-theoretic and B-theoretic conceptions of time. The A-theoretic determinations of events give rise to an inconsistency when we take an atemporal standpoint (cf. Dummett 1960: 503) and collect together the ways events are throughout time: the same event is then past, present, and future. The A-theorist can reply to this that the mistake is to resort to an atemporal perspective: if one assumes an atemporal perspective one will indeed attribute incompatible determinations to the same event (which cannot subsequently be explained away), but this just shows that we should never assume an atemporal perspective in the first place and say that an event is past and present and future; any event is only past, or present, or future (Prior 1967: 5–6). The objection raised in section 1 plugs this hole in McTaggart’s argument: if we merely describe the way things are now, so that any given event is either only past, or only present, or only future, then our description turns into a snapshot of a single moment, within which no passage or change is to be found. Passage occurs from one moment to the next; we must stand back from a single moment of time and assume an atemporal perspective if we are to make room for a real passage of time. Against the B-theoretic conception of time McTaggart famously insisted on the essential connection between passage and change, arguing that B-theoretic conceptions do not allow for genuine change (McTaggart 1908: 459). I agree. It’s a necessary condition for passage that at least some of the passage of time constitutes genuine change. If an alleged notion of passage did not make for any change, it would not be passage. But to make for change, there must be passage between contrary facts—and this the B-theory allows no room for. McTaggart’s assumptions about the passage of time have all been taken on board as necessary conditions that a primitive notion of passage must meet in order to qualify as genuine passage. The crucial question is whether we can conceive of the world in such a way that passage—thus understood—can obtain. Fragmentalism delivers such a world and so, pending other ways of making sense of a world harboring incompatible facts, it seems that fragmentation across time is thereby a necessary condition for genuine passage. We

order relation O in the model theory. This is how we also explore various systems of tense logic; see e.g. Burgess (2002).

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arrive at a metaphysical view that is not logically incoherent and yet offers everything that McTaggart demanded of a temporal world.³² University of Amsterdam

R E F E REN C E S Broad, C. D. (1938/1976). Examination of McTaggart’s Philosophy. Octagon Books. Burgess, J. P. (2002). ‘Basic Tense Logic’. In D. Gabbay and F. Guenther, eds., Handbook of Philosophical Logic, Vol. 7. Dordrecht: Kluwer. Dummett, M. (1960). ‘A Defense of McTaggart’s Proof of the Unreality of Time’. Philosophical Review 69.4: 497–504. Earman, J. (1974). ‘An Attempt to Add a Little Direction to “the Problem of the Directionality of Time” ’. Philosophy of Science 41: 15–47. Ehring, D. (1997). ‘Lewis, Temporary Intrinsics, and Momentary Tropes’. Analysis 57.4: 254–8. Fine, K. (2001). ‘The Question of Realism’. Philosophers’ Imprint 1.2: 1–30. Fine, K. (2005). ‘Tense and Reality’. In K. Fine, Modality and Tense: Philosophical Papers, pp. 261–320. Oxford: Oxford University Press. Gödel, K. (1949/1990). ‘A Remark about the Relationship between Relativity Theory and Idealistic Philosophy’. In S. Feferman et al., eds., K. Gödel, Collected Works. Oxford: Oxford University Press. Haslanger, S. (2003). ‘Persistence Through Time’. In M. J. Loux and D. W. Zimmerman, eds., The Oxford Handbook of Metaphysics, pp. 315–54. Oxford: Oxford University Press. Hofweber, T. and J. D. Velleman (2011). ‘How to Endure’, Philosophical Quarterly 61.242: 37–57. Jaśkowski, S. (1948/1969). ‘Propositional Calculus for Contradictory Deductive Systems’. Studia Logica 24: 143–57. Johnston, M. (1987). ‘Is there a Problem of Persistence? I’. Proceedings of the Aristotelian Society (Supplementary Volume) 61: 107–35. Kamp, H. (1971). ‘Formal Properties of Now’. Theoria 37: 227–73. Lewis, D. (1982). ‘Logic for Equivocators’. Noûs 16: 131–41. Lewis, D. (1986). On the Plurality of Worlds. Oxford: Blackwell. Lipman, M. A. (2015). ‘On Fine’s Fragmentalism’. Philosophical Studies 172.12: 3119–33. Lipman, M. A. (2016). ‘Perspectival Variance and Worldly Fragmentation’. Australasian Journal of Philosophy 94(1): 42–57.

³² Many thanks to Aaron Cotnoir, Kit Fine, Katherine Hawley, Bruno Jacinto, Colin Johnston, Gabriel Uzquiano, Sander Werkhoven, and Tobias Wilsch, attendees of the MMM seminar at Arché, attendees of the SPA 2013 conference at Stirling, Dean Zimmerman, and the referees of this volume, for many helpful discussions and helpful feedback on earlier drafts.

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Lowe, E. J. (1988). ‘The Problems of Intrinsic Change: Rejoinder to Lewis’. Analysis 48.2: 72–7. MacBride, F. (2001). ‘Four New Ways to Change Your Shape’. Australasian Journal of Philosophy 79.1: 81–9. McTaggart, J. E. (1908). ‘The Unreality of Time’. Mind 17.68: 457–74. Maudlin, T. (2007). The Metaphysics within Physics. Oxford: Oxford University Press. Mellor, D. H. (1981). Real Time. Cambridge: Cambridge University Press. Mellor, D. H. (1998). Real Time II. New York: Routledge. Meyer, U. (2013). The Nature of Time. Oxford: Clarendon Press. Norton, J. D. (2010). ‘Time Really Passes’. Humana Mente: Journal of Philosophical Studies 13: 23–34. Park, D. (1971). ‘The Myth of the Passage of Time’. Studium Generale 24: 19–30. Price, H. (2011). ‘The Flow of Time’. In C. Callender, ed., The Oxford Handbook of the Philosophy of Time, pp. 276–311. Oxford: Oxford University Press. Priest, G. (2007). ‘Dialetheism and Paraconsistency’. In D. Gabbay and J. Woods, eds., Handbook of the History of Logic, pp. 129–204. Amsterdam: North Holland. Priest, G. (2008). ‘Jaina Logic: A Contemporary Perspective’. History and Philosophy of Logic 29: 263–78. Prior, A. N. (1962/2003). ‘Changes in Events and Changes in Things’. In P. Hasle et al., eds., Papers on Time and Tense, new edition, pp. 7–19. Oxford: Oxford University Press. Prior, A. N. (1967). Past, Present and Future. Oxford: Oxford University Press. Rescher, N. and R. Brandom (1980). The Logic of Inconsistency: A Study in NonStandard Possible-Worlds Semantics and Ontology. Oxford: Basil Blackwell. Restall, G. (1997). ‘Ways Things Can’t Be’. Notre Dame Journal of Formal Logic 38.4: 583–97. Rodriguez-Pereyra, G. (2003). ‘What is Wrong with the Relational Theory of Change?’ In H. Lillehammer and G. Rodriguez-Pereyra, eds., Real Metaphysics: Essays in Honour of D. H. Mellor. Abingdon: Routledge. Savitt, S. (1995). Time’s Arrow Today. Cambridge: Cambridge University Press. Savitt, S. (2002). ‘On Absolute Becoming and the Myth of Passage’. Time, Reality & Experience 50: 153–68. Sider, T. (2001). Four Dimensionalism: An Ontology of Time and Persistence. Oxford: Oxford University Press. Smart, J. J. C. (1949). ‘The River of Time’. Mind 58.232: 483–94. Tallant, J. (2013). ‘A Heterodox Presentism: Kit Fine’s Theory’. In R. Ciuni, K. Miller, and G. Torrengo, eds., Contributions to Philosophy, pp. 2–26. Munich: Philosophia Verlag. Varzi, A. (1997). ‘Incoherence Without Contradiction’. Notre Dame Journal of Formal Logic 38.4: 621–38. von Wright, G. H. (1965). ‘And Next’. Acta Philosophica Fennica 18: 293–304. Williams, D. C. (1951). ‘The Myth of Passage’. The Journal of Philosophy 48.15: 457–72. Zeh, H. D. (2007). The Physical Basis of the Direction of Time, 5th edition. Berlin: Springer.

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4 Fragmenting the Wave Function Jonathan Simon A key insight of Fine (2005)’s fragmentalism is that there is a symmetric coordination relation between facts, such that facts that are pairwise incompatible (like Hugh’s being happy and Hugh’s being sad) can both obtain provided that they are not related by this relation.¹ Here, I will use the term ‘fragmentalism’ to describe any view that incorporates this insight. In this paper, I will present a new fragmentalist account of B-theoretic endurantism and a new fragmentalist account of the metaphysics of the quantum state, and I will highlight the deep parallels between the considerations that motivate them. Along the way, I will make clear that these new accounts do not rely on the further details of Fine’s (or Lipman’s) fragmentalism— e.g. on the claim that reality is (logically) incoherent, in the sense that P can obtain in one fragment while : P obtains in another. The new fragmentalist account of B-theoretic endurantism is motivated by a strengthened version of Sider (2001)’s problem of exotica, which I present below. The new fragmentalist account of the quantum state, which I will call conservative realism about the quantum state, is motivated by the desire to reconcile three desiderata: first (completeness), that the universal quantum state says (more or less) all there is to say about the universe; second (anti-holism), that the universal quantum state is grounded in its branches; and third (familiarity), that the branches are grounded in local states of affairs involving the positions of particles in space-time. Most abandon at least one of these desiderata. As we will see, there is a striking parallel between the strengthened version of the problem of exotica and the considerations that have led most to abandon one of these three desiderata for the quantum state.

¹ Fine (2005) calls this relation coherence and Lipman (2015a; 2015b) calls it co-obtaining. I follow Lipman in assessing the key insight of Fine’s view. Following both authors, I stress that talk of facts may be regarded as loose speaking, shorthand for talk that can be couched in terms of sentential operators.

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I will proceed as follows. In §1, I distinguish between three strategies for B-theoretic endurantists: relativizing, outsourcing, and defusing, and I motivate fragmentalism as the most promising development of the defusing strategy. In §2, I consider two cases, the cases of spooky isolation and spooky coincidence, which grow out of Sider (2001)’s argument from exotica against endurantism, and I argue that no other endurantist theory can handle these cases as well as fragmentalist endurantism. In §3, I present the considerations that seem to force the rejection of conservative realism, highlight the striking parallel between these considerations and the spooky arguments against endurantism that I consider in §2, and then present the fragmentalist implementation of conservative realism. Along the way, I will make it clear that fragmentalism need not be “jagged” (i.e. facilitate contradiction) in order to underwrite either fragmentalist B-theoretic endurantism or conservative realism about the quantum state.

1 . S T R A T E G I E S F O R B -T H E O R E T I C E N D U R A N T I ST S : R E L A T I V I Z I N G, O U T S O U R C I N G , A N D D E F U S I N G I take a B-theoretic endurantist view to be a view according to which one and the same entity can instantiate both of a pair of apparently incompatible properties (like the property of being happy and the property of being sad, or the property of being red and the property of being green, or the property of being round and the property of being square), and do so by being wholly present “twice over”: once at each instantiation—or anyway not by having temporal parts or counterparts.² Here I will distinguish three strains of B-theoretic endurantism that fit this description, strains that differ over how they make sense of the idea that one and the same thing can have apparently incompatible properties at different times: relativizing accounts, outsourcing accounts, and defusing accounts. We will see that fragmentalism yields a cogent defusing account.

² My aim here is not to offer an all-encompassing characterization of endurantism per se. I am not concerned with what endurantism requires at worlds where things do not change their properties as they persist. And there are views that are neither stage-theoretic nor worm-theoretic, but which may not qualify as endurantism in my sense (e.g. Nolan 2014). Also, my characterization allows a B-theoretic endurantist to countenance enduring perdurants—things that have multiple temporally extended locations, which others will resist. For more on the general characterization of endurantism see Parsons (2007), Donnelly (2011), Effingham (2012), and Gilmore (2014).

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1.1. Relativizing accounts According to relativizing accounts, the endurant directly instantiates two properties that are apparently incompatible, but there is no real incompatibility, because the properties that the endurant instantiates, or the episode of instantiation itself, constitutively involve further variables: there is some parameter of persistence—presumably, time—and to be happy or sad is really to be suitably related to one or another value of this parameter. There are important debates among relativizers: Mellor (1981) and Van Inwagen (1990) say that even apparently intrinsic properties, if they are temporary, must really be relations to times. Johnston (1987) and Lowe (1988) hold that there can be truly intrinsic temporary properties but in instantiating them we are related to times. Similarly Haslanger (1989) says that propositions which report on the instantiation of temporary intrinsics are true relative to times.³ I note that the challenges I will consider below do not hinge on questions about intrinsics.

1.2. Outsourcing accounts According to outsourcing accounts, the endurant does not directly instantiate both of the pair of incompatible properties. On such accounts there are two genuinely incompatible properties and they are directly instantiated by distinct individuals, at least one of which is not the endurant, though the endurant takes the credit. On such accounts either it is strictly true that the endurant instantiates the properties, though it does so indirectly, i.e. by proxy, or it is not strictly true but we nevertheless have pragmatic reasons to talk as though it is true. Standard worm-theoretic perdurantist accounts involve outsourcing: the worm takes the credit for the properties instantiated by its temporal parts. But endurantist outsourcers say that the thing that takes the credit is wholly located at the proxies that do the instantiating. Outsourcing accounts call for an ontology that takes in more than endurants (or they bottom out in relativizing or defusing accounts). For example, Eagle (2010) defends an outsourcing account in which an endurant e counts as having property P at region r insofar as e is exactly located at region r and region r has property P. Giordani and Costa (2013) and Costa (forthcoming) defend a rival outsourcing account in which for an ³ A sophisticated recent variation is Spencer (2016). Parsons (2007)’s distributional property approach is a borderline case. Neither of these views fare any better than standard relativizing approaches against the challenges I raise below.

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endurant e to have property P at time t is for there to be some event in which e participates in some suitable way, which is located at t and instantiates P.⁴

1.3. Defusing accounts According to defusing accounts, the endurant directly instantiates two properties simpliciter that are genuinely incompatible, but there is a mitigating factor which somehow defuses this incompatibility.⁵ A successful defusing account must provide an analysis of property incompatibility that explains how it can be defused (i.e. explain what incompatibility is, such that one and the same thing can instantiate (simpliciter) a pair of incompatible properties if defused), and it must also provide an analysis of defusing, one which makes it neither too rare nor too ubiquitous. There are three questions that confront a defuser. The first is how to understand the relevant kind of property incompatibility. On one extreme, the relevant kind of property incompatibility is logical incompatibility. This route leads to dialethism. On the other extreme, we restrict our attention exclusively to pairs of properties that are not really incompatible (like Thinking about Tennis and Thinking about Golf ). But then we cannot account for change between genuine incompatibles. A middle ground focuses on nonlogical but genuine incompatibilities, such as that of Happy and Sad. The second question is how to identify the conditions that do the defusing. Plausibly, this is a role played by time. But as we will see below, other conditions may serve as well. The third is to identify the mechanism of defusing. If the defusing strategy is to be an alternative to relativizing or outsourcing strategies, it is not enough to say that Harry can be both happy simpliciter and also sad simpliciter provided that he is happy at one time and sad at another. What is it for Harry to be happy at one time and sad at another, if it is not what relativizers or outsourcers say it is? At the time of writing I am aware of no authors who endorse nonfragmentalist defusing strategies explicitly. Authors that come close to adopting the defusing strategy, but who do not quite do so, include Ehring

⁴ See Miller (2013) for critique and Costa and Giordani (2016) for reply. Another outsourcing theory says that endurants are haecceities instantiated by perdurants. See Benovsky (2011). ⁵ Here I take it that in having a property simpliciter, there is no mode with which one has it, and nothing to which one’s having it is a relation (my usage corresponds with that of Lewis 2002). In the sense I have in mind, relativizers must deny that endurants have ordinary properties simpliciter. Compare Miller (2005) and Miller and Braddon-Mitchell (2007).

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(1997), Mellor (1998), and Hansson (2007).⁶ Ehring, Mellor, and Hansson all reify property instances (as tropes, in Ehring’s case, and facts in Mellor’s and Hansson’s). All three also suppose that property instances can have temporal locations, and deny that it follows that the property in question is a relation to a temporal location. However, none of these authors accept that it follows from the existence or obtaining of the property instance of Hugh’s being happy that ‘Hugh is happy’ is true simpliciter, at least not if the ‘is’ in question is the B-theoretic, fundamentally untensed ‘is-simpliciter’.⁷ All three are therefore relativizers in my broad sense. But these authors, at least, embrace ontologies hospitable to the defusing strategy. As I will explain just below, fragmentalism emerges as a compelling implementation of the defusing strategy. But any workable defusing approach has an advantage over relativizing and outsourcing accounts. Any such account would permit us to infer from ‘x is F at t’ that ‘x is F’ is true simpliciter in the fullest possible sense, where nothing is relativized and nothing is outsourced. To those influenced by Lewis (1986)’s argument from temporary intrinsics this is virtue enough.⁸

1.4. Fragmentalism Defusers must identify both the conditions under which incompatibility is defused and the mechanism of defusing. In the “Mellorian” analysis, the condition is time: two properties are incompatible just in case nothing can instantiate them at the same time. And to make sense of “instantiation at a time” one countenances property instances (like Hugh’s being happy) and then takes these to themselves have temporal-locational properties. But time is obviously parochial here: a more ecumenical analysis would speak generally of whatever parameters of persistence may vary at the world ⁶ Honorable mention goes to Carroll (2011), who suggests that the endurantist’s best response to time travel cases like the spooky coincidence case I discuss in section 2.2 is to accept that properties like sitting and standing are compatible after all. Carroll does not address the more general question of what makes it true that, e.g. the apple is red at t1 but brown at t2 . If Carroll were to seek to answer this question (without invoking a relativist or outsourcing account after all) his answer would likely lead to a defusing account. ⁷ Hansson (2007) comes closer than Ehring or Mellor to a defusing strategy. He holds that the fact that Hugh is happy makes ‘Hugh is happy’ true, where the ‘is’ is understood as tenseless. But in his (2007) he states and in his (2010 [Hansson Wahlberg]) he clarifies that the tenseless copula he has in mind can be elucidated in terms of a disjunction of tensed copulae; roughly: ‘was, presently-is, or will be’, and he denies that the fact that Hugh is happy makes ‘Hugh is happy’ true if the ‘is’ is a strict simpliciter copula. (See especially Hansson Wahlberg 2010: §VII, and Hansson Wahlberg 2013: 243.) ⁸ But see Eddon (2010) for reasons to doubt the argument from temporary intrinsics.

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in question or of whatever kinds of locations the relevant states of affairs can have. In those terms we might say: what it is for two properties P and Q to be incompatible is for it to be impossible for there to be any x such that the state of affairs Px and the state of affairs Qx share a location. The fragmentalist goes a step further. Instead of thinking of states of affairs as having locational properties or standing in locational relations, the fragmentalist introduces a primitive relation, co-obtaining (terminology creditable to Lipman 2015b), which can relate states of affairs without relating them spatiotemporally. As we will soon see, it is this feature which allows fragmentalism to solve problems which other endurantist theories (including other defusing theories) cannot solve. I stress (following Fine and Lipman) that for those who eschew an ontology of relations between facts, the structure we wish to capture here may be expressed with sentential operators. Following Lipman we may use ‘  ’ as a binary sentential operator expressing co-obtainment, readable as ‘insofar as’: ‘Fa  Gb’ says that a is F insofar as b is G.⁹ But for ease of exposition I will continue to quantify over facts. A fragment is then a maximal collection of states of affairs that mutually co-obtain. Something can instantiate incompatible properties provided that the instantiations do not co-obtain, and what it is for two properties P and Q to be incompatible is for it to be impossible for anything to be P insofar as it is Q. We generally suppose that when two properties are incompatible it is impossible that they be co-instantiated simpliciter. Fragmentalists (and defusers more generally) must qualify this claim. But there is a choice point here—a question of how much incompatibility the fragmentalist thinks we can live with, or in other words, a question of how “jagged” one’s fragmentalism is. Broadly there are three grades to consider. What we might call dialethic fragmentalism revises our logic itself, allowing for true (first-order) logical contradictions to obtain without quodlibet. Loss (2017) endorses such a view. What we might call jagged fragmentalism allows that fragments may fail to cohere with one another, in the sense that there is some notion of obtaining-in-a-fragment such that P can obtain in one fragment while : P obtains in another, but this does not engender genuine contradiction. Both ⁹ Lipman (2015b). Note that temporal defusers also need a primitive notion of location for states of affairs. Hansson (2007: §6) points out that perdurantists, insofar as they countenance states of affairs, must also take these to have locations. However the perdurantist can treat a state of affairs’ location as derivative on the location of the particular entity (the perdurant) that participates in it.

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Fine (2005; 2006) and Lipman (2015a; 2015b; 2016) endorse versions of jagged fragmentalism, though they implement it in different ways.¹⁰ What I will call smooth fragmentalism, in contrast, denies that there is any genuine incoherence, within the relevant operators or without. On this approach, logic remains classical, reality is coherent, and obtaining is a pre-condition for co-obtaining with something or other. On this approach, we do not claim to defuse any logical inconsistency in any guise. Instead we focus exclusively on those cases of metaphysical incompatibility that do not generate logical incompatibility—i.e. distinct determinates of a common determinable, like being scarlet (all over) and being crimson (all over). I tentatively endorse this view, though here, my focus will be on motivations for the fragmentalist framework broadly construed.¹¹ Distinct versions of fragmentalism face distinct challenges. The smooth fragmentalist faces a semantic challenge: we ordinarily talk as though ‘sad’ implies ‘not happy’. Suppose that Hugh is happy in one fragment and sad in another. Dialethic fragmentalists may allow that the fact that Hugh is not happy obtains, while jagged fragmentalists may allow that it obtains-in-afragment. But the smooth fragmentalist denies this. The smooth fragmentalist appeals to pragmatics: in relevant circumstances, ‘Hugh is not happy’, though literally false (because it means that the fact that Hugh is not happy obtains), conveys that Hugh’s being happy is not a part of the relevant fragment.¹² There are some challenges faced by both smooth fragmentalists as well as jagged fragmentalists (like Lipman) who distinguish between facts that obtain and those that merely co-obtain. For example, shape properties like

¹⁰ Fine takes the logic of predication to involve an In Reality operator, such that it does not follow from ‘In Reality : P’ that ‘ : In Reality P’, but such that contradictions would ensue if this inference could be drawn. Lipman allows that some (non-atomic) facts may co-obtain without obtaining simpliciter. Hugh’s not being happy co-obtains with Hugh’s being sad, but if Hugh’s being happy also obtains (in a distinct fragment) then Hugh’s not being happy does not obtain. ¹¹ Tractarian accounts seek to reduce all cases of metaphysical incompatibility to cases of logical incompatibility. See again Moss (2012), Turner (2016) for recent defenses. If the Tractarian can show that Scarlet and Crimson really are logically incompatible, this is a problem for the smooth fragmentalist. But fragmentalism can be a boon for the Tractarian who concedes that there is no logical incompatibility here, because fragmentalism allows us to deny that Scarlet and Crimson are incompatible with metaphysical necessity. Of course the fragmentalist holds that it is impossible that their instantiations co-obtain, but further resources are available here. The logic of co-obtainment may shed light on the logic of determinable relations. Moreover, in some cases anyway, the fragmentalist might hold that incompatibilities are nomological rather than metaphysical. ¹² For example, if we treat ‘At-t’ as a sentential operator, such that ‘At-t P’ is true iff the fact that P obtains and is located at the t-situation, then we might say that ‘Hugh is not happy’ uttered at t generally communicates that : At-t (Hugh is happy), though it means that At-t ( : Hugh is happy).

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‘round’ and ‘square’, or ‘straight’ and ‘bent’ may seem like basic, positive properties, but on closer inspection there are reasons to suspect otherwise. Arguably, part of what it is to be round is to not have any right angles in one’s boundary, and part of what it is to be square is for there to not be any central point such that all of one’s boundary points are equidistant from it, on any relevant metric. But then these fragmentalists cannot allow that something that changes from being round to being square is both round simpliciter and also square simpliciter. Which is it? Similar problems arise with mereological predicates like ‘overlap’ and locative ones like ‘exact location’.¹³ But these limitations also help to illustrate why smoother forms of fragmentalism do not go too far. Just as pain is your body’s way of telling you that it needs fixing, so inconsistency is a theory’s way of telling you that it needs fixing. A worry about dialethic fragmentalism, and perhaps some jagged fragmentalisms, is that they foster one with an all-purpose painkiller, a way of living with something that needs fixing, rather than taking measures to repair it. But smooth fragmentalism requires that there is no inconsistency or incoherence between any fact that obtains in a fragment and any other fact that obtains in any other fragment. Thus, for example, the smooth fragmentalist cannot resolve the paradox of the statue and the clay by affirming the identity of the two while maintaining that the statue can survive shattering in one fragment but not in another (but see Pickup 2016). Thus the applicability of smooth fragmentalism, anyway, must be evaluated on a case by case basis, by considering its implications for what facts would end up obtaining simpliciter. Now, for some applications, it matters which version of fragmentalism we embrace. On Fine’s approach each fragment’s pronouncements about what is past and what is future are inconsistent with one another, and on Lipman’s approach real change requires passage from one thing’s being the case to its negation being the case. But the considerations I present below motivate smooth fragmentalism as well as its more rough-hewn cousins. What is the logic of the co-obtainment relation? This depends on the version of fragmentalism one prefers. All should agree that it is symmetric: the relation is not order-sensitive. Reflexivity is trickier: if it is true simpliciter that Hugh is happy, but also true simpliciter that Hugh is sad, then if ¹³ The beginning of a reply: a shape specification can be separated into a positive and a negative component. The positive component specifies the parts a thing has or the points or regions it occupies (and the metric and topological connections among these); the negative component states that the thing has no more parts, or occupies no further points or regions, beyond these. The negative components get the same treatment as ‘Hugh is not happy’. An alternative is to deny the relevant implications, for example by taking the relevant properties or ideology as primitive, though it would be a weakness of the view if this were compulsory.

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co-obtainment is transitive we must deny that the conjunction of these facts co-obtains with itself. On the other hand there are advantages to denying transitivity. If co-obtainment is not transitive, we can think of one and the same state of affairs as involved in different fragments. Suppose there is one fragment in which Caspar is happy and hungry, and another in which he is happy and sated. If the fact that Caspar is happy is identical in both cases then by transitivity Caspar’s being hungry would co-obtain with his being sated.¹⁴ Note also that co-obtainment is not the same as compresence: x’s being F may co-obtain with y’s being G. How does fragmentalism accommodate ordinary cases of change? It depends on how things and states of affairs turn out to be located at the world in question. I will develop this theme at greater length below. But assuming there are times, and objects are located at them, what it is for Hugh to be happy at t1 is for the state of affairs of Hugh’s being happy to co-obtain with the state of affairs of Hugh’s being temporally located at t1 . More generally, if this is a classical Newtonian world at which time travel is impossible, a single fragment might comprise all of the goings on at t1 . If Hugh is happy and Donald is sad at t1 , then Hugh’s being happy and Donald’s being sad and Hugh’s being at t1 and Donald’s being at t1 all co-obtain. Fragmentalism gets more interesting as the worlds in which we want to accommodate endurance get more interesting. Insofar as we are concerned only with classical Newtonian worlds (and we are comfortable quantifying over times) fragmentalism and a temporal defusing strategy more or less coincide. But fragmentalism shines when we confront scenarios that do not offer up other entities obviously suited to serve as parameters of persistence. I turn my attention now to some of these.

2 . S P O O KY IS O L A T I O N AN D S P O O K Y C O I N C I D E N CE Sider (2001: §4.7)’s argument from exotica targets the same aspect of endurantism as the argument from temporary intrinsics: viz. the endurantist’s need to index property instantiation to a parameter of persistence. But where the argument from temporary intrinsics identifies a special difficulty with this, one that only arises if the relevantly indexed properties are intrinsic, the argument from exotica identifies a general difficulty, one ¹⁴ Both Mellor (1998) and Fine (2005) duplicate states of affairs. Ehring (1997) similarly duplicates tropes, but that is arguably a more familiar move.

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which arises no matter what the arity of the properties involved: namely, that it can vary from world to world what parameters of persistence there are but it should not vary from world to world what it is to instantiate the relevant properties. Sider’s argument centers around two cases, the case of timeless worlds and the case of time-traveling worlds. At timeless worlds, Sider argues, endurantists must deny that being round (for example) is a relation to a time (or a temporal mode of instantiating roundness). This suggests an uncomfortable duality: how can there be two species of roundness, or two modes of instantiation of roundness? At time travel worlds, Sider argues, endurantists must take it that there is some further parameter of persistence in addition to time. That step may be anyways motivated in a relativistic world, but it seems to lack any independent motivation in the non-relativistic case. Others have more recently noted that time travel carries other costs for the endurantist: for example it suggests a tension with the axioms of minimal mereology.¹⁵ These arguments are important, but as I will discuss, endurantist replies are available. However, we can add details that make the cases far more difficult for the endurantist to respond. The only version of endurantism that can comfortably respond to these modified cases is fragmentalist endurantism (and there are no further modifications that make it difficult for the fragmentalist to respond). Incidentally, these modifications of the argument from exotica make for an argument that is strikingly similar to an argument against conservative realism about the quantum state. As we will see in section 2.1 it is therefore no accident that fragmentalism offers a novel response to that argument as well.

2.1. Spooky isolation Endurantists can bite the bullet and allow that there are two ways of being round: the temporal way which we employ at our world, or the atemporal way employed at a timeless world. As Sider points out, some endurantists who also countenance perdurants, but deny that either grounds the other, may have to accept that there are two distinct ways of being round anyway. Endurantists can also insist that even at timeless worlds there is some parameter or other to which properties may be indexed. Endurantists may say that what is necessary is that properties are indexed to locations, while what is contingent is that the indices at our world are times. Thus, a ¹⁵ See, for discussion, Effingham and Robson (2007), Gilmore (2007; 2009; 2014), Donnelly (2010), Effingham (2010), and Kleinschmidt (2011).

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relativizer may say that I am round-at t, while my counterpart is round-at l , where l is some suitable location at the timeless world. The outsourcer may take the endurant’s proxies to be located at locations like l , and a nonfragmentalist defuser might hold that two states of affairs are incompatible only if it is impossible that they have the same location. But all of this is only as compelling as the claim that a timeless world must involve locations (or distance relations). What if we stipulate, against the endurantist, that the problem case is a timeless world in which there are no distance relations? Maybe the endurantist can insist that the only possibility in the vicinity is a world where there is only one token location (or equivalently, a world where everything is zero distance from everything else). But even this is no help if we stipulate that one and the same thing instantiates both of a pair of (apparently) incompatible properties at such a world. Of course the endurantist may refuse to countenance such an outlandish possibility. But if the endurantist countenances non-empty location-free worlds at all, the endurantist has some reason to countenance such a world where a thing instantiates incompatibles. If we allow, as B-theoretic endurantists do, that something can have multiple whole locations at a world, we have some reason to embrace an at-a-location haecceitistic permutation principle, which holds we can permute haecceities whole location by whole location rather than endurant by endurant. If we allow that things can have properties at location-free worlds we need a more general notion than “whole location”—call them manifestations. But if we make it this far, we should say that the permutation principle extends to all manifestations. Now consider a location-free world where Caspar is happy and Homer is sad. Then by the permutation we obtain a world where Caspar is happy and Caspar is sad. Call this the case of spooky isolation. But the spooky isolation case does not contain enough things for standard endurantists to differentiate the two manifestations of Caspar. Either both manifestations share a single location or there are no locations at all for them to share.¹⁶ ¹⁶ Non-fragmentalist defusers (who appeal to difference in location rather than co-obtainment structure) may say that the relevant states of affairs lack locations, which means they do not share a location, which defuses the incompatibility. However, this strategy entails that no states of affairs may be coordinated with one another at such a world. Maybe Caspar can be happy and feel pleasure at one manifestation while being sad and feeling pain at another. The non-fragmental defuser cannot differentiate this scenario from one at which Caspar is happy and feels pain at one manifestation and is sad and feels pleasure at another. We might describe the former scenario thus: jHappyic jPleasureic þ jSad ic jPainic, and describe the latter thus: jHappyic jPainic þ jSad ic jPleasureic.

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In contrast, the fragmentalist can allow that a spooky isolation case is coherent as described, simply by maintaining that Caspar’s being happy and Caspar’s being sad are not part of the same fragment: that is, that Caspar’s being happy does not co-obtain with Caspar’s being sad.¹⁷ This counts in fragmentalism’s favor, at least insofar as we have reason to countenance the permutation principle noted above. But I do not raise this case because it is decisive; I raise it because it foreshadows what is to come. As we will see just below, the case of spooky coincidence is better motivated, and it leads to an analogous challenge for endurantism. Moreover, as we will see shortly thereafter, there is a striking parallel between this case and standard cases of quantum superposition which lead to a challenge for conservative realism about the quantum state.

2.2. Spooky coincidence As with the problem of timeless worlds, there are various ways that endurantists can respond to the problem of time travel as Sider presents it. As above, properties may be taken to be relations to locations, where the exact nature of the locational relata varies from world to world. Modes of instantiation might also correspond to types of locations generally rather than times in particular. The outsourcer is fine as long as there is a category of proxies to appeal to, and the non-fragmental defuser is fine as long as incompatible states of affairs must not share a location.¹⁸ Sider notes that matters are more difficult if the time travelers lack spatial locations.¹⁹ If the time travelers lack spatial locations the time travel case behaves much like the case of spooky isolation—at least if we also specify that these aspatial beings do not occupy “non-spatial” locations or stand in some “non-spatial” distance relation that can serve as a parameter of persistence or change. But we can achieve the same troubling effect even in worlds where all beings have spatiotemporal locations, provided that some of these beings are capable of both time travel and interpenetration. These are the kinds of cases that I will call spooky coincidence cases. ¹⁷ And unlike the non-fragmental defuser I discuss in note 16, the fragmentalist can with equal comfort differentiate between the jHappyic jPleasureic þ jSad ic jPainic scenario and the jHappyic jPainic þ jSad ic jPleasureic scenario by differentiating their co-obtainment structure. ¹⁸ This is to suppose that the real problem for the endurantist is having to add a second parameter to the index, not the problem of breaking down the disanalogy between space and time (on which see Miller 2006) or the problem of possibilities that the endurantist cannot differentiate (on which see Simon 2005). ¹⁹ Sider (2001: 105).

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At one such world Caspar is happy in his youth but melancholy as time passes. He travels back in time and passes right through his younger self (coincidentally he has kept his figure!), but this does not allay his sorrows: his journey to the past is a mournful one. Thus there is an exact region of space-time at which Caspar is wholly present twice over: happy on the first pass and sad on the second.²⁰ The case of spooky coincidence raises the same difficulty for endurantists as the case of spooky isolation. It is a further challenge, even for endurantists who embrace the need to index property instantiation to place as well as time or to spatiotemporal region, because in this case there is a pair of a place and a time, or a spatiotemporal region, such that Caspar is happy at it and also sad at it. The case is equally a problem for outsourcers, because it suggests that they must find two distinct proxies that share a spatiotemporal location. And it is a problem for non-fragmentalist defusers who say states of affairs are incompatible if they cannot share a location, because by the lights of that strategy, spooky coincidence cases involve incompatible states of affairs sharing a location. As with the case of spooky isolation, we can deny the possibility of this case. We can deny the possibility of time-traveling ghosts outright, for example. Alternatively we can deny that Caspar has two coincident manifestations. But what, then, is Caspar feeling at the moment of apparent selfinterpenetration: is he happy or sad, or both? Proposals holding that it is one or the other seem arbitrary: does he experience a fleeting spark of joy in his haunted old age, or a somber spell of sorrow in his spirited youth? And we have maintained that by ‘happy’ and ‘sad’ we mean to denote properties that are genuinely incompatible.²¹ So it looks as though, unless they are willing to go further and deny the possibility of all things that can travel back in time and pass through their former selves, endurantists have some reason to countenance the possibility of spooky coincidence cases. In disanalogy with the spooky isolation case, I have not stipulated any restrictions on what else there is at the world in question. The endurantist therefore has the option of countenancing the case by appealing to further basic ontology (or ideology), for example a ²⁰ This example traces back at least to Gilmore (2004: 190). Hawthorne discussed it in a commentary on Sider at the 2006 Pacific APA. It is also discussed in Carroll (2011). ²¹ A possible reply: mental properties are a special case. We can make sense of one and the same thing instantiating ‘incompatible’ mental properties insofar as that thing’s conscious experience is not unified (see Bayne and Chalmers 2003 for discussion). So perhaps the answer is that at the moment of self-interpenetration Caspar’s consciousness momentarily becomes disunified. However, some (including Bayne and Chalmers 2003) argue that a subject’s conscious experience is necessarily unified at a time, on which more below.

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dimension of hyperspace,²² or a sui generis dimension of personal time.²³ But ought we really expand our metaphysics just to handle cases like this? And shouldn’t we be able to say that if such cases are possible they are possible in worlds with no hyperspace and only one temporal dimension?²⁴ It might seem that the only cases at issue are nomologically impossible anyway. But here is an argument that there might be a spooky coincidence case even at a wholly material, ghost-free world. Consider a commisurotomy (split-brain) patient living at the nearest possible world to ours which is both non-orientable and contains closed time-like curves: i.e. such that there are subluminal trajectories that leave one in the “past” and “mirror reversed” (but note that neither notion is globally defined at such a world). Say also that in such a commisurotomy case there are two subjects: one, L, whose cognition centers on the left hemisphere and the other, R, whose cognition centers on the right hemisphere. But each subject extends beyond the center of its core cognition, just as you or I do. Suppose in particular that L and R mereologically and spatiotemporally coincide: a supposition to which an interpenetration-friendly endurantist should be open. Now at t1 surgeons perform an operation: they remove L and replace it with NL, an alternative left hemisphere that houses a subject. After the surgery NL relates to R in the same way that L did before the surgery: each is a distinct center of consciousness: NL sees only the word ‘able’ where R sees only the word ‘tax’, etc. Much later, at t100 , NL and R are separated. R is taken along a path that leaves it mirror reversed in the surgery room at t0 and hey presto, it turns out that R ¼ NL—or anyway, it does assuming an endurantist account of persistence. We then have a case of spooky (though not ghostly) coincidence in the time between the two surgeries, that is, between t1 and t100 .²⁵

²² Hudson (2005), Lockwood (2005). ²³ See Gilmore (2016) and Valaris and Matthew (2015). Note that many give reductive accounts of personal time. But if personal time is reducible it is hard to see how Caspar might have his different mental states in relation to different values of it (cf. Sider 2001: §4.7). ²⁴ See Kleinschmidt (2011: ch. 4) for a discussion of alternative strategies for the endurantist that avoid appealing to personal time. None are cost free. ²⁵ This case relies on the rejection of the various ‘single mind’ interpretations of splitbrain cases (see again Bayne and Chalmers 2003). This is controversial: many (including Bayne and Chalmers) defend such interpretations. But there is much to be said for a ‘two minds’ approach (or more generally a ‘not exactly one mind’ approach: Nagel (1971) suggests that the number of minds is fractional). It coheres with the phenomenal unity thesis without making us hostage to the existence of asymmetries accounting for which hemisphere dictates conscious experience at any given moment. The implication that two minds can (perfectly) coincide might be anathema to some, but recall that our challenge is aimed at endurantists, and many endurantists accept at least some cases of perfect coincidence (see, for discussion, Gilmore 2014).

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As with the spooky isolation case, the spooky coincidence case is a challenge for all endurantists. Unless extra ontology is introduced, relativizers have nothing to relativize to, outsourcers have nothing to outsource to, and temporal defusers have nothing to defuse with: both states of affairs have the same spatiotemporal location.²⁶ But introducing extra ontology is problematic in its own right. Does this extra ontology (e.g. sui generis personal time) only exist at worlds where there actually are cases of spooky coincidence, or must it also exist at worlds where there could be such cases? If the latter, then the endurantist position may force us to actually countenance that extra ontology, since as I note above versions of the case may be possible at worlds relevantly like ours. The fragmentalist, in contrast, can allow that a spooky coincidence case is coherent as described insofar as Caspar’s being happy and Caspar’s being sad are not part of the same fragment, or in other words, insofar as Caspar’s being happy does not co-obtain with Caspar’s being sad.²⁷ Of course in the spooky coincidence case there is more to say about what makes up a fragment. This points to another incidental virtue of fragmentalism: the fragmentalist is not compelled to identify a single parameter of persistence even at a world. At a spooky coincidence world, other things equal, fragments in which no one is doing any time travel may correspond to goings on at times or spatiotemporal regions. It is only in the special case of Caspar’s coincidence that we have two ‘co-located’ fragments. These will each comprise all of the other goings on at the relevant region, and one or the other of Caspar’s mood state of affairs, but not both. But it does not follow that every property instantiation must be located at the pair of a region and a mood. Fragmentalism therefore offers the endurantist a very comfortable response to the problems of exotica, a response that accommodates even the most extremely exotic versions of these problems with no extra ontological or ideological accumulation.²⁸ I conclude that even B-theoretic endurantists have good reason to consider fragmentalism. ²⁶ Sider (2001: §4.8) notes one “desperate reply”: to relativize property coinstantiations to one another. This makes property instantiation more holistic than some will find palatable, but more generally, what does it add to say that being happy is a relation to not being sad? ²⁷ Gilmore (personal communication) raises a challenge. Presumably it is true that young Caspar is happier than old Caspar. But if young Caspar and old Caspar do not share a fragment, where can this fact obtain? One solution, noted above, is to deny the reflexitivity of co-obtainment. We may then deny that this fact co-obtains with any other fact. Alternatively, we may think of it as inhabiting a fragment of its own, a fragment which is a counterexample to the transitivity of co-obtainment. ²⁸ A caveat: I have not addressed here the special mereological questions that time travel raises for endurantism. It might seem that the fragmentalist is bound to say that

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I turn my attention now to another selling point of fragmentalism: it provides us with a novel analysis of the metaphysics of the non-relativistic quantum state, one that may appeal to perdurantists or endurantists alike. Along the way we will observe the striking analogy between the arguments from exotica that I have just considered and the argument against the theory of the wave function that I am about to develop. And I reiterate that nothing here turns on the dialethic or jagged facets of some versions of fragmentalism: even smooth fragmentalism can foster the novel analysis promised here.

3. SPOOKY ACTION AT A DISTANCE Measurements of the spin of an electron along a given axis—say, the z-axis—will result in observations of one of two values: ‘up’ (‘"’) or ‘down’ (‘#’). However, if the electron has not very recently been polarized along the z-axis, then the quantum state of its spin will not be one of these values or the other: it will be a non-trivial superposition of both, describable in Ket notation, with c1 and c2 standing for non-zero complex numbers, as: ð1Þ

c1 j "i þ c2 j #i

Philosophers dispute the metaphysical nature of the quantum state. If quantum mechanics were our final theory, what should we understand an expression like (1) to be telling us about the world? The expression mentions two simple states that an electron might be in. These states are apparently, in some sense, incompatible. But on a flatfooted reading, incompatibility aside, the equation seems to be telling us that both states obtain (and also that there is a parameter, amplitude, designated by a complex number, that somehow modifies the way that they obtain). Indeed, the equation is a quantum analogue of the spooky isolation scenario considered above. According to this flat-footed reading, what the equation tells us is that the electron is in both of these states, and its being in the state of having up-spin along the z-axis has amplitude c1 while its being in the state of having downspin along the z-axis has amplitude c2 (the reading is neutral on what it is for parthood is really a two-place relation, a relation that does not involve locations or times. The fragmentalist may say this, but need not. The fragmentalist may alternatively follow, e.g. Gilmore (2009) in holding the fundamental parthood relation to be four-place, relating a whole, the whole’s region, the part, and the part’s region. The relevant state of affairs would co-obtain both with the whole being located at the whole’s region and the part being located at the part’s region.

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a state of affairs to have amplitude c, but more on this below. It is at minimum clear that values of amplitude cannot serve as parameters of persistence or locations for the states of affairs, since c1 may equal c2 ). Why be dissatisfied with the flat-footed reading? Well, for one thing, up-spin along the z-axis and down-spin along the z-axis are incompatible, or anyway, they are at least as incompatible as being happy simpliciter and being sad simpliciter. For another thing, quantum states can become entangled. For example, in general if a spin-zero particle decays into a pair of two electrons, those electrons (call them a and b) are in the singlet state: ð2Þ

1 pffiffiffi ðj"a #b i  j#a "b iÞ 2

It is less obvious what a flat-footed reading of (2) would say. It had better at least say that a has up-spin and b has down-spin and also that a has downspin and b has up-spin. But the singlet state tells us more: in particular, it tells us that somehow the first two states of affairs are coordinated as are the latter two (in such a way that any measurement of the system is bound to yield either the first coupling or the second). In other words, it tells us that it is not the case that a and b have the same spin-state (along the z-axis). So even if we abandoned our scruples about the co-instantiation of incompatible properties to allow that an isolated electron can have both up-spin and down-spin along the z-axis, this is not enough to make sense of entanglement. If we allow for the instantiation of incompatibles we can say that a has both up-spin and down-spin, and so does b. But this is not enough to accommodate the further facts of coordination: that is, the facts about entanglement as such. For this and closely related reasons, many authors conclude that the quantum state must be, in one sense or other, holistic: we must take equations like (1) and (2) to be describing properties of the system that cannot be metaphysically explained in terms of properties of the system’s components—at least not in any ordinary sense of component. Indeed this is sometimes presented as a direct consequence of the fact that quantum states like the one described in (2) are non-separable, i.e. entangled.²⁹ But the fragmentalist can say otherwise. Setting the question of amplitude temporarily aside, the fragmentalist can countenance the face value reading of (1): the state of affairs of the electron’s having up-spin along the z-axis obtains, and so does the state of affairs of that same electron’s having down-spin along the z-axis: but these two states of affairs do not co-obtain,

²⁹ See, for discussion, Teller (1986), Darby (2012), Calosi (2013), Esfeld (2014), and Ismael and Schaffer (2016).

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and indeed, as they are incompatible, they cannot co-obtain. With the same tools the fragmentalist can accommodate (2): it tells us that a has upspin insofar as b has down-spin, and also that a has down-spin insofar as b has up-spin. That is, the first and second states of affairs co-obtain, as do the third and fourth, but neither the first nor the second co-obtains with the third or the fourth: we have two distinct fragments. In other words, fragmentalism offers a precise answer to a vexing question, one that many take to afford only imprecise answers: what is a quantum mechanical ‘branch’? The fragmentalist answer is that a branch is a fragment. But where standard theories of branches take them to be derivative of the quantum state as a whole, characterized primarily as terms in equations like (2), the fragmentalist may take branches to be built up out of states of affairs involving the things that equations like (2) purport, on the flat-footed reading, to be talking about. I call this view conservative realism about the quantum state (‘conservative’ in the sense that it conserves our intuitive conception of what things and properties are fundamental, ‘realist’ in that it shares with quantum state realism the idea that the quantum state is (more or less) all there is). Conservative realism combines the key element of primitive ontology approaches, namely that the fundamental or primitive ontology contains local ‘beables’: observable entities that instantiate more or less familiar properties and can be localized in space or space-time—with the key element of no-primitive-ontology approaches, namely that there is no more (or anyway not much more) to reality than what is comprised by the quantum state.³⁰ At the same time, conservative realism is flexible concerning the answers to the big interpretive questions. There can be conservative realist collapse theories and conservative realist no-collapse theories, since this is ultimately just a question of the dynamics of amplitude across branches. And speaking of amplitude, the conservative realist has options concerning what it is: it might be a primitive property of fragments or a primitive relation between them, or we can give it some kind of analysis—e.g. we can think of its modulus squared, following Vaidman (1998) as standing in for a measure of

³⁰ Conservative realism compares to Sebens (2015)’s Newtonian Quantum Mechanics. But conservative realism is not committed to an interpretation of amplitude as branch-density, and more importantly, conservative realism offers a metaphysical alternative to the two that Sebens (2015: §11) considers: that worlds are points in a high dimensional configuration space whose structure is grounded in their dynamics, and that worlds are parameters instantiated by particles (or, perhaps, that serve as parameters relative to which particles instantiate properties). But one could develop Newtonian QM within a conservative realist (i.e. fragmentalist) framework.

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existence of situations, or following Sebens (2015) as standing in for a measure of the density of copies of a given situation.³¹ There is also no default conservative realist story about survival, or how to individuate branches over time. The conservative realist may appeal to various accounts currently on offer.³² But fragmentalism, as we have seen, also offers a framework for endurantist theories. And this suggests new possibilities. In particular it suggests novel ways for a theory of survival in a quantum mechanical world to accommodate further facts about identity over time, should one want to do so. For example, one possibility is to think of the macroscopic entities at each branch as counterparts of one another, but nevertheless take the macroscopic entities at each branch to survive into future branches (e.g. a unique one for each time). Paths of macroscopic survival might then be determined by the flow of amplitude. This could allow the conservative realist to deploy a frequentist account of the Born rule, just as the Bohmian may.³³ Another option is to think of branches as fundamentally static, and think of the only real changes as changes in the distribution of amplitude.³⁴ The conservative realist theory compares to other theories of quantum reality that take the notion of a branch, or world, to be primitive (rather than derived from the decoherence structure of the wave function).³⁵ But difficult questions arise for such theories. What exactly are branches and what differentiates them? Here, we confront parallels of questions that arise in the endurance/perdurance debate. Can one and the same thing exist on more than one branch? If so, do we relativize the having of properties to branches? The fragmentalist approach offers us the same elegant solution here that it offers to the analogous questions about time: it allows us to account for branching structure, just as it allows us to account for temporal structure, without having to deny that things can exist at more than one branch (time), and also without having to relativize (or outsource) the having of properties ³¹ This would require that we follow Mellor (1998) and Fine (2005) in allowing duplicates of one and the same state of affairs; moreover, here we cannot take these duplicates to have different spatiotemporal locations. ³² See, for example, Albert (2015), Wallace (2012). ³³ If this is just to say that the fragmentalist may add additional structure, why cannot others do the same? The answer is that the conservative realist’s branches are suited to accommodate this sort of additional structure in ways that branches that derive from the quantum state as a whole are not. ³⁴ This account would in some respects resemble that in Albert (2015). The world consists of a multitude comprising a branch for every way that things might be, and the dynamics of the universe is given entirely by the specification of how amplitude redistributes (along hydrodynamic lines, but without a world-density analysis). On this view the facts about ‘persistence lines’ are all derivative of the hydrodynamics. ³⁵ See again Sebens (2015).

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to branches (times). Moreover, on the fragmentalist account branches still are derivative entities: it is just that they are grounded in individual facts and their co-obtainment relations, rather than the quantum state as a whole. Of course there are challenges for the conservative realist approach. How if at all does it generalize to the relativistic case? And since it denies that branch structure emerges out of (the dynamics of) the quantum state, mustn’t it posit further structure beyond that contained within the quantum state after all? It would require more than a few paragraphs to address these issues in the depth they merit.³⁶ But my aims here are relatively modest. I do not seek to show here that the conservative realist approach to quantum mechanics is superior to its rivals in all matters. My aim has been to show that the theory offers a novel alternative, one which allows us to continue thinking of moreor-less familiar, non-holistic facts about the location of particles as fundamental, without adding primitive ontology beyond the wave function.

4 . C O N C LU S I O N I hope to have highlighted the interest and utility of the fragmentalist framework, beyond its role in the analysis of A-theoretic tense, passage, and related phenomena. To this end I have argued that it yields a novel account of B-theoretic endurance, and also a novel account of the metaphysics of the quantum state, each of which merits further consideration.³⁷ New York University ³⁶ There are some interpretations of relativistic quantum reality for which fragmentalism could serve as a platform. For example, on the wave functional interpretation, fragments could correspond to field configurations (but see Baker 2015 for worries about wave functional interpretations). Concerning the problem of extra structure, one option is to invoke a privileged basis, for example the position basis. Here the fragmentalist would be far from alone. But the fragmentalist may also directly countenance the noncommutativity of observables by taking the relevant observables to participate in distinct situations. On one version of this picture, there is some kind of deep equivalence between a universe of fragments involving facts about position and a universe of fragments involving facts about momentum. On another version of this picture, there is a set of position-fragments and a distinct set of momentum-fragments, and a systematic correlation between the amplitude distributions across the two (and so for the elements of any set of non-commutable observables). ³⁷ Thanks to David Chalmers, Paul Daniels, Cody Gilmore, Dana Goswick, Martin Lipman, Kristie Miller, Mike Raven, Ted Sider, Alex Skiles, Barry Smith, Tobias Hansson Wahlberg, Jen Wang, the editors of Oxford Studies in Metaphysics, the Sanders Prize Committee, and especially Kelvin McQueen.

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BIBLIOGRAPHY Albert, D. (2015). After Physics. Cambridge, MA: Harvard University Press. Baker, D. (2015). ‘The Philosophy of Quantum Field Theory’. http://philsci-archive. pitt.edu/11375/1/QFToup.pdf. Bayne, T. and D. Chalmers (2003). ‘What is the Unity of Consciousness?’ In Axel Cleeremans (ed.), The Unity of Consciousness, pp. 23–58. Oxford: Oxford University Press. Benovsky, J. (2011). ‘Endurance and Time Travel’. Kriterion Journal of Philosophy 24: 65–72. Calosi, C. (2013). ‘Quantum Mechanics and Priority Monism’. Synthese 191(5): 1–14. Carroll, J. W. (2011). ‘Self-Visitation, Traveler Time, and Compatible Properties’. Canadian Journal of Philosophy 41(3): 359–70. Costa, D. (forthcoming). ‘The Transcendentist Theory of Persistence’. Journal of Philosophy. Costa, D. and A. Giordani (2016). ‘In Defence of Transcendentism’. Acta Analytica 31(2): 225–34. Darby, G. (2012). ‘Relational Holism and Humean Supervenience’. British Journal for the Philosophy of Science 63(4): 773–88. Donnelly, M. (2010). ‘Parthood and Multi-Location’. In D. Zimmerman, ed., Oxford Studies in Metaphysics, vol. 5, pp. 203–43. Oxford: Oxford University Press. Donnelly, M. (2011). ‘Endurantist and Perdurantist Accounts of Persistence’. Philosophical Studies 154: 27–51. Eagle, A. (2010). ‘Perdurance and Location’. In D. Zimmerman, ed., Oxford Studies in Metaphysics, vol. 5, pp. 53–94. Oxford: Oxford University Press. Eddon, M. (2010). ‘Three Arguments from Temporary Intrinsics’. Philosophy and Phenomenological Research 81(3): 605–19. Effingham, N. (2010). ‘Mereological Explanation and Time Travel’. Australasian Journal of Philosophy 88(2): 333–45. Effingham, N. (2012). ‘Endurantism and Perdurantism’. In N. Manson and R. Barnard, eds., The Continuum Companion to Metaphysics, pp. 170–97. London: Continuum. Effingham, N. and J. Robson (2007). ‘A Mereological Challenge to Endurantism’. Australasian Journal of Philosophy 85(4): 633–40. Ehring, D. (1997). ‘Lewis, Temporary Intrinsics, and Momentary Tropes’. Analysis 57(4): 254–8. Esfeld, M. (2014). ‘Quantum Humeanism, Or: Physicalism Without Properties’. Philosophical Quarterly 64(256): 453–70. Fine, K. (2005). Tense and Reality. Modality and Tense: Philosophical Papers. Oxford: Oxford University Press. Fine, K. (2006). ‘The Reality of Tense’. Synthese 150(3): 399–414. Gilmore, C. (2004). ‘Material Objects: Metaphysical Issues’. Doctoral Dissertation, Princeton University.

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Gilmore, C. (2007). ‘Time Travel, Coinciding Objects, and Persistence’. In D. Zimmerman, ed., Oxford Studies in Metaphysics, vol. 3, pp. 177–98. Oxford: Oxford University Press. Gilmore, C. (2009). ‘Why Parthood Might Be a Four-Place Relation, and How it Behaves if it Is’. In L. Honnefelder, E. Runggaldier, and B. Schick, eds., Unity and Time in Metaphysics, pp. 83–133. Berlin: de Gruyter. Gilmore, C. (2014). ‘Location and Mereology’. The Stanford Encyclopedia of Philosophy, ed. Edward N. Zalta. https://plato.stanford.edu/archives/fall2014/ entries/location-mereology/. Gilmore, C. (2016). ‘The Metaphysics of Mortals: Death, Immortality, and Personal Time’. Philosophical Studies 173(12): 3271–99. Giordani, A. and D. Costa (2013). ‘From Times to Worlds and Back Again: A Transcendentist Theory of Persistence’. Thought: A Journal of Philosophy 2(1): 210–20. Hansson, T. (2007). ‘The Problem(s) of Change Revisited’. Dialectica 61(2): 265–74. Hansson Wahlberg, T. (2010). ‘The Tenseless Copula in Temporal Predication’. Erkenntnis 72(2): 267–80. Hansson Wahlberg, T. (2013). ‘Dissolving McTaggart’s Paradox’. In Christer Svennerlind, Jan Almäng, and Rögnvaldur Ingthorsson, eds., Johanssonian Investigations, pp. 240–58. Frankfurt: Ontos Verlag. Haslanger, S. (1989). ‘Endurance and Temporary Intrinsics’. Analysis 49(3): 119–25. Hawthorne, J. (2008). ‘Three-Dimensionalism vs. Four-Dimensionalism’. In J. Hawthorne, T. Sider, and Dean Zimmerman, eds., Contemporary Debates in Metaphysics, pp. 263–82. Oxford: Blackwell. Hudson, H. (2005). The Metaphysics of Hyperspace. Oxford: Oxford University Press. Ismael, J. and J. Schaffer (2016). ‘Quantum Holism: Nonseparability as Common Ground’. Synthese: 1–30. https://doi.org/10.1007/s11229-016-1201-2. Johnston, M. (1987). ‘Is there a Problem about Persistence?’ Proceedings of the Aristotelian Society 61: 107–35. Kleinschmidt, S. (2011). ‘Multilocation and Mereology’. Philosophical Perspectives 25: 253–76. Lewis, D. (1986). On the Plurality of Worlds. Oxford: Blackwell. Lewis, D. (2002). ‘Tensing the Copula’. Mind 111(441): 1–14. Lipman, M. (2015a). ‘On Fine’s Fragmentalism’. Philosophical Studies 172(12): 3119–33. Lipman, M. (2015b). ‘A Fragmented World’. Doctoral Dissertation, University of St Andrews. Lipman, M. (2016). ‘Perspectival Variance and Worldly Fragmentation’. Australasian Journal of Philosophy 94(1): 42–57. Lipman, M. (2018). ‘A Passage Theory of Time’. In Karen Bennett and Dean Zimmerman, eds., Oxford Studies in Metaphysics, vol. 11, pp. 95–122. Oxford: Oxford University Press. Lockwood, M. (2005). The Labyrinth of Time. Oxford: Oxford University Press.

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Loss, R. (2017). ‘Manuscrito – Rev. Int. Fil’. Campinas 40(1): 209–39. Lowe, E. J. (1988). ‘The Problems of Intrinsic Change: Rejoinder to Lewis’. Analysis 48(2): 72–7. Mellor, D. H. (1981). Real Time. Cambridge: Cambridge University Press. Mellor, D. H. (1998). Real Time II. London: Routledge. Merricks, T. (1995). ‘On the Incompatibility of Enduring and Perduring Entities’. Mind 104(415): 521–31. Miller, K. (2005). ‘A New Definition of Endurance’. Theoria 71(4): 309–32. Miller, K. (2006). ‘Travelling in Time: How to Wholly Exist in Two Places at the Same Time’. Canadian Journal of Philosophy 36(3): 309–34. Miller, K. (2013). ‘Times, Worlds and Locations’. Thought: A Journal of Philosophy 2(3): 221–7. Miller, K. and D. Braddon-Mitchell (2007). ‘There is No Simpliciter Simpliciter’. Philosophical Studies 136(2): 249–78. Moss, S. (2012). ‘Solving the Color Incompatibility Problem’. Journal of Philosophical Logic 41(5): 841–51. Nagel, T. (1971). ‘Brain Bisection and the Unity of Consciousness’. Synthese 22: 396–413. Ney, A. and D. Albert, eds. (2013). The Wave Function: Essays in the Metaphysics of Quantum Mechanics. Oxford: Oxford University Press. Nolan, D. (2014). ‘Balls and All’. In S. Kleinschmidt, ed., Mereology and Location, pp. 91–116. Oxford: Oxford University Press. Parsons, J. (2007). ‘Theories of Location’. In D. Zimmerman, ed., Oxford Studies in Metaphysics, vol. 3, pp. 201–32. Oxford: Oxford University Press. Pickup, M. (2016). ‘A Situationist Solution to the Ship of Theseus Puzzle’. Erkenntnis 81(5): 973–92. Sebens, C. (2015). ‘Quantum Mechanics as Classical Physics’. Philosophy of Science 82(2): 266–91. Sider, T. (2001). Four Dimensionalism: An Ontology of Persistence and Time. Oxford: Oxford University Press. Simon, J. (2005). ‘Is Time Travel a Problem for the Three-Dimensionalist?’ The Monist 88(3): 353–61. Spencer, J. (2016). ‘Relativity and Degrees of Relationality’. Philosophy and Phenomenological Research 92(2): 432–59. Teller, P. (1986). ‘Relational Holism and Quantum Mechanics’. British Journal for the Philosophy of Science 37(1): 71–81. Turner, J. (2016). The Facts in Logical Space: A Tractarian Ontology. Oxford: Oxford University Press. Vaidman, L. (1998). ‘On Schizophrenic Experiences of the Neutron or Why We Should Believe in the Many-Worlds Interpretation of Quantum Theory’. International Studies in the Philosophy of Science 12(3): 245–61. Valaris, M. and M. Michael (2015). ‘Time Travel for Endurantists’. American Philosophical Quarterly 52(4): 357–64. Van Inwagen, P. (1990). Material Beings. Ithaca: Cornell University Press. Wallace, D. (2012). The Emergent Multiverse. Oxford: Oxford University Press.

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PART III RECOMBINATION, RELATIONS, AND SUPERVENIENCE

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5 Possible Patterns Jeffrey Sanford Russell and John Hawthorne 1 . T W O CO M B I N A T O R I A L I D E A S A famous Humean slogan has it that there are no necessary connections between distinct existences. (“There is no object, which implies the existence of any other if we consider these objects in themselves”, Hume 1978, 1.III.vi.) Many contemporary philosophers have endorsed this kind of “combinatorialist” idea: “there should be no arbitrary limits to what’s possible” (Sider 2009), there are “no gaps in logical space” (Lewis 1986, 87), “there are no brute necessities” (Dorr 2008; see also e.g. Kleinschmidt 2015). This picture might be motivated epistemologically: brute necessities would make trouble for any tight connection between what’s conceivable and what’s metaphysically possible. Or it might be motivated metaphysically: arbitrary-looking constraints on metaphysical possibility, whether arising from primitive essences, powers, laws, or the necessary will of God, seem occult. We won’t be evaluating these motivations here: rather, we’ll be examining logical limits on what such a view could consistently say. Here’s one way of trying to articulate the Humean idea: Any pattern of instantiation of any fundamental properties and relations is metaphysically possible (Wang 2013, 538; Saucedo 2011; see also Armstrong 1989, 49). This “pattern” idea is still not completely clear. If “any pattern” means “any actually instantiated pattern”, then this says no more than the truism that what is actual is possible; and if it means “any pattern which is metaphysically possible to instantiate, it says no more than the tautology that what is possible is possible.¹

¹ Wang and Saucedo both recognize these difficulties. They each consider ways of articulating the slogan in terms of the logical consistency of certain sentences; we’ll come back to this idea in section 5. Note also that Wang puts the slogan forward as a target, not as her own view: see footnote 10.

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We’ll be exploring ways of spelling out this “pattern” idea. But in response to similar difficulties, David Lewis concluded that similar principles “cannot be salvaged as principles of plenitude”² and “we need a new way to say . . . that there are possibilities enough, and no gaps in logical space” (1986, 87). So Lewis proposed a second, mereological way of articulating the combinatorial idea—the “cut and paste” idea. Possibility is governed by a combinatorial principle. We can take apart the distinct elements of a possibility and rearrange them. We can remove some of them altogether. We can reduplicate some or all of them. We can replace an element of one possibility with an element of another. When we do, since there is no necessary connection between distinct existences, the result will itself be a possibility. (2009, 209)³

(In Lewis’s earlier work (1986) he mainly applies this combinatorial idea to spatiotemporal parts of possible worlds. But in this 2009 presentation, he also tries to capture some of the pattern idea within the cut and paste idea— rearranging, as he puts it, “not only spatiotemporal parts, but also abstract parts—specifically, the fundamental properties.”) It turns out that there are serious problems for straightforwardly unifying these two different combinatorial ideas: natural ways of spelling out the “pattern” idea and the “cut and paste” idea are inconsistent with one another. Peter Forrest and David Armstrong (1984) attempted to show this (though this isn’t exactly how they put it); Daniel Nolan (1996) showed that their argument was dialectically ineffective, and the idea has since been neglected. But different arguments do successfully reveal conflict between the two combinatorial ideas. We’ll show the inconsistency of a certain combinatorialist package; we’ll go on to also show how to devise consistent alternative packages based on the “pattern” idea. Our main technical tools for this project come from model theory: we’ll deploy the mathematical theory of relational structures to regiment the intuitive notion of patterns of instantiation. We should note that these results don’t rely on Lewis’s brand of modal realism, nor indeed on any particular commitments about the nature of possibilia. Neither do they rely (as some arguments have) on the idea that possible worlds or possible objects form a set. Furthermore, these consistency and inconsistency results have philosophical ramifications for many views other than a full-blown Humean picture of metaphysical possibility. They apply even if just certain special aspects of reality are freely recombinable—a single relation, perhaps. They also apply to other modalities besides metaphysical possibility. For instance, parallel issues arise for views that say certain ² He attributes the point to Peter van Inwagen, specifically concerning the slogan: “absolutely every way that a world could possibly be is a way that some world is”. ³ See also Lewis (1986, sec. 1.8); Nolan (1996); for critical discussion see Wilson (2015).

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qualitative patterns of instantiation are epistemically possible. (Recall that Hume’s original target was a priori necessary connections.) So our results should not just be of interest to the metaphysician, but also to the epistemologist.

2. PATTERNS OF PROPERTIES A well-known argument due to Forrest and Armstrong (1984) reveals that a certain kind of combinatorialism is inconsistent. We’ll present a variant of this argument. The variant is close to the original in spirit, but while the original version only targeted Lewis’s specific modal realist conception of possible worlds, our version abstracts away from those commitments: it doesn’t depend on any particular view of the nature of worlds. We’ve also taken the opportunity to put things in terms that are continuous with arguments and ideas we present later in this paper.⁴ The Forrest–Armstrong argument targets the combination of two principles. The first principle is a version of the “pattern” idea: given a possible world W, There will then be some property, F-ness . . . which each electron in W may or may not have, and may or may not have independently of whether the other electrons in W have it. For each sub-set of the N electrons it will be possible that precisely the electrons in that sub-set have the property F-ness. (1984, 165)

To make it more transparent that “electron” is a placeholder in the argument, we’ll instead talk about “marbles”: these are supposed to be some kind of concrete particulars that are candidates for instantiating recombinable properties. We’ll make this “pattern” principle precise as follows:

⁴ Throughout this paper we freely appeal to the truth at every possible world of ZFCU—standard set theory adapted to a setting with urelements. (We do not assume Urelement Set—that there is a set of non-sets.) We also make the simplifying assumptions that set-membership is rigid, and that it is not contingent what pure sets there are. We make some additional assumptions about the logic of possible worlds. First, we assume that truth-at-a-world is closed under logical consequence: Closure If ψ is a logical consequence of ϕ₁, ϕ₂ . . . , and at W, ϕ₁, and at W, ϕ₂, and . . . , then at W, ψ. But we will qualify this principle later in this section: see footnotes 8 and 14. We also assume for simplicity that it is not contingent what worlds there are, nor what is true at them: that is, for any world W at which ϕ, at every world V, W is a world at which ϕ. (Given natural background assumptions about the connection between modal operators and possible worlds, this is tantamount to assuming the modal logic S5.)

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Property At any world, if D is the set of marbles and X is a subset of D, then at some world: the marbles are precisely the elements of D, and the F marbles are precisely the elements of X. (This principle is in fact a little stronger than Forrest and Armstrong’s, since it requires the “pattern world” to be one in which there are no other marbles besides the D-marbles. Nothing important turns on this, but it will simplify some connections later on.) The second principle is a version of Lewis’s “cut and paste” idea: [G]iven any number of possible worlds, W ₁, W ₂ . . . , there exists a possible world, having wholly distinct [i.e. non-overlapping] parts, such that one of these parts is an internally exactly resembling duplicate of W ₁ [ . . . ], another a duplicate of W ₂, and so on. (Forrest and Armstrong 1984, 164)

Their argument is presented in terms of duplication (specifically targeting Lewis’s early formulations of recombination), but we’ll put this a bit more abstractly.⁵ If Ω is a set of possible worlds, then where Forrest and Armstrong say that W þ has distinct duplicates of the worlds in Ω as parts, we’ll instead say “W þ disjointly embeds Ω”. In due course we’ll give a definition of disjoint embedding, but for the moment all that is important is this fact about it: Enough Marbles If W þ disjointly embeds Ω, and at each world W in Ω there is at least one marble, then at W þ there are at least as many marbles as elements of Ω. So our version of the principle says: Paste For any set of worlds Ω there is a world W þ that disjointly embeds Ω. Note that when Forrest and Armstrong say “any number of possible worlds” they mean it literally: their “paste” principle only applies to worlds that have a cardinal number. With orthodoxy, we suppose this requires that they are not too numerous to form a set, and we’ve made this explicit in our formulation of Paste, by saying “for any set of worlds”.⁶ The argument against Property and Paste relies on several background assumptions. Forrest and Armstrong apply their “paste” principle to the plurality of all possible worlds, which requires: World Set

There is a set of all worlds.

⁵ This bypasses some concerns about the part–whole structure of possible worlds, and whether the recombinable properties are intrinsic. ⁶ Forrest and Armstrong alternatively consider applying the argument to some broader notion of an “aggregate” of all worlds. It’s not clear what they have in mind, but they might be gesturing at a version of the argument using plural quantification: we take up this idea in section 4.

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Similarly, when they say “the N electrons” they are implicitly assuming that the electrons also have a cardinal number, and thus they assume: Marble Set

At any world, there is a set of all marbles.⁷

There are powerful arguments against these set-theoretic assumptions, and accordingly we will relax them in section 4. But since that introduces extra technical complications, we will assume World Set and Marble Set for now. The argument uses two further auxiliary premises. Possible Marble At some world, there is at least one marble. Possibilities For any sets X and Y, if at W the F marbles are precisely the elements of X, and at W the F marbles are precisely the elements of Y, then X¼ Y. (Equivalently, for any distinct sets X and Y, if at W the F marbles are the elements of X and at W 0 the F marbles are the elements of Y, then W and W 0 are distinct worlds.) This final premise is about the uniqueness of what is true at a possible world. We’ll return to this in a moment.⁸ Here is the main Forrest–Armstrong result, in our setting: Given Possible Marble, Marble Set, World Set, and Possibilities, it follows that Property and Paste are not both true. Let Ω be the set of all worlds at which there is at least one marble. (This set exists, given World Set and the axiom of Separation.) Then by Paste, there is some world W þ that disjointly embeds Ω, and so at W þ there are at least as many marbles as there are worlds in Ω. By Marble Set these marbles form a set D, and by Possible Marble D is non-empty. Then Property tells us that at W þ , for each subset X of D there is some world in Ω at which the F marbles are precisely the elements of X; and by Possibilities these worlds are numerically distinct from one another. Thus, at W þ , there are at least as many ⁷ Assuming that (at every world) marbles and worlds are non-sets, Marble Set and World Set are consequences of the necessity of the more familiar Urelement Set axiom, which says that there is a set of all non-sets. Of course, some hold that possible worlds are sets—for example, sets of propositions. In this case World Set would not follow from Urelement Set. There are challenges to Urelement Set that don’t apply to Marble Set. For example, suppose that for each marble m and any distinct sets A and B, the fusion of m and A is a distinct non-set from the fusion of m and B. Then there will, after all, be as many non-sets as sets, violating Urelement Set (see Uzquiano 2006). But this argument makes no trouble for Marble Set on its own. ⁸ In fact, we can derive Possibilities from our background principles, using the necessity of identity. Suppose that at at W, X contains the F marbles, and at W, Y contains the F marbles. Then by the unqualified version of Closure (footnote 4), together with the fact that the axiom of Extensionality is true at W, it follows that at W, X ¼ Y .

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worlds in Ω as subsets of D. So there are at least as many members of D as subsets of D. But this contradicts a standard result of set theory: Cantor’s Theorem of D.

There are strictly more subsets of D than members

In short, at the world W þ we would have to have D  Ω  2D > D which is impossible. QED. The Forrest–Armstrong argument establishes an interesting result, but it is not one that has dialectical force against their original target, namely David Lewis’s system: for Lewis rejects some of the assumptions on which the argument relies (Nolan 1996).⁹ At a crucial point in the argument we infer that there are as many distinct possible worlds as sets of marbles, since for each set of marbles X there is some world at which just the X-marbles are F (let’s say “red”). Take the case of two singleton sets {Mary} and {Marvin}. Then in particular, the argument requires that, given that Mary and Marvin are distinct, the world at which just Mary is red is distinct from the world at which just Marvin is red. But Lewis is a counterpart theorist, and a counterpart theorist can resist this. For it to be true at W that just Mary is red, it suffices that Mary have some counterpart which is the only red marble in W. The same goes for Marvin. Crucially, Lewis allows that the very same marble can be a counterpart for both Mary and Marvin—and thus the very same world can do double duty for both possibilities (see Lewis 1986, 232ff.). So, for the counterpart theorist, Property does not imply that there are as many distinct possible worlds as there are sets of marbles. In particular, given this counterpart-theoretic gloss on truth-at-a-world, the background premise Possibilities fails: it can be true at W1 that Mary is red, and also true at W2 that Mary is not red, and yet it does not follow that W1 and W2 are distinct. A more general lesson is that the Forrest–Armstrong argument is only effective against the haecceitist combinatorialist: someone who accepts both Property—which says there are possible worlds witnessing propertydistributions that differ merely with respect to what individual marbles are ⁹ Nolan puts the point like this: “[T]hey talk as if there is trans-world identity of electrons. [This is bad] because Lewis does not think that there is any such thing, and they are supposed to be discussing a problem for Lewis’ theory” (1996, 243, original emphasis). Strictly speaking, trans-world identity isn’t exactly what’s at issue, but rather the haecceitistic principle that isomorphic possibilities, which merely differ regarding which particular marbles are F, are witnessed by distinct possible worlds. But the best-known way of rejecting this haecceitistic principle is Lewis’s, which goes by way of rejecting trans-world identity. (It’s a bit surprising that Lewis himself did not seem to notice this problem with the argument: for his response see Lewis 1986, sec. 2.2.)

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like—as well as Possibilities—which guarantees the distinctness of these possible worlds. It would be nice, then, to try to rehabilitate a version of the argument with more general application, by only appealing to qualitatively distinct possible worlds. Moreover, the slogan we began with— Any pattern of instantiation of any fundamental properties and relations is metaphysically possible —is much more plausible when read as a principle concerning qualitative patterns, rather than as a constraint on de re possibilities. The de re reading is not just haecceitistic, but in fact radically anti-essentialist. For example, suppose that “marbles” include both photons and electrons, and suppose that each electron is essentially an electron, and each photon is essentially a non-electron. Then reading “is F ” as “is an electron” would make Property false—but this essentialist picture is still compatible with a qualitative gloss on the combinatorialist slogan.¹⁰ From here on, we’ll be focusing on principles that concern the possibility of certain sorts qualitative patterns, rather than haecceitistically loaded combinatorial principles. (That isn’t to say we are assuming anti-haecceitism, nor are we assuming Lewisian doctrines associated with counterpart theory. For the purposes of this paper we aim to stay neutral on such questions.) In particular, rather than Forrest and Armstrong’s Property, which is about distinct de re possibilities for the F-ness of particular marbles, we can explore the prospects for a principle about different qualitative patterns of F-ness over the marbles. Here’s a natural principle to try (Nolan 1996, 243): Property Pattern At any world, if D is the set of marbles and X is a subset of D, then at some world: the number of marbles is the same as the cardinality of D, and the number of F marbles is the cardinality of X. Daniel Nolan points out that, unlike Forrest and Armstrong’s Property principle, Property Pattern does not lead to any contradiction with Paste.

¹⁰ Wang (2013, 539–40) raises a related objection to combinatorialism: if every pattern of instantiation of fundamental properties and relations is metaphysically possible, and is located at is fundamental, then something which is not a region could have something located at it. Qualitative pattern principles do not have this de re modal consequence. Wang also raises other objections which apply equally well to qualitative principles: the best contenders we currently have for fundamental properties and relations don’t look especially well-suited to free recombination. For instance, determinate properties like having 1kg mass and having 2kg mass are incompatible with one another, and distance relations obey geometric constraints. In this paper our focus is on the logical limits of recombination theses: we do not really rest anything on whether recombinable properties and relations are fundamental, or which ones they might be.

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We can show this with a simple model (Nolan 1996, 243–4). There is a countable infinity of worlds, each of which contains a countable infinity of marbles. For each natural number n there is a world containing just n red marbles, and another world containing just n non-red marbles. There is also one world containing infinitely many red marbles and infinitely many nonred marbles. (To see that this respects Paste we need only recall the standard fact that the disjoint union of countably many countable sets is countable.) The combinatorial idea of Property Pattern can also be generalized to patterns of arbitrarily many properties. Doing this precisely takes a bit of work—our next task. In what follows, in order to talk about different sorts of patterns, we will appeal to the standard model-theoretic notion of a structure.¹¹ Suppose P is a set of monadic properties.¹² (We’ll generalize this in the next section.) A P-structure is a pair of a set D, called the domain, and a function that takes each property F in P to some subset of D, called the extension of F in that structure. One of the overarching ideas of this paper is that model theory—the mathematical theory of structures—is a valuable tool for modal theorizing.¹³ A marble structure is any structure whose domain contains every marble and nothing else. For example, suppose the properties in P are just redness and circularity, and there are just two marbles Mary and Marvin. Then there are sixteen marble structures (with signature P). For instance, one structure S₁ has these extensions: S₁

circular red

↦ ↦

∅ fMaryg

S₂

circular red

↦ ↦

fMarving fMary, Marving

Another has these:

Among these sixteen abstract marble structures, one is special: the one that assigns redness just to marbles which are really red, and that assigns circularity ¹¹ For reference, see e.g. Hodges (1997). Philosophers are most likely to be familiar with structures in the context of the semantics for first-order logic (e.g. Sider 2010, sec. 4.2). In that setting, the signature of a structure—our set of properties P—is usually left implicit, and relationships between different structures are not emphasized. Another minor difference is that in that setting structures with empty domains are not usually allowed. ¹² Model theorists are usually neutral about what sort of things are the elements of a signature—they might, for instance, be symbols or numbers. But nothing in the standard formalism prevents us from using properties for this purpose, and we’ll find this choice convenient. ¹³ The idea that combinatorial principles can be articulated in terms of structures has been suggested from time to time, but not worked out at the level of detail required for the results we will be investigating (e.g. Bricker 1991, 608; Hazen 2004, 332).

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to those marbles which are really circular. Call this the real marble structure. In general: If S is a marble P-structure, S is real iff, for each property F in P, the S-extension of F includes just the F marbles. At any world there is exactly one real marble P-structure (given Marble Set), though generally there are many abstract marble structures. Structures provide a way of precisifying the idea of a “pattern of properties”, and articulating claims about their possibility. What we still need to spell out is what it is for a possible world to realize a certain pattern. Remember, realizing a pattern shouldn’t require that any particular marbles instantiate these properties: the pattern principle we are articulating is not haecceitistic. For example, for the pattern represented by S1 to be metaphysically possible requires that at some world there are two things, neither of which is circular and just one of which is red. For S2 , we require a world at which there are two red things, just one of which is circular. What is it for two structures with different domains to represent the same qualitative pattern? The standard answer is isomorphism. Structures are isomorphic iff there is a one-to-one correspondence between their domains which respects the extension of each property in P. (See Appendix A for more official statements of standard definitions.) But also, for a world W to realize the pattern represented by S1 shouldn’t even require that Mary and Marvin exist at W. If they don’t, then it’s plausible that the structure S1 also does not exist at W. In general, we want to make “cross-world” structural comparisons, while allowing that particular marbles and marble-structures may only exist contingently. Our strategy is to appeal to an intermediary structure whose existence is not contingent. For this purpose we will make the natural assumption that the size and structure of the universe of pure sets does not vary from world to world.¹⁴ This assumption is not unassailable; but if it fails, we take this to mainly make trouble for

¹⁴ In addition to assuming that the existence, elements, and identities of pure sets do not vary from world to world, we also make another more technical assumption. Counterpart theorists typically reject Closure (see footnotes 4 and 8). Perhaps at W, Marvin is red (thanks to one of Marvin’s counterparts), and at W, Marvin is green (thanks to another counterpart), but it’s not the case that at W, Marvin is red and green. But even the counterpart theorist should accept Closure for the special case of qualitative statements, which make no reference to any particular individuals: in that case, counterparts are inert. More generally, we also will assume Closure for the special case of statements which make reference to no particular objects other than pure sets, possible worlds, or properties and relations. The counterpart theorist may still wish to resist this assumption: perhaps even abstracta bear non-trivial counterpart relations. But exploring this view would raise extra technical complications; we leave it to others.

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expressing the idea of cross-world isomorphism. Presumably this is an idea that one would like to make sense of somehow or other.¹⁵ Let a pure structure be a structure whose domain is a pure set. Given that the universe of pure sets is fixed, the existence, size, and isomorphism facts for pure structures are also fixed. Putting these ideas together, we can now say what it is for a possible world to realize a certain pattern of properties. A world W realizes a P-structure S iff, for some pure structure S 0 which is isomorphic to S, at W, the real marble P-structure is isomorphic to S 0 . Using this definition, we can do what we set out to do, making precise the generalization of Property Pattern to arbitrarily many properties. Let P be any set of properties. P-Pattern realizes S.

At each world, for any marble P-structure S, some world

Note that, using our definition of “realizes” and our background assumptions about pure sets, we can derive a useful principle which is analogous to Possibilities, without the tendentious de re commitments: each possible world tells a story which is unique up to isomorphism.¹⁶ Structural Possibilities At any world, for any world W and P-structures S₁ and S₂, if W realizes S₁ and W realizes S₂, then S₁ and S₂ are isomorphic. Forrest and Armstrong’s “paste” principle was about duplication and parts. We’ve noted that for our purposes it’s more perspicuous to discuss a different, more abstract relation between worlds: disjoint embedding. An embedding is, intuitively, an isomorphism between one structure and part of another. This is a one-to-one (but not necessarily onto) function from the domain of one structure to the domain of another, which respects the extension of each P-property. If there is an embedding from S₁ to S₂ then we say S₂ embeds S₁. A structure S þ disjointly embeds a family of structures iff there are embeddings of each of them into S þ , such that the ranges of the embeddings of different ¹⁵ An alternative kind of non-contingent structure is available to Necessitists, who hold that (necessarily) everything exists necessarily (e.g. Williamson 2002). Necessitists can make sense of cross-world isomorphism without any need for special pure structures. For related discussion see Fritz (2013); Fritz and Goodman (2017). ¹⁶ Suppose that W realizes each of S₁ and S₂. That is, there are pure structures T ₁ and T ₂ isomorphic to S₁ and S₂, respectively, such that at W there is a real marble structure isomorphic to T ₁, and also at W there is a real marble structure isomorphic to T ₂. We also know that at W, there is at most one real marble structure. Since these truths-at-W make reference to no particular objects other than pure structures, by our qualified version of Closure (see footnote 14), it follows that at W, T ₁ and T ₂ are isomorphic to one another. Since isomorphism facts for pure structures do not vary from world to world, T ₁ is isomorphic to T ₂, and thus S₁ is isomorphic to S₂.

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structures in the family have no elements in common. (Again, see Appendix A for more careful statements.) We can also naturally extend these notions from structures to worlds. A world W embeds a world V (with respect to P) iff W realizes some Pstructure S and V realizes some P-structure T such that S embeds T. A world W þ disjointly embeds a set of worlds Ω (with respect to P) iff W þ realizes some P-structure S þ , each Ω-world W realizes some P-structure SW , and S þ disjointly embeds the family of structures SW for W 2 Ω. (Earlier we appealed to Enough Marbles: if W þ disjointly embeds Ω, and at each Ω-world there is at least one marble, then at W þ there are at least as many marbles as Ω-worlds. This can now be derived from the definition of disjoint embedding.) Now we can generalize Paste to arbitrary sets of properties: P-Paste For any set of worlds Ω, there is a world W þ that disjointly embeds Ω (with respect to P). With these definitions, we can state a possibility result that generalizes Nolan’s observation. It turns on the existence of a certain sort of “universal” structure: Theorem 1 Let P be any set of monadic properties. There is a P-structure U which disjointly embeds isomorphic copies of every P-structure no larger than U. (To be explicit, the size of a structure is the cardinality of its domain.) Here’s the idea of the proof of Theorem 1. Each element of a structure has a certain profile of properties—a certain subset of P which includes just the properties that apply to that element. We can characterize a P-structure (up to isomorphism) just by specifying how many elements it has with each different profile of properties. Call this specification—a function from subsets of P to cardinal numbers—the structure’s global profile. We can find a suitable infinite cardinal κ so that there are only κ different global profiles for structures no bigger than κ. Then we can glue together one representative structure for each of these κ different global profiles in one big structure, whose size is κ  κ ¼ κ. See Appendix A for further details. We can use Theorem 1 to give a model for P-Pattern and P-Paste (along with the other background assumptions). The idea is that there is a world W þ that realizes the “universal” κ-sized P-structure given by Theorem 1. The set of all possible worlds includes one representative from each isomorphism type of structure with at most κ elements—satisfying P-Pattern. The structures realized by any set of worlds in this model can be disjointly embedded in the universal structure realized by W þ —satisfying P-Paste.

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Thus: If P is a set of monadic properties, then P-Pattern and P-Paste are jointly consistent (together with World Set, Marble Set, and Possible Marble).¹⁷ The Forrest–Armstrong result relied on the fact that there strictly more subsets of D than elements of D. Theorem 1 shows that this does not carry over from particular sets to qualitative patterns of properties. That is, there exist sets D for which there are no more isomorphism-types of P-structures on D than elements of D. It’s an important feature of this model that the grand world W þ has even more individuals than there are properties in the set P. Indeed, the possibility result is blocked if we add a further principle: P-Plenitude At each world, there are at least as many properties in P as there are marbles. Then we can argue (for any set of monadic properties P): Given World Set, Marble Set, and Possible Marble, it follows that P-Plenitude, P-Pattern, and P-Paste are not all true. We can use similar reasoning to the Forrest–Armstrong argument: let Ω be the set of all worlds containing marbles, and let W þ disjointly embed Ω. Possible Marble and P-Pattern ensure that, for each P -profile, Ω includes some world at which some marble has that profile. So at W þ there must be at least one representative marble with each P-profile, and thus at W there are at least as many marbles as there are P-profiles. Since there are strictly more P-profiles than properties in P, this contradicts P-Plenitude. D  2P > P Note that the assumption of disjoint embeddings is dispensable for this argument. If W þ merely embeds each world in Ω, with no regard for disjointness, this still ensures that W þ includes at least one marble with each P-profile. So (given the background assumptions) P-Plenitude and P-Pattern also conflict with this weaker principle, which drops the disjointness condition: ¹⁷ By “consistent” we mean having a (Kripke) model. A model assigns an arbitrary extension to each property in P in each world in the model: each P-property plays the role of a primitive predicate. Of course, some of these extensions may not represent realistic possibilities. For example, even if P includes both red and colored, there is no guarantee that the extension of red is a subset of the extension of colored. Likewise, even if P includes self-identical, there are models that leave some individuals out of its extension in some worlds. Many absurd theses count as consistent in this formal sense. This caveat applies to all of our consistency claims.

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Overlapping P-Paste For any set of worlds Ω, there is a world W that embeds each Ω-world. But P-Plenitude is not obviously a well-motivated constraint. For instance, if P is a set only containing fundamental qualitative monadic properties, then there might well not be that many of them.¹⁸ Here are the main observations so far. Forrest and Armstrong’s argument raised a problem for a haecceitistic package of recombination principles. These haecceitistic principles are alien to Lewis’s worldview—and in any case, it’s more natural to understand combinatorialist slogans about “patterns of fundamental properties and relations” as concerning qualitative patterns. Moreover, we’ve seen that qualitative patterns of monadic properties don’t lead to the kind of cardinality explosion that drives the Forrest– Armstrong argument. So far, this style of impossibility result doesn’t make serious trouble for qualitative combinatorialist views.

3 . P A T T E R N S O F RE L A T I O N S There are no obvious logical difficulties for qualitative recombination of monadic properties. But when we extend these combinatorial ideas to relations, things look quite different. We’ll now let the signature P contain not just monadic properties, but relations of any finite adicity. (We’ll count properties as “monadic relations”.) Call P monadic iff it contains only monadic properties; otherwise, if P contains at least one relation of adicity at least two, P is polyadic. In this more general case, a P-structure is a pair of a domain together with a function that takes each n-adic relation F in P to some set of ordered n-tuples of elements of D (the extension of F ). Extending the notions of isomorphism and embedding to polyadic signatures is routine; explicit definitions are provided in Appendix A for reference. With these more general definitions in place, the statements of the principles in section 2— specifically, P-Pattern, P-Paste, and Overlapping P-Paste—make sense for for a polyadic signature P without any further modification. But it turns out that these generalized principles stand in importantly different relationships. In fact, whereas the Forrest–Armstrong-style argument involving monadic properties didn’t end up presenting any problem for qualitative pattern and

¹⁸ The distinction between Overlapping P-Paste and P-Paste is analogous to the distinction Nolan (1996, 241–2) draws between the “Principle of Recombination” and the “Stronger Principle of Recombination”.

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paste principles, the same kind of argument using relations raises serious trouble.¹⁹ The argument turns on the following mathematical fact, which is a kind of generalization of Cantor’s Theorem. Theorem 2 Let P be a polyadic signature. For any set D there are strictly more non-isomorphic P-structures with domain D than elements of D. In short, IsoP D > D This points to a crucial contrast between the monadic and polyadic cases: as we have seen, Theorem 1 implies that this statement is false if we replace “polyadic” with “monadic”. We’ll sketch two different proofs, using different ideas. (The two proofs also lead to interestingly different strengthenings of Theorem 2.) Details are again provided in Appendix A. For a warm-up, suppose D is countably infinite, consider a single dyadic relation zapping, and consider a structure S where zapping has the structure of the ordering of natural numbers (so a “zero” marble zaps everything in D, including itself, another marble “one” zaps everything in D except zero, and so on). This order structure is rigid, in the sense from model theory: there is only one isomorphism from S to S—namely, the identity function, which takes each element of D to itself.²⁰ Next, suppose that in addition to the relation of zapping, P also contains one monadic property, redness. How many ways are there to distribute redness over the order structure given by S? That is, how many different ways are there of extending the {zapping}-structure S to a {zapping, redness}structure? One way assigns redness to just the first thing, another assigns redness to just the odd-numbered marbles, and so on. In general, for each set X of marbles in S, there is a corresponding P-structure SX which has the same zapping structure as S, and which has X as the extension of redness. Furthermore, each of these structures is qualitatively distinct.²¹ So there are ¹⁹ Kit Fine pointed out in discussion that in this respect recombination principles are closely analogous to abstraction principles of the sort that play a role in neo-Fregean philosophy of mathematics. “Monadic” abstraction principles, such as Hume’s Principle for cardinal numbers, are consistent, while similar “dyadic” abstraction principles, such as the analogous principle for ordinal numbers, are inconsistent. Furthermore, the proofs of both of these facts are closely related to ours. (See Boolos 1998, 184 and 222.) ²⁰ In other words, a rigid structure is one that has no non-trivial automorphisms, where an automorphism is an isomorphism from a structure to itself, and the trivial automorphism is the identity function. This use of “rigid” is unrelated to the modal meaning common among philosophers. ²¹ This is because any isomorphism f from SX to SY has to be, in particular, an isomorphism of the underlying zap-structure, which means that f must be the identity function, and so, since f preserves the extension of redness, X ¼ Y .

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at least as many isomorphically distinct P-structures as subsets of D—and thus, by Cantor’s Theorem, strictly more such structures than elements of D. We can get from here to Theorem 2 in two steps. The first step is to notice that the extra monadic property of redness wasn’t really needed. Instead of varying structures according to which elements are red, we can do the same thing with harmless modifications of the zap-ordering itself. In the original ordering based on the natural numbers, every element zaps itself. For each set X, we can come up with a modified ordering SX where, instead of coloring the individuals in X red, we switch off their self-zapping. This modified ordering still naturally matches marbles up with natural numbers. (The marble at position n is the one which zaps everything except for the marbles in positions before n, and perhaps itself.) In particular, SX is still rigid, which is what we needed for the argument. The second step is to note that the assumption that there are only countably many marbles was inessential. In fact, a standard fact of set theory—the Well-Ordering Principle (which is equivalent to the axiom of Choice)—says that any set can be ordered in a way which has the same rigidity property as the natural numbers ordering.²² The second proof of Theorem 2 uses a different fundamental theorem from set theory, concerning ordinals (for this formulation see e.g. Clark 2016): Burali-Forti’s Theorem For any cardinal κ, there are strictly more than κ isomorphically distinct well-ordered sets with at most κ elements. For example, let κ be countable infinity, and consider the dyadic relation of zapping. There are many ways of putting non-isomorphic well-ordering zapstructures on countable sets. It could be an ω-sequence (structured just like the natural numbers), or an ω-sequence with an extra element at the tail, or two copies of an ω-sequence end to end, or an ω-sequence of end-to-end ω-sequences, and so on. What we learn from Burali-Forti is that in fact there are uncountably many qualitatively different order structures; and furthermore this generalizes from the countable case to any size. Theorem 2 has this important consequence for recombination. Let P be polyadic. Given World Set, Marble Set, and Possible Marble, then P-Pattern and P-Paste are not both true. This follows from Theorem 2 the same way that the Forrest–Armstrong result followed from Cantor’s Theorem. If Ω is the set of worlds containing ²² As it turns out, the principle we really need here—that every set is the domain of some rigid relational structure—is strictly weaker than Choice. Like Choice, though, it is independent of ZF set theory. See Hamkins and Palumbo (2012).

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marbles, and W þ disjointly embeds Ω, then at W þ the set of marbles D is at least as large as Ω, which (by P-Pattern and Structural Possibilities) has at least as many elements as there are non-isomorphic structures with domain D, which by Theorem 2 is strictly larger than D. In short: D  Ω  IsoP D > D Contradiction, QED.²³ So there’s an important disanalogy between recombination for relations and recombination for monadic properties. Any set allows more qualitative patterns of a dyadic relation than it has members; but the same isn’t true for qualitative patterns of monadic properties. So while qualitative recombination principles may remain well-behaved for monadic properties, analogous principles for relations lead to combinatorial explosion. It’s striking that the requirement of disjoint embeddings in P-Paste plays an essential role in this impossibility result (unlike the P-Plenitude result in section 2). In particular, the argument that at W þ there are at least as many marbles as Ω-worlds (i.e. D  Ω) requires that the “paste” world has at least one distinct marble representing each pattern. If the embedded structures are allowed to overlap, then this is not guaranteed. In fact, we can show that if P-Paste is weakened to Overlapping P-Paste, combinatorial explosion is once again averted. This is due to another striking mathematical fact. Theorem 3 If P is finite, then there is a countable P-structure that embeds every countable P-structure. Since the proof of this result is more involved, we defer it to Appendix A. The proof uses a construction from model theory called a Fraïssé Limit.²⁴ In the special case of a single dyadic relation, this “universal” structure is called the random graph, on account of one of its striking properties (Erdős and Rényi 1963; Rado 1964). Let D be a countable set. Suppose an angel visits each ordered pair of elements ðd ₁, d ₂Þ in D and flips a fair coin: if it comes up heads, d ₁ zaps d ₂, and otherwise not. Once every pair has been visited, we have a certain zapping-structure with domain D. It turns out that with ²³ Note that the proof of Theorem 2 using Burali-Forti’s Theorem shows something stronger: a weaker principle than P-Pattern also conflicts with P-Paste. Namely: Order Pattern At any world, for any marble structure S which is a well-ordering, some world realizes S. The proof of Theorem 2 using Cantor’s Theorem also shows something stronger: not only is IsoP D > D, but in fact IsoP D  2D . ²⁴ Thanks to Alex Kruckman on http://math.stackexchange.com for pointing us in the right direction.

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probability one, the resulting structure is isomorphic to the random graph. (This property can also be used to provide an alternative existence proof for the structure in question—using the fact that a set of measure one must be non-empty!) Using Theorem 3, we can argue: If P is finite, then P-Pattern and Overlapping P-Paste are consistent (together with World Set, Marble Structure, and Possible Marble). Once again, we can provide a model by letting the worlds include one representative from each isomorphism type of countable P-structures—thus satisfying P-Pattern. These include a world W þ that realizes the “universal” structure from Theorem 3, which embeds every world—thus satisfying Overlapping P-Paste. In fact, a stronger version of Theorem 3 also holds, which generalizes beyond the finite case. (See Appendix A.) Theorem 4 If κ is an inaccessible cardinal and P < κ, there is Pstructure of size κ that embeds every P-structure of size at most κ. So, for an arbitrary set of relations P, given the existence of a sufficiently large inaccessible cardinal, P-Pattern can be maintained consistently with Overlapping P-Paste.

4. PLURALITIES OF WORLDS So far our arguments have relied heavily on two set-theoretic background assumptions: World Set and Marble Set. Nolan (1996) and Sider (2009) have thought that these were the key culprits that make trouble for recombination—and it’s true that rejecting them does evade one kind of combinatorial argument (see also Pruss 2001). But as we’ll see in this section, the main impossibility result of section 3 does not essentially turn on these set-theoretic assumptions. Let’s begin by examining Daniel Nolan’s independent argument against the combination of World Set and Marble Set. His basic observation (adapted to our terminology) is that World Set and Marble Set are jointly inconsistent with this principle: Sizes For any cardinal κ, at some world there are at least κ marbles. Here’s a version of the argument. Suppose Marble Set: so for each world W, there is a certain cardinal which at W is the number of marbles. Call any such cardinal a “world cardinal”. Suppose World Set: then there is a set of all

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world cardinals.²⁵ For any set of cardinals, there is a cardinal κ strictly greater than each of them.²⁶ Then no cardinal greater than or equal to κ is itself a world cardinal—contradicting Sizes, QED.²⁷ Why might the combinatorialist accept Sizes? As Nolan presents things, Sizes follows from a combinatorial principle that Lewis entertained (without quite endorsing): “for any objects in any worlds, there exists a world that contains any number of duplicates of all of those objects” (Nolan 1996, 239, our emphasis; paraphrasing Lewis 1986, 89). This leaves off Lewis’s caveat: “size and shape permitting”. What Nolan recommends is that we don’t impose the caveat, and instead drop the combination of World Set and Marble Set. (There are two ways this could go. One way would be to keep Marble Set, but say that for each cardinal, there is a world with that many marbles—and thus too many worlds to form a set. The other way would be to say that there is at least one vast world containing too many marbles to form a set— which all by itself would suffice to make Sizes true.) An alternative way of motivating Sizes uses our ideology of structures, rather than the Lewisian ideology of duplication. Sizes follows from this principle: Copy For any cardinal number κ and any world W, there is a world W þ that disjointly embeds at least κ isomorphic copies of the structure realized by W. Just as with Sizes, the Copy principle is fine as long as we don’t try to maintain both Marble Set and World Set.

²⁵ By the axiom of Replacement. ²⁶ This follows from a combination of the axiom of Unions—getting us an upper bound for the cardinals—and Power Set—getting us a cardinal strictly greater than that upper bound. ²⁷ This version of the argument has the advantage of not relying on either Lewisian modal realism or Williamsonian Necessitism (compare Uzquiano 2015). Nolan’s version of this argument was a bit simpler, but depends on more contentious background assumptions. Suppose that there is a set of all possible marbles. Then this set has a certain cardinality, and there is another cardinal κ which is greater yet. Then Sizes says there is some world in which there are at least κ marbles—and so there are at least κ possible marbles, contradiction, QED. Note that this version of the argument relies on this inference: If at some world W, there are at least κ marbles, then there are at least κ possible marbles. This inference is unproblematic in the Lewisian framework in which Nolan is working. It is similarly unproblematic in the Necessitist framework defended by Timothy Williamson, according to which, if there could have been at least κ marbles, then there are at least κ things which could have been marbles (see Sider 2009). But the relevant inference is rejected by Contingentists, who hold that it is contingent which things are anything at all. The version of the argument we presented doesn’t rely on this contentious inference. (It does still rely on a restricted and less contentious Necessitist assumption: namely, that it is not contingent what cardinals there are.)

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So giving up one or both of these set assumptions—World Set and Marble Set—is a perfectly good way of escaping Nolan’s problem of Sizes. But the problem of recombinable relations we presented in section 3 is not like that: as we’ll now show, it does not rely in any essential way on either of these set-theoretic assumptions. We can restate a generalized version of the argument using plural quantification.²⁸ Our formulations of P-Paste and P-Pattern both involved structures whose domains and extensions were sets. But it’s natural to extend the underlying idea to versions which—in the absence of World Set and Marble Set—are stronger. Consider a dyadic relation of zapping. A standard structure for zapping specifies a set of ordered pairs as its extension. If there is no set of all marbles, then there are ways of distributing zapping over pairs of marbles where those pairs are too numerous to form a set—and thus they don’t comprise a zapping-extension in any structure, in the standard sense we presented in section 2. But we can still naturally extend the idea of a zapping-pattern to this case, by replacing singular quantification over structures with plural quantification over ordered pairs. And we can show that, in a naturally extended sense, it’s still true that there are strictly more isomorphically distinct patterns of pairs of marbles than there are marbles. The rest of this section will provide more detail about how this works. What we need to do is translate standard structure-theoretic talk into plural language. This requires some care, but it is essentially straightforward. (In what follows, we’ll use the plural quantificational expression “there are zero or more”, rather than the alternative “there are one or more”. Nothing essentially turns on this choice, but it makes certain results easier to state. Apparent singular quantification over “pluralities” is just a heuristic shorthand for more serious plural talk.) A structure, as we defined it before—or to be more explicit in what follows, a set structure—was defined as a pair of a certain set—the domain—and an extension function—a function from relations to sets of n-tuples. Instead of quantifying over these set-theoretic objects, we can instead quantify plurally over the things in the domain and the tuples in the extensions, directly. Where before we said “there is a structure S such that . . . ”, instead we can say “there are some things, the S-domain-things, and there are some tuples, the S-extension-tuples, such that . . . ”. For a general signature, we can think of a “plural structure” as an indexed family of pluralities: a plurality for the domain, and a plurality of n-tuples for each n-place relation.

²⁸ We use standard plural logic (with full impredicative comprehension; see e.g. Linnebo 2014). We also assume that pluralities of abstracta are fixed, in this sense: if each of the X ’s is a pure set or a possible world, then x is one of the X ’s iff at every world x is one of the X ’s.

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While sets of sets are unproblematic, orthodox plural logic does not provide us with the resources to straightforwardly quantify over “pluralities of pluralities”. But there is a coding trick that lets us get around this obstacle in certain cases.²⁹ Suppose that for each i among the I ’s, there are certain things, the Xi ’s: then we can think of this as a family of pluralities indexed by the I ’s. (This is also called a class-valued function from the I’s.) We can encode an indexed family of pluralities like this using a plurality of pairs. In the case at hand, for each n-place relation F (in P) we want to represent a “plural extension” for F, which consists of certain n-tuples, the SF ’s. We can do this using a plurality of ordered pairs, the S ’s, such that each of the S’s is an ordered pair of some n-place relation F,
and some n-tuple.  Then the SF ’s are those n-tuples ðd ₁, . . . , dn Þ such that F , ðd ₁, . . . , dn Þ is one of the S ’s. We also want to represent a domain: we can do this by picking some canonical object Dom which is not a relation (for example, the word “domain” or the number 0), and include among the S’s some ordered pairs whose first element is Dom. Then the S-domain consists of those things d such that ðDom, d Þ is one of the S’s. In general, we’ll say the X ’s code a family of pluralities indexed by the I’s iff the X ’s are ordered pairs each of which has one of the I’s as its first element. Then for any i among the I’s, we can let the Xi ’s be those things x such that ði, xÞ is one of the X ’s. So, in particular, the S ’s code a plural P-structure iff they code a family of pluralities indexed by the relations in P together with Dom, where for each n-adic relation F in P, each of the SF ’s is an n-tuple of things among the SDom ’s. The SDom ’s are the S-domain, and the SF ’s are the S-extension of F. In what follows, it will sometimes be convenient to speak singularly, saying “there is a plural P-structure S such that . . . ”. But it’s important to bear in mind that, like talk of “pluralities” or “families”, this is intended to be cashed out plurally, not as singular quantification over any kind of object which is itself a plural structure. We can similarly extend other structure-theoretic notions to the plural case, such as isomorphism and disjoint embedding. For instance, if S and S 0 are plural structures, we’ll say the X ’s code an isomorphism from S to S 0 iff each of the X ’s is an ordered pair ðd , d 0 Þ where d is in the S-domain and d 0 is in the S 0 -domain, and these pairs satisfy suitable conditions. The details are straightforward but tedious, so we’ll relegate them to Appendix B. In order to compare structures across worlds, it will again be helpful to appeal to “fixed” structures. Once again, we will deploy a fixed universe of

²⁹ This coding trick, from Paul Bernays, takes advantage of the Curry correspondence between I ! 2X and 2I X . (See Uzquiano 2015, 9.)

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pure sets for this purpose. Let a pure plural structure be a plural structure each of whose domain-things is a pure set. Then we can define realization as before. At any world, the real marble plural structure S has as its domain all of the marbles, and for each relation F in P, the S-extension of F consists of just the n-tuples of marbles that stand in F. Then if W is a world, and the S ’s code a plural P-structure: W realizes S iff, for some pure plural P-structure S 0 which is isomorphic to S, at W, the real marble plural structure is isomorphic to S 0 . Now we can state our generalized pattern principle: Plural Pattern At any world, for any S ’s that code a plural structure whose domain consists of the marbles, some world realizes S. We can also straightforwardly extend the definition of disjoint embedding (with respect to a signature P ) to pluralities of worlds. (Again, this is in Appendix B.) As in the set case, the definition has the following important consequence. Let a marble-world be a world at which there is at least one marble. Enough Marbles If each of the W ’s is a marble-world, and W þ disjointly embeds the W ’s, then at W þ there are at least as many marbles as the W ’s. (As is standard, plural cardinal comparisons can be spelled out in terms of pluralities of pairs: at W þ there are some ordered pairs that code a one-toone function from the marbles to the W-worlds.) Now we can state a plural generalization of our Paste principle: Disjoint Plural Paste For any worlds, the W ’s, there is some world W þ such that W þ disjointly embeds the W ’s. The key point is that these plurally generalized recombination principles face exactly the same difficulty as the set-theoretic versions. We can now adapt our impossibility result from section 3 to show: Global Choice, Possible Marble, Plural Pattern and Disjoint Plural Paste are not all true. The main idea of the argument is the same as before. Consider a single dyadic relation of zapping. Given Disjoint Plural Paste, there is a world W þ that disjointly embeds all of the marble-worlds—those worlds at which there is at least one marble. By Enough Marbles, at W þ there are at least as many marbles as marble-worlds. Possible Marble ensures that there is at least one marble-world, and so at W þ there is at least one marble. Plural Pattern says that at W þ , for any pairs of marbles, some world realizes that zapping pattern—and in particular, this is a marble-world. So there is a

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distinct marble-world for each isomorphically distinct way of choosing pairs of marbles.³⁰ Thus at W þ there are at least as many marbles as patterns of pairs of marbles. But as we’ll show, this is impossible. What remains to be shown is that there are strictly more patterns of pairs of marbles than there are marbles. To show this, we can use a plural generalization of Cantor’s Theorem (Bernays 1942). This theorem formalizes the intuitive idea that there are strictly more pluralities of marbles than there are marbles. We can state this using the same trick for coding indexed families of pluralities. The idea is that no family of pluralities indexed by D’s can include every plurality of D’s. Cantor–Bernays Theorem Let the X ’s code a family of pluralities indexed by the D’s. Then there are (zero or more) D’s which are not the Xd ’s for any d. Bernays’ proof is an easy application of the usual Cantor–Russell trick: let the R’s be the (zero or more) things d such that d is not among the Xd ’s. Suppose for reductio that for some d, the Xd ’s are the R’s. Then it follows that d is not among the R’s. That is, d is among the Xd ’s, and so by construction d is one of the R’s, which is a contradiction. We can extend Bernays’ result about pluralities to an analogous result about plural structures. This formalizes the intuitive thought that there are more isomorphism-types for a relation on marbles than there are marbles. Theorem 5 Let P be a polyadic signature. Let the S’s code a family of plural P-structures indexed by the D’s. (That is, for each d among the D’s, the Sd ’s code some plural structure.) If the D’s can be well-ordered, then some plural structure on the D’s is not isomorphic to Sd for any d. The proof is a straightforward “pluralization” of the proof of Theorem 2 we gave in section 3. For details, see Appendix B.³¹ ³⁰ This step relies on the plural analogue of Structural Possibilities, which can be shown in the same way as the set version from our background assumptions about the fixity of pluralities of pure sets. ³¹ Here is a technicality. (Thanks to Daniel Nolan for very helpful discussion.) To derive our impossiblity result from Theorem 5, we need the further claim that, in any possible world, the marbles can be well-ordered. While the fact that any set can be wellordered is equivalent to the set-theoretic axiom of Choice, it turns out that the plural generalizations of these principles—Global Well-Ordering and Global Choice—are not equivalent: in fact, there are models of Global Choice without Global Well-Ordering (Howard, Rubin, and Rubin 1978). Still, Global Choice does imply that any plurality of pure sets can be well-ordered (see e.g. Linnebo 2010, 161–2). It follows that, if there are no more X ’s than pure sets, then the X ’s can be well-ordered. Furthermore, recall that we defined “realizes” in terms of isomorphism with a pure plural structure. It follows from this definition that any plural structure with a

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5 . UN R E S T R IC T E D PA T T E RN S So far we’ve been exploring the difficulties that arise for the combination of two different combinatorial ideas. First, the “pattern” idea: Any pattern of instantiation of any fundamental properties and relations is metaphysically possible. Second, the “cut and paste” idea: For any objects in any worlds, there exists a world that contains any number of duplicates of all of those objects. Let’s now consider the prospects for the “pattern” idea taken on its own. This turns out to be very powerful. Let P be some arbitrary polyadic signature: some set of properties and relations, including at least one relation which is not a monadic property. In this section “structures” are to be understood as structures with signature P. We have considered two ways of spelling out the “pattern” slogan. First: Marble Set Pattern At any world, for any (set) structure S whose domain is the set of marbles, some world realizes S. (In section 2 we called this “P-Pattern”.) The second way (which in section 4 we called Plural Pattern) generalizes from set structures, which are limited in size to what can be contained in a single set, to a pluralized version that is not so limited. Marble Plural Pattern At any world, for any plural structure S whose domain consists of all marbles, some world realizes S. (Recall that apparently singular quantification over plural structures, like “plurality”-talk, is cloaked plural quantification: the variable S here is really a plural variable.) Each of these Pattern principles has consequences that go beyond those of Lewis’s “cut and paste” duplication principle. The duplication principle does not guarantee that, if there could be a red marble and a square marble, then there domain outnumbering the pure sets is unrealizable. So in fact, Plural Pattern implies that there are no more marbles than pure sets. (See our discussion of Limitation of Size in section 5.) Given this, Global Choice implies that the marbles can be well-ordered. More generally, Howard, Rubin, and Rubin (1978) show that many different formulations of Choice-like principles whose set-theoretic formulations are equivalent can subtly come apart in the context of plural logic. Fortunately for us, these subtleties shouldn’t matter so long as there are no more marbles than pure sets: any standard plural formulation of Choice should do as far as Theorem 5 is concerned.

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could be a red square marble; nor does it guarantee that if there could be a zapping pair of marbles, then there could be marbles that don’t zap one another, or that any marble could zap itself (see also Wilson 2015, 148). By contrast, Marble Set Pattern and Marble Plural Pattern each imply all of these conditionals (assuming the signature P includes redness, squareness, and zapping). That said, the duplication principle has consequences that the Pattern principles by themselves do not secure: if there could be one marble, then there could be three, or infinitely many, or indeed κ-many for any cardinal κ. That is, if there could be at least one marble, then this principle we discussed in section 4 follows: Sizes

For any cardinal κ, at some world there are at least κ marbles.

But neither Marble Set Pattern nor Marble Plural Pattern implies Sizes. In fact, for any non-zero cardinal κ, both Marble Pattern principles are consistent with there being at most κ marbles at any world. Unlike “cut and paste”, these Pattern principles (on their own) don’t give us any way of getting larger worlds from smaller ones. This suggests that they don’t entirely do justice to the picture that motivated the combinatorial slogan about “any pattern of instantiation”. We can do better by slightly modifying the Pattern principles. Notice that in the statement of these principles, marbles are really playing two distinct roles. One role is as possible “realizers” of structures. The other role is as “generators” of structures. The pattern principle roughly says: any abstract structure could be concretely realized. The abstract structures are generated using a domain of objects, and a set of properties and relations. But there is no obstacle to using, say, numbers to generate an abstract structure that can be concretely realized by, say, people. For example, an abstract structure for the loving relation with a domain of numbers just amounts to a set of ordered pairs of numbers. Notice that the existence of such structures has nothing to do with whether numbers are capable of love. For some people to realize this structure just requires that the pattern of loving among them be isomorphic to those pairs of numbers. The key point is that even if we are using marbles as “realizers”, this doesn’t preclude us from using different things as “generators”. So we might choose generators that exist in great multitudes—like numbers. If every structure generated by such a multitude can be realized by marbles, then in particular, there can be a multitude of marbles. This motivates strengthening Marble Set Pattern as follows: Unrestricted Set Pattern world realizes S.

At any world, for every (set) structure S, some

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(Compare Bricker’s principle (P1) 1991, 612.) This formulation simply drops the words “whose domain is the set of marbles” from Marble Set Pattern. The principle says that any structure with any set-sized domain will do. Note that Sizes immediately follows from this. Recall that in section 2 we defined “realizes” in terms of pure structures (to allow cross-world comparisons), and we are assuming that it is not contingent what pure sets there are. This means we can make two simplifications to Unrestricted Set Pattern without losing any power: we can restrict the structure-quantifier to pure structures, and we can drop the words “at any world”. Given our background assumptions, this version is equivalent: Pure Set Pattern

For every pure set structure S, some world realizes S.

There are as many distinct isomorphism types of pure set structures as pure sets.³² Thus Pure Set Pattern implies that there are at least as many worlds as pure sets, and thus the worlds are themselves too plentiful to form a set (by the axiom of Replacement). So Unrestricted Set Pattern is inconsistent with the principle World Set. Even so, Unrestricted Set Pattern is consistent taken on its own. It is also consistent together with the principle Marble Set (which, recall, says that at each world there is a set of all marbles). One way this could be is if for each pure set structure S there is a possible world WS that realizes S, and these are all of the possible worlds. This would clearly satisfy Pure Set Pattern, and thus Unrestricted Set Pattern. In previous sections we’ve considered four different “paste”-style principles. (In section 2 we used the names “P-Paste” and “Overlapping P-Paste” for the first two.) Disjoint Set Paste For any set of worlds Ω, some world realizes a set structure that disjointly embeds set structures realized by each world in Ω. Overlapping Set Paste For any set of worlds Ω, some world realizes a set structure that embeds a set structure realized by each world in Ω. Disjoint Plural Paste For any worlds, the W ’s, some world realizes a plural structure that disjointly embeds a family of plural structures realized by the W ’s. Overlapping Plural Paste For any worlds, the W’s, some world realizes a plural structure that embeds a plural structure realized by each W-world. What is the upshot of Unrestricted Set Pattern for these various principles? ³² Indeed, there is an isomorphically distinct structure for each ordinal, and there are as many ordinals as sets.

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First, suppose Marble Set is true. Unrestricted Set Pattern and Marble Set together imply that both Plural Paste principles are false: no single set of marbles is big enough to embed every set-sized marble pattern. But Unrestricted Set Pattern and Marble Set also jointly imply that both Set Paste principles are true: each set of worlds can be disjointly embedded in some world. (This is because any set of set-structures has a disjoint sum: see Appendix A.) Things are a bit messier if Marble Set is false. In that case, some world doesn’t realize any set structure at all—it has too many marbles for that. So both Set Paste principles come out trivially false. Also, without Marble Set, Unrestricted Set Pattern does not imply either of the Plural Paste principles, nor their negations. Let’s now consider the stronger plural version of this unrestricted pattern principle: Unrestricted Plural Pattern (UPP) ture S, some world realizes S.

At any world, for any plural struc-

Again, because we defined “realizes” in terms of non-contingent pure plural structures, UPP implies that every plural structure is isomorphic to some pure plural structure. In particular, UPP implies Limitation of Size

At any world, there are no more things than pure sets.

Against our set-theoretic background, this principle is equivalent to (the necessitation of) an influential proposal from Von Neumann: some things form a set iff they are not in one-to-one correspondence with everything.³³ (For discussion of this principle’s merits, see Hawthorne and Uzquiano 2011, sec. 6.3.) In fact, UPP is equivalent to the conjunction of Limitation of Size with a restricted pattern principle (given our background assumption that it’s not contingent what pure sets there are): Pure Plural Pattern For any pure plural structure S, some world realizes S. We noted that Unrestricted Set Pattern is inconsistent with World Set. Unrestricted Plural Pattern has more radical consequences yet for worlds:

³³ Note also that Limitation of Size implies Global Choice and Global Well-Ordering. (Limitation of Size puts everything in one-to-one correspondence with the ordinals; we can use this correspondence to define a global choice function.) In the presence of the Urelement Set axiom (that there is a set of all non-sets) the converse implication from Global Choice to Limitation of Size holds as well (Linnebo 2010, 151–2 and 161–2). But since we are not assuming Urelement Set, in our context Limitation of Size is in fact a stronger claim than Global Choice.

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indeed, on one natural way of understanding it, the principle is inconsistent. Theorem 5 tells us that there are strictly more isomorphically distinct plural structures than things. But if worlds are things, then since UPP requires that there are as many worlds as isomorphism types, this would imply that there are strictly more worlds than worlds, which cannot be. The issue here arises from the fact that UPP lets us use any sort of thing as generators—and so in particular, if worlds are things, then they can generate patterns themselves. There are two natural ways to respond to this. One is to back off from the fully unrestricted principle, and stick to a restricted principle that doesn’t allow worlds as pattern-generators: for example, Pure Plural Pattern is a natural fallback principle. As we noted, if Limitation of Size holds, then Pure Plural Pattern is just as strong as Unrestricted Plural Pattern. But (putting this another way) if worlds are things, then in fact Pure Plural Pattern implies that Limitation of Size is false: there are strictly more worlds than pure sets. The second response is to understand quantification over worlds as a façon de parler—just as we have done with quantification over pluralities or plural structures or families or isomorphism types. The idea is that there aren’t any such objects as worlds; but rather, this is a convenient shorthand for some other more perspicuous idiom. If our goal is just to restate UPP, then this could be the idiom of familiar modal operators (“boxes and diamonds”): Necessarily, for any plural structure S, possibly S is realized. If worlds aren’t things, then they can’t be used as generators for structures, and so collapse is averted. Paraphrases using modal operators won’t work for every use of worldquantifiers in this paper—in particular, plural quantification over worlds and cardinal comparisons pose special challenges. For a more general solution, one might invoke some higher-order idiom, such as quantification in sentence or operator position (see Fine 1977, 137ff.). For consistency, we’ll understand the principles we discuss to be officially expressed in a sorted language that distinguishes world-quantifiers from first-order objectquantifiers.³⁴ Similar issues may arise not just for worlds, but also for other plenitudinous domains, such as propositions, properties, events, facts, etc. (We’ll return to this shortly.)

³⁴ We’ll also need plural quantifiers for both individual and world types, as well as a sort of quantifier for “cross-categorial ordered pairs”, where one element is of world-type and the other is of individual-type—or at least some surrogate for these quantifiers, such as even-higher-order relational quantification. We’ll suppress these technical details to keep things readable.

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Like Unrestricted Set Pattern, Unrestricted Plural Pattern implies Sizes: for each cardinal, some world has at least that many marbles. But the plural principle also generates even larger patterns yet, whose domain-things are more numerous than any cardinal. For instance, since there are plural patterns whose domains include all the pure sets, UPP implies that there could be as many marbles as there are pure sets. Thus UPP is inconsistent with Marble Set.³⁵ Even though UPP is inconsistent with World Set, and also inconsistent with Marble Set, it’s still consistent taken on its own (given the caveat about worldquantifiers). One way this could be is if there is one possible world realizing each pure plural structure, and no other possible worlds. This guarantees Pure Plural Pattern. If furthermore at each world there are no objects besides marbles and sets, then since at each world there are no more marbles than pure sets, Limitation of Size follows.³⁶ Finally, as we noted, Limitation of Size and Pure Plural Pattern together imply Unrestricted Plural Pattern. Now let’s examine how this plural-structure-based way of articulating recombination interacts with the “cut and paste” idea. It follows directly from the Cantorian argument we presented in section 4 that Unrestricted Plural Pattern is inconsistent with Disjoint Plural Paste. But all three of the other Paste principles we’ve considered—Overlapping Plural Paste, Disjoint Set Paste, and Overlapping Set Paste—not only are consistent with UPP, but in fact follow from UPP. Disjoint Set Paste and Overlapping Set Paste each follow from this stronger Paste principle: Disjoint Paste for Not Very Many Worlds For any worlds the W ’s, if the W ’s are not more numerous than the things, there is some world W þ that disjointly embeds the W ’s. By “Many” we will mean as numerous as the things, and by “Very Many” we will mean even more numerous than the things. (Remember, “worlds” are not things themselves—and indeed, taken all together they are more numerous than the things. Limitation of Size says that there are Many pure sets—but not Very Many.) The basic reason why Disjoint Paste for Not Very Many Worlds follows from Unrestricted Plural Pattern is that—putting things a bit roughly—any not-Very-Many pluralities of things have a disjoint union, ³⁵ This depends on our background assumption that there couldn’t be more pure sets than there are. Note also that if we assume that (at every world) no marble is a set, then UPP is inconsistent with the necessity of the Urelement Set axiom. This may also put further pressure on Limitation of Size: for instance, if at each world distinct pluralities of marbles have distinct fusions, then there could be strictly more fusions of marbles than pure sets (see Hawthorne and Uzquiano 2011). ³⁶ This relies on Global Choice (see Uzquiano 2015, proposition 2 in the appendix).

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which is another plurality of things. Thus any not-Very-Many pure plural structures can be disjointly embedded in another pure plural structure (their “disjoint sum”), which suffices for Disjoint Set Paste—given Limitation of Size, since in that case every world realizes some pure plural structure. The derivation of Overlapping Plural Paste is less obvious: this relies on a plural generalization of Theorem 4 based on Fraïssé’s construction. This generalization shows the following remarkable fact (see Appendix B): Theorem 6 Given Global Choice, there is a pure plural structure that embeds every pure plural structure. Again, since Limitation of Size implies that every world realizes a pure plural structure, this suffices for Overlapping Paste. The plural structure guaranteed by Theorem 6—we’ll call it the universal plural structure—is a kind of mathematical pluriverse: an abstract universe that, in a sense, includes every abstract universe. If we lived in a world that realized this structure, then something very much like Lewisian modal realism would be true.³⁷ Note also that the universal plural structure contains many copies of itself—in fact, as many copies as things.³⁸ So we also have two strong duplication-style principles that follow from Unrestricted Plural Pattern with Limitation of Size. Let a plural part of a world be a plural-substructure of the plural structure realized by that world, and say that a world embeds a plural structure S iff it realizes a structure that embeds S. (Again quantification over “plural parts” is really shorthand for a plural quantification.) Overlapping Plural Copy For any plural parts of any worlds, some world embeds Many isomorphic copies of each of them. Disjoint Plural Copy for Not Very Many Parts For any not-VeryMany plural parts of any worlds, some world disjointly embeds Many isomorphic copies of each of them. These principles are very similar in spirit to Lewis’s duplication principle— but these are not extra postulates, but rather consequences of Unrestricted Plural Pattern. And unlike Lewis’s version, there is no pressure to tack on any extra caveats like “size and shape permitting” to this package.

³⁷ Except Lewis holds that the concrete universes are isolated, in the sense that no fundamental relations—or at least no “spatio-temporally analogous” fundamental relations—link non-world-mates (1986, 75–8). In contrast, the universal plural structure is not divisible into relationally isolated substructures. ³⁸ The basic reason for this is that “Many times Many equals Many”: we can divide up a plurality of Many things into Many disjoint subpluralities of Many things. Then we can paint the universal plural structure onto each of these subpluralities.

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Unrestricted Plural Pattern looks to us like a promising articulation of the combinatorialist idea—“there are no gaps in logical space”. But we should note that this version wasn’t available to Lewis: it’s integral to his vision that worlds are genuine concrete individual objects, and as we noted earlier, this conception of possible worlds is incompatible with UPP—since then worlds themselves would be generators of patterns. Putting this point another way, we have a vindication of Forrest and Armstrong’s original idea, understood broadly: a combinatorial argument against Lewis’s theory of possible worlds. Unrestricted Plural Pattern also makes trouble for other metaphysical views besides Lewis’s. Notice first that the argument against Lewisian modal realism doesn’t essentially rely on construing worlds as concrete: it also applies to any view according to which worlds are particular sets, or sui generis abstract objects (whether these are structured “states of affairs” or unstructured simples), as long as they are something. More generally, UPP conflicts with any view according to which there are at least as many objects as marble-worlds. For example, you might think that for each world W there is a certain necessarily existing state of affairs which obtains just at W. Any view like this is incompatible with UPP. Here’s another example. Some philosophers, having become convinced that statues and lumps of clay can be distinct while entirely coinciding, go on to embrace “bazillion-thingism”: in addition to familiar objects like statues and lumps of clay, there are many less familiar coincidents. Some are more modally fragile—like Tate-Museum-statues that are destroyed by transport—and some are more modally robust—like clay-aggregates that can survive utter dispersal (Bennett 2004, 356; see also Yablo 1987; Hawthorne 2006; Leslie 2011; Fairchild 2018). One ambitious version of bazillion-thingism says that each marble M coincides with a distinct thing for each way of choosing either a marble or nothing from every non-actual possible world. In that case there are even more objects coinciding with M than there are marble-worlds. So this kind of plenitudinous ontology is also at odds with Unrestricted Plural Pattern. Of course, our point here is just to point out the tension between Unrestricted Plural Pattern and this kind of plenitude—we take no stand on which way it should be resolved. We’ve been examining combinatorial principles based on the modeltheoretic ideas of structures and isomorphisms. It’s worth taking a moment to compare this to a different approach. Raul Saucedo (2011, 242–3) makes the following proposal for explicating the idea that every pattern of certain properties and relations is metaphysically possible: Suppose that L is a first-order language with standard logical vocabulary (the truthfunctional connectives, first-order variables and quantifiers, and the identity symbol), whose non-logical vocabulary consists of only a stock of predicates. Let’s assume that every n-place predicate of L expresses exactly one n-place property or relation,

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and that every n-place property or relation is expressed by exactly one n-place predicate of L. Then we may formulate recombination principles for properties and relations as follows: Any such-and-such sentence of L is true at some metaphysically possible world.

Saucedo discusses several ways of filling in “such-and-such”; here is one: Any sentence of L that has a model is true at some metaphysically possible world. (2011, 245)

(This is a bit stronger than his favored version of the principle, which adds in some extra qualifications. Some of these can be handled simply by restricting which properties and relations are expressed by predicates in L.) As Saucedo acknowledges (his footnote 14), an ordinary first-order language is expressively limited. So this recombination principle is accordingly weaker than one might wish. For example, standard metalogical results about first-order logic show that Saucedo’s principle is compatible with there being no world where a certain binary relation expressed by a predicate in L has the structure of an ω-sequence. Likewise, it is compatible with every property expressed by a predicate in L having just countably many instances at every world. So this principle does not guarantee the metaphysical possibility of many perfectly good infinite patterns. One might try to overcome these limitations (as Saucedo also suggests) by switching to a more expressive language (see also Dorr and Hawthorne 2013, 14ff.). In fact, in order to get a principle as strong as Unrestricted Set Pattern, one would have to go all the way up to a language which includes Many sentences—no ordinary set-sized language will do. To get a principle with the strength of Unrestricted Plural Pattern we have to go further yet— for instance, by considering the Very Many pluralities of sentences in such a large language.³⁹ But these complications are avoidable. These are efforts to find languages which are expressive enough to characterize every structure. But why not simply talk about structures themselves directly, as we have done, without this detour through transfinite syntax? The structures were in the background of the sentential approach anyway, since having a model (that is, being true in some structure) was our test all along for which sentences are logically consistent, and thus, according to the sentential recombination principle, true at some world. Furthermore, as we hope we’ve demonstrated throughout this paper, examining structures directly can provide us with illuminating insights into the space of possible patterns.

³⁹ Unrestricted Plural Pattern is thus plausibly a counterexample to Dorr and Hawthorne’s (2013, footnote 23) conjecture.

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Jeffrey Sanford Russell and John Hawthorne APPENDIX A: SET STRUCTURES

In this appendix we’ll present proofs of three key model-theoretic facts we used in this paper. (1) For any set of monadic properties, there is a structure that disjointly embeds every structure of its size. (2) For polyadic relations, there is no non-empty structure that disjointly embeds every structure of its size. (3) In either case, there is a structure that embeds each structure of its size, if we don’t require disjointness. We begin with some standard definitions, for reference. Definition 1 (a) A signature is a set P each element of which has some adicity (which is some positive natural number). As in the main text, we’ll call the elements of P relations. A signature P is monadic iff each of its members has adicity one. Otherwise P is polyadic. (b) A (set) P-structure is a pair of a set D, the domain, and a function that takes each n-place relation F in P to a subset of Dn , the extension of F. (c) An element of a structure S is an element of its domain. The size of S (written jSj) is the number of its elements. (d) Let S₁ and S₂ be P-structures with domains D₁ and D₂, respectively. An embedding of S₁ in S₂ is a one-to-one function f : D₁ ! D₂ such that, for each n-place F in P, and for each n-tuple of elements ðd ₁, . . . , dn Þ in D, ðd ₁, . . . , dn Þ is in the S₁-extension of F iff ðfd ₁, . . . , fdn Þ is in the S₂-extension of F. The structure S₂ embeds S₁ iff there is an embedding of S₁ in S₂. (e) An isomorphism is an embedding which is also an onto function. S₁ and S₂ are isomorphic (S₁ ffi S₂) iff there is an isomorphism from S₁ to S₂. (f) S₁ is a substructure of S₂ (written S₁  S₂) iff the domain of S₁ is a subset of the domain of S₂, and the function the takes each element of S₁ to itself is an embedding. Note that an embedding of S₁ in S₂ is an isomorphism between S₁ and some substructure of S₂. If S₂ embeds S₁ then clearly jS₂j  jS₁j. Definition 2 A structure S þ disjointly embeds a family of structures Si indexed by I iff for each i 2 I there is an embedding fi : Si ! S þ such that for i 6¼ j, the ranges of fi and fj have no elements in common. Some of our arguments will use the sum of some structures Si , which “glues together” a family of structures without overlap. This is the minimal structure

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that disjointly embeds the family Si . The domain of the sum-structure has as its domain a disjoint union of the original domains, and its extension for each relation F is the corresponding disjoint union of the Si -extensions for F. Definition 3 Let Si be a family of P-structures, for i 2 I . The disjoint a sum i2I Si is the P-structure whose domain consists of all ordered pairs ði, d Þ for i 2 I and d in Si , and whose extension for each n-place F in P is the set of all n-tuples ðði, d ₁Þ, . . . , ði, dn ÞÞ for i 2 I and ðd ₁, . . . , dn Þ in the Si -extension of F. The following facts about sums are clear from the definitions. Lemma 1 A structure S þ disjointly a embeds the family of structures Si iff S þ embeds their disjoint sum i Si . a Lemma 2 Let S þ ¼ i2I Si and let λ ¼ jI j. If Si is non-empty for each i 2 I , then jS þ j  λ. If jSi j  κ for each i 2 I , then jS þ j  κ  λ. Definition 4 (a) A P-structure U is weakly universal iff U embeds each structure which is no larger than U. (b) A P-structure U is strongly universal iff U disjointly embeds a representative of each isomorphism type of structure no larger than U. Clearly any strongly universal structure is also weakly universal. Theorem 1 If P is a monadic signature, there exists a strongly universal P-structure. Proof. When P consists of just monadic properties, we can fully describe a P-structure by specifying how many elements it has with each profile of P-properties. (In the monadic case, we don’t have to keep track of any connections between different elements.) If S is a P-structure, then for each d in S, let the individual profile of d be the set of F 2 P such that d is in the extension of F. Let the global profile for S be the function that takes each subset Q  P to the number of elements in S which have individual profile Q: this is some cardinal which is at most jSj. Two P-structures are isomorphic iff they have the same global profile. We can choose a cardinal κ such that there are no more than κ different global profiles corresponding to structures of size at most κ. In particular, let π ¼ jPj, and let κ be an infinite cardinal such that π

κ2 ¼ κ (If π is finite, this equation holds for any infinite κ. More generally, it holds for any inaccessible κ > π. It also holds for κ ¼ 2μ for any infinite μ  2π .) Let Φ be the set of all functions from subsets of P to cardinals which are at π most κ. Then Φ clearly has at most κ2 ¼ κ elements.

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For each function f 2 Φ, we can choose a representative structure Sf whose global profile is f. So every structure of size ata most κ is isomorphic to S . Since Φ has Sf for some f 2 Φ. Finally, let U be the sum f 2Φ f cardinality κ and jSf j  κ for each f 2 Φ, by Lemma 2, jU j  κ  κ ¼ κ. (In fact, jU j ¼ κ.) So U disjointly embeds a representative of each isomorphism type of structure no larger than U, which means that U is strongly universal. ∎ Now suppose P is a polyadic signature. Let κ > 0, and let Si for i 2 I be a family of structures including one representative of each isomorphism type of structure with size κ. Let λ ¼ jI j. In section 3 we proved Theorem 2: λ > κ. Moreover, if U is a structure that disjointly embeds the family Si , by Lemma 1 and Lemma 2, jU j  j

a

Si j  λ > κ

i

Corollary 1 If P is a polyadic signature, there is no non-empty strongly universal P-structure. Next we’ll prove the main “possibility” result from section 3: there are structures that embed every structure that is not too big. First, recall the following definitions. Definition 5 (a) κ is regular iff any union of strictly fewer than κ sets, each of which has strictly fewer than κ elements, has strictly fewer than κ elements. (b) κ is inaccessible iff κ is regular and for any λ < κ, we also have 2λ < κ. Theorem 4 Let P be any signature. If κ is an inaccessible cardinal strictly larger than P, then there is a weakly universal P-structure of size κ. This fact follows from a theorem by Roland Fraïssé. Since the existing presentations of the proof we’ve been able to find are either insufficiently general for our purposes (for the countable case see Hodges 1997, 158–64) or else use forbiddingly high-powered technical machinery (e.g. Caramello 2014, and references therein), we’ll sketch a proof here. This proof sketch will also help us generalize to the plural case. It’s worth noting that the theorem does not rely on any large cardinal axioms; but if large enough inaccessibles do not exist, then it is vacuous. To use Theorem 4 to deduce that there exist weakly universal P-structures for each signature P requires the additional premise that every cardinal is exceeded by some inaccessible. This is independent of ZFCU. In what follows, let P be any signature, and let κ > jPj be an inaccessible cardinal. A small structure is a P-structure with size strictly less than κ.

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The proof of Theorem 4 turns on a certain homogeneity property. Our strategy will be to prove two main lemmas: (1) If U is homogeneous, then U embeds every structure of size at most κ. (2) There exists a homogeneous structure of size κ. Very roughly, a homogeneous structure looks basically the same everywhere. More precisely, if U is homogeneous, then any small substructure of U can be extended however we like to bigger small structures.⁴⁰ Definition 6 If A  B and f : A ! U and g : B ! U are embeddings, then g extends f iff gðd Þ ¼ f ðd Þ for every d in the domain of A:

A

B f

g

U Definition 7 A structure U is homogeneous iff for any small structures A  B, and any embedding f : A ! U , there is some embedding g : B ! U that extends f. Here’s the idea of the first step. Suppose U is homogeneous, and let A be any structure of size at most κ. We can build A up as the limit of an infinite expanding chain of small substructures A0  A1  A2  . . . . Start with the trivial embedding of the empty structure in U. Then we can use the homogeneity property to extend this embedding to A0 , and then extend it further to A1 , and so on. This expanding chain of embeddings also has a limit, and this is an embedding of A in U. Let’s make this idea more precise. Definition 8 Let α be an ordinal. An α-chain is a sequence of structures Ai for each ordinal i < α, such that Ai  Aj whenever i  j < α. Lemma 3 For any α-chain of structures Ai , there is a unique limit structure Aþ such that, for any structure B, Aþ  B iff Ai  B for every i < α. The limit of a chain of structures is denoted limi