The Foundation of Reality: Fundamentality, Space, and Time 0198831501, 9780198831501

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OUP CORRECTED PROOF – FINAL, 22/2/2020, SPi

The Foundation of Reality

OUP CORRECTED PROOF – FINAL, 22/2/2020, SPi

OUP CORRECTED PROOF – FINAL, 22/2/2020, SPi

The Foundation of Reality Fundamentality, Space, and Time

E 

David Glick, George Darby, and Anna Marmodoro

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Great Clarendon Street, Oxford,  , 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  The moral rights of the authors have been asserted First Edition published in  Impression:  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  Madison Avenue, New York, NY , United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number:  ISBN –––– Printed and bound by CPI Group (UK) Ltd, Croydon,   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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Contents List of Figures List of Contributors Introduction David Glick

vii ix 

Section . The Metaphysics of Fundamentality . Fundamentality and Non-Symmetric Relations Ralf M. Bader



. Classifying Dependencies Alastair Wilson



. Ontic Structuralism and Fundamentality Matteo Morganti



. Fundamental and Derived Quantities J. E. Wolff



. Privileged-Perspective Realism in the Quantum Multiverse Nora Berenstain



Section . Quantum Mechanics and Fundamentality . Super-Humeanism: The Canberra Plan for Physics Michael Esfeld . What Entanglement Might Be Telling Us: Space, Quantum Mechanics, and Bohm’s Fish Tank Jenann Ismael





. Wave Function Realism in a Relativistic Setting Alyssa Ney



. In Defense of the Metaphysics of Entanglement David Glick and George Darby



Section . Spacetime Theories and Fundamentality . On the Independent Emergence of Space-time Richard Healey



. Duality, Fundamentality, and Emergence Elena Castellani and Sebastian De Haro



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. Radical Structural Essentialism for the Spacetime Substantivalist Tomasz Bigaj



. When the Actual World Is Not Even Possible Christian Wüthrich



Bibliography Index

 

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List of Figures . The set-up. . Front and side views of a fish.

 

. Examples of DAGS.



. Pre-general-relativistic physics is conceived on spacetime. The recent developments of string theory, with bulk physics described in terms of a boundary theory, are a step towards the same direction. Genuine full quantum gravity requires no spacetime at all.



. Duality relations vs. renormalization group flow. . Emergence and fundamentality.

 

. As shown in this Hasse diagram of (a section of) a causal set, events p and q are connected by a chain of causal relations (bold lines), yet do not stand in a direct causal relation. A Hasse diagram represents the antisymmetry of the partial ordering by vertical stacking.



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List of Contributors R M. B is a Fellow of Merton College and an Associate Professor in the Philosophy Department at the University of Oxford. His research primarily focuses on ethics, meta-ethics, metaphysics, Kant, political philosophy and decision theory. He is also interested in neo-Kantian and early analytic philosophy. He is currently finishing a book on Kant’s Theory of Modality for Cambridge University Press and a book on Person-Affecting Population Ethics for Oxford University Press. N B is Associate Professor at the Department of Philosophy at the University of Tennessee, Knoxville. She is also co-director of UTK’s Intersectionality Community of Scholars (ICOS). Her research in the metaphysics of science focuses on questions surrounding the role of mathematics in the empirical sciences and the relationship of math to modality and laws of nature. She is also interested in structuralisms in the philosophy of mathematics and philosophy of physics. Her work in feminist epistemology focuses on epistemic exploitation and epistemic oppression. She is currently working on a project explicating the notions of modal structure and modal profile in the social world in order to track the stability and predictability of patterns of relations among structures of oppression. Her recent work includes Implicit Bias and the Idealized Rational Self (Ergo, Vol. , , pp. –), The Applicability of Mathematics to Physical Modality (Synthese, Vol. , No. , , pp. –), and Epistemic Exploitation, (Ergo, Vol. , , pp. –). T B is Professor of philosophy at the Institute of Philosophy, University of Warsaw, Poland. His areas of specialization include philosophy of physics, philosophy of science and analytic metaphysics. He focuses also on philosophical logic, history of science, philosophy of language and cognition, critical thinking and epistemology. His recent publications include On quantum entanglement, counterfactuals, causality and dispositions, Synthese, , https://doi.org/./s-–., On some troubles with the metaphysics of fermionic compositions, Foundations of Physics, (), , pp. -, and Essentialism and modern physics (in: Bigaj, T., Wüthrich, C. (eds.), Metaphysics in Contemporary Physics, Poznań Studies in the Philosophy of the Sciences and the Humanities, vol. , , pp. –. E C is Associate Professor at the department of Philosophy at University of Florence. Her research and teaching areas are philosophy of science and philosophy of physics. In particular, her research focuses on symmetry and symmetry breaking, ontological aspects of physical theories, reductionism and emergence, structuralism in physics, history and philosophy of String Theory, dualities, and models in science. Her recent work includes Scientific methodology: A view from early string theory (in: R. Dawid, R. Dardashti, K. Thébault (eds). Epistemology of

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fundamental physics: Why trust a theory?, , Cambridge University Press, pp. –), Duality and ‘particle’ democracy (Studies in history and philosophy of modern physics, vol. , , pp. –) and The practice of naturalness: A historical-philosophical perspective (with A. Borrelli, in Foundations of physics, forthcoming). G D is currently a Teaching Fellow in Philosophy at Oxford Brookes University. He has previously been a Postdoctoral Research Fellow at the University of Oxford, a Lecturer at Birkbeck, University of London, and a Leverhulme Early Career Fellow at the University of Kent. His research concerns issues at the intersection of logic, the philosophy of physics, and metaphysics. Publications include “Quantum Mechanics and Metaphysical Indeterminacy” in the Australasian Journal of Philosophy and “Relational Holism and Humean Supervenience” in the British Journal for the Philosophy of Science. M E is full Professor of Philosophy of Science at the University of Lausanne. His research and teaching areas are the metaphysics of science (in particular laws of nature); philosophy of physics (in particular quantum physics and space-time relationalism), and the philosophy of mind (in particular mental causation, freedom and the relationship between naturalism and normativity). His recent work includes A minimalist ontology of the natural world (with Dirk-André Deckert, New York: Routledge ), La philosophie des sciences. Une introduction (Lausanne: Presses polytechniques et universitaires romandes, , Italian translation Filosofia della natura, Torino: Rosenberg & Sellier ), and Metaphysics of science as naturalized metaphysics, (in Anouk Barberousse, Denis Bonnay and Mikaël Cozic (eds.). The philosophy of science. A companion. Oxford: Oxford University Press, , pp. –). D G is currently an Honorary Associate in Philosophy at the University of Sydney. He has previously been a Visiting Assistant Professor at the University of Rochester, Postdoctoral Research Fellow at the University of Oxford, and a Lecturer at Ithaca College. His research concerns issues at the intersection of the philosophy of science, the philosophy of physics, and metaphysics. His two main areas of interest are: structural realism and the interpretation of quantum theory. Recent publications include “Against Quantum Indeterminacy” (Thought), “Generalism and the Metaphysics of Ontic Structural Realism” (British Journal for the Philosophy of Science), and “Timelike Entanglement for Delayed-Choice Entanglement Swapping” (Studies in History and Philosophy of Modern Physics). S D H is Lecturer in philosophy, mathematics and theoretical physics at the Amsterdam University College, University of Amsterdam, and Tarner scholar in Philosophy of Science and History of Ideas at Trinity College, Cambridge. His research interests include philosophy of physics, equivalence of theories, emergence, history of modern physics, philosophy of science, and scientific understanding. His recent work includes The Heuristic Function of Duality’ (Synthese, pp. –) and Interpreting theories without a spacetime (with HW De Regt, European Journal for Philosophy of Science, , pp. –).

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R H is Professor of philosophy at the University of Arizona. He has worked mostly in the philosophy of physics, especially quantum theory. Earlier his focus was on metaphysical implications of fundamental physical theories, assuming what he now regards as a rather uncritical realism. He won the Lakatos Prize for his book Gauging What’s Real, on the conceptual foundations of contemporary gauge theories in physics. More recently, ideas of contemporary pragmatist philosophers have prompted him to question the nature and extent of such metaphysical implications and to reexamine the roles of probability, causation, explanation, meaning and objectivity in quantum theory and elsewhere. He recently published The Quantum Revolution in Philosophy (, Oxford University Press: paperback ). J I is Professor of Philosophy at Columbia University and a member of the Foundational Questions Institute (FQXi). Her areas of specialization are philosophy of physics, metaphysics, philosophy of science and the philosophy of mind. In particular, she focuses on the structure of space and time, the foundations of quantum mechanics, the role of simplicity and symmetry in physics, questions about the nature of probability, natural laws and causal relations. She is also interested in cognition, phenomenology, and the nature of perspective. Her recent work includes: “Passage, Flow, and the Logic of Temporal Perspectives” in The Nature of Time, The Time of Nature, University of Chicago Press, edited by Christophe Bouton and Philippe Hunemann, ; and “Why Study the Humanities?” in Making Sense of the World: New Essays on the Philosophy of Understanding, (ed.) Stephen Grimm, Oxford University Press, . A M holds the Chair of Metaphysics at the University of Durham, and is concomitantly a Research Fellow at Corpus Christi College at the University of Oxford. Her research interests are in metaphysics; ancient, late antiquity and medieval philosophy; philosophy of mind; and philosophy of religion. She has published monographs, edited books and journal articles in all these areas. Among her recent publications there are: Metaphysics: An introduction to Contemporary Debates and Their History, co-authored with Erasmus Mayr, OUP, ; Everything in Everything. Anaxagoras’s Metaphysics, OUP, ; Aristotle on Perceiving Objects, OUP, . Anna has been the recipients of several research grants from the European Research Council; the Templeton World Charity Foundation, the Leverhulme Trust, the British Academy, and the Arts and Humanities Research Council, among others, totalling a combined sum of £,M. She is also the co-founder and co-editor of the peer-reviewed journal Ancient Philosophy Today: DIALOGOI, published by Edinburgh University Press. M M is Associate Professor at the Department of Philosophy, Communication and Performing Arts, University of Rome ‘Tre’. His research and teaching areas are the philosophy of science, metaphysics and the philosophy of physics. He particularly focuses on the connections between physical theories and metaphysical issues. His recent work includes Interpreting Quantum Entanglement: Steps towards Coherentist Quantum Mechanics (with C. Calosi, in the British Journal for the Philosophy of Science, ), The Structure of Physical Reality: Beyond Foundationalism, in Bliss, R. and Priest, G. (eds.): Reality and its Structure, Oxford University Press,  and Combining Science and Metaphysics (Palgrave Macmillan, ).

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A N is Professor in the department of Philosophy at University of California, Davis. Her recent research has been focused on interpretations of quantum theories. She explores and defends the view that quantum entanglement may suggest that the world we inhabit is not fundamentally constituted by a collection of objects in threedimensional space, but rather a field spread out in a much-higher-dimensional space. She also continues to explore how to fit facts about the mind and consciousness into the picture presented to us by our best physical theories. She defends a staunch reductionism about mental phenomena, however, argues that this leaves open a number of interesting metaphysical questions. She is the author of Metaphysics: An Introduction (Routledge, ), and co-editor with David Z. Albert of The Wave Function: Essays on the Metaphysics of Quantum Mechanics (Oxford, ). A W is Professor of Philosophy at the University of Birmingham. He works in metaphysics and the philosophy of science, with particular interests in the metaphysics of modality, Everettian quantum mechanics, chance, laws of nature and fundamentality. He is Principal Investigator on FraMEPhys, a major project on explanation in physics funded by a European Research Council Starting Grant. This project explores how physics helps us to understand the world in ways that go beyond the familiar model of causal explanation, and will cast light on some of the deepest puzzles of physics including the role of spacetime in explaining the motion of matter, the behaviour of ‘time-travelling’ systems in the presence of closed timelike curves, and the mysterious connections between entangled quantum systems. His recent work includes The Nature of Contingency: Quantum Physics as Modal Realism (forthcoming , Oxford University Press) and The Routledge Companion to the Philosophy of Physics (with Eleanor Knox; forthcoming , Routledge). J. E. W is a Senior Lecturer in the Department of Philosophy at the University of Edinburgh. Before joining the University of Edinburgh in , Jo was a Lecturer in Philosophy of Science at King’s College London. As an Alexander von Humboldt fellow, Jo has been a regular visitor to the MCMP at LMU Munich. Jo primarily works on questions at the intersection of philosophy of science and metaphysics. Recent work includes The Metaphysics of Quantities (under contract with OUP), “Heaps of Moles—Mediating macroscopic and microscopic measurement of chemical substances” (Studies in History and Philosophy of Science, , in press) and “Representationalism in Measurement Theory – Structuralism or Perspectivism?” In: Understanding Perspectivism: Scientific challenges and methodological prospects. Michela Massimi (ed.), Routledge (). C Wü is Associate Professor of Philosophy at the University of Geneva. He works in philosophy of physics, philosophy of science, and metaphysics. The primary focus of his research has long been the philosophy of quantum gravity, but also includes the philosophy of space and time, time travel, modality and laws of nature, emergence and reduction, as well as general methodological issues arising in fundamental physics. He is Co-Director of the ‘Beyond Spacetime’ research project, funded by NSF, FQXi, John Templeton Foundation, and ACLS (http://www. beyondspacetime.net).

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Introduction David Glick

In recent years, fundamentality and emergence have come to occupy a central place in both metaphysics and the philosophy of physics. Many metaphysicians now think that, in giving a complete account of reality, saying what exists is only part of the story—we also need to say how everything “hangs together.” Meanwhile, philosophers of physics have begun to appreciate that much of physics—including current so-called “fundamental physics”—in fact concerns effective or emergent levels. A point of intersection between these two areas is the status of spacetime. Is it possible that spacetime itself is non-fundamental? What would this mean for our understanding of reality? The diverse collection of chapters that follow have the common aim of clarifying the nature of fundamentality and emergence as they relate to spacetime. Both the interpretation of quantum theory and developments in quantum gravity prompt us to reconsider the status of spacetime in our understanding of physical reality. In so doing, we should make use of the latest developments in the metaphysics of fundamentality and emergence. Even if one is skeptical of certain claims made by contemporary analytic metaphysicians, the concepts they deploy, and the distinctions they draw, are likely to prove useful in making precise the challenge posed by emergent spacetime. Thus, by combining the efforts of philosophers of physics and metaphysicians, we stand a better chance of solving the novel challenges presented by contemporary physics. An investigation into the fundamentality of spacetime first requires getting clear about the concepts of emergence and fundamentality in play. The first section of this introduction will sketch some of the ways these concepts are used in contemporary metaphysics and philosophy of physics. Next, we will turn to two important motivations for taking spacetime to be non-fundamental: first, interpretative issues in quantum theory, and second, emergent spacetime in quantum gravity. In Section I., we will consider the argument that characteristic features of quantum theory—non-locality and entanglement—motivate moving to a highdimensional fundamental space. In Section I., we will briefly introduce approaches to quantum gravity that are alleged to render spacetime non-fundamental.

David Glick, Introduction In: The Foundation of Reality: Fundamentality, Space, and Time. Edited by: David Glick, George Darby, and Anna Marmodoro, Oxford University Press (2020). © David Glick. DOI: 10.1093/oso/9780198831501.003.0001

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  

I. Concepts of Fundamentality and Emergence I.. Fundamentality and Emergence in Metaphysics There is no single concept of fundamentality to be found in contemporary metaphysics. A familiar approach in trying to give an account of the metaphysically fundamental is mereological. It seems to be a conceptual truth about parthood that parts are more fundamental than the wholes they compose, which leads to the thought that this might provide an account of fundamentality in terms of parthood: Mereological Fundamentality (MF): parts are more fundamental than the wholes they compose and to be fundamental is to lack parts.¹ MF is closely related to atomism, as it suggests a metaphysics in which indivisible atoms occupy the most fundamental level of reality. Indeed, MF’s commitment to such a picture is at odds with certain non-atomistic (or holistic) metaphysical proposals. For example, priority monism is the view that there is only one fundamental thing—the Cosmos or the universe as a whole—and its parts have a derivative status (Schaffer ). In order for priority monism to be coherent, the relation between mereology and fundamentality embodied by MF must be revised. Some philosophers have sought to articulate a non-mereological concept of fundamentality. One such approach is based on the notion of metaphysical dependence, which is often cast in terms of grounding. This gives rise to the following formulations: Grounding Fundamentality (GF): grounds are more fundamental than that which they ground and to be fundamental is to be ungrounded. Dependence Fundamentality (DF): that on which something depends is more fundamental than that which depends on it and to be fundamental is to be independent. GF and DF allow for the possibility of priority monism, which would amount to the view that the Cosmos as a whole is ungrounded or independent and everything else depends on it. Indeed, this understanding seems to capture the idea that (to use the popular metaphor) all God had to do is make the Cosmos, and everything else came for free. GF is widely held, but the nature of grounding and how it differs from other dependence relations is not entirely clear. In Chapter , Alastair Wilson considers a number of criteria for distinguishing grounding from another well-known dependence relation: causation. Ultimately, he argues, by appeal to the cases of gravitation and quantum entanglement, for the criterion that causation is dependence mediated by laws of nature while grounding is dependence that is not so mediated. This minimal criterion narrows the gap between grounding and causation, allowing for fruitful connections to be drawn between the two concepts.

¹ While there may be few defenders of MF in contemporary metaphysics, it is an intuitive and familiar conception, and one that is frequently presupposed in discussion of fundamentality elsewhere in philosophy and science. For a discussion of MF, see Schaffer (, p. ).

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



In Chapter , Ralf Bader takes up a version of GF. He argues that on such a conception, there are no fundamental asymmetric relations, as allowing for them would lead to problematic indeterminacy and redundancy. However, he does allow for derivative asymmetric relations grounded in monadic properties and symmetric relations. One might wonder about the status of grounding itself, which looks both fundamental and asymmetric. Bader claims that these considerations recommend viewing grounding as a generative operation rather than an asymmetric relation. Once one settles on a version of DF, there is a further question of the direction of dependence, and hence of what counts as fundamental. As we saw above, some believe that dependence can come apart from mereology, allowing for holistic metaphysics such as priority monism.² Another view in the vicinity is ontic structural realism, which holds that (in some significant sense) structure is metaphysically fundamental (Ladyman ). In Chapter , Matteo Morganti argues that ontic structural realism should be understood in terms of metaphysical coherentism, according to which dependence relations can be symmetric. In particular, Morganti maintains that ontic structural realists should say that objects depend on structures (as structuralists often do), but also that structures depend on objects (as structuralists often don’t). This, he argues, captures the motivations for structuralism from contemporary physics with the least amount of conceptual revision. Each of these concepts of fundamentality³ suggests a hierarchical picture comprising levels of reality, ordered by relations of relative fundamentality. One challenge to this traditional hierarchical picture comes from the possibility of emergence. A common distinction is between strong and weak varieties (Bedau ): Strong Emergence: an emergent entity or feature is novel in that its existence or nature cannot be reduced to the base; it is “something over and above” that from which it emerges. Weak Emergence: an emergent entity or feature is novel in the sense of not being practically knowable, predictable, or explicable in terms of the base, but is in principle reducible to it; it is not “something over and above” that from which it emerges. Metaphysicians have generally been concerned with strong emergence, and the threat it poses to physicalism. In the philosophy of mind, strong emergentists claim that mental properties involve something “over and above” the physical (neuroscientific) properties that give rise to them. A historically significant test case for strong emergence may be found in the understanding of biological life: Is life in principle

² Schaffer’s priority monism is holistic in that it takes a whole (the Cosmos) to be fundamental. The approach to entanglement discussed by Glick and Darby (Chapter , section ..) is also holistic in this sense—it posits wholes (entangled joint systems) that are more fundamental than their parts (the entangled particles themselves). ³ There are several other notions of fundamentality in metaphysics besides those discussed here. For instance, some characterize the fundamental in terms of perfectly natural properties, or as providing a complete minimal basis, or take the notion as primitive. For a survey of accounts of fundamentality in metaphysics, see Tahko ().

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   reducible to physical (chemical) processes or does it involve something “over and above” them as the vitalists supposed?⁴ Of course, in this case the emergentists were defeated, but the case of strong emergence with respect to the mental remains open to debate. That said, there are serious challenges facing strong emergence, in particular, those involving downward causation (Kim ).⁵ Moreover, as the case of vitalism shows, strong emergentism leaves hostages to fortune by betting on the failure of in principle deduction. In what follows, we are concerned primarily with the fundamentality of spacetime in physical reality, so physicalism is not at issue here.⁶ Given this, it’s unsurprising that the discussion that follows will be centered on weak forms of emergence. Thus, claims that spacetime is emergent should not be taken to imply that spacetime cannot be in principle deduced from the fundamental nonspatiotemporal description of reality. One interesting variant of weak emergence with broad application is perspectivalism. A given phenomenon may be emergent in that it only becomes apparent from a particular perspective, even though taking such a perspective doesn’t require positing anything “over and above” the base ontology. A good example is the approach to time presented in Ismael (), according to which the phenomenology of temporal passage results from taking the perspective of an agent embedded in a block universe. Nora Berenstain addresses perspectivalism in Chapter . She argues against privileging the embedded “frog’s eye view” over the Archimedean “bird’s eye view” in general. This, according to Berenstain, has implications for the philosophy of time (recommending eternalism over presentism) and the foundations of quantum theory (recommending Everettianism over single-world interpretations).

I.. Fundamentality and Emergence in the Philosophy of Physics As in metaphysics, there are several distinct notions of fundamentality used in (the philosophy of) physics, which disagree on particular applications. Moreover, they don’t connect to the metaphysical notions discussed above in any straightforward manner. One important notion concerns energy or distance scales. This is generally cast in terms of the (relative) fundamentality of physical theories, rather than objects or properties:⁷ ⁴ Vitalism is the view, vigorously debated in the eighteenth and nineteenth centuries, that life requires some novel property such as an entelechy or élan vital over and above familiar physical properties. Today, while puzzles remain about the nature of life, vitalism is no longer considered to be a serious hypothesis by biologists. See Bechtel and Richardson (). ⁵ Downward causation occurs when a higher-level entity causes something to occur at a lower level. The problem is that if the lower level is supposed to be causally complete, downward causation would require causal overdetermination: To be causally relevant, strongly emergent entities would seem to require causing things that already have sufficient causes at their own level. ⁶ As mentioned above, strong emergence is often discussed in connection to physicalism, the thesis that “everything is physical.” Strong emergence may be seen as a way to understand the idea that the physical can give rise to something non-physical (e.g., the mind). But those who discuss emergence in physics do not intend to claim that physicalism is false—emergent entities and phenomena in physics are still “physical” in the relevant sense. ⁷ Of course, there may be a simple connection between the two. For instance, Wüthrich (Chapter ) maintains that an entity is fundamental iff a fundamental theory entails its existence.

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Energy (Distance) Fundamentality (EF): higher-energy (shorter-distance) theories are more fundamental than the lower-energy (longer-distance) theories to which they converge at a low-energy (long-distance) limit and to be fundamental is to apply at arbitrarily high energies (short distances). EF is widely deployed in the context of particle physics, where effective theories are distinguished from fundamental ones by requiring an energy/distance cutoff beyond which their equations break down. As stated, EF carries no metaphysical implications. That is, to be fundamental in the sense of EF need not entail being fundamental in any of the metaphysical senses. One must further assume that (a) the physical theories involved should be interpreted in a realist manner such that one is committed to the existence of the properties and objects they mention, and (b) the entities that appear in higher-energies/shorter-distances are more metaphysically fundamental in one of the senses sketched above. This mismatch between the characterization of fundamentality in physics and metaphysics can also be seen by looking at the notion of a fundamental quantity. After dismissing the idea that physical laws (and physical theories more generally) tell us which quantities are fundamental, J.E. Wolff considers in Chapter  whether measurement theory might be able to help. The distinction between base and derived quantities in a system of measurement suggests the following criterion: Unit Fundamentality (UF): base units correspond to fundamental quantities and derived units correspond to non-fundamental quantities. For example, in classical physics one may regard position and time as fundamental and velocity as derivative given that the latter can be defined in terms of the former. There is, however, an obvious problem with UF from the perspective of metaphysics: What counts as a base or derived quantity is relative to the system of units one adopts. Thus, like EF, one cannot read off metaphysical fundamentality from UF. Let us now briefly turn to emergence in physics. Ever since Anderson’s () influential proclamation that “more is different,” philosophers of physics have appealed to emergence to better understand a diverse range of phenomena including chaos, symmetry-breaking, virtual particles, and critical phenomena. Unlike metaphysicians concerned with the viability of physicalism, philosophers of physics have tended to employ weak varieties of emergence in these cases. There are a variety of understandings of weak emergence relevant to physics. In the context of emergence in string theory, Elena Castellani and Sebastian de Haro (Chapter ) deploy a weak epistemic form of emergence, which involves the appearance of a novel description rather than genuine ontological novelty. Other cases may recommend different notions of emergence. For instance, spontaneous symmetry-breaking can suggest a dynamical process unfolding in time, and complex systems are often characterized by autonomy from the details of the underlying micro-physics. However, it’s not clear that such features apply to the case of spacetime emergence. It seems clear that the emergence of spacetime cannot be a process in time and, while there may be some measure of autonomy to spacetime, part of the challenge of quantum gravity lies in specifying the details of the underlying physics to ensure that one can recover ordinary (general relativistic) spacetime as a low-energy

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   approximation. Similarly, it’s incumbent on those interpretations of quantum theory that make spacetime derivative to provide a clear link between particular happenings at the fundamental level and our observations in ordinary spacetime. Thus, while there is some room for debate, the notion of emergence involved in these two cases seems to be (a) weak rather than strong, (b) non-dynamical, and (c) not strongly autonomous.

I. Substantivalism and Relationalism There is another well-known approach to the status of spacetime that is relevant here: the debate between substantivalism and relationalism: Substantivalism: it contains.

spacetime is a substance; it exists independently of the entities

Relationalism: spacetime is a way of encoding the collection of distance relations between material bodies. It’s not entirely clear, however, how to update this approach in light of developments in physics and philosophy. For example, the hole argument was offered by Earman and Norton () as an argument against substantivalism, but has been taken by others to recommend a more nuanced position. One such position is spacetime structuralism, which contends that spatiotemporal structure—rather than bare points or regions—is the appropriate target of ontological commitment. In Chapter , Tomas Bigaj argues for a version of spacetime structuralism he calls radical structural essentialism. On this view, to be a spacetime point p is to occupy a certain place in a spacetime structure essentially. This means that it’s impossible for p to occupy a different place or for something other than p to occupy the particular place it does. Such a view invites worries with counterfactuals and symmetric spacetimes, but Bigaj argues that his view does better than similar positions in the literature (Maudlin ; Glick ) at responding to these challenges. One may reasonably wonder how the substantivalism/relationalism debate bears on the question of spacetime fundamentality. Indeed, some have recommended recasting the traditional debate in terms of (DF) fundamentality: either spacetime depends on spatiotemporal relations between material bodies or spacetime is independent (North ). And yet, it’s hard to see how this understanding can be applied to the possibility of emergent spacetime. The cases of spacetime emergence discussed in this volume do not purport to do away with the concept of a fundamental space in which reality unfolds, but rather, suggest that such a space does not have the usual features ascribed to spacetime.⁸ Thus, it remains open for one to be a relationalist, substantivalist, or structuralist about these non-spatiotemporal structures. The interest of the cases that follow is to show how spacetime may be

⁸ For instance, the fundamental space may have more or less than  dimensions, or lack the topological or metrical structure required by general relativity. In such cases, it seems natural to distinguish the fundamental space from ordinary spacetime.

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non-fundamental even if one commits to the fundamental existence of whatever space our best physics requires.

I. Quantum Theory and Spacetime Non-fundamentality Quantum theory is often taken to have radical metaphysical implications: indeterminism, indeterminacy, action at a distance, ontological non-separability, holism, etc. Unsurprisingly, some have sought to find ways to avoid or diminish these implications and thereby preserve a more common sense metaphysical picture. Such metaphysical maneuvers can have important implications for the status of spacetime; either making it derivative (Section I..) or necessarily fundamental (section I..).

I.. Wavefunction Realism One approach to the metaphysics of quantum theory is wavefunction realism, which posits the existence of a fundamental wavefunction located in a high-dimensional space (Albert ). In the case of ordinary, non-relativistic quantum mechanics, the wavefunction is a field on configuration space, a n-dimensional space (where n is the number of particles) in which each point corresponds to an entire configuration of particles in -dimensional space. There are a number of motivations for wavefunction realism, but a particularly influential one concerns the explanation of non-local phenomena it affords. Quantum theory is non-local in spacetime; it predicts correlations between distant measurements that do not admit of local causal explanations of the usual sort. One could posit some new entity or process in spacetime to account for non-locality—e.g., “spooky action at a distance”—or, alternatively, one can move to a new space in which locality is restored. In Chapter  Jenann Ismael suggests that non-local phenomena may be viewed as distinct “images” of a single entity in a higher-dimensional reality. As an analogy, she considers two cameras displaying, on different (twodimensional) screens, distinct perspectives of a fish in a fish tank. One might notice correlations between the images—movements of one sort on the left screen always co-occur with movements of another sort on the right screen—and be led to posit some connection between the screens. But, of course, the correct explanation is that the images are not ultimately distinct; they are -dimensional reflections of the same fish in -dimensions. The analogy with quantum theory is clear: measurements of particles in -dimensional space display correlations because the particles involved are low-dimensional reflections of some single entity in higher-dimensional reality. Thus, one may argue that wavefunction realism provides an explanation of nonlocal phenomena without being saddled with non-local metaphysics. However, wavefunction realism faces a number of challenges. Perhaps the most pressing is that the high-dimensional space in question—n-dimensional configuration space— is only available in non-relativistic quantum mechanics. In particular, it’s not clear that any such space is available once we move to quantum field theory, which is formulated in special relativistic spacetime. In Chapter , Alyssa Ney responds to this

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   challenge on behalf of the wavefunction realist. She maintains that the core motivation for wavefunction realism—to avoid non-locality—does not depend on the existence of configuration space, and so carries over to the case of quantum field theory. Suppose, for example, that one takes up Wallace and Timpson’s () spacetime state realism, according to which there is a density operator (quantum state) associated with each region of spacetime. Such a view contains non-locality at the level of spacetime as the density operator of an extended (or scattered) region need not supervene on the density operators of its subregions. Ney claims that the wavefunction realist may modify their view by positing a new space in which each point corresponds to a complete configuration of density operators on spacetime. In this infinite-dimensional space, locality would be restored. In sum, wavefunction realism seeks to avoid the metaphysical consequences of non-locality by relegating spacetime to a derivative status. But what exactly is the relation between the high-dimensional space and spacetime on wavefunction realism? Given that the relation is supposed to have explanatory import, and be consistent with quantum theory, it seems that it should be a non-causal dependence relation. One might reasonably wonder, then, whether a non-causal dependence relation between a high-dimensional space and spacetime is preferable to a non-causal dependence relation between the entangled particles themselves. At any rate, the upshot is that if wavefunction realism is correct, spacetime is not (DF) fundamental, but rather depends on a high-dimensional, non-spatiotemporal space.

I.. Super-Humeanism A quite different approach to the metaphysics of quantum theory is the Humean approach put forward by Michael Esfeld in Chapter . Before turning to his approach directly, it’s worth reviewing the apparent tension between Humeanism and quantum theory. Contemporary Humeanism is closely associated with the work of David Lewis, whose thesis of Humean supervenience maintains that there are only “local matters of fact” and everything else supervenes on these. However, there is a clear tension between Humean supervenience and quantum entanglement.⁹ Roughly, entanglement occurs when the quantum state of a composite system cannot be factored into quantum states of the component systems that compose it.¹⁰ This suggests that the local matters of fact (the individual quantum states) are insufficient to account for the facts concerning their joint state. There have been a number of attempts to resolve this tension, including the approach just discussed, wavefunction realism (Loewer ).¹¹

⁹ For background on David Lewis’ metaphysics, which figures in several of the chapters below, see Hall (). ¹⁰ For instance, if two systems are jointly assigned a (pure) quantum state |Ψ>₁₂, then they are entangled N just in case |Ψ>₁₂ 6¼N|Ψ>₁ |Ψ>₂, where |Ψ>₁ and |Ψ>₂ are (pure) quantum states of subsystems  and  (respectively) and is the standard tensor-product operation used to combine quantum states. If one takes the quantum state to provide a complete description of the physical properties of a system, then entanglement suggests that joint systems can have properties that are irreducible to the properties of their constituents. For a discussion, see Glick and Darby (Chapter , Section .). ¹¹ For other resolutions to the (apparent) conflict between Humean supervenience and quantum entanglement, see Darby (), Bhogal and Perry (), and Glick and Darby (Chapter , Section .).

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In contrast to the wavefunction realists, Esfeld aims to maintain Lewis’ conception of local matters of fact in spacetime forming the supervenience base for all else. He argues for a methodology of naturalistic metaphysics according to which one posits a primitive ontology of entities consistent with contemporary physics and then locates everything in that ontology. The ontology he favors contains only bare matter points and the spatial relations between them. This ontology may then be combined with the Bohmian approach to quantum mechanics to yield a consistent picture of the quantum world free from fundamental non-locality, indeterminacy, and indeterminism. In such a picture, spacetime is fundamental and, in fact, spatiotemporal relations are the only fundamental properties or relations. Everything else supervenes on facts about the mosaic of matter points in accordance with Humean supervenience. In one sense, then, wavefunction realism and Esfeld’s super-Humeanism are opposites: the former regards spacetime as derivative and the latter takes it to be fundamental. However, in Chapter , David Glick and George Darby argue that both positions are instances of a common strategy: one which seeks to eliminate entanglement. As noted above, wavefunction realism does away with fundamental non-locality by moving to a higher-dimensional space where locality is preserved. This means that entanglement—the feature of particles that display non-local correlations—is eliminated from the picture, at least at the fundamental level. On super-Humeanism, there are just matter points and spatial relations at the fundamental level, so entanglement—like everything else—is once again derivative rather than fundamental. Glick and Darby argue that instead of adopting such an eliminativist strategy, it is better to embrace entanglement in one’s metaphysical picture, either by including new relations in the ontology, or by adopting ontological holism for entangled systems. Doing so, they contend, allows one to better explain non-local phenomena (e.g., Bell-type correlations in measurement results) and is more in line with a naturalistic approach to metaphysics.

I. Quantum Gravity and Emergent Spacetime The conceptual difficulties involving non-relativistic quantum mechanics and quantum field theory have been much discussed by philosophers. By contrast, quantum gravity—which seeks to reconcile quantum theory and general relativity—has received significantly less attention. One major obstacle is that we do not yet have a theory of quantum gravity, but only a variety of approaches to arriving at one. Nevertheless, in these approaches one can already see a number of challenging suggestions for philosophers to consider.

I.. Some Approaches to Quantum Gravity There are two broad families that capture the best-known approaches to quantum gravity, which proceed in opposite directions toward their goal: Canonical Quantum Gravity (CQG) begins with a particular formulation of general relativity and applies standard quantization techniques to arrive at quantum gravity.

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   String Theory begins with a novel theory that posits quantized -dimensional strings (or higher-dimensional objects) and seeks to recover general relativity as a low-energy limit of this quantum theory. Interestingly, both approaches seem to involve “emergent spacetime” in one sense or another. In the context of CQG, Butterfield and Isham () argue that spacetime is emergent insofar as it’s only an approximation, valid for sufficiently large scales of time and length. String theory shares this general approach (of recovering general relativity as a low-energy limit), but like other quantum theories, it presupposes a fixed background space that resembles classical spacetime in some respects. However, matters are complicated by the presence of dualities—symmetry mappings that are taken to preserve physical equivalence—between string theories with different background spaces. One particularly important duality is the AdS/CFT duality, which relates string theory in -dimensional space to quantum field theory (in  dimensions). This seems to suggest another sense of “emergence” in that a theory containing gravity (string theory) can be transformed via a duality into a (nongravitational) quantum field theory. In their contribution, Elena Castellani and Sebastian De Haro refer to the emergence involved here as horizontal emergence and distinguish it from the vertical emergence that corresponds to novel phenomena appearing at different energy scales. They argue that horizontal emergence must be understood as weak emergence, as the theories related by a duality are physically equivalent. Moreover, horizontal emergence can come apart from (EF) fundamentality: a high-energy theory can (horizontally) emerge from a low-energy one. This leads Castellani and De Haro to the surprising claim that the more fundamental can emerge from the less fundamental. There are other approaches to quantum gravity that aren’t instances of either the CQG or string theory approaches. Christian Wüthrich, in Chapter , considers causal set theory, which posits a fundamental ontology of discrete events ordered by an asymmetric causal relation. Continuous classical spacetime, and all spatiotemporal notions, are taken to emerge from these causal sets at a suitable low-energy limit (given an appropriate dynamics).

I.. Problems of Emergent Spacetime Taking up the idea that quantum gravity describes fundamental physical reality leads to a number of challenges for philosophers. The problem of time, often discussed in the context of CQG, concerns whether it’s even coherent to suppose that fundamental reality doesn’t involve temporal evolution. Another problem is raised by Wüthrich in his chapter, namely: the apparent incompatibility between causal set theory and David Lewis’s influential possible worlds account of modality. According to Lewis, a possible world is unified by sharing a common spacetime, but in causal set theory, there is no fundamental spacetime, just discrete events standing in causal relations. Thus, combining these theories, one seems forced to the paradoxical conclusion that the actual world is not possible. A theory of quantum gravity is, in the end, supposed to be a quantum theory. This means that it is likely to inherit the familiar difficulties of quantum interpretation. Richard Healey (, a) has defended a pragmatist interpretation of quantum

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theory in which it lacks a distinctive ontology of its own, but rather, functions primarily to advise agents about how to navigate the (non-quantum) world. This creates a puzzle for understanding the emergence of the classical world from quantum theory. In Chapter , Healey proposes a notion of independent emergence according to which something can emerge without an accompanying relation of ontological dependence. Thus, if spacetime emerges from a quantum theory of gravity, it emerges from nothing. This would seem to imply that emergence comes apart from (DF) fundamentality: Spacetime would be emergent (in Healey’s sense) while being (DF) fundamental insofar as it is independent. As is by now clear, the question of spacetime’s fundamentality is multi-facetted and intersects with a diverse range of philosophical issues. Most importantly from our perspective, it serves as a potential point of contact between philosophers of physics, metaphysicians, and others in the field. We hope this volume will help to promote fruitful cross-disciplinary engagement on the critical question of spacetime’s place in our understanding of physical reality.

Acknowledgement Many thanks to George Darby and Anna Marmodoro for their valuable feedback on this Introduction.

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SECTION 1

The Metaphysics of Fundamentality

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1 Fundamentality and Non-Symmetric Relations Ralf M. Bader

. Introduction Non-symmetric relations allow for differential application.¹ A binary relation R can hold of a and b in two different ways: . aRb and . bRa. Different states of affairs result from completing R by means of a and b, depending on the order in which a and b are combined with R. The extension of a binary non-symmetric relation is, accordingly, not to be understood in terms of a set of unordered pairs. One has to operate with a structured conception of the extension of a relation, for instance in terms of ordered pairs, that not only considers which things R relates, but also the order in which it relates them. Differential applicability can straightforwardly be explained if relations have directions: R can then either go from a to b, or from b to a. Yet, if relations have directions, then non-symmetric relations would appear to have distinct converses. For instance, an asymmetric binary relation R, such as ‘better than’, seems to have a distinct converse R
1 , namely ‘worse than’, where R holds of x and y in a given order whenever R
1 holds of them in the opposite order. Non-symmetric relations and their converses can This   fail to be interchangeable. implies that these relations are distinct: ◇9x9y xRy∧:ðxR0 yÞ ! R 6¼ R0 . If a stands in R to b, then b stands in the converse relation R
1 to a. Given that R is not symmetric, b can fail to stand in R to a. Yet, if b can stand in R
1 to a without standing in R to a, then R is distinct from R
1 . . . .

aRb :ðbRaÞ bR
1 a

∴

R 6¼ R
1

¹ Non-symmetric relations are those that are not symmetric. They are either asymmetric or neither symmetric nor asymmetric. Ralf M. Bader, Fundamentality and Non-Symmetric Relations In: The Foundation of Reality: Fundamentality, Space, and Time. Edited by: David Glick, George Darby, and Anna Marmodoro, Oxford University Press (2020). © Ralf M. Bader. DOI: 10.1093/oso/9780198831501.003.0002

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  .  The first part of this chapter argues that there are no non-symmetric relations at the fundamental level (Sections . and .). The second part identifies different ways in which asymmetry and order can be introduced into a world that only contains symmetric but no non-symmetric fundamental relations (Section .). The third part develops an account of derivative relations and puts forward identity criteria that establish that derivative non-symmetric relations do not have distinct converses. Instead of a plurality of relations, there are only different ways of picking out the same relation (Section .). The final part provides an account of how generative operations can induce order and argues for a reconceptualisation of grounding as an operation rather than as a relation (Section .).

. Converse Relations Difficulties arise if non-symmetric relations have distinct converses. .

 

If R is distinct from R
1 , then it is indeterminate whether a relational expression ‘X’ refers to R or its converse R
1 (cf. Williamson 1985; van Inwagen 2006). Williamson illustrates this point by means of the following languages: language L : bXa = b stabs a : aXb = b stabs a language L0 aXb = a is stabbed by b language L00 : Language L uses the expression ‘bXa’ to refer to the fact that b stabs a. Language L0 , by contrast, uses the expression ‘aXb’ to refer to this very same fact. L and L0 use the same relational expression ‘X’ to pick out the stabbing relation but employ different conventions. Language L00 uses the expression ‘aXb’ to refer to the fact that a is stabbed by b, i.e. that a stands in the converse of the stabbing relation to b. The problem now is that L0 and L00 are indistinguishable. These languages are for all extents and purposes the same. This implies that there is nothing that makes it the case that we are speaking L0 as opposed to L00 . If we grant that relations are distinct from their converses, then it would seem to be indeterminate which relations our relational expressions refer to. Given all the facts about how we use relational expression ‘X’, it could just as well refer to R as to its converse R
1 . There is nothing to differentiate between these two candidates.² Distinguishing these languages is to draw a distinction without a difference. We should, accordingly, reject the idea that relations have distinct converses. .   If R and R
1 are distinct relations, then we end up with brute necessities amongst distinct existences. Relations and their converses go together as a matter of

² This indeterminacy is ineliminable since there is no way of picking out a relation rather than its converse in order to stipulatively fix reference (cf. n. ).

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  - 



necessity. It will necessarily be the case that x and y stand in R in a given order iff they stand in R
1 in the opposite order. □8x8yðxRy $ yR
1 xÞ This necessary biconditional would seem to involve a brute necessity that violates Humean strictures (cf. Dorr ). . -   The existence of both a relation R and its distinct converse R
1 leads to a proliferation of facts. If relations enter into relational facts and play a role in fixing the identity conditions of such facts, then distinct relations give rise to distinct facts. In addition to the fact [xRy], there would then also be a further fact, namely [yR
1 x]. For instance, the relation ‘on top of ’ and its converse ‘beneath’ would enter into the two facts: i. the cat is on top of the mat ii.

the mat is beneath the cat

Yet, intuitively there should only be one state of affairs, one fact about how the cat and the mat are related to each other (cf. Russell , pp. –; Fine ). The problem is not simply that recognising distinct converses leads to a non-parsimonious theory that countenances too many entities and thereby contravenes against Ockham’s razor. Rather, the objection is that distinguishing these facts amounts to drawing a distinction without a difference. Intuitively, the fact that the cat is on top of the mat is the very same fact as the mat being beneath the cat. There is only one fact concerning the relative placement of these two objects. The underlying problem is that relations and their converses are so closely and intimately connected that it is not plausible to consider them to be distinct.³ Treating them as being distinct generates reference problems insofar as it is indeterminate which one we are picking out, bruteness problems insofar as they are not separable but of necessity go together, and a bloated ontology since they give rise to distinctions that do not make a difference. These problems multiply as the arity of the relation increases, since a relation of arity n has n!
1 converses. Whereas binary relations have one converse, ternary relations have five converses, and quaternary relations have twenty-three converses. The greater the number of converses, the greater the amount of indeterminacy, the larger the number of brute necessities and the more extensive the over-abundance of states of affairs.

.. Relational Properties It is often suggested that converse relations generate difficulties for the theory of relations and that there are no analogous difficulties in the theory of properties. The relevant contrast is taken to be one between properties and relations, between

³ Even though R and R
1 have different extensions across modal space when these are construed in terms of ordered n-tuples, they have the same extension understood in terms of unordered n-tuples.

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  .  the monadic and the polyadic (e.g. van Inwagen ). However, these problems do not only affect relations but also relational properties. If a relation R is distinct from its converse R
1 , then the relational properties λx½xRa and λx½aR
1 x are also distinct. For instance, if x has the property of being taller than some particular object a, then x also has the converse property of being such that a is shorter than it. In the same way in which one complex cannot be the completion of two distinct relations (cf. the ‘uniqueness’ claim put forward by Fine 2000, p. 5), one relational property cannot result when λ-abstracting one and the same object from two relational complexes involving distinct relations.⁴ If the relations are distinct, then the corresponding relational properties are likewise distinct.⁵ The problems (of indeterminacy, brute necessities, and proliferation of facts) that derive from relations having distinct converses carry over to relational properties that have distinct converse relational properties. The source of the problem consists in relationality, not in polyadicity. In particular what generates the problems is the differential applicability of non-symmetric relations. This carries over to relational properties, since there will be two relational properties corresponding to the two different applications of a non-symmetric binary relation. The fundamental distinction is not between the polyadic and the monadic, but between the relational and the non-relational.

.. Converses: Strict and Loose In order to address these problems, one has to deny that there are non-symmetric relations that have distinct converses. . The most radical approach denies that there are non-symmetric relations. Since symmetric relations are identical to their converses, there will not be any distinct converses if there are no non-symmetric relations. The denial of non-symmetric relations, however, is implausible since the world is full of asymmetry and order (though cf. Dorr , sections –). Alternatively, one can accept that there are non-symmetric relations, but deny that they have distinct converses. . Non-symmetric relations seem to have directions (what Russell , p.  calls their “from-and-to character”). This is what allows them to introduce order and to apply differentially to their relata. If relations have directions, then there is a meaningful notion of a converse relation. In order to ensure that relations do not have distinct converses, one has to adopt a sparse theory of relations and deny the existence of converses. Although the idea of the converse of a relation is intelligible and well-defined, no such relations exist according to this approach. As ⁴ If the relational properties were identical, then λ-conversion would not be functional, since it would then be one-many. One would get two distinct relational complexes from the same relational property. ⁵ This means that those who consider relations to be distinct from their converses end up with hyperintensional commitments. The relational properties derived from these two different relations will be necessarily co-extensive: □8x8yðλx½yRxx $ λx½xR
1 yxÞ, yet distinct: λx½yRx 6¼ λx½xR
1 y. These properties differ merely hyperintensionally. They are distinct despite being necessarily co-extensive. Anyone who countenances distinct converses is thus committed to a hyperintensional theory of properties.

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  - 



MacBride suggests: “Assuming universals are sparse, it doesn’t follow from the fact that the notion of the converse of a relation is definable that there is a corresponding converse relation” (MacBride , ). Adopting a sparse theory does not imply that one is committed to claiming that only one of ‘taller than’ and ‘shorter than’ refers to an existing relation whereas the other fails to refer. This is because one can follow MacBride and consider both of them to be impure referring terms that refer to the same relation (whichever of R and R
1 it is that happens to exist). Nevertheless, such a sparse theory would seem to involve a significant degree of arbitrariness. It recognises that there are two meaningful candidates that are on a par, yet privileges one of them.⁶ . One can adopt a revisionist view about the nature of relations that does not consider them to have directions. By denying that relations have directions, one can reject the idea that there is a meaningful notion of a converse relation. The two main revisionist approaches are positionalism (cf. Williamson ) and antipositionalism (cf. Fine ). Positionalism attempts to explain differential application, not in terms of relations having directions, such that R can either go from a to b or from b to a, but in terms of relations having argument places. There are two ways in which objects can be assigned to the argument places α and β of a binary relation R, each giving rise to a distinct completion: a can be assigned to α and b to β or, alternatively, a can be assigned to β and b to α. Antipositionalism, by contrast, attempts to explain differential application in terms of the idea of different completions of a relation being co-mannered. A completion of R by a and b can either be co-mannered or non-co-mannered to another completion of R by c and d. The relation of being co-mannered gives rise to equivalence classes of completions that correspond to the different applications of the relation in question. Differential application is then not understood in terms of the internal structure of a completion considered by itself, but in terms of how different completions relate to each other.⁷

⁶ Rather than arbitrarily privileging one of them, one can claim that there is a privileged relation yet that it is indeterminate which of them is privileged. This, however, implies a commitment to ontic indeterminacy. Moreover, sparseness has to be understood at the level of existence (rather than, say, naturalness) if one is to address all the problems deriving from the existence of distinct converses. It is not enough to accept an abundant conception of relations and then privilege one relation over its converse by taking the former to be more natural or fundamental than the latter. One has to deny that converses exist. Accordingly, one would be committed to existence being indeterminate if one were to attempt to mitigate arbitrariness by invoking indeterminacy. (Privileging one relation over its converse in terms of naturalness may well be sufficient for addressing the problems of uniqueness and redundancy (identified in Section ..) and possibly also the problem of referential indeterminacy when invoking reference magnetism to secure determinate reference. Yet, it will neither address the problem of brute necessities nor the objectionable proliferation of states-of-affairs.) ⁷ The antipositionalist mirrors the relationalist approach to chiral properties which denies that objects are endowed with intrinsic handedness properties and only stand in same- or opposite-handed relations, on the basis of which one can partition them into equivalence classes. One important disanalogy is that for the relationalist there are no cross-world same- or opposite-handed relations, since one would otherwise end up with there being two worlds, each with a lonely hand, that differ merely in terms of their chiral properties. Yet, the antipositionalist is committed to completions in different worlds being co-mannered: cf. “we must be able to identify the manner of completion from one world to another” (Fine , p. ). This, however, is deeply problematic since the relation of being co-mannered is an external rather than an internal relation, which contravenes the ban on cross-world external relations.

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  .  On these approaches, there is no such thing as the converse of a relation, and, a fortiori, no such thing as a non-symmetric relation having a distinct converse. Problems arise, however, once we distinguish a strict from a loose sense of converse. Converses in the strict sense are characterised in terms of the notion of direction. A relation R
1 is the converse of a relation R iff these relations only differ in terms of their directions.⁸ Converses in the loose sense (which we will also call ‘duals’), by contrast, are characterised in terms of relations merely differing in terms of whatever it is that allows for differential application. Revisionist theories also need to account for differential application and hence need to find some substitute that plays the role that is played by directions. If a relation R* involves the very same objects as R and differs only with respect to whatever it is that accounts for differential application, then these relations are mere duals. Such relations hold of the same things, i.e. they have the same extension (understood in an unstructured way) such that □8x8yðxRy $ yR*xÞ, and only differ in terms of their argument places (in the case of positionalism) or in terms of their equivalence classes of co-mannered completions (in the case of anti-positionalism). For the positionalist, for instance, a dual of R is such that necessarily for any objects x and y, x will be assigned to α and y to β in R iff y is assigned to γ and x to δ in R*. Whilst a binary relation can have one converse, since there are only two possible directions in which two relata can be related, it can have any number of duals. For instance, one can also have a further relation R** where x is assigned to ϵ and y to ζ. (van Inwagen seems to be invoking the loose notion of a converse when he questions why a binary relation has “just one converse” (van Inwagen , p. , n. ).) If one denies that relations have directions, then R* does not classify as the converse of R, i.e. as a relation that merely differs in terms of its direction from R. Nevertheless, R* gives rise to the very same difficulties of referential indeterminacy, brute necessities, and a proliferation of states-of-affairs as the converse R
1 of R. One cannot simply claim that R and R* are identical on the grounds that they are necessarily co-extensive. Given the differential applicability of relations, we need to distinguish aRb from bRa. This means that the extension of a binary relation cannot be understood as a set of unordered pairs: {{a,b}, {c,d} . . . }. The extension has to specify not only the relata, but also their order. This, however, means that R and R* are not co-extensive. Although they hold of the same things, they do not hold of them in the same order (where the notion of order can be understood in terms of directions, assignments to argument places, or co-mannered equivalence classes). The relevant sense of order is traditionally understood in terms of the direction of a relation and the extension is, correspondingly, construed in terms of ordered pairs: {⟨a,b⟩, ⟨c,d⟩ . . . }. Two binary relations are then necessarily co-extensive iff they hold of the same ordered pairs across all of modal space. The positionalist, by contrast, invokes assignments of objects to argument places. The extension of a relation is understood More generally, the parallels between the debate about chiral properties and the debate about relations are worth noting. For instance, the modal scenarios that Dorr invokes directly mirror those found in the chiral case. ⁸ Cf. A converse relation in the strict sense “is one that differs from the given relation merely in the order of its arguments” (Fine: , p. , n. ).

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  - 



in terms of assignments to argument places: {{⟨a,α⟩, ⟨b,β⟩}, {⟨c,α⟩, ⟨d,β⟩} . . . }.⁹ This, however, means that R and R* will fail to be co-extensive and hence turn out to be distinct. Given that R and R* have different argument places, they cannot be identical. Yet, since neither of them is privileged over the other, it would be arbitrary to adopt a sparse theory of relations and favour one of them, say, by claiming that it is only relation R which involves α and β that exists and that R* which involves γ and δ, though being perfectly well-defined, does not exist. In order to consider R and R* to be co-extensive, one has to identify argument places across relations.¹⁰ Yet, the intelligibility of such an identification is precisely what Fine denies in order to ensure that the notion of a converse is not definable for the positionalist. “We may indeed ask whether, for given argument-places α, β, α0 and β0 , the relation R0 holds under the assignment of a to α0 and b to β0 just whenever R holds under the assigment [sic] of a to α and b to β. But this merely tells us whether the relations are coextensive under the given alignment of argument-places. To obtain the notion of converse, we also need to assume that α0 = β and β0 = α. But I doubt that there is any reasonable basis, under positionalism, for identifying an argument-place of one relation with an argument-place of another” (Fine , p. ). Fine’s suggestion of denying that argument-places can be identified across relations allows the positionalist to deny the existence of distinct converses, however doing so opens up the possibility of distinct duals. Revisionist approaches face a dilemma. If they deny the possibility of crossrelation comparisons,¹¹ they can deny that there is a meaningful notion of a converse relation that merely differs in terms of direction. Whilst successfully denying the intelligibility and hence existence of converses in the strict sense, such an approach cannot provide informative identity criteria for relations and hence cannot establish the identity of a relation with any of its duals. It cannot even provide an informative account of what it is for two relations to be co-extensive. This means that neither positionalism nor antipositionalism, construed in this way, is able to rule out distinct duals in a principled way. The original difficulties, however, arise just as much in the case of duals. There is nothing to distinguish a relation from its dual in the same way as there is nothing to distinguish a relation from its converse. Given that there is nothing to privilege one of them, defenders of this approach likewise end up with a sparse theory of relations that only countenances some relations but rejects the existence of others (in a seemingly arbitrary manner). Rejecting duals and countenancing only one of R and R* does not seem to be any less arbitrary and problematic than rejecting converses and only countenancing one of R and R
1 . In each case, there are a number of

⁹ Similarly, for the antipositionalist extensions are understood in terms of sets of objects together with membership in an equivalence class of co-mannered completions. ¹⁰ Similarly, in order to establish co-extensiveness the antipositionalist needs to accept that completions of different relations can be co-mannered, i.e. that a completion of R can be co-mannered to a completion of R*. ¹¹ For the positionalist this amounts to rejecting the possibility of identifying argument places across relations. For the antipositionalist it amounts to rejecting cross-relation co-manneredness facts.

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  .  distinct yet intimately related relations that the theory deems to be intelligible, yet only one of which it deems to exist.¹² Alternatively, if they accept the possibility of cross-relation comparisons, then they can meaningfully speak of different relations being co-extensive. This allows them to specify informative identity criteria for relations, for example by considering relations that are necessarily co-extensive to be identical. Distinct duals that merely differ in terms of the identity of their argument places can then be ruled out. However, one can also reintroduce the notion of a converse in the strict sense. A relation R* will classify as a converse of R for the positionalist if the assignment of objects to argument places in R is necessarily the reverse order of the assignment of objects to argument places in R*, whereas for the antipositionalist it will be the converse if necessarily there is a completion of R by x and y iff there is a completion of R* by the same objects such that the two completions fail to be co-mannered. This, however, means that revisionist approaches effectively collapse into the traditional approach and face the very same difficulties that are involved in either countenancing distinct converses or adopting a sparse theory of relations that arbitrarily privileges some relations over others. A satisfactory account needs to not only rule out the existence of distinct converses in the strict sense but also in the loose sense. In order to do this, it needs to provide informative identity criteria for relations that preclude, in a principled way, any excessive proliferation of relations. This will be achieved, on the one hand, by rejecting non-symmetric relations at the fundamental level whilst individuating fundamental relations in terms of necessary co-extensiveness and, on the other hand, by developing an account of derivative non-symmetric relations that are individuated in terms of a theory of hyperintensional equivalence that does not allow for converses in either the strict or the loose sense.

. Fundamental Symmetric Relations Two further problems arise if fundamental non-symmetric relations have distinct converses, namely the problem of uniqueness and the problem of redundancy. These problems suggest that fundamental theorising is only allowed to invoke symmetric relations. They can be nicely illustrated by means of the difficulties that arise for Carnap’s Aufbau due to the fact that he operates with an asymmetric basic relation. In the Aufbau, Carnap puts forward the outlines of a logical construction of the world. Starting with an auto-psychological basis, consisting of () a domain of elementary experiences (‘Elementarerlebnisse’) and () a unique basic relation Rs (= recollected similarity), Carnap logically constructs the entire world: from the subject’s inner mental life, to the intersubjective physical world, to cultural objects and ethics. This construction proceeds by giving pure structural descriptions of all items in the world. In virtue of being structural, such descriptions are objective. They

¹² In addition, we will see in Section . that an unrestricted ban on cross-relation comparisons is problematic for independent reasons.

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  - 



are communicable, scientific, and can constitute knowledge, unlike what is merely subjective and can only be identified by ostension (cf. Carnap , § ). After having outlined his constructional system, Carnap returns in § to the problem of the basic relation. The goal of the Aufbau is to put forward a logical construction. Every scientific statement is to be transformed into a purely logical structural statement. Carnap notes that this goal has not been achieved at that point. The problem is that we only have a logical construction of the world if the construction proceeds on the basis of logical relations. The Aufbau, however, has not transformed anything into a purely logical statement. Instead it has produced constructions that contain one extra-logical primitive, namely the basic relation Rs. This relation is not a logical but a psychological relation. As a result, it undermines the purity of the system and prevents the Aufbau from classifying as a logical construction. The basic relation Rs needs to be eliminated from the constructional basis. It itself has to be logically constructed. Only then will the Aufbau be a purely logical construction. The problem, however, is that any attempt to construct Rs would seem to require another extra-logical primitive, in which case the original problem would simply arise again. It is for this reason that Carnap proposes to provide, not an explicit definition of Rs in terms of some more basic concept, but an implicit definition in terms of the constructional system: Rs is the unique binary relation R such that R allows us to construct the Aufbau (more precisely, he picks a sufficiently high-level object in the constructional system). We use Rs to construct the system and then define this relation away as that relation which allows us to perform this logical construction of the world. Once we have constructed the Aufbau by means of Rs, we can kick away the ladder that got us there. This suggestion runs into serious trouble. As Carnap notes in §, it will trivially be the case that there is a very large number of relations that allow one to construct the Aufbau unless restrictions are imposed on what R can be, i.e. what R ranges over. If we have a domain D with an abstract structure in terms of Rs over it, then a permutation f of the members of the domain that is not an identity-mapping gives rise to a relation R* such that R*f(x)f(y) iff Rsxy. This means that if Rs allows one to construct the Aufbau, then so does R*. These two relations have the very same structure and hence can construct the same system. As a result, uniqueness will fail and Rs cannot be implicitly defined.¹³ Carnap attempts to resolve this problem in § by introducing a new primitive: foundedness. The failure of uniqueness is due to the fact that no restrictions were imposed on R, i.e. R can simply be understood in terms of a set of ordered pairs. Any such set of ordered pairs is as good as any other. This is what allows us to construct relations, such as R*, that have the same structure as Rs by means of arbitrary permutations. In order to ensure uniqueness, we need to rule out gerrymandered relations and narrow down the possible candidates that can satisfy the implicit definition. Foundedness is precisely meant to fill this role. Carnap suggests that Rs

¹³ This problem is closely related to the Newman problem for structural realism (cf. Newman ), as well as to Putnam’s model-theoretic argument (cf. Putnam ).

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  .  is the unique founded binary relation R such that R allows us to construct the Aufbau. The random permutations giving rise to R* and the like do preserve the structure that is required for constructing the Aufbau; however, they are not founded relations.¹⁴ For this suggestion to work, foundedness has to be a logical concept. Otherwise, the construction would not classify as being a purely logical construction. After all, the reason for trying to eliminate Rs by means of an implicit definition was that Rs is an extra-logical primitive that renders the constructional system impure. Replacing one extra-logical primitive (Rs) by another extra-logical primitive (foundedness) would not be an improvement. What needs to be done is to devise an implicit definition of Rs that only employs logical notions. For this reason, Carnap has to insist that foundedness is a purely logical primitive. It is this commitment that usually has been singled out as being the point where Carnap’s project founders. “The idea that [the implicit definition of Rs] is a purely logical formula is absurd” (Demopoulos and Friedman , p. ). However, Carnap’s construction fails even if one grants that foundedness is a purely logical primitive. The problem is that the basic relation Rs is an asymmetric relation (cf. Carnap , §). As such, it has a distinct converse Rs
1 . Foundedness cannot differentiate between a relation and its converse. It is implausible to think that one of them is privileged and more basic, say that ‘worse than’ is more fundamental than ‘better than’, or that ‘taller than’ is more fundamental than ‘shorter than’. This means that Rs is founded iff Rs
1 is also founded. Foundedness, naturalness and the like do not differentiate between a relation and its converse—either both are founded/natural or both are gerrymandered.¹⁵ Foundedness thus does not rule out all permutations. It only allows us to exclude various gerrymandered permutations that do not preserve foundedness. Yet the permutation that maps a relation into its converse is precisely one that does preserve foundedness. By simply permuting the relata of Rs, we ensure that Rsxy iff Rs
1 f(x)f(y), which is equivalent to Rs
1 yx. The resulting two structures are isomorphic and are equally founded. Even when the structure is augmented by means of foundedness, we are still left with a failure of uniqueness due to there being non-trivial structure-preserving permutations that are also foundedness-preserving. There is no unique founded relation that allows us to construct the Aufbau. As a result, uniqueness fails and it is not possible to implicitly define Rs even when appealing to foundedness.

¹⁴ This corresponds to the suggestion that Newman considers, namely that certain relations are ‘important’, and which he rejects as being absurd (cf. Newman , p. ). It also corresponds to Lewis’s proposal to appeal to ‘naturalness’ (cf. Lewis ). ¹⁵ Whilst many non-symmetric relations seem to be equally natural as their converses, there are some cases in which one can plausibly privilege a relation over its converse. For instance, when R is active and its converse R
1 is passive, it is plausible to consider R to be prior to R
1 . This difference, however, is not to be explained at the level of relations. These cases are best construed not in terms of one relation being more natural than or prior to another, but instead in terms of there being a generative operation that has an input-output structure (cf. Section 1.6). The input of such an operation is prior to its output and actively generates it, whereas the converse operation is not generative. It is for this reason that the active is privileged over the passive.

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  - 



.. Fundamental Theorising The lesson to be learnt from these difficulties for the Aufbau is that fundamental theorising is to invoke only symmetric and no non-symmetric relations (given that, if there were to be fundamental non-symmetric relations, they would have distinct converses or at least distinct duals). .

  

Pinning down reference and ruling out unintended interpretations is a central role that the notion of fundamentality is meant to fulfil, not only in the Aufbau, but also in the context of the Newman problem and Putnam’s model-theoretic argument. It can only fulfil this role if there are no fundamental non-symmetric relations that have distinct converses. Since fundamentality/naturalness cannot differentiate between a relation and its converse, it cannot achieve uniqueness when dealing with non-symmetric relations. If one is fundamental/natural, then so is the other. This means that the notion of fundamentality only succeeds in fulfilling one of the important roles that it is meant to fulfil if the fundamental level does not contain any non-symmetric relations that have distinct converses. Otherwise, the notion of fundamentality, though ruling out various gerrymandered candidates, will not enable us to achieve uniqueness. How problematic one considers the failure of uniqueness to be depends on whether one considers implicit definitions to lack a denotation or to have an indeterminate denotation in cases in which there are multiple candidates.¹⁶ Lewis, for instance, held each of these views at some point and ultimately settled on an intermediate position: “In cases where there is a unique x such that T[x], Lewis says that t denotes that x. What if there are many such x? Lewis’s official view in the early papers is that in such a case t does not have a denotation. In ‘Reduction of Mind’, Lewis retracted this, and said that in such a case t is indeterminate between the many values. In ‘Naming the Colours’ he partially retracts the retraction, and says that t is indeterminate if the different values of x are sufficiently similar, and lacks a denotation otherwise” (Weatherson , section .). Whilst outright reference failure is particularly troubling, indeterminate reference is likewise not all that palatable. Either way, complete determinacy can only be achieved by denying that there are relations that have distinct converses at the fundamental level. .  If a fundamental relation R has a distinct converse R
1 , then the fundamental level will exhibit a radical form of redundancy. There will be two fundamental relations, but it will not be necessary to appeal to both of them. Since Rs and Rs
1 have the very same structural features, i.e. they satisfy the same Ramsey sentences, it follows that whatever can be constructed by means of the one can also be constructed by means of the other. As a result, one of them will suffice and the other will be redundant. ¹⁶ Although Carnap insists on uniqueness (cf. Carnap , §§–), Friedman and Demopoulos suggest that he could have put forward an existential claim and not incurred any commitments to uniqueness, cf. Demopoulos and Friedman , p. .

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  .  Fundamental relations should not be dispensable in this way. One should not be able to do without them when giving a fundamental characterisation of the world. Something has gone wrong if Rs
1 is deemed to be fundamental, yet the entire world can be constructed without recourse to this relation. Part of what it is to be a fundamental relation, one might think, is to play an ineliminable role in fundamental theorising. Fundamental theorising should satisfy a non-redundancy constraint and, accordingly, has to operate only with symmetric rather than nonsymmetric relations. This means that fundamental theorising (of which Aufbau-style theorising is a paradigmatic instance) has to proceed by means of symmetric relations. Nonsymmetric relations have to be excluded from the fundamental level and instead have to be banished to derivative levels.¹⁷

. Derivative Non-symmetric Relations Fundamental theorising is to eschew non-symmetric relations. Restricting the fundamental level to symmetric relations makes it difficult to explain where order comes from. Symmetric relations seem to contain too little structure, or at any rate the wrong kind of structure, to account for all the order and asymmetry that is to be found in the world. This section explains how asymmetry and order can be introduced into a world that only contains symmetric but no non-symmetric fundamental relations.

.. Asymmetric Structures The relata of a symmetric relation are not ordered. The relation goes in both directions: xRy and yRx. As a result, permuting the relata does not result in a different state of affairs. Nevertheless, symmetry can fail when operating only with symmetric relations. Even though one symmetric relation by itself does not order its relata, a network of such relations can do so. The pattern of instantiation of symmetric relations can give rise to an asymmetric structure. If there is some permutation of the elements of the domain that does not preserve the structure, then there is some asymmetry and order. The three element structure xRy and yRz, though symmetric in x and z, fails to be symmetric in y, i.e. any permutation that non-trivially involves y will not be structure-preserving. This means that the ternary relation R0 ¼ λxyz½xRy∧yRz is not fully symmetric, even though the relation R out of which it is constructed is symmetric. If the structure does not allow for any non-trivial automorphisms, then it is possible to provide unique structural identifications of all elements in the structure. This can be achieved by combining different symmetric relations. For instance, the structure xRy and yR*z allows us to distinguish the different elements. Each element can be uniquely identified: x is the unique element that is R-related without

¹⁷ This means, for example, that one cannot posit a fundamental temporal arrow, but instead either has to explain the direction of time in terms of a generative operation, or consider it to be a derivative feature of the world that is grounded in the global distribution of property instantiations.

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  - 



being R*-related, whereas z is the unique element that is R*-related without being R-related, and y is the unique element that is both R and R*-related. This means that the polyadic property connecting x, y and z, namely the ternary relation R00 ¼ λxyz½xRy∧yR* z is non-symmetric (if R and R* are incompatible, i.e. if xRy implies :(xR*y) then it will be asymmetric). This relation is internal to the structure, i.e. it is intrinsic to the triple. By derelativising the property, we can construct a non-symmetric binary property that is extrinsic to the pair, namely R000 ¼ λxz½9yðxRy∧yR* zÞ. This is a non-symmetric binary relation connecting x and z that is grounded in the symmetric relations in which x and z stand in to some y. The relative product of two symmetric relations can thus be nonsymmetric.¹⁸ Symmetry can fail globally even when working with a single symmetric relation, as long as the structure is sufficiently complex. In particular, the structure has to form an asymmetric graph. In that case, one can uniquely identify each point in the structure in terms of the place that it occupies within the network of symmetric relations. For instance, an asymmetric connected graph that is based on only one binary symmetric relation will have to have at least six nodes (cf. Dipert 1997, p. 347).¹⁹

.. Internal Relations Derivative relations can fail to be symmetric if they are grounded in properties that are asymmetrically distributed amongst their relata. In this case, we are not concerned with the positions that the relata occupy in an asymmetric structure, but with the properties that the relata instantiate. The properties in which the relation is grounded can be intrinsic as well as extrinsic. Moreover, they can be monadic as well as polyadic, depending on whether the relata of the relation in question are individual objects or pluralities of objects. For instance, the asymmetric ‘taller than’ relation is grounded in the heights of its relata. What makes it the case that x is taller than y is nothing but the heights of x and y. The monadic properties instantiated by the relata, Fx and Gy, collectively ground xRy. Similarly, the ‘further apart than’ relation is grounded in the distances of its relata. What makes it the case that x and y are further apart from each other than are v and w is nothing but the distances between x and y as well as between v and w. The polyadic properties instantiated by the relata, namely xRy and vR0 w, collectively ground xyR*vw. These derivative relations, unlike those grounded in asymmetric structures, are internal relations in the sense that they can hold across worlds. That x is F whereas y is G grounds x’s being taller than y, independently of whether x and y are to be found

¹⁸ We can generate a binary non-symmetric relation that is intrinsic to the pair from a structure consisting of only two elements that stand in xRy and xR*x, i.e. R0000 ¼ λxy½xRy∧xR*x (though it is not clear whether the reflexive relation R* is nothing but a monadic distinguishing feature had by x, in which case the failure of symmetry would not be due to how the relata are connected to each other, but due to the properties that they have). ¹⁹ Much of the Aufbau precisely consists in construing the world as such an asymmetric graph, i.e. a graph in which there are no homotopic points (cf. Carnap , §). Although Carnap’s fundamental relation Rs is asymmetric, most of the constructions proceed via the symmetric closure of Rs, i.e. Rs \ Rs
1 (cf. Carnap 1928, §104).

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  .  in the same world or not. Asymmetric structures, by contrast, are constructed out of external relations that can only hold amongst world-mates.²⁰ This understanding of internal relations is sometimes questioned on the basis that the properties by themselves do not imply any sense of order. All that we get is a difference between the objects, yet the notion of a difference is symmetric. Russell, for instance, claims that “the mere fact that the adjectives are different will yield only a symmetrical relation” (Russell , §). The asymmetric relation xRy, it is argued, can only be grounded in Fx and Gy because there exists an asymmetric higher-order relation R* such that FR*G (cf. Bigelow and Pargetter , pp. –; MacBride , section ). For instance, it is only because the heights F and G stand in an asymmetric second-order relation that it is possible for x and y to stand in the first-order ‘taller than’ relation in virtue of having these heights, i.e. being F and G respectively.²¹ The properties instantiated by the relata, on this view, do not suffice but need to be supplemented with an asymmetric relation holding amongst those properties. This would mean that instead of explaining how asymmetry emerges, we would in fact be presupposing asymmetry. This suggestion, however, is incorrect. The relation xRy ¼ λxy½Fx∧Gy is nonsymmetric and allows for differential application (if F and G are incompatible, then R is asymmetric). Although the difference between F and G is symmetric in that these two properties differ from each other, the way in which these properties are distributed amongst x and y is asymmetric. Put differently, even though we only have a symmetric difference at the level of the properties, at the level of the property instantiations we have an asymmetric distribution. When we are operating with numerical representations of, say, heights, then we are in fact presupposing an ordering over the real numbers by means of which we represent the members of the domain. This, however, is at the level of measurement theory, not at the level of metaphysics. The qualitative ‘taller than’ relation (which we are representing by means of the numerical ‘greater than’ relation) can be grounded in the intrinsic properties of the relata. This relation can be represented by means of a relation amongst numbers but is not to be identified with (nor is it grounded in) such a relation. The intrinsic properties of the relata, namely their heights, do not presuppose any ordering. As long as they are distributed asymmetrically, the resulting derivative relation will fail to be symmetric. This is particularly clear when considering relations that are internal not to the relata but to the pair. Such relations are grounded not only in properties of the relata but also in external relations connecting the relata, where the failure of symmetry is due to the way in which the properties are asymmetrically distributed amongst the relata.²² If a symmetric relation R* is combined with a distinguishing intrinsic feature ²⁰ This construal of internal relations is a generalisation of the way in which this notion is usually understood, namely in terms of a relation that supervenes on the intrinsic properties of the relata. For our purposes there is no reason to restrict the grounds to intrinsic properties. Extrinsic betterness, for instance, can be treated as an internal relation in the same way as intrinsic betterness. That x is extrinsically better than y is grounded in the extrinsic values of x and y, i.e. it is grounded not only in the intrinsic but the extrinsic properties of the relata. This relation can unproblematically hold across worlds. ²¹ For a higher-order theory of quantities cf. Mundy . ²² Accordingly, they cannot hold across possible worlds.

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  - 



F that can be had by only one of the relata, then one ends up with a derivative non-symmetric relation R. The relation xRy ¼ λxy½Fx∧R*xy, where R* is a symmetric relation, is a non-symmetric relation that is grounded in R*xy together with Fx. For example, the relation that holds between x and y in case x is a brother of y is a non-symmetric relation that is grounded in the symmetric sibling relation and the intrinsic property of being male, i.e. x stands in the ‘brother of relation’ to y iff x and y are siblings and x is male. This case clearly shows that one is not presupposing any asymmetric relations or orderings in the background. There is no need for any higher-order relation amongst the properties involved in grounding this non-symmetric relation. Instead, the failure of symmetry derives from the distinguishing feature F had by x. This property does not have to be related in any distinctive way to anything else; in particular it does not have to stand in any non-symmetric higher-order relations. All that is required is that it is not a trivial property relative to R*, i.e. it has to be such that not all relata of R* are automatically F. If it is such that only some of the relata of R* are F, then those will be asymmetrically related via R. In short, what is at issue is not any non-symmetric relation amongst properties but a non-symmetric distribution of property instantiations amongst the relata.²³

.. Absolutism v. Comparativism This account of how order and asymmetry are introduced into a world that only contains symmetric relations at the fundamental level has important implications for the debate between absolutism and comparativism. This debate concerns the question as to whether comparative or absolute notions are more fundamental when it comes to providing an account of quantities/magnitudes. For instance, is goodness prior to betterness, or is betterness prior to goodness? Which of these is the more fundamental notion? Whereas comparativists take betterness to be basic and analyse goodness in terms of it (cf. Broome ), absolutists take goodness to be the basic notion and consider betterness to be analysable.

²³ To give a fully satisfactory account of derivative relations, one not only has to explain how they can fail to be symmetric but also give an explanation of their structure. It might be thought that, even though non-symmetric higher-order relations amongst properties are not in general required in order to ground non-symmetric relations, they are nevertheless required in order to adequately account for the structure of these relations, for instance when it comes to explaining the transitivity of the taller than relation. How can the monadic non-relational properties of x, y and z ensure that if x is taller than y and y is taller than z that x is also taller than z? One way of explaining this is to operate at the level of the grounds of the relevant properties. In the case of extensive magnitudes (where these are construed in the traditional sense, i.e. as magnitudes that have a spatial or temporal extension) this can be done in terms of the parthood structure of the relevant interobject grounds. If x is F, y is G and z is H, then this is because the xx’s that compose x satisfy condition Γ, the yy’s that compose y satisfy condition Δ and the zz’s that compose z satisfy condition Λ. Now if the xx’s can be partitioned into two sub-pluralities such that Γ(xx) consists in Γ1 ðx1 x1 Þ and Γ2 ðx2 x2 Þ, whereby the condition satisfied by the x2 x2 ’s is the same as that satisfied by the yy’s, i.e. Γ2 ¼ Δ, and if in addition, the yy’s can be partitioned into two sub-pluralities such that ΔðyyÞ consists in Δ1 ðy1 y1 Þ and Δ2 ðy2 y2 Þ, whereby the condition satisfied by y2 y2 ’s is the same as that satisfied by the zz’s, i.e. Δ2 ¼ Λ, then we can explain why xRy (which is grounded in Fx and Gy) and yRz (which is grounded in Gy and Hz) implies xRz (which is grounded in Fx and Hz).

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  .  An important problem for comparativism is that the relevant comparative notion is asymmetric. As a result, it has a distinct converse. If betterness is not grounded in goodness, but is instead taken to be prior to goodness, does this mean that worseness is likewise prior to goodness? Symmetry-reasoning would suggest so. All the arguments that speak in favour of betterness being prior to goodness also speak in favour of worseness being prior in this way. Privileging one of them would be arbitrary. This, however, means that there are two primitives: betterness and worseness. The resulting theory will be far from parsimonious and will give rise to an excessive proliferation of states of affairs. Its fundamental base, moreover, will be characterised by redundancy, since there will be two relations that can perform the very same work. In addition, the comparativist would seem to incur brute necessary connections in order to ensure that betterness and worseness are co-ordinated in the right way. The absolutist, by contrast, does not face these problems. Goodness is prior.²⁴ Both betterness and worseness are grounded in the properties of their relata. Given that they have the very same grounds, one can explain the identity of these two relations and can hence avoid ontological profligacy as well as redundancy. In short, the absolutist can simply appeal to the internal relations framework sketched above. Not only do comparativists fail to give a satisfactory account, due to ignoring worseness and instead focusing exclusively on betterness, they are also unable to adequately deal with the ‘equally good as’ relation. Is this relation likewise prior to goodness? For the purposes of measurement theory, one usually operates with a single primitive, namely the weak betterness relation . This allows one to define all the ordering relations, e.g. equally good can be defined in terms of weak betterness (i.e. x ¼ y iff x  y∧y  x). It is, however, implausible to consider  to be a fundamental notion. This notion is used in measurement theory because it is the most convenient notion to work with. It is for this reason that it is used as a primitive in formal systems. Whilst convenient, it does not correspond to any metaphysical primitive. Instead, it is a disjunctive notion. We do not have any independent grasp on weak betterness except as the disjunction of strict betterness and equal goodness. (This is analogous to the situation in mereology where the notion of proper or improper parthood is used when constructing a formal mereology, yet where this notion is not understood to be metaphysically fundamental but to be a disjunction of identity and proper parthood.) The comparativist thus has to treat the ‘equally good as’ relation as yet another primitive alongside the ‘better than’ relation and its converse the ‘worse than’ relation. Given completeness, one can define equal goodness in terms of strict betterness/ worseness, i.e. x ¼ y iff :ðx>yÞ∧:ðy>xÞ. Or alternatively, in terms of x ¼ y iff 8z½ðz>x $ z>yÞ∧ðx>z $ y>zÞ. Completeness, however, cannot simply be assumed in this way. On the one hand, there are plausible cases of incompleteness. On the ²⁴ One might worry that in the same way that focusing on betterness leads one to ignore worseness, focusing on goodness likewise leads to a problematic neglect of badness. However, goodness is understood in terms of the entire value field, not just its positive segment, i.e. goodness refers to absolutist value properties whether they be positive, neutral, or negative.

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  - 



other, it should be a substantive question whether the axiological ordering is complete, not one that is settled by stipulation. Difficulties arise, in addition, for the comparativist once we note that completeness is domain-relative. When we say that a relation induces a complete ordering, this is always an ordering of a given domain. The question now, however, is how one is to specify the domain in question. This domain has to be restricted to objects that are value-bearers. Otherwise, completeness will be guaranteed to fail since objects that are not value-apt will end up being classified as being equally good as each other (given that they will vacuously satisfy the condition) and as non-comparable to all those things that are value-apt. Restricting the domain to value-bearers, however, is problematic for the comparativist since the most natural characterisation of an object as being a value-bearer is an absolutist characterisation.²⁵ Measurement theory takes a qualitative relational structure as basic and then establishes a homomorphism into a suitable numerical structure. Comparative notions, however, are metaphysically derivative. Monadic non-relational properties are metaphysically prior and give rise to asymmetric comparative relations. (When dealing with distance functions and the like, the relata of the comparative ordering are not individual objects but n-tuples. In that case it is absolutist symmetric relations connecting the members of n-tuples that are prior and that give rise to asymmetric comparative relations amongst the n-tuples. E.g. that the distance between x and y is greater than that between v and w is grounded in the absolutist distances, i.e. xRy and vR0 w ground xyR*vw, rather than R* being a fundamental comparative relation that connects the pairs.) Although comparativism is very natural from a measurement-theoretic point of view, we should not take measurement theory to be a guide to metaphysics. Metaphysical commitments should instead be decided on the basis of metaphysical considerations. These considerations strongly favour an absolutist approach. Whilst the comparativist runs into serious difficulties due to operating with nonsymmetric comparative relations, the absolutist is working within the framework of internal relations and does not have any of these difficulties. There is no redundancy in the base, no indeterminacy of reference, no brute necessities and no overabundance of states of affairs. Accordingly, we should consider goodness to be prior to betterness.

. Identity Criteria for Derivative Relations Order and asymmetry can arise at the derivative level, even when all fundamental relations are symmetric. Derivative non-symmetric relations, however, would seem to have distinct converses, or at least distinct duals. As a result, the original problems

²⁵ If one were to treat ‘equally good as’ as a primitive, then one could characterise value-bearers as things that stand in the ‘equally good as’ relation to themselves. (However, even this is questionable since one might well construe reflexive instantiations of ‘equally good as’ in absolutist terms, i.e. as simply amounting to the instantiation of a monadic non-relational property, since the reflexive restriction of this relation is not an external relation and hence not in the requisite sense a comparative primitive.)

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  .  of indeterminacy, brute necessities and the proliferation of states of affairs still seem to loom large and it is only the problems of uniqueness and non-redundancy that are avoided, given that they apply only to fundamental relations. We can address these problems by showing that derivative relations neither have distinct converses nor distinct duals. Instead of there being distinct relations, we simply have different ways of picking out one and the same relation. In order to do this, we have to provide identity criteria for derivative relations. These criteria have to establish that whenever there is a derivative relation R, there is no distinct relation R* that is intimately connected to R in the way that a converse or dual is connected to R. By establishing that it is not possible for distinct derivative relations to coincide in this way, we can address all the original problems. Derivative properties and relations (i.e. both monadic and polyadic properties) can be individuated in terms of a criterion of hyperintensional equivalence.²⁶ What it is to be a given property is to be grounded in certain ways in certain things. This is what makes a derivative property the particular property that it is. Accordingly, sameness of grounds implies sameness of property. Derivative properties, whether monadic or polyadic, are identical iff they have the same grounds, i.e. they are grounded in the same ways in the same things across all of modal space. If we understand the identity criteria for derivative relations in terms of hyperintensional equivalence, then there can neither be distinct converses nor distinct duals. In order for two relations to be connected in the relevant way as to classify as converses/duals, they would have to derive from the same base. Yet, in that case the criterion of hyperintensional equivalence classifies them as being identical. For instance, there is only one relation that holds between x and y on the grounds that x is F and y is G, namely xRy ¼ λxy½Fx∧Gy ¼ yR
1 x ¼ λyx½Gy∧Fx. Both are grounded in x being F and y being G. Whatever grounds xRy also grounds yR
1 x and vice versa. Fx plays the very same grounding-role in xRy as it does in yR
1 x, and likewise for Gy. A derivative relation and its converses thus have the very same grounds. They are grounded in the same ways in the same things. Accordingly, there is only one relation that is picked out in different ways. Given that there is only one relation, there is no proliferation of facts. For example, what it is for x to be taller than y is the very same thing as what it is for y to be shorter than x. There is no difference in terms of which aspects of the world make it the case that the fact [x is taller than y] obtains and those that make it the case that [y is shorter than x] obtains. These facts involve the same objects standing in the same relation and are grounded in the same distribution of underlying monadic non-relational properties amongst x and y. They are hence identical. There is only one fact about the relative heights of x and y that can be represented in two ways.

²⁶ Cf. “Hyperintensional equivalence” (Bader: manuscript) for the resulting hyperintensional logic.

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  - 



These hyperintensional identity criteria allow us to establish that there is only one relation in any given case.²⁷, ²⁸ Instead of there being two intimately connected binary relations, there are two different ways of picking out one and the same relation. The distinction between a relation and its converse, accordingly, is a merely nominal rather than a real distinction. Relations and their converses are distinguished only in language and thought but not in the world.²⁹

.. Individuating Fundamental Relations By individuating relations in terms of their possible grounds, one can establish that derivative non-symmetric relations do no not have distinct converses or duals. Since they have the same grounds, they are identical. This strategy for avoiding an excessive proliferation of derivative relations cannot be applied to fundamental relations. Since they do not have any grounds, they cannot be individuated in terms of their grounds. In order to avoid a proliferation of relations at the fundamental level, one has to deny the existence of fundamental non-symmetric relations. Unless one is willing to endorse a sparse theory that privileges some relations in a seemingly arbitrary manner, one has to reject non-symmetric relations at the fundamental level. Since symmetric relations cannot have distinct converses/duals, there will then not be any distinct converses or duals at the fundamental level. Moreover, once non-symmetric relations are ruled out at the fundamental level, one can reject the differential applicability of fundamental relations. This allows one to individuate fundamental relations in terms of their unstructured extensions across modal space. A structured conception of extensions, for instance in terms of ordered pairs or assignments to argument places, is only required when relations allow for differential applicability. If there are no non-symmetric fundamental relations, then the extension of a fundamental relation can be understood in terms of unordered n-tuples. In that case fundamental relations cannot be had in different ways. Differential applicability does not apply to them. It is only when dealing with derivative relations that we need to consider differential applicability and bring in a structured construal of extensions that takes into consideration not only which objects instantiate the relation but also which grounding-roles are played by which of the relata. Fundamental relations R and R* are thus identical iff they are necessarily co-extensive. Fundamental relations that apply to the very same things cannot differ merely in the way in which they apply to those things and hence are identical. As a result, one can avoid any

²⁷ This does not mean that we end up with a sparse theory of relations, where this is understood relative to an abundant background of possibilities that we can make sense of. It is not the case that we recognise a number of relations and deem them to be intelligible, yet only consider some of them to exist. Instead, we have a theory according to which there is no room for distinct converses/duals of derivative relations. There is hence no need for arbitrarily privileging some relations over others. ²⁸ Whilst ruling out distinct converses and duals, the theory is nevertheless fine-grained and allows for hyperintensional differences, distinguishing relations that hold of the very same things across modal space yet nevertheless differ in terms of their grounds. ²⁹ Cf. Ben-Yami , pp. – for a proposal as to why we have names for both relations and their converses in natural language.

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  .  over-abundance of relations at the fundamental level. There is no proliferation of relations and no need to invoke a sparse theory of relations that draws invidious distinctions.

.. Relational Expressions and Conventions Relations do not have distinct converses. Instead of two relations, there are two ways of picking out one and the same relation. ‘R’ and ‘R
1 ’ co-refer. This, however, conflicts with the original argument for distinctness: . . .

aRb :ðbRaÞ bR
1 a

∴

R 6¼ R
1

Given that ‘R’ and ‘R
1 ’ are not interchangeable, it would seem that these relational expressions do not refer to the same relation after all. This conflict can be resolved by appealing to Williamson’s approach to relational expressions and the role that conventions play in establishing reference to relations (as well as MacBride’s related proposal in terms of impure referring terms). The reason why the relational expressions ‘R’ and ‘R
1 ’ are not interchangeable is due to the fact that they only achieve reference in combination with a convention: “one must know not just which relation [the relational expression] stands for, but which way round its flanking terms are to be fed into that relation” (Williamson 1985, p. 249; also cf. p. 257). Although these two expressions refer to the very same relation, they do so via different conventions. Accordingly, one needs to adjust for differing conventions when substituting co-referring relational expressions. Since the convention associated with the converse relational expression is the converse of the convention of the relational expression, substitution requires permuting the flanking terms of the expression. When substituting ‘R
1 ’ for ‘R’, one has to reverse the order of the flanking terms. This defuses the original argument for distinctness and avoids referential indeterminacy. Although there are two relational expressions, namely ‘R’ and ‘R
1 ’, there is only one relation. Both expressions refer to the very same relation, where this is possible on the grounds that they achieve reference in combination with different conventions. The role of the convention is to indicate which flanking term plays which role in grounding the derivative relation.³⁰ For instance, in the case of the relation that holds between x and y on the grounds that Fx and Gy these relational expressions differ in that the relatum that plays the F-role is mentioned first in the case of ‘R’ but second in the case of ‘R
1 ’, whereas the G-role is played by the second flanking term in the case of ‘R’ yet by the first in the case of ‘R
1 ’.³¹ ³⁰ For the positionalist the role of conventions is to assign the flanking terms to argument places. ³¹ When it comes to fundamental relations one cannot specify conventions in terms of the groundingrole that the various flanking terms of a relational expression play. This is unproblematic in the case of fundamental symmetric relations. However, it means that those who believe in fundamental

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  - 



.. Reflecting Differences v. Inducing Order A relation that allows for differential application can hold in different ways: e.g. xRy as well as yRx. If there were to be fundamental non-symmetric relations, then it would be a primitive fact that such a relation holds of the relata in a given way. There is nothing about the relata that makes it hold one way rather than the other. The relata by themselves do not fix the way in which the relation holds, the manner in which R applies. They do not order themselves. The relation is an external relation. It is added to the relata and thereby induces order. Order does not derive from the relata but is imposed on them. If there were to be fundamental non-symmetric relations, then they would impose order externally. It would then be possible to add, not just one such relation to a given set of relata, but several such relations that merely differ in terms of their directions. If x and y do not order themselves but have their order imposed upon them, then inducing order on them by means of R, such that R goes from x to y, is perfectly compatible with there being another fundamental relation, namely the converse relation R
1 , that goes from y to x.³² Since these relations order their relata differently, they are distinct despite holding of the very same things. Moreover, since they are distinct relations, there is no conflict in them ordering the same relata in opposite ways. That way one would end up with distinct fundamental relations that merely differ in terms of how they order the same relata. By contrast, this cannot happen if order is not externally imposed by a relation that is superadded to the relata, but instead derives from the relata. In that case, one cannot add different relations that merely differ in direction. Rather than inducing order, the derivative relation only reflects differences amongst the relata that are antecedently given. The relation is then internal. In such cases we do not need to induce order. There is no need to add a fundamental relation that applies one way rather than another. The relata (or the system of fundamental symmetric relations) already take care of this. Instead of order being externally imposed, the relation is fixed by the differences amongst the relata. The derivative relation R derives from x and y instantiating certain properties and/ or standing in certain relations, such that any relation that derives from the same basis will be identical to R. Put differently, R reflects certain underlying differences,

non-symmetric relations will not be able to specify what conventions are operative in our language. For instance, the positionalist would have to be able to identify and differentiate the various argument places of fundamental relations in order to specify how conventions operate. This does not seem to be possible. The problem is particularly pressing when working with a sparse theory of relations of the kind proposed by MacBride. In that case, conventions specify which flanking term refers to the relatum from which the relation proceeds and which one refers to the one to which it proceeds. Yet, which of the two possible relations, namely R or its converse R
1 , exists would seem to be an inaccessible fact about the sparse realm of relations. Accordingly, we cannot identify which conventions are operative, i.e. whether the first flanking term refers to the relatum that plays the ‘from-role’ or to the one that plays the ‘to-role’. This means that if there were to be fundamental non-symmetric relations, then the conventions that would be operative in the case of these relations would not be transparent to us. ³² Or if there is a relation R with assignment x to α and y to β, then this is likewise compatible with there being another fundamental relation, namely a dual relation R*, with assignment x to γ and y to δ (or for that matter any number of dual relations with distinct argument places).

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  .  such that any relation that reflects the same differences will be identical to R. For each way of grounding order amongst the relata, there is only one relation that is grounded in this way, where this relation can be picked out in different ways, namely by means of different relational expressions. As a result, one does not end up with a proliferation of relations. Differential application is then not to be explained in terms of the possibility of inducing order in different ways. Instead, differential application is to be explained in terms of there being different ways in which the underlying properties and relations can be distributed. By changing which relata play which roles in grounding the derivative relation, one generates different applications of the same derivative relation. One and the same binary relation can be instantiated in different ways by the same objects, i.e. aRb and bRa, since there are different possibilities as to which object plays which grounding role. For instance, whereas Fa and Gb ground that a is taller than b, Fb and Ga ground that b is taller than a. We can explain differential application without having to invoke the idea that relations have directions or argument places. Nor do we need to consider differential applicability to be an unexplained primitive (as is done by the antipositionalist). Instead, we can explain it in terms of there being different grounding roles that can be played by the relata of derivative non-symmetric relations. On this approach, relations can be understood as being completely unstructured. Differential application is not explained in terms of the internal structure of nonsymmetric relations. In particular, it is not explained in terms of the idea that relations have directions, argument places or the like, which means that there is no need for a reification of directions or argument places. Unlike anti-positionalism, which also rejects a conception of relations according to which they have internal structure, the grounding account does not locate the complexity in the external connections amongst different instantiations of a relation. Instead, it locates the complexity that accounts for differential applicability in the grounds of the relation, thereby ensuring that the way in which a relation applies to its relata is an internal rather than external matter. That which makes it the case that the relation is instantiated explains the particular manner in which the relation is instantiated. The structure is not in the relation but in its grounds. Since fundamental relations are ungrounded, the extension of such relations is to be understood in terms of a set of unordered n-tuples. (As we have seen, this does not leave any room for distinct converses or duals, given that the fundamental level is restricted to symmetric relations.) The extension of a derivative relation, by contrast, needs to be understood in terms of assignments to grounding roles: {⟨a,F⟩, ⟨b,G⟩}. Given the possibility of differential application, it is not enough to be told which objects are related by the relation in question. In addition, we need to be told in which way the objects are related. This amounts to being told which object plays which grounding role. Accordingly, the extension is a set of assignments to grounding roles. This ensures that relations and their converses are necessarily co-extensive (and, in fact, even hyperintensionally equivalent) since they involve the very same objects fulfilling the same grounding roles.

.. Symmetric Instantiations The grounding account construes extensions in terms of assignments to grounding roles. This is somewhat analogous to positionalism, which operates with assignments

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

to argument places {⟨a,α⟩, ⟨b,β⟩}. There is, however, an important difference when it comes to symmetric relations. A major problem for positionalism is that {⟨a,α⟩, ⟨b,β⟩} will be distinct from {⟨b,α⟩, ⟨a,β⟩} even when R is symmetric (cf. Fine , pp. –). A binary symmetric relation will have two distinct argument places and hence will allow for two distinct completions. The problem for positionalist is, in fact, more general. Difficulties arise not only in the case of symmetric relations. Positionalism is also unable to provide a satisfactory account of symmetric instantiations of relations that are neither symmetric nor asymmetric. Such non-symmetric relations allow for two different kinds of symmetric instantiations. On the one hand, a symmetric instantiation of a binary relation can consist in a single two-way instantiation. For example, the weak betterness relation  which is the disjunction of the symmetric ‘equally good as’ relation and the asymmetric ‘strictly better than’ relation can hold symmetrically of objects that are equally good. In this case there is only one state of affairs. When a and b are equally good, the fact that a is weakly better than b and the fact that b is weakly better than a are the very same fact, i.e. [aRb] is identical to [bRa]. Although there are two ways in which [aRb] can obtain (i.e. two ways in which R can be instantiated by a and b in that order) and two ways in which [bRa] can obtain, there is only one way in which both [aRb] and [bRa] can co-obtain, namely the one that is shared by both of them. Even though it is possible to have [aRb] without [bRa] (and vice versa), they are identical when both of them obtain together. Such a two-way instantiation involves the same objects instantiating the same relation on the basis of one and the same ground. There is hence only one fact to the effect that a and b are symmetrically R-related.³³ (This means that facts are to be individuated in terms of their constituents and their grounds, not in terms of the possible ways in which they can be grounded or the different ways in which they can obtain. Facts can be identical even though they admit of different possible grounds, as long as they have the same constituents and the same actual grounds.³⁴ Object-individuation and property-individuation are, accordingly prior. Facts are constructed entities that are individuated in terms of their constituents and grounds.) On the other hand, a symmetric instantiation of a binary relation can consist in two one-way instantiations. For example, the ‘brother of ’ relation is grounded in the symmetric sibling relation together with the distinguishing feature of being male that can but need not be had by only one of the relata. If the distinguishing feature is had by both relata, then this relation holds in both directions. The fact that a stands in the brother of relation to b and the fact that b stands in the brother

³³ Accordingly, it is not only possible for there to be different instantiations of the same ‘fact’ insofar as one and the same property can be instantiated multiple times over by the same object on the basis of different grounds, but also for different ‘facts’ to share one and the same instantiation. ³⁴ Both sameness of constituents and sameness of grounds is required for identity of facts. On the one hand, [(F ∨ G)x] in virtue of [Fx] is distinct from [(F ∨ G)x] in virtue of [Gx], even though they involve the same constituents, due to obtaining in virtue of different grounds. On the other, [(F ∨ G)x] and [(F ∨ H)x] are distinct due to involving different constituents, even though they can obtain in virtue of the very same ground [Fx].

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  .  of relation to a are two different facts. Although they involve the same objects and the same relation, [aRb] and [bRa] have different grounds and hence are distinct. Even though both are partially grounded in a and b instantiating the symmetric sibling relation R*, [aRb] is partially grounded in Fa whereas [bRa] is partially grounded in Fb. The positionalist is unable to account for single two-way instantiations. Since R has two argument places and holds symmetrically of a and b, there will be two completions corresponding to the two assignments of a and b to the two argument places α and β. Accordingly, it treats all symmetric instantiations as two one-way instantiations.³⁵ This problem is particularly clear when there is a change from a single-two way instantiation, e.g. from a  b and b  a, to an asymmetric instantiation of R, i.e. to a  b yet :ðb  aÞ. Positionalism is unable to make sense of the way in which a  b differs across the two cases. It construes both situations as involving an assignment of a to α and b to β. The only difference is that a is in addition assigned to β and b to α in the former situation, whilst this is no longer the case in the latter situation. Yet, intuitively these two situations do not share anything in common. In the former case there is a single ground for the two-way instantiation, whereas there is a different ground for the asymmetric one-way instantiation in the latter case. Since there is no ground that persists across the two cases, the way in which a  b is instantiated in these cases is importantly different. Whereas positionalism mistakenly classifies states of affairs as being distinct when they are identical and thus leads to an excessive multiplication of completions, antipositionalism faces the opposite problem. It treats all symmetric instantiations as single two-way instantiations and is unable to adequately account for the possibility of two one-way instantiations. If aRb and bRa yet cRd though :(dRc), then one cannot consider the application of R to c and d to be co-mannered with that of R to a and b, since co-manneredness is an equivalence relation (cf. Fine , p. , n. ). This, however, means that the antipositionalist is unable to account for cases involving two one-way instantiations. All symmetric instantiations will be treated as single two-way instantiations. Relations that can but need not hold in both ways are thus mistakenly treated in the same way as relations that are disjunctions of symmetric and asymmetric relations. Moreover, a change from a situation in which aRb and :(bRa) to one in which both aRb and bRa will involve a change from aRb being co-mannered with cRd to them no longer being co-mannered. However, aRb is unchanged and should remain in the same equivalence class. None of the facts that are involved in grounding aRb have changed: it is still grounded in aR*b together with Fa. All that has changed is that b has changed from :F to F, thereby making it the case that the second partial

³⁵ It has been suggested that the positionalist can address the problem of symmetric relations by claiming that such relations do not involve two argument places but instead only one argument place that is filled by both relata. (Cf. MacBride , pp. – and Fine , p.  for critiques of this response.) The problem of symmetric instantiations that we have identified clearly shows that this approach cannot succeed. A non-symmetric binary relation has to have two distinct argument places, yet can be symmetrically instantiated in such a way as to allow for only one two-way rather than two one-way completions.

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

ground of bRa also obtains. This, however, is not relevant to aRb but only to bRa. Put differently, in the case of relations that allow for two one-way instantiations, the way in which aRb holds should be independent of whether bRa holds as well.³⁶ This requirement cannot be satisfied by the antipositionalist.³⁷ The grounding account has none of these difficulties. It can distinguish between cases in which there is one ground for a symmetric two-way instantiation and cases in which there are two different grounds for two one-way instantiations. The grounding approach neither gives rise to an objectionable multiplication nor to an objectionable collapse of completions. If the grounds of aRb and bRa are the same, then one is dealing with a single two-way instantiation. In that case the relata play the same grounding role: {⟨a,F⟩, ⟨b,F⟩}. Permuting the relata does not generate a distinct completion of this relation. There is only one grounding role that is played by both relata and hence only one assignment of relata to grounding roles. By contrast, if they have different grounds, then there are two one-way instantiations of R by a and b. In that case the relata play multiple grounding roles such that there are two distinct completions of the same relation: {⟨a,F⟩, ⟨b,G⟩} and {⟨a,G⟩, ⟨b,F⟩}.³⁸

.. Cross-Relation Comparisons A further advantage of the grounding account is that it provides a satisfactory account of cross-relation comparisons. It neither accepts such comparisons across the board, nor deems them to always be unintelligible, but instead accepts crossrelation comparisons in a restricted range of cases. At one extreme, one can accept what MacBride  calls the third degree of relatedness and countenance comparisons across arbitrary relations. For any relation R there will be a fact of the matter whether it orders some objects a and b in the same way that a different relation R0 orders c and d, e.g. that the shorter than relation applies to a and b in the same way that the brighter than relation applies to c and d. This approach is based on the problematic idea that there is an absolute ordering of the way in which a relation applies to its relata, allowing for comparability of these absolute positions across arbitrary relations. This approach is implausible. At the other extreme, one can reject all cross-relation comparisons and hold that one can only compare the way in which a relation R applies to a and b with the way in which that very same relation applies to c and d. Positionalism and antipositionalism are committed to this approach. They have to preclude the possibility of cross-relation

³⁶ Two one-way instantiations differ crucially from a single two-way instantiation, where aRb has to change when there is a change from :(bRa) to bRa. E.g. in the case of the weak betterness relation, there is a change from a>b when :(bRa) to a ¼ b when bRa. ³⁷ Similarly, a case in which aRb, bRa, aRc yet :(cRa) will be treated by the anti-positionalist in such a way that aRb and aRc are not co-mannered. This, however, means for instance that the way in which a is a brother of b (who is also a brother of a) will be fundamentally different than the way in which a is a brother of c (who is not a brother but a sister of a). ³⁸ Symmetric relations are those that are always instantiated symmetrically. Since there are two ways in which a relation can be symmetrically instantiated, namely via a single two-way instantiation or via two one-way instantiations, there are important hyperintensional differences amongst symmetric relations that are not recognised when working with ordered n-tuples and the like.

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  .  comparisons in order to ensure that the notion of a converse in the strict sense is not well-defined. This, however, is also troubling. Even though unrestricted comparability is implausible, there are a number of important cross-relation comparisons and implications that any satisfactory theory of relations will have to make room for. First, constructed disjunctive relations such as the weak betterness relation should be comparable to the relations out of which they are constructed. If a and b are ordered by the strict betterness relation in a given way, then they are also ordered in that way by the weak betterness relation. Second, relations that merely differ in terms of their adicity should be comparable. For instance, we have to make sense of the idea that objects are related in the same way by the binary betterness relation as by the ternary betterness relation: a and b are ordered in the same way in the case of >a,b as in the case of >a,b,c. In each case, one has to explain both () the way in which the instantiation of R implies the instantiation of R0 , and () the way in which these two relations order certain objects in the same way. On the one hand, we need to account for the fact that a>b implies a  b. Similarly, that the ternary betterness relation holds: >a,b,c implies that the binary relation holds: >a,b (as well as >b,c and >a,c). On the other hand, we need to make sense of the idea that if a>b, then a and b are ordered by  in the same way as by >. Likewise, if >a,b,c, then a and b are ordered by the ternary betterness relation in the same way as by the binary betterness relation. Both desiderata are satisfied by the grounding account. It explains implication across relations in terms of the notion of hyperintensional implication.³⁹ A (monadic or polyadic) property hyperintensionally implies another if for every ground Γ of the former there is a ground Δ of the latter such that Δ is either identical to Γ or is a partial ground of Γ. The strict betterness relation, accordingly, implies the weak betterness relation, since the possible grounds of the former form a subset of those of the latter: every ground of > is identical to some ground of . Similarly, the ternary betterness relation implies the binary betterness relation, since the grounds of the binary are amongst the grounds of the ternary: the grounds for >a,b,c form a subplurality of the grounds for >a,b. The grounding account can also explain the way in which the different relations order objects in the same way in terms of the objects playing the same grounding role in each case. Even though the two relations > and  are distinct, it is nevertheless the case that a and b play the same role in grounding a>b as in grounding a  b. The binary and ternary betterness relations are, likewise, distinct yet nevertheless allow for a and b to play the same grounding role: >a,b,c and >a,b do not differ in terms of how a and b are implicated but only differ in that the ternary ordering involves an additional object c, where the grounding role played by c does not have any effect on those played by a and b. The grounding account can, accordingly, make sense of both cross-relation comparisons and cross-relation implications amongst relations that involve the same grounding roles (whereby the grounds of the one are either amongst the grounds of the other or are partial grounds thereof), without invoking the problematic idea of an

³⁹ This notion is characterised in detail in “Hyperintensional equivalence” (Bader: manuscript).

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

absolute ordering of the relata that would give rise to comparability across completely unrelated relations.

. Generative Operations Derivative non-symmetric relations can be grounded in a fundamental level that does not contain any non-symmetric relations. The proposed account makes use of grounding both to generate derivative asymmetry and to specify the hyperintensional identity criteria for derivative relations. The problem now is that the notion of grounding is asymmetric. Even if one can account for asymmetry at the derivative level by developing a theory of derivative relations that does not countenance distinct converses or duals, the very relation that connects the fundamental to the derivative and by means of which one can explain how order arises at the derivative level, namely the grounding relation, is itself asymmetric. Since grounding is not part of the derivative but what gets us to the derivative, the asymmetry of grounding itself cannot be merely derivative. As a result, it would appear that failures of symmetry cannot be restricted to the derivative level. More generally, there seem to be a number of other asymmetric relations besides grounding, such as causation and composition, that play an indispensable role in fundamental theorising that would similarly constitute counter-examples to the claim that there are no non-symmetric relations at the fundamental level. Whilst one might be able to adopt a Humean approach to, say, causation and consider it to be a derivative matter that can be accounted for in terms of the internal relations model, non-reductive approaches will have to consider these asymmetric relations to be fundamental. This problem can be addressed by arguing that grounding, causation, composition, and the like are not relations but are instead generative operations.⁴⁰ Whereas relations presuppose the existence of their relata, generative operations generate outputs from inputs and are thus prior to their outputs. Such operations do not merely reflect underlying differences but induce order. The relata of a relation are in an important sense prior to the instantiation of the relation. In the same way that individual objects are prior to the instantiation of monadic properties, relata are likewise prior to the instantiation of polyadic properties. A property instantiation [Fa] results from combining an object a and a property F. Similarly, a relation instantiation [aRb] results from putting together the objects a and b (= the relata) and the relation R. We start out with the relata and then connect them by means of the relation to form a relational complex. Generative operations, by contrast, only presuppose their inputs. The outputs are not presupposed by the operation. They are instead generated by applying the operation to the inputs. We do not start out with both the outputs and the inputs and then connect them by means of a relation. Instead of inducing order by ⁴⁰ This is not to be confused with the idea that a logic of ground should be formulated by means of a sentential operator. For arguments in favour of operationalism in the case of composition cf. Fine . Fine also uses the term ‘generative operation’, though in a somewhat different way and for different purposes.

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  .  connecting up pre-existing objects,⁴¹ generative operations induce order by generating outputs from the independently given inputs. The input is presupposed and has to be independently given.⁴² For example, in the case of grounding, we start out with what is fundamental. That is all that we have to begin with and hence the only thing that can enter as input into the operation. We then generate the derivative out of the fundamental. Since we only get the output from the input by applying the operation, it is not possible to start with the output. The input must come first, whereas the output that is generated comes second. Although operations can have converses that invert the inputs and outputs, it is not the case that the converse of a generative operation is likewise generative. Once the output has been generated, one can ask from what it was generated. We can retrace our steps back to the input. This, however, does not amount to generating the input, i.e. the converse operation is not generative. For instance, if some parts compose a whole then the whole can be decomposed into the parts, yet if composition is generative then the whole owes its existence to the parts that compose it but not vice versa.⁴³ The parts are independently given. They are not generated by the application of the decomposition operation to the whole. The whole, by contrast, only exists because it has been generated via an application of the composition operation to the independently given parts. This ensures that the application of the converse of a generative operation is not independent but parasitic on the generative operation. We first need to generate the output in order to then be able to (nongeneratively) go back to the input via the converse operation. This is what allows us to privilege one operation, namely the generative one, over its converse. The fact that we are dealing with operations rather than with relations is, accordingly, not sufficient for addressing the various problems concerning converses. What is crucial is that the operations in question are generative operations.

.. Causation as a Generative Operation The difference between relations and operations can be illustrated by considering presentism. On a presentist approach, the causal unfolding of the world amounts to a series of successive replacements. Given that only the present exists, cause and effect cannot both co-exist. Instead, the one gets replaced by the other. There is no time when both of them are present, which implies that they cannot co-exist. Since relations (at least external relations) are existence-entailing, one cannot have a relation without its relata: aRb cannot obtain unless both a and b exist. If R is a trans-temporal relation, aRb cannot obtain unless one countenances the existence of objects that are located at different times. This contradicts the presentist’s thesis that only the present exists. The problem is thus that if causation is a trans-temporal

⁴¹ The notion of a ‘pre-existing object’ does not have to be understood temporally but can also be understood in terms of metaphysical priority. ⁴² What is independently given can either be understood in an absolute sense, i.e. independent of any application of the operation, as happens in the case of that which is fundamental, or in a relative sense independent, i.e. relative to a particular application of the operation. ⁴³ Which operation is generative is not settled by adopting an operational approach, i.e. operationalism is neutral between holism, which privileges decomposition, and atomism, which privileges composition.

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

relation that necessitates its relata, then both relata have to exist. Yet at most one of them is present, which means that at most one of them exists, according to the presentist. Since it is never the case that both cause and effect exist, they cannot stand in a trans-temporal relation to each other. Whilst the impossibility of trans-temporal causal relations has traditionally been considered to be a problem for presentism (cf. Bigelow , pp. –), we can resolve this difficulty by adopting an operational approach. Causation, on this approach, is not a trans-temporal relation that connects things existing at different times, but a trans-temporal operation that has inputs and outputs that exist at different times. Unlike a causal relation that presupposes the existence of both relata, an operation takes inputs and gives rise to outputs. Accordingly, it is not necessary for the inputs and outputs to co-exist. There is no need for both the cause and effect to co-exist. Instead, the input can exist at time t and give rise to the output that exists at a subsequent time t0 . Reality, on this approach, is considered to evolve dynamically, whereby one state of the world generates and is replaced by the next. This presentist approach illustrates that causation, construed as a generative operation, can be operative, even when it is not possible for there to be causal relations.⁴⁴

.. Grounding as a Generative Operation A related problem arises in the context of grounding. Here the problem is not that of connecting things across time, but across levels of the fundamentality hierarchy. The problem is that it is impossible to account for the idea that the derivative owes its existence to the fundamental in terms of a grounding relation. Since the existence of the derivative is presupposed in order for a grounding relation connecting the fundamental to the derivative to hold, one cannot make sense of the derivative owing its existence to the fundamental when operating with grounding relations. That y owes its existence to x cannot consist in x and y standing in a grounding relation. The instantiation of the relation by x and y would be dependent on the relata, in particular it would be dependent on the relatum y, given that the relation presupposes its relata. Yet, y’s existence is meant to be explained in terms of and hence be posterior to the holding of this grounding relation.⁴⁵ ⁴⁴ This approach to causation naturally gives rise to a construal of persistence in terms of immanent causation. Rather than adopting the traditional ‘only x and y’ rule that treats persistence as an intrinsic trans-temporal relation connecting up relata existing at different times, one should adopt the ‘only x’ rule, which specifies the intrinsic conditions that x has to satisfy at t in order for it to persist until t0 . Cf. “The fundamental and the brute” (Bader: forthcoming section 3.2). ⁴⁵ A further problem is that any connection between the fundamental and the derivative would seem to violate the purity constraint identified by Sider (cf. Sider , sections .– and ..). If a fundamental fact stands in a grounding relation to a derivative fact, then the question arises whether this grounding fact is a fundamental or a derivative fact. For instance, it would seem that the fact [[Γ] grounds [Δ]] is either derivative or fundamental. Since fundamental facts only contain fundamental constituents (= purity constraint), it cannot be a fundamental fact. After all, it involves a derivative fact, namely [Δ], as one of the relata of the grounding relation, i.e. the relational fact cannot be more fundamental than its relata. It would thus have to be a derivative fact. Yet, in that case one has to find a ground of this fact. This, in turn, would constitute a further grounding fact that would likewise need to be grounded and so on. Superinternalists attempt to ground all these grounding facts in the fundamental fact that one started out with: [Γ] not only grounds [Δ] but also [[Γ] grounds [Δ]], as well as [[Γ] grounds [[Γ] grounds [Δ]]] and so on (cf. Bennett ; deRosset ). This, however, conflicts with our account of hyperintensional

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  .  This problem can be addressed by adopting an operational approach. Instead of presupposing the output, the operation generates the output from the input. All that there is to begin with is the fundamental input, which then generates the derivative output. The application of the operation is, accordingly, prior to the output, allowing fundamental inputs to generate derivative outputs. We can thus make sense of the idea that the derivative owes its existence to the fundamental. On this approach, there are no grounding relations and hence no grounding facts. Grounding is not a relation that connects different items in the world, but something that generates the derivative from the fundamental. The operation itself is neither fundamental nor derivative. It is not part of the fundamentality hierarchy but stands outside it. It is what gets us from the fundamental to the derivative. It does not occupy a position in the grounding order but instead gives rise to this order. It generates the hierarchy and induces its structure. In the same way that the instantiation ‘relation’ cannot be instantiated (for that way lies Bradley’s regress), so the grounding ‘relation’ cannot be grounded. The only things that are apt to be grounds or be grounded are instantiations of monadic as well as polyadic properties. That the fundamental is the input into an operation that has a certain derivative output (and correspondingly that the derivative was generated from a certain input) is, accordingly, not a fact in the world, but at best a fact about the world.⁴⁶ As such, it is to be located outside the grounding hierarchy. The question of what grounds the grounding facts can, accordingly, be circumvented altogether.⁴⁷

. Conclusion All fundamental relations are symmetric. Nevertheless, there is order and asymmetry in the world. On the one hand, there is order at the derivative level. Non-symmetric derivative relations can be grounded in asymmetric networks of symmetric relations as well as in asymmetric distributions of properties amongst objects. Such derivative relations are individuated in terms of their grounds, thereby precluding the existence of distinct converses as well as distinct duals.

equivalence (i.e. if [Δ] and [[Γ] grounds [Δ]] have the very same grounds, namely [Γ], then they are identical). Moreover, it does not explain how the derivative derives from the fundamental. If the grounding fact is a derivative fact, then we have not yet been given a satisfactory account of how the grounding relation connects the fundamental to the derivative. Positing another derivative fact, namely a grounding fact, that itself is in need of explanation does not amount to an explanation of the derivative in terms of the fundamental. Grounding is meant to be the connection that gets us from the fundamental to the derivative. Grounding relations (on the superinternalist account), however, do not establish a bridge between the fundamental and the derivative but instead only generate additional derivative facts. ⁴⁶ This is analogous to the way in which the ‘fact’ that a certain collection of facts is the totality of facts is not itself a fact in the world but only a fact about the world. (Otherwise, the relevant collection of facts would not in fact be the totality of facts.) ⁴⁷ This does not mean that one cannot explain grounding connections. One can explain why Fx grounds Gx in terms of the possible basic grounds of F forming a subset of those of G. Alternatively, one can explain grounding connections in terms of laws that govern grounding operations. These explanations of grounding connections are not themselves grounding explanations. One explains why Γ grounds Δ, yet one does so without identifying a ground of the supposed grounding fact that [[Γ] grounds [Δ]].

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On the other hand, order arises in the world as a result of the application of various generative operations. The derivative level is generated from the fundamental level by means of grounding, later times are dynamically generated from earlier ones by means of causation, and complex objects are generated from simple ones by means of composition.⁴⁸

⁴⁸ For helpful comments and discussions I would like to thank audiences at Edinburgh, Princeton, the Foundation of Reality conference at Oxford, the Objects and Properties conference at Cambridge, the Metaphysics and the Theory of Content conference at Tübingen, and Nick Jones, Christoph Michel, Katherine Hong, Gonzalo Rodriguez-Pereyra, Martin Pickup and Alex Roberts, as well as an anonymous referee.

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2 Classifying Dependencies Alastair Wilson

. Introduction It is no surprise that the literature on causation is primarily concerned with the kinds of causal relationships that are discovered and exploited in mainstream empirical science and in ordinary planning and decision-making. Philosophers of causation usually focus on paradigm cases of practical relevance such as the dependence of a window’s shattering on an incoming missile, the dependence of climatic trends on carbon dioxide emissions, and the dependence of an industrial injury on an employer’s negligence. In comparison, the outer limits of causation are not often explored, and in particular the precise relation between causation and temporal priority is rarely questioned. Do causes always precede their effects? Is causation across a temporal gap possible? Is simultaneous causation possible? The neglect of such questions means that we still lack a clear view of the underlying nature of causation. In this chapter I will probe the limits of causation by first investigating the surprisingly slippery distinction between causing and grounding, then arguing that we should draw this distinction in terms of the status of the principles that mediate the dependency, and finally exploring some initial implications for the possibility of simultaneous causation in physics. These days, metaphysical questions are frequently cast in terms of the ideology of grounding. This notion is usually introduced in explicit contrast to causation: ground is supposed to be a non-causal dependency relation that supports metaphysical explanations, just as causal relations support causal explanations. But the distinction between causation and grounding has never been very clear-cut, and recent work (Schaffer ; A. Wilson a) has highlighted how deep the structural similarities between the notions run. Schaffer concludes that causation and grounding are merely closely analogous, whereas I have defended the more radical view that grounding is a specific type of causation; however, I set that heterodox view aside for the purposes of this chapter and proceed on the assumption that there is a coherent distinction to be drawn between the two notions. In any case, the structural similarities between them show that we cannot afford to take the distinction between grounding and causation for granted; those who would wield both notions owe us a substantive account of the way in which they differ. Tracing the contours of the distinction between grounding and causation accordingly promises to cast valuable Alastair Wilson, Classifying Dependencies In: The Foundation of Reality: Fundamentality, Space, and Time. Edited by: David Glick, George Darby, and Anna Marmodoro, Oxford University Press (2020). © Alastair Wilson. DOI: 10.1093/oso/9780198831501.003.0003

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light on both notions. In this chapter I will consider and reject six natural-seeming demarcation criteria, and endorse a seventh. First, Section . provides some relevant background on causation and grounding. Then in Sections .–. I examine six obvious criteria by which to distinguish these two notions. I argue that each of the criteria is problematic in some way or other, which motivates the search for a better criterion. In Section . I offer my own account of the distinction between grounding and causation in terms of how the dependency is mediated. This mediation criterion can explain the appeal of the next best candidate criteria—the temporal criterion and the modal criterion—without suffering from their problems. Section . provides further support for the mediation criterion by arguing that it makes the classification of dependencies in physics appropriately sensitive to the interpretation of the physical theories involved. Section . is a conclusion.

. Causation and Grounding So, back to the guiding question: what is the difference between grounding and causation? Numerous possible ways of drawing the distinction spring to mind, but (surprisingly enough) few of these have received much explicit defence. Grounding is often simply introduced as a non-causal form of explanatory connection (see e.g. Fine , p. ) without much attention to exactly what might prevent it from being causal. If this way of introducing grounding is not to be badly misleading, no single dependency relation can count both as grounding and as causation (although of course two relata may be related both by a grounding relation and by a causal relation). And if this way of introducing the relation is to be informative, there must be more to be said about how the two relations differ. In informal fieldwork, I have encountered the following six criteria most frequently: • Perhaps causation relates distinct entities, while grounding relates not fully distinct entities? (Call this the distinctness criterion.) • Perhaps causation has no connection to fundamentality, while the ground fact is always more fundamental than the grounded fact? (Call this the fundamentality criterion.) • Perhaps causation relates events, while grounding relates facts? (Call this the categorial criterion.) • Perhaps causation holds diachronically, while grounding holds synchronically? (Call this the temporal criterion.) • Perhaps causal connections are those which can in principle be exploited for purposes of manipulation and control? (Call this the intervention criterion.) • Perhaps causal connections hold contingently, while grounding connections hold non-contingently? (Call this the modal criterion.) The next six sections will argue against these criteria in turn. Before diving in, we should note that it remains controversial whether there is any clear or univocal notion of ground. Outright sceptics about ground, including Chris Daly () and Thomas Hofweber () argue that the notion is ‘unintelligible’ or

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   ‘esoteric’. More qualified sceptics, such as Naomi Thompson (, ), have suggested that we should understand grounding talk in a deflationary way, perhaps by appeal to fictionalist or expressivist machinery. If there is no coherent notion of ground, then there is no coherent grounding /causation distinction either. A different sort of objection to the project of this chapter doesn’t deny that the grounding/ causation distinction is coherent, but instead questions whether it is exhaustive. For example, Karen Bennett () acknowledges numerous different ‘building relations’ alongside and akin to grounding and causation; these include set formation, composition, and property realization. Other pluralists about dependence, including Kathrin Koslicki (, ), Jessica Wilson (), and David Kovacs () have argued, in different ways, that grounding fails to capture a theoretically interesting category of dependence relations. Empiricist philosophers of science including David Lewis (a), Bradford Skow () and Michael Strevens () have in various ways attempted to give deflationary accounts of all non-causal explanation in science, typically treating apparent non-causal explanations as highly abstract causal explanations. If causation is the only kind of objective dependence relation we need, my target distinction fails to correspond to any real difference. Both grounding sceptics and dependence pluralists, however, typically still agree that causation itself is a clear and theoretically interesting category of dependence relation. Dependence pluralists can therefore still contrast causation in a useful way with non-causal dependence relations, whatever such there be. Eliminativists about non-causal dependence can similarly ask what features characterize causation. If you have misgivings about how clean the grounding/causation distinction is, then you can still think of this chapter as potentially casting light on the limits of causation taken by itself. While I will assume that causation is at least a relatively unified phenomenon, I will not presuppose that there is any unified phenomenon on the non-causal side of the target distinction. I will aim to remain neutral on a number of theoretical controversies concerning grounding. For example, some grounding theorists take ground itself to be a form of explanation, while others think of it as a kind of worldly relation that can back or support explanation (for further discussion, see Raven ). And, some grounding theorists deny that (full) grounding entails necessitation, while others maintain that a genuine full ground must necessitate the grounded fact (for further discussion, see Skiles ). I aim for my arguments to remain neutral on all such controversies; however, the question of necessitation will inevitably come into play in Section . when I evaluate the modal criterion. I will also aim to remain neutral on whether ground is best expressed by a relational predicate or a sentential connective (for further discussion, see Raven ). I have needed to pick a particular conception of the relata of grounding relations in order to frame my arguments throughout. In an attempt to be as neutral as possible on the metaphysics of individuals, I treat both grounding and causation in full generality as relating facts, in the sense of states of affairs: real, worldly, ways things are. Facts in the intended sense are not representational entities: they do not depend in any way on us or on our linguistic practices, and they are not in any interesting sense abstracted from reality. While there is certainly a fact that I am human, and this fact is as concrete as can be, I don’t assume anything about its underlying

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metaphysics, for example about how it is composed out of me and of humanity. I will presuppose, however, that facts have an algebraic structure sufficient to define basic logical operations: facts can be negated, conjoined, and disjoined to generate new facts. My preferred theoretical model for facts, as developed by Lewis (a, b), links them to cells of partitions over possible worlds. Lewis’s notion is intensional, but Yablo () generalizes it to a hyperintensional framework of facts which can be modelled by cells of partitions over impossible worlds in the sense of Priest () and Nolan (a). An advantage of all of these approaches is that facts are treated as fundamentally answers to questions, and so picking out a fact picks out a relevant contrast class of alternative answers. (Structural-equations treatments of causation also have this advantage.) With preliminaries out of the way, we can now focus on the demarcation question: what exactly is the difference between grounding and causation? What we would ideally want from our account of this distinction is a decision procedure which would allow us to determine, for each particular instance of dependence, whether it is a case of grounding or causation. No such decision procedure may be forthcoming— perhaps not every concept can be given necessary and sufficient conditions—but before we conclude that no such procedure is available in this case, we should explore and assess all the plausible candidates. That is the purpose of the next six sections of this chapter.

. Against the Distinctness Criterion The first of the criteria that I will be criticizing is the distinctness criterion. Fine hints at this criterion in the following passage: “It will not do, for example, to say that the physical is causally determinative of the mental, since that leaves open the possibility that the mental has a distinct reality over and above¹ that of the physical” (Fine , p. ). Depending on how ‘distinct reality’ is to be understood, the distinctness criterion may converge with one of the criteria to be discussed below: perhaps with the categorial criterion, the concreteness criterion, or the modal criterion. However, I’m sure that Fine would resist any such ways of understanding ‘distinct reality’, and I understand he would prefer to take distinctness of realities to be a basic and unanalysable notion. This would fit with his general approach to our question of taking the distinction between grounding and causation to be basic and unanalysable (Fine, p.c.). Still, we might reasonably flesh out Fine’s suggestion into an account of the grounding-causation distinction as follows: Distinctness Criterion: Causation relates facts corresponding to fully distinct realities, while grounding relates facts not corresponding to fully distinct realities. My primary objection to the distinctness criterion is that we lack an independent grip on the notion of ‘distinct reality’. Indeed, there is a strong suspicion that any notion which bears the appropriate relation to grounding will have to be tailored specifically ¹ It is possible that Fine intends the work to be done by a notion of ‘over-and-aboveness’ rather than by a notion of distinct reality. I assume that the former is to be explained in terms of the latter, but I think that my arguments would still go through if we assumed the contrary.

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   for the purpose. The best way to cash out the notion of distinctness involved will then itself appeal to the notion of grounding: two facts having distinct realities will coincide with lack of any chains of grounding (including connections of common ground) linking the two facts. We accordingly face the question of whether to explain distinct reality in terms of ground, or to explain ground in terms of distinct reality. Ideological parsimony dictates that we do one or the other; and grounding seems to be much more apt as a theoretical primitive than distinctness of realities. Perhaps there is another way to explain what it is for two realities to be distinct, but Fine gives us no hint as to what it might be and the terms in which such an explanation might be given remain unclear. Some such explanations (for instance, a modal explanation in terms of the possibility of one reality existing without the other) would make the criterion coincide with criteria discussed later in this chapter (in our instance, the modal criterion). I shall consider one example of this phenomenon: Lewis’s account of distinctness, which is only informative insofar as it collapses the distinctness criterion into the modal criterion. Lewis makes it a necessary condition of one event standing in a causal relation to another that the two events must be distinct, contrasting causation here with implication: “We may take it as a general principle that when one event implies another, then they are not distinct and their counterfactual dependence is not causal” (Lewis a, p. ).² Lewis provides a modal gloss on implication as necessitation; but to characterize distinctness in this way would be simply to adopt a disguised version of the modal criterion and I think it is not what he intended. Later, in Postscript F to “Causation”, Lewis provides a gloss on distinctness in terms of identity and parthood: “two events are distinct if they have nothing in common: they are not identical, neither is a proper part of the other, nor do they have any common part” (Lewis a, p. ). The problem now is that the notion of parthood is unspecified, if we are not to understand it in modal terms. Lewis points out that mere spatiotemporal overlap for events doesn’t exclude distinctness—two distinct events (a conference and a goblin battle) can occur at the same spacetime region. And once we move beyond events to include dependencies between facts including negative facts (as I shall argue in Section . we must), the parthood relations involved become more obscure still. Absent a general theory of parthood for facts, the Lewisian account of distinctness fails to give it an independently graspable content. While I cannot canvass all conceivable accounts of distinctness here, the most natural way to account for (the relevant kind of )³ distinctness of facts seems to be to characterize distinct facts as facts with non-overlapping grounds. This account, while ² This doesn’t cover cases where events fail to be distinct despite neither implying the other, as with two partially overlapping events such as my teenage years and my adulthood. Presumably Lewis would want to distinguish between partial implication and full implication, but still to account for each of them in modal terms. ³ A referee has pointed out (correctly, I think) that my theorizing in terms of facts in the first place requires some grip on how facts are individuated. But fact individuation by itself is not enough to account for fact distinctness: as Lewis emphasizes, distinctness goes beyond mere non-identity to non-identity of any parts, and so we plausibly can get a grip on fact individuation while still lacking any grip on fact parthood.

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very plausible, renders the Distinctness criterion uninformative for the broader purposes of this chapter. What I am searching for is a criterion that allows us to distinguish between grounding and causation in terms on which we have an independent grip, and (at least absent further explication) the Distinctness criterion fails to deliver this. This type of difficulty—lack of independent graspability—will also affect the criterion that I shall consider in the next section.

. Against the Fundamentality Criterion The fundamentality criterion in its most plausible form relies on the thought that causation bears no special relation to relative fundamentality, while grounding does bear a distinctive relation to relative fundamentality. Fundamentality Criterion: Causation has no connection to fundamentality, while the ground fact is always more fundamental than the grounded fact. A variety of fundamentality-based criteria are conceivable, of which this particular one is amongst the strongest, in the sense of being the most specific concerning the connection between grounding and fundamentality. Weaker criteria which are less specific about the grounding-fundamentality connection are also viable, but they are subject to the same line of criticism that I shall develop in this section against the stronger criterion, and so I set them aside here. The stronger criterion is the most plausible-looking, and it is the one which has been recently advocated by Alex Skiles: [W]hat contingentists should point to [when characterizing grounding] are the different implications of grounding and causation for relative metaphysical fundamentality. If a fact is grounded, then it must also be metaphysically less fundamental than each of the facts that partially ground it; yet an effect may be more, less, or equal in relative fundamentality with respect to its causes. (Skiles , p. )

My reason for rejecting the fundamentality criterion has the same structure as my reason for rejecting the distinctness criterion: both criteria explain the obscure in terms of the even more obscure. One of the main applications of the notion of ground is to characterizing a variety of notions in metaphysics, especially the notion of relative fundamentality. By using relative fundamentality itself to characterize the nature of ground, we give up on the prospect of a reductive account of relative fundamentality in terms of ground. That is bad news.⁴ Proponents of grounding should, I suggest, endorse a reduction of fundamentality (both relative and absolute) to grounding. The case for employing grounding (whether as a unified theoretical category or merely as a useful catch-all term) rests on it being able to do theoretical work in a variety of areas; a reductive account of fundamentality in terms of grounding is a key component of this explanatory work. If ⁴ A different perspective (e.g. Turner ) sees this instead as good news, since it opens up logical space for heterodox views on the relation between fundamentality and dependence such as that of Barnes (). I’m not convinced that the explanatory benefits of these heterodox views is worth the costs of the more complex ideology required to get such views onto the table, but can’t address the question properly here.

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   we endorse such an account, then we should not presuppose fundamentality when demarcating grounding from causation. Bennett (, ch. ) devotes an extensive discussion to accounts of relative fundamentality, defending what she calls the “deflationist” view that “relative fundamentality facts just are complex patterns of building” (p. ). Her project is not to reduce relative fundamentality to grounding, but to reduce relative fundamentality to relations from her larger category of building relations which includes causation and grounding. She thus accepts that causes are (at least in some sense) more fundamental than their effects, a conclusion that most will be unwilling to swallow.⁵ Stripping that unpalatable consequence from Bennett’s account yields what I think is the most plausible reduction of relative fundamentality to grounding. I will not here dwell on the details of the resulting reduction, but since we want to be able to make comparisons of relative fundamentality between isolated individuals it will need to be more complex than simply ‘A is more fundamental than B iff A grounds B’. Bennett (ibid.) has argued—convincingly, in my view—that such a reduction is viable, and has explored a number of potential ways of developing it. In summary, since the best way to understand relative fundamentality is itself in terms of chains of grounds, the fundamentality criterion should be rejected as uninformative. What we would like is a criterion that allows us to distinguish between grounding and causation in terms on which we have some independent grip. In the next section I will consider one such criterion: a criterion that appeals to the difference between facts and events.

. Against the Categorial Criterion According to the categorial criterion grounding and causation differ with respect to the categories of their relata: grounding is always a relation between facts, whereas causation is always a relation between events. Categorial Criterion:

Causation relates events, whereas grounding relates facts.

My first objection is that this combination of views about the relata of dependence relations is dialectically unstable. The arguments that motivate a conception of grounding as relating facts also motivate a conception of causation as relating facts, and the arguments that motivate events as causal relata also motivate events as grounding relata. However, the real problem with the categorial criterion is more basic. A criterion that gets to the bottom of the difference between grounding and causation ought to specify something distinctive not merely about the relata but about the connection between the relata. To simplify the discussion, I will set aside more exotic views of the causal relata, such as those envisaged by proponents of agent causation. Some have proposed views of the causal relata that, while event-like, are more fine-grained than events. Examples are L.A. Paul’s aspect causation (Paul ) or Jonathan Schaffer’s contrastivism (Schaffer ⁵ A. Wilson () and Schaffer () criticize Bennett’s assimilation of causation to the other building relations.

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). Most of these views can be fitted into the fact-causation framework on which I will settle. My first line of argument against the categorial criterion—call it the parity argument—is that the asymmetry it relies on does not in fact obtain. The main arguments that motivate facts as grounding relata also motivate facts as causal relata, and the main arguments that motivate events as causal relata also motivate events as grounding relata. As I will now argue, considerations of dependence upon absences motivate treating both causation and grounding as relating facts, while considerations of preserving the grammatical form of dependence attributions motivate treating both of them as relating objects, including events. There seems to be no principled basis for accepting either argument in the causal case but not in the grounding case, or vice versa. A central argument for facts as causal relata, heavily relied upon for example by Mellor (), is the argument from absence causation. Absences can cause and be causes. The absence of beer causes dismay, the absence of a hat causes sunburn, and the absence of air causes suffocation. As Schaffer () has pointed out, even such a paradigmatic causal process as the execution of a prisoner by firing squad presupposes absence causation, since there are steps in the operation of typical gun mechanisms that rely on an absence. But absence causation, treated as event causation, is a most peculiar phenomenon. It requires negative events— not-happenings—which are metaphysically problematic. In contrast, fact causation has no problem with absences: the fact that something does not occur is no more mysterious than the fact that it does, and it can be unproblematically captured by the partition-based framework for facts (see Section .).⁶ Since facts already have an algebraic structure, it is natural to negate them; events, lacking any algebraic structure, cannot be naturally negated. The argument from absences for facts as relata applies equally just as strongly to grounding as it does to causation. The absence of any unicorns grounds the emptiness of the set of unicorns; the absence of any sodium ions grounds the water’s zero salinity; the absence of any brain activity in the comatose patient grounds the absence of any conscious states. In each of these cases, grounding between facts is preferable to grounding between events, and for an analogous reason to the causal case: facts can be naturally negated, while events cannot be. So there is a fact of absence, which is available to do causal work or grounding work, but there is no absent event to do that work: it isn’t around to stand in any relation at all. Grounding and causation are thus on a par when it comes to the argument from absences for facts as relata. When combined with the observation that every case of apparent causation between events can be captured in terms of causation between facts, this case for fact causation becomes rather strong. For every event that could be a cause or an effect, there is a corresponding fact: the fact that that event occurs. So the view that facts are causal relata is more flexible than the view that events are causal relata. Everything event causation can do, fact causation can do also. Once again, the same ⁶ Some accounts of the nature of facts do not give them this sort of algebraic structure. I take this to be a reason to prefer the partition-based account of facts, rather than a problem for the overall argument of this section. The reader’s mileage may vary.

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   goes for grounding: for any putative case of grounding between events or grounding between objects, we can understand that case in terms of grounding between facts about events’ occurrences or about objects’ existences. Of course, the argument from absence causation to causation as relating facts remains very controversial. Cases of apparent absence causation can be explained away, perhaps by locating suitable surrogates for negative facts amongst the events, or by attributing them some non-obvious logical form. While I am myself convinced by the above arguments and accordingly prefer to think of both causation and grounding as relating facts in full generality,⁷ not all are convinced; for example, Noordhof () criticizes Mellor’s case for fact causation. Faced with this sort of resistance, for the purposes of the parity argument we can concede the point and instead rest the case for parity on the idea that objects—including events—can ground and be grounded. Schaffer (a) claims that entities of arbitrary ontological category can stand in the grounding relation, motivating his claim through examples such as the grounding of modes by the substances they modify, of singleton sets by their members, of abundant properties by sparse properties, and of truths by their truthmakers. Consider the singleton set containing a workshop, and the truth that this workshop occurred. Each of these, according to Schaffer, is grounded in an event: the workshop itself. Furthermore, Schaffer would say that events can ground other events, for example when individual talks ground a conference. I think of this line of thought as an argument from the grammatical form of grounding claims. In each case the grounding relation could be reformulated as connecting facts (the fact that a singleton set exists, the fact that a truth is true, and so on); however, such reformulations require a positive motivation, which appears not to be forthcoming. Hence we should accept at face value our intuitive judgments of ground between events. Putting all this together, we are left with no principled basis for distinguishing grounding from causation via the nature of their relata. There are good arguments for treating causation as relating facts in at least some instances, and the best reasons for rejecting these arguments tend to motivate treating grounding as relating events in at least some instances. While we could insist on taking the logical form of the dependency claims at face value in the one case but not in the other, unless we can find some independent motivation for doing this then such a move looks ad hoc. The parity argument might be resisted by rejecting Mellor’s arguments, endorsing Schaffer’s arguments, and locating the difference between grounding and causation in a modified categorial criterion: grounding can relate things of any ontological category, while causation always relates events. Here is the first instance of an approach that I will call occasionalism, and which we will encounter again in later sections. Occasionalist approaches characterize the difference between two relations by specifying that all instances of one relation have some property, while only some instances of the other relation have that property. I will reject occasionalist approaches throughout, since they do not always allow for an answer to the basic ⁷ It is also worth observing at this point that it is quite orthodox to treat grounding as always relating facts; see Rosen () and Audi (). It is also mandated by the approach to ground which represents ground by a sentential operator (see e.g. Fine ); thanks to Nicholas Jones for pointing this out.

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 

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diagnostic question I ask: is this particular instance of dependence a case of grounding or causation? Suppose we are presented with a case of grounding between events: the present approach gives us no way to say what makes it grounding and not causation. I have a second line of argument against the categorial criterion, which I think runs deeper; call it the perspicuity objection. There is an underlying problem with the strategy of characterizing relations by their relata. Even if instances of the relations of grounding and causation do take different relata, this is something that ought to be explained in terms of the kinds of relations of which they are instances, rather than vice versa. If grounding and causation are genuinely different relations, this difference should amount to more than just a different pattern of instantiation. My discussion in this section has presupposed a view of both grounding and causation as relational. How do things look if we instead treat grounding as a sentential connective? The view that ground is best expressed by a sentential connective rather than by a relation will tend to undermine the categorial criterion as stated, but the criterion can easily be revived by contrasting the sentences on either side of the connective. However, this improved version of the categorial criterion does not escape the problems I raised in this section. The case for a sentential connective is just as strong in the causal case as in the grounding case—indeed, Mellor has explicitly argued that causation should be thought of as a connective—so the parity argument still tends to undermine the categorial criterion.⁸ And the perspicuity argument applies in just the same way as in the relational approach: we shouldn’t classify dependencies expressed by a connective in terms of the type of sentences that they connect, but in terms of the type of connection that is imposed between these sentences. It is time to move on.

. Against the Temporal Criterion According to the temporal criterion, causes are temporally prior to their effects, while grounds are not temporally prior to the facts they ground: Temporal Criterion: non-diachronically.

Causation holds diachronically, while grounding holds

The temporal criterion seems more initially promising than the criteria I have discussed up to this point. Indeed, if there is any orthodoxy in this domain then the temporal criterion probably constitutes that orthodoxy. It has been explicitly defended by Stephan Leuenberger (Leuenberger ), and is apparently lurking in the background of many informal discussions including that of Schaffer (a). Something very like the temporal criterion is endorsed as an account of causation by Bader (Chapter , this volume), who recognizes classes of ‘generative operations’ corresponding to kinds of dependence—causal, grounding, and compositional—such that causation is “a trans-temporal operation that has inputs and outputs that exist at ⁸ It will also tend to undermine any attempt to build a demarcation criterion on the idea that grounding is best expressed by a connective while causation is best expressed by a relational predicate.

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   different times”.⁹ Paradigm cases accord with the temporal criterion; the throwing of a rock is temporally prior to the breaking of a window, but Socrates is not temporally prior to Singleton Socrates. However, once we move beyond paradigm cases, the criterion begins to look more doubtful. A preliminary line of argument against the temporal criterion appeals directly to the possibility of particular cases of simultaneous causation. Examples might be drawn from mechanics: perhaps the net applied force on an object causes its instantaneous acceleration within Newtonian mechanics, even though the application of the force and the acceleration are simultaneous (see Section . for further discussion of this case). Or, examples might come from various kinds of mental causation: perhaps an interactionist dualism, where mind can act instantaneously on matter. There is a ready response to this kind of argument: modal epistemology is hazy, and our judgments about particular cases at the borderline are not fully reliable. If a compelling theoretical principle demands it, revision to our intuitions about a reasonable proportion of borderline cases is to be expected. So appeal to the possibility of these sorts of cases may be ineffective against the denier of simultaneous causation, who will either seek to reinterpret Newtonian mechanics in a way that doesn’t require simultaneous causation, or will deny that this theory corresponds to a genuine metaphysical possibility. I would prefer to have an argument against the temporal criterion that does not rely on controversial judgments about the possibility of particular cases at the limits of our modal knowledge; but readers who are convinced of the intuitive possibility of simultaneous causation may skip the remainder of this section. A more interesting-looking line of argument for simultaneous causation is indirect, going via the possibility of time travel loops. If a contemporary time-traveller goes back in time to hand over the blueprints for their time machine to their earlier self, who goes on to construct the machine and complete the loop, the activation of the time machine is caused (at least in part) by the activation of the time machine. Indeed, if time is cyclical (in the sense of having the topology of a circle, not in the sense of including an endless sequence of distinct indiscernible epochs) then every event is both temporally prior and temporally posterior to its effect, and so causation of events by themselves might turn out to be endemic. Perhaps these are esoteric scenarios, whose metaphysical possibility a defender of the temporal criterion may deny, but importantly for present purposes they don’t seem to be ruled out by the nature of causation. If they are metaphysically impossible at all, presumably they are rendered impossible by metaphysical necessities concerning the temporal structure of reality rather than by metaphysical necessities concerning the nature of causation. Although the argument against the temporal criterion from time travel has some prima facie force, I don’t think it’s ultimately decisive. This is because a defender of the temporal criterion can legitimately respond by distinguishing between immediate ⁹ Bader combines the temporal criterion as a positive account of causation with the fundamentality criterion as a positive account of grounding, seeing both as distinct from composition and maybe from other forms of dependence.

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simultaneous causation and simultaneous causation with diachronic intermediaries. This move appeals to a distinction between local time and global time.¹⁰ Local time is elapsed time along a worldline, time from the perspective of an object undergoing change, while global time is time from the perspective of the universe. Armed with this distinction, we can sharpen the temporal criterion to say that causation is always diachronic with respect to local time, even if it’s not diachronic with respect to global time. Causation, on this picture, is always directed forwards along a worldline even if that worldline ends up reversing direction or coming full circle with respect to the global temporal ordering. The temporal loop argument against the temporal criterion is therefore inconclusive. A better argument against the temporal criterion takes a different tack. Rather than arguing that causation is sometimes non-diachronic, we can approach the problem from the opposite direction and argue that grounding is sometimes diachronic.¹¹ As far as I know, cases of this sort were first discussed by Allen Hazlett (, ). An initial class of cross-temporal grounding relations are those that relate entities to the causal histories that are responsible for their kind-membership. On some plausible views of what it takes to be human, for example, humans must be descended from the ancestral Homo sapiens; a swampman (a perfect duplicate of a human formed by pure chance) would not qualify as a human. So the fact that I am a human is grounded in the fact that some past events occurred: those constituting my ancestral lineage. Cases of this general sort can be proliferated. We might think that to be a church is (in part) to have been consecrated at some prior time; so the fact that this building is a church is partly grounded in a particular past event of consecration.¹² Perhaps the fact that I’m in the mental state of believing that Montana is beautiful has to be partly grounded in the fact that I’ve had some prior causal interaction with Montana. And perhaps anyone’s holding of a PhD degree is grounded in some prior event of degree conferral. More generally, being an ex-convict—or an ex-anything—requires having been some way in the past, and being a future president—or a future anything—requires being some way in the future. Being ‘the once and future king’, like T.H. White’s King Arthur, is grounded in both past and future. Could the opponent of cross-temporal grounding find some principled reason to reject these cases? Perhaps they could maintain that the cases all appeal to abundant properties rather than to sparse properties, or that the cases all appeal to extrinsic properties rather than intrinsic properties, and then go on to argue that when the facts involved are appropriately restricted then the ban on cross-temporal grounding remains in force. However, I’m not sure what motivation there would be for imposing such restrictions. And in addition, some of the examples—being

¹⁰ This is distinction is more commonly referred to as personal time vs. external time, but the cases under consideration here need involve no persons. ¹¹ This way of putting things assumes a broadly B-theoretic approach to time. Presentists and other A-theorists can substitute ‘facts obtaining at different times’ for ‘facts about different times’ as necessary. ¹² It is also relevant in this case that the church was not deconsecrated at some later time. I defer discussion of these delicate issues to the next section, where cases of contingent grounding take centre stage.

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   human, having beliefs about objects—certainly seem at least relatively sparse, while some of them—being human, being a church—seem potentially intrinsic. I won’t pursue these details here, since I don’t want to put much weight on these cases. Instead I want to draw attention to a more systematic motivation for rejecting the temporal criterion: the coherence of metaphysical views that posit systematic crosstemporal grounding. The claim that cross-temporal grounding is ubiquitous has been defended in multiple ways in the recent literature on the metaphysics of time. Sam Baron () argues that the best form of presentism—a view that he calls priority presentism— should not deny that past entities exist but instead should maintain that past entities are grounded in the present. And Ross Cameron can be construed as arguing that presentists (Cameron ) and moving spotlight theorists (Cameron ) should each say that facts that either are or purport to be about the past are grounded in distributional properties of the world as it presently is. Now, these views about time are obviously controversial. Tim Williamson has written dismissively that such views are “obviously false: what I was yesterday is not grounded in what I am today, in any useful sense” (Williamson , p. ). Still, I think such views give us reason to look for an alternative to the temporal criterion if any is available. Three alternative demarcation criteria remain to be addressed.

. Against the Intervention Criterion According to the intervention demarcation criterion, what is essential to causation is its connection to the possibility of intervention, manipulation, or control. It is of course a platitude that some causal relations can be exploited for practical purposes. The intervention criterion makes this into the defining feature of causation: all causal relations can (at least in principle) be exploited, whereas grounding relations cannot be. Intervention Criterion: Causation in principle permits manipulating the dependent fact by intervening on the fact on which it depends; grounding does not. A clear example of the intervention criterion is the view of Price and Menzies (), who defend the view that “the ordinary notions of cause and effect have a direct and essential connection with our ability to intervene in the world as agents” (p. ). Something very like the intervention criterion has played a prominent role in recent work on causal explanation within philosophy of science. In particular, interventionist accounts of causal explanation characterize causation (albeit in a non-reductive manner, since interventions themselves are counted as causings) in terms of the counterfactual consequences of interventions. The locus classicus of this approach is Woodward (), who traces a number of historical anticipations of his interventionist approach (p. ) and whose eventual ‘manipulability theory’ of the notion of direct cause (p. ) is given in terms of possible interventions on the cause variable, holding fixed suitable other variables, which changes (the probability distribution over) the effect variable. Interventionist accounts of causal explanation are enlightening and fruitful. However, they are not well suited for characterizing my target in this chapter,

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

the distinction between causation and grounding. It is difficult to make sense of interventions on some apparent causal relata, such as the dimensionality of spacetime (Woodward , p. ), but I shall set that concern aside here. My primary objection to the intervention criterion is that relations of ground also support manipulation, in the sense that we may intervene on the grounding fact and thereby manipulate the grounded fact. Consider intervening on the colour of a red house, by painting it emerald green. Since (we may grant for the sake of argument) the fact that the house is emerald green grounds the fact that the house is green, by intervening to paint it emerald green we have thereby manipulated whether it is green. Without further restrictions on the relevant interventions, then, manipulability accounts of causation fail to distinguish causation from grounding. Woodward is aware of the problem. In the specific context of mental causation and the causal exclusion argument, Woodward imposes a condition he calls Independent Fixability on the variables that can be included in causal models. [F]or each value it is possible for a variable to take individually, it is possible (that is, “possible” in terms of their assumed definitional, logical, mathematical, or mereological relations or “metaphysically possible”) to set the variable to that value via an intervention, concurrently with each of the other variables in V also being set to any of its individually possible values by independent interventions. (Woodward , p. )

While Independent Fixability does prevent relations of ground from counting as causal (at least on the conventional view according to which full grounds necessitate what they ground), it collapses the intervention demarcation criterion into (some form of ) the modal criterion that is the focus of the next section. It is not manipulability per se, but the possibility of independent fixing of variables, that is now bearing the theoretical load. So it is time to set aside the intervention criterion, and consider the modal criterion directly.

. Against the Modal Criterion According to the modal criterion, the difference between grounding and causation concerns their modal force. Causation, so this line of thought runs, holds contingently, in that it is possible for the cause and the effect both to occur without the causal relation holding between them. Grounding on the other hand holds non-contingently: necessarily, if the ground fact and the grounded fact both obtain, then the former grounds the latter. This view about grounding is often called ‘grounding internalism’. We can distill grounding internalism into the following demarcation criterion: Modal Criterion: The causal relation between cause and effect holds contingently; the grounding relation between ground and grounded holds necessarily. The modal criterion seems reasonably popular. Gideon Rosen identifies the modal status of the dependency as a respect of difference between grounding and causation: [That grounds necessitate] is one respect in which the grounding relation, which is a relation of metaphysical determination, differs from causal and other merely nomic forms of determination. There is a difference between the materialist who holds that facts about phenomenal

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   consciousness are grounded in (hence necessitated by) the neurophysiological facts directly, and the dualist who think that facts about the brain cause or generate conscious states according to contingent causal laws. (Rosen , p. )

The modal criterion relies on orthodox views about the modal status of laws of nature and of grounding. For those who accept nomic necessitarianism—the view that the laws of nature are necessary—the modal criterion seems unsuitable. Likewise, the modal criterion presupposes grounding internalism in order to work as intended. Here I am disagreeing with Alex Skiles, who has argued that the modal approach to demarcation is compatible with the rejection of grounding internalism: For the distinction between the two might simply be that in at least some cases a fact is necessitated by its ground, while an effect is never necessitated by its causes, given that the metaphysically contingent laws of nature governing causal interaction could have differed. (Skiles , p. )

I have already argued that this kind of occasionalist approach to distinguishing causation from ground is unsatisfactory. What we are after is a criterion that allows us to classify any given case of dependence as causation or as grounding. If it is to correctly classify all cases, the modal criterion does require grounding internalism. Still, the modal criterion is intuitively quite appealing. Causation is a connection that can be interfered with and re-routed; there are various different possible causal pathways, we might think, between two events. Grounding, on the other hand, looks to be harder to interfere with; no external contingencies seem capable of disrupting the dependency between a thing and its grounds. The criterion also gets paradigm cases right. It is possible for a brick to be thrown at the window, and for the window to break, without the former causing the latter (for example, suppose some other missile breaks the window first). But it is impossible for Socrates to exist, and for Singleton Socrates to exist, without the former grounding the latter. So far, so good. Unfortunately, I think that the modal criterion is untenable because it appeals to a false view of ground. The problem is that the modal criterion builds in grounding internalism: the view that grounding connections hold necessarily when their relata do. This view has been the target of a number of recent critiques, including those of Skiles () and Leuenberger ();¹³ a related line of argument in the context of truthmaking dates back to Parsons () and has been defended by Briggs (). I think that the counterexamples provided by these authors—while one might quibble here and there—are collectively compelling. But, in addition, I think there are important further cases of failures of grounding internalism that have not attracted as much attention; they involve grounding connections that are mediated by contingent principles of ground. For an example of a contingent grounding connection, consider first legal grounding; the illegality of a particular act is at least partly grounded in whichever particular features of that act are forbidden by law. Or consider grounding as it figures in rule-governed activities such as sports; the location of the ball relative to the

¹³ These authors primarily target the closely-related principle of grounding necessitarianism: that full grounds necessitate the facts they ground.

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

goalposts partly grounds the fact that a goal has been scored. To work through a particular example, consider my current status as an EU citizen. Given the actual legal framework of the EU, my citizenship of the EU is grounded in my citizenship of a member country. However, the EU legal framework could at some point be changed to introduce birthright citizenship, in which case my citizenship of the EU might become grounded instead in my having been born in the UK. I would still be a UK citizen, and still be an EU citizen, but the latter would no longer ground the former. Still, the EU would persist through the change: altering membership regulations does not replace an institution with a different institution. Similar counterexamples can be generated for any cases of grounding mediated by contingent grounding principles, including all kinds of conventional principles. An obvious line of response for the grounding internalist is to build the apparently contingent grounding principles into the full grounds in all such cases. Then, we might say, the full ground of my EU membership includes both my UK citizenship and the rules of the EU connecting member-state citizenship with EU citizenship. Along the same lines, the fact that a goal has been scored might depend partly on the position of the ball and partly on the laws. However, this move should be resisted, since recourse to it tends to undermine the modal criterion. If we can build the supposed grounding principles into the grounds of social facts, then we can likewise build the laws of nature into the causes of particular effects— unless of course some relevant disanalogy between grounding and causation rules this out, but as yet we have found no such disanalogy. So if the laws of football may legitimately be counted among the grounds of the goalscoring, it looks like the law of gravity may legitimately be counted among the causes of the apple’s falling. But this manoeuvre, at least in the case of deterministic laws, means that dependency governed by laws of nature would be judged as grounding rather than causation by the modal criterion. The connection between the conjunction of the initial conditions and the deterministic laws on the one hand, and the later state of the universe on the other, becomes non-contingent: if the laws are deterministic, then all later states of the universe are logically entailed by the conjunction of the laws and the initial conditions. This conception of causation as involving necessitation may have been popular in the Early Modern period, but few have any truck with it today. Again, in arguing that grounding and causation should be treated in the same way unless we can identify a non-question-begging reason to do otherwise I am drawing on the closeness of the grounding-causation analogy. Just as the same arguments concerning relata apply in each case (Section .), so here the same arguments concerning the inclusion of general principles in the dependence base apply in both cases. I have defended the use of this grounding-causation analogy to draw substantive conclusions about the relations involved elsewhere (A. Wilson b), and it has also been emphasized by Jonathan Schaffer (Schaffer ). Essentially the only place where Schaffer thinks the grounding-causation analogy breaks down is with respect to indeterministic causation. He maintains that grounding cannot be indeterministic, but causation can. We need not adjudicate here on whether this is correct, since even if correct it cannot provide the kind of demarcation criterion we are seeking. At most, the appeal to indeterminism can provide an occasionalist

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   criterion unable to distinguish instances of deterministic causation from instances of grounding. We must continue our search for a more general criterion. There are of course ways in which my line of argument against the modal criterion can be resisted. Perhaps there is after all some non-question-begging reason why we should build contingent grounding principles into the grounds in the case of social grounding, but resist building contingent laws into the causes in the case of physical causation. Or perhaps deterministic laws of nature are not metaphysically possible (maybe because the actual laws are indeterministic, and a modal necessitarian account of the laws is correct). However, I take it that the arguments of this section at least provide us with some reason to continue our search. In the next section I shall present my own preferred demarcation criterion, which accounts for the intuitive appeal of the temporal and modal criteria while avoiding the counterexamples to them that I have discussed.

. A Mediation Demarcation Criterion In many cases, the temporal criterion seemed to produce the right results. Most causal relations are diachronic and most grounding relations are synchronic. Similarly, the modal criterion got things largely correct: causal connections do seem to be contingent, and the paradigm cases of grounding connection do seem to be noncontingent. Perhaps these successes are due to some deep conceptual connection between causation and time, or between grounding and modality? I have an alternative diagnosis: perhaps what distinguishes causation from grounding is whether or not the holding of dependency itself partly depends on the laws of nature. To help formulate the mediation criterion, it is useful to introduce the notion of a mediating principle. In the case of causation, the mediating principles are laws of nature; in the case of grounding, the mediating principles are something like laws of metaphysics.¹⁴ Either way, the mediating principles are the principles responsible for the substantive connection between ground and grounded: those general facts that explain the holding of more specific explanatory connections. In Wilson (a) I explain the notion of a mediating principle in terms of the structural-equations modelling framework that is there used to unify both grounding and causation. Schaffer also makes appeal to mediating principles in his treatment of the analogy, using the term formative principles (Schaffer , p. ). Here I will rest content with the intuitive notion of a principle that explains the holding of the dependency connection. With the notion of a mediating principle in hand, we can set up our contrast between causation and grounding as a contrast between two different ways in which a dependence relation can hold. We have clear cases where the dependence is mediated by the laws of nature and cases where it is not. That the throwing of the stone is ¹⁴ In this category I would intend to include synthetic principles such as principles of mathematics as well as principles with an obviously metaphysical subject-matter. There is a substantive further debate to be had about the unity or diversity amongst the principles in this category, and it will feed directly into what we say about the unity of grounding. This doesn’t matter for present purposes; what matters is the contrast with laws of nature.

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

sufficient for the breaking of the window is to be explained by reference to the laws of nature that mediate the process; that the existence of Socrates is sufficient for the existence of Singleton Socrates is not to be explained by reference to any laws of nature. (Presumably the connection is mediated instead by principles of set theory.) Mediation Criterion:

Causation is mediated by laws of nature; grounding is not.

The mediation criterion gives correct results in paradigm cases, and it avoids the objections to the criteria described in previous sections. Facts about dependencies between facts, including facts about dependencies involving absences, can be properly assessed for what they themselves depend on; so can facts about dependencies between events. Some dependency relations between both facts about concreta and facts about abstracta depend on the laws of nature, while others do not. There are at least some dependency connections between facts about different times that do not depend on the laws of nature: it’s not because of any law of nature that it’s impossible to be an ex-president without having been president at some earlier time. The mediation criterion can account for the apparent plausibility of some of the other criteria we have considered. In particular, the mediation criterion can explain why the temporal criterion gets paradigm cases right. According to the mediation criterion, causal relations, but not grounding relations, are mediated by laws of nature. So, we can explain our intuitions about the different temporal properties of grounding and causation by relying on the widespread conception of laws of nature as diachronic constraints that relate events at a later time to events at an earlier time. Since the laws of nature typically (but not always) entail diachronic connections between facts, while principles of logic or metaphysics typically (but not always) entail non-diachronic connections between facts, the mediation criterion supports the corresponding intuitions about the temporal properties of grounding and causation. A recurring theme in recent interlevel metaphysics has been to emphasize the dynamic and interactive aspects of the dependence between phenomena at different levels.¹⁵ Material objects may initially look static and unchanging, but zoom in on the smaller-scale structure of their matter and we find a complex and active interplay of intermolecular, interatomic, and nuclear forces. Complex systems with many degrees of freedom may be in overall equilibrium, yet at all but the lowest energies this equilibrium is a dynamic one maintained through constant causal interactions between and within subsystems. We must accordingly take care in the application of the mediation criterion to interlevel connections. For example, where some property (say, magnetization) of a macroscopic object (say, an iron girder) is grounded in the configuration of the microscopic parts of the girder (atoms of iron in a lattice arrangement), a full story about the dependence of overall magnetization on the individual iron atoms will cite causal relations between adjacent atoms in the cubic lattice. So the laws of nature that mediate the causal connections between atoms are at least complicit in the grounding of the overall magnetization. However,

¹⁵ This point is emphasized for example by Bennett (), who refers to the phenomenon as ‘causal taint’ in various other building relations.

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   that is no threat to the mediation criterion, since a systematic treatment of these cases of ‘dynamic composition’ is available: distinguish between the causal processes operating at some lower level and the grounding relation which connects the operation of those causal processes with the higher-level phenomenon. The causal relations at the lower level are mediated by the causal laws of electromagnetism, while it is not dynamical laws but some kind of level-connection principles that mediate the grounding relations between the lower-level pattern of causal relations (atomic electromagnetic interactions) and the higher-level phenomenon (overall magnetization). What the level-connection principles actually are is typically a hard question, and one that is answered by work on specific reductive programs from the philosophy of the special sciences.¹⁶ But proponents of the mediation criterion may point to a systematic division of labour between level-connection principles and the laws of nature, such that the former take as input patterns of instances of the latter. Importantly, the proposed distinction between grounding and causation is conceptually conservative. It relies primarily on the notion of a law of nature, a notion to which most metaphysicians and philosophers of science are independently committed and which can, if desired, be given a deflationary Humean analysis.¹⁷ What the mediation proposal achieves, then, is to bring our understanding of the distinction between causation and grounding up to the same level of our understanding of the distinction between laws of nature and other synthetic principles that characterise reality. This does not, of course, by itself provide a full account of the nature of grounding and of what grounding facts there are: for that, we still need to better understand the range of metaphysical laws. But my intention here has been only to cast light on the difference between causation and grounding by connecting that distinction to the well-understood notion of laws of nature, and that much does seem to have been successfully achieved. We are now in a position to classify arbitrary putative dependencies as causal or grounding, even if we are not yet in a position to know which grounding dependencies there in fact are. Perhaps the notion of a law of nature is not after all as clear and unproblematic as I have been supposing. We could imagine borderline cases, where it is clear that some dependence is mediated by a particular principle, but not clear whether that status has the status of a law of nature. This situation might arise in the face of disagreement about what it is to be a law of nature, but it might also arise within the context of a specific theory of laws if that theory does not support a thoroughgoing demarcation into laws and non-laws. While this would be bad news for our ability to distinguish causation from grounding according to the mediation criterion, it would not necessarily be bad news for the criterion itself. If the distinction between law and non-law is unclear in just the same cases as the distinction between causation and

¹⁶ I make some remarks about how we might begin to identify these principles in the specific context of emergent spacetime in A. Wilson (forthcoming). In general, the way in which individual theoretical principles are classified will be sensitive to the details of interpretation of the physical theory involved, as I argue below. ¹⁷ If (as contemporary Humeans maintain) the notion of a law of nature does not run metaphysically deep, then the distinction between causation and grounding will likewise fail to run deep. I take it that this is a feature rather than a bug.

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 



grounding, then the mediation criterion in fact delivers the correct results in those cases. And consider a potential interpretation of quantum theory according to which the principle mediating the dependency between the measurements is a hybrid, halfway between a metaphysical principle and a law of nature. The mediation criterion quite reasonably predicts that the dependency involved is likewise a hybrid between grounding and causation.¹⁸ Beyond the notion of a law of nature, the proposal makes use of the notion of a mediating principle. In this chapter I have characterized mediating principles as neutrally as possible in order not to make unnecessary assumptions about the controversial question of how grounding itself is grounded. But I take it that any plausible account of grounding will include some link between general metaphysical truths and particular instances of metaphysical dependence. Grounding facts are not scatterered randomly and arbitrarily around reality—their pattern should reflect underlying general metaphysical principles. Any approach to grounding capable of doing justice to this thought will give us suitable candidates for mediating principles. For an extended discussion of what the principles mediating grounding connections might be like, see Wilsch (). In making this appeal to mediating principles, the mediation criterion reuses the notion of dependence. The explanatory connection between the mediating principle and the instance of dependence that it mediates is itself an instance of dependence of the sort that that is already presupposed in the very terms of the debate. This is benign, and does not give rise to any circularity. If we can make sense of one fact depending on another in order to pose the question of whether that dependence is causal or grounding, then we can make sense of the second-order dependence of that first-order dependence fact on some further fact. (The mediation criterion will plausibly count this second-order dependence as grounding, since it is plausible that laws of nature do not tell us anything about what dependence facts depend on. But this is not our present concern.¹⁹) It should be noted that the mediation criterion does not automatically render all dependence involving laws of nature as causation. For example, the set of all laws of nature is presumably grounded in the laws, but still this connection is not itself mediated by laws. (Again, it is likely mediated by principles of set theory.) A worry raised by Jessica Wilson may be addressed along similar lines. Functionalist physicalists typically suppose that when a mental state M is grounded in a physical state P that plays the role R associated with M, whether P plays the role R will depend on laws of nature. This may look like it risks making functional realization into a causal relation. But these laws of nature are mediating a different connection, the connection between P and R; they are not mediating the connection between M and P. So the mediation criterion does after all give the desired conclusion: M is grounded in P rather than caused by P. It will be helpful to see how the mediation criterion plays out in applications. To that end, the next section considers its implications for some cases of dependence ¹⁸ Thanks to Suzy Kilmister for pressing me on this point. ¹⁹ See A. Wilson (MS) for discussion, within the current general framework, of what dependencies depend on.

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   in physics. The mediation criterion makes the type of dependence involved sensitive to details of the interpretation of the physical theory, which I think is exactly the right result.

. Implications for Simultaneous Causation Some cases of simultaneous dependence are assessed as clear cases of grounding by the mediation criterion: for example, the EU membership and goal-scoring cases from Section ., as well as the relation between Socrates and his singleton set. But does the criterion allow for cases of simultaneous causal dependence? If simultaneous dependence is to count as causation according to the mediation criterion, then the holding of the dependency between two facts about simultaneous events must be able itself to depend on the laws. Whether this is possible turns in part on what laws are. On the conception of laws as generative, producing later states of the world out of earlier states (described forcefully, for instance, by Maudlin ), then laws do not directly support dependencies between distinct facts about any single time. Such facts may be common effects of some earlier cause, but there will be no direct causal dependency between them. (The dependence counterfactuals are then analogous to back-trackers; see Lewis .) On alternative conceptions of laws, for example the Humean view of laws as efficient summaries of the occurrent facts (Lewis ), there is no obvious barrier to the laws directly entailing relations between simultaneous events. Absent barriers of this sort, we need to look at particular physical theories and their interpretations to establish whether they really do involve simultaneous causation. The demarcation criterion makes the status of physical dependencies, as grounding or causation, dependent on the interpretation of the physical theories involved. A simple example comes from Newtonian mechanics. Newton’s so-called second law, expressible by the familiar equation F = ma, relates the resultant force vector applied to a body to the mass and acceleration vector of that body. It is highly plausible to think that accelerations depend on the applied force, rather than vice versa; and, if F = ma expresses a law of nature according to Newtonian mechanics, then this dependency will be classified as a case of causation according to the mediation criterion. However, the fact that F is the resultant force applied to the body and the fact that a is the instantaneous acceleration of the body seem to be facts about the same instant. So, as I suggested above, this case apparently involves simultaneous causation.²⁰ The relation between matter distribution and gravitation presents another example. In general relativity, gravitation is interpreted as a manifestation of curved spacetime; according to Wheeler’s memorable description of general relativity, “spacetime tells matter how to move; matter tells spacetime how to curve” (Wheeler ). In contrast, in Newtonian theory gravitation is interpreted as a manifestation of a force law acting at a distance. However, these interpretations are ²⁰ There may be legitimate reasons for doubting the existence of instantaneous accelerations in classical mechanics, but these considerations seem to be completely orthogonal to current concerns.

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 



not forced on us by the bare mathematics. Gravitation can be understood as a manifestation of curved spacetime even in classical physics, which gives us Newton-Cartan theory. Conversely, the theory of teleparallel gravity is mathematically closely related to general relativity but it involves interpreting gravitation as resulting from the operation of forces.²¹ It is natural to expect that these interpretive differences are relevant to how we classify the dependencies involved, and the mediation criterion ratifies this expectation. In classical mechanics the dependency of gravitational motion on the distribution of matter can be interpreted nomically (yielding Newtonian mechanics) or geometrically (yielding Newton-Cartan theory). In relativistic mechanics the dependency of gravitational motion on mass-energy distribution can be interpreted nomically (yielding teleparallel gravity) or geometrically (yielding general relativity). The mediation criterion parlays these interpretive differences into differences in the status of the dependence between matter distributions and gravitational motions. If a geometric formulation of the physics involved is correct, then it is incompatible with the nature of space (in the classical case) or of spacetime (in the relativistic case) to pull apart the curvature of space(time) from the motion of matter. Since facts about the nature of space(time) are not usually regarded as laws of nature, the mediation criterion will classify the relation between matter distribution and gravitational motion as one of grounding. But if a nomic (force-law) understanding of gravitation is correct, then the mediation criterion will classify the link between matter distribution and gravitational motion as causal. So, if Newtonian mechanics and/or teleparallel gravity describe ways the world could have been, then the mediation criterion will give us possible cases of simultaneous causation. Limitations of space prevent me from considering further examples in any detail, but two additional cases are worth mentioning briefly: the dependence in some quantum gravity theories between spacetime and a more fundamental non-spatiotemporal reality, and the relationship between entangled systems in quantum theory. The emergent spacetime example I discuss in a companion chapter to this one (A. Wilson forthcoming); my preliminary conclusion in that chapter is that a promising way to understand emergent spacetime is as being grounded in the operation of causal (although non-spatiotemporal) processes at the fundamental level. The mediation criterion enables this plausible conclusion through its focus on mediating principles rather than on any relationship with time. The invocation of mediating principles permits both grounding and causation to be achronic, or timeless, rather than being (respectively) synchronic and diachronic as envisaged by friends of the temporal criterion. The quantum entanglement example I hope to address in future work. A preliminary conjecture is that the status of entanglement dependencies as causation or as grounding

²¹ For detailed discussion of both cases, see Knox (). While in the relativistic cases the relationship between matter and gravitation is not simultaneous in the sense that it holds at a specific instant, it is still simultaneous in the sense relevant to this discussion, since there is no temporal priority between mass distribution and spacetime curvature.

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   will turn out to be highly sensitive to the chosen interpretation of quantum theory, particularly to questions concerning the metaphysical status of the quantum state.

. Conclusion I have proposed a criterion for determining whether particular cases of dependence are causation or grounding. Causal dependencies are those dependencies the holding of which partly depends on a law of nature, while grounding dependencies are those dependencies the holding of which does not depend on any law of nature. I have argued that this criterion avoids various difficulties facing other criteria, that it classifies core cases correctly, that it explains the initial appeal of the more plausible alternative criteria, and—aptly—that applying it within physics typically requires adopting a specific interpretation of the physical theory describing the dependence of interest.

Acknowledgements I am grateful to audiences at the Philosophy Mountain Workshop, Leeds, Gothenburg, Birmingham, and Monash for feedback. Particular thanks to Antony Eagle, Shamik Dasgupta, Nina Emery, Dana Goswick, Nicholas Jones, David Kovacs, Dan Marshall, Kristie Miller, Martin Pickup, Michael Raven, Gonzalo Rodriguez-Pereyra, and two OUP referees for extensive and helpful comments on previous drafts, and to David Glick for both extensive and helpful comments and extensive and helpful patience. This work forms part of the project A Framework for Metaphysical Explanation in Physics (FraMEPhys), which received funding from the European Research Council (ERC) under the European Union’s Horizon  research and innovation programme (grant agreement no. ). Funding was also provided by the Australian Research Council (grant agreement no. DP).

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3 Ontic Structuralism and Fundamentality Matteo Morganti

. Introduction Ontic structural realism (OSR) is the view that () in spite of the discontinuities that characterise the historical development of science we can be realist about something, i.e., the concrete counterpart of certain theoretical structures that remain preserved across theory-change; and () such structure is all there is in the actual world, at least at the fundamental level. OSR was introduced by Ladyman () and has been intensively discussed since then (see, e.g., Ladyman and Ross ; French ; and, for further references, Ladyman , esp. section ). The question is still open whether sense can be made of claim () above in a way that does more than just identify the ‘right’ structures in hindsight. As for (), there is the issue whether contemporary science, especially physics, truly urges an ontological revision whereby relational structure becomes fundamental. Conceding, at least for the sake of argument, that there is enough structural continuity in the development of physics, that it suffices for establishing a plausible form of scientific realism, and that it is to be interpreted in an ontic sense, a further query—directly relevant for our present purposes—emerges: what does it mean exactly to consider relational structures ontologically basic? In order to provide an answer to this question, and for it to be as articulated and informative as possible, the way in which the debate about OSR impinges on that concerning fundamentality, and vice versa, needs to be precisely identified. In this chapter, I will first of all claim that, in the present context, the issue of fundamentality is in fact twofold: on the one hand, it is the issue concerning which entities are truly fundamental (if any); on the other, it is the issue concerning how reality is structured based on the fundamental items (if any). Accordingly, I will contend that OSR is a thesis about fundamentality in a dual sense: in addition to the claim that relations are ontologically basic, it must say something concerning the ‘architecture of reality’, to the effect that the latter is, in some sense to be specified, essentially structural. As a matter of fact, while the relation between objects and structures has been extensively studied already (see, e.g., French , ; McKenzie ; Wolff ), and continues to be explored by philosophers, the connection between the debate about OSR and that concerning the metaphysical structure of reality is still underdeveloped. Matteo Morganti, Ontic Structuralism and Fundamentality In: The Foundation of Reality: Fundamentality, Space, and Time. Edited by: David Glick, George Darby, and Anna Marmodoro, Oxford University Press (2020). © Matteo Morganti. DOI: 10.1093/oso/9780198831501.003.0004

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   Here, I will attempt to at least make a start on bridging this gap. In doing so, I will assume a mildly naturalistic perspective (along the lines of Morganti and Tahko ): one in which science and philosophy are in a two-way relationship such that it is not only the case that indications coming from science and the philosophy of science should be brought to bear on the metaphysical debate; at the same time, metaphysical reflection should also be employed, primarily if not exclusively, to gain a better understanding of our best scientifically grounded views of reality. (As will become clear in what follows, my primary focus will be on the latter element, which is perhaps less commonly associated with naturalism). The structure of the chapter is as follows: in Section . I will concisely introduce the key notions employed in extant discussions of metaphysical fundamentality that will play a role here, and in Section . I will do the same with OSR, identifying with more precision the two fundamentality-related questions that arise in connection to it. In Section . I will present and discuss the various ontologies that can be and, partly, have been associated with OSR—with special attention to the quantum domain and the nature of space-time—bearing in mind the two different questions of fundamentality. In Section . I will critically assess these ontologies based on the indications coming from both a priori metaphysical reflection and current science. In particular, I will point out potential difficulties for structuralism as a metaphysical thesis. In Section . I will conclude by briefly presenting an alternative view, which arguably preserves the essential insights of structuralism while at the same time offering a different take on fundamentality.¹

. Grounding, Ontological Dependence, etc. Several philosophers have maintained that there is a sui generis ‘in virtue of ’ relation that does explanatory work that, arguably, cannot be done by any other relation, and cannot be reduced to purely scientific facts and claims. This relation is what many philosophers refer to as ‘grounding’. Here, I will side with supporters of metaphysical explanation, hence disagree with grounding sceptics, according to whom grounding talk adds nothing of value to philosophical discourse (see, e.g., Daly  and Wilson ). Although grounding is often said to be akin, yet not identical, to ontological dependence, in what follows I will more or less equate grounding and ontological dependence,² taking the key idea to be that the relevant relations—whichever is at work, and whatever one may want to call it, in specific cases—enable us to formulate informative philosophical claims concerning the way in which reality is structured. Putting it slightly ¹ In the course of the discussion, I will try to avoid sweeping claims concerning ‘physics’ or ‘quantum theory’ (let alone ‘science’), and offer instead a more nuanced representation of the way in which the various metaphysical views may be interpreted and successfully applied in particular areas of physical inquiry—different ontologies possibly being required for appropriately accounting for the nature of, say, space-time, non-relativistic quantum mechanics or algebraic quantum field theory. It goes without saying, however, that the chapter will remain mostly at the general level and, as far as ‘local ontologies’ are concerned, it will only be possible to hint at possibilities and directions for future research. ² In what follows, that is, I will presuppose that ‘a grounds b’ is more or less a synonym of ‘b depends on a for its existence/properties/identity’ and of ‘a determines the existence/properties/identity of b’. For a useful discussion of grounding and notions of dependence, see Wilson (Chapter , this volume).

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   



differently, allowing for the possibility that ‘grounding’ is just a term that refers to certain typical aspects of philosophical explanations, I will focus on the putative real-world relations that determine the ‘architecture of reality’—independently of whether they actually are sui generis grounding relations, or something more specific.³ The idea that will be scrutinised in what follows is that facts concerning relational structure ground facts concerning objects, and that the architecture of reality is in some sense essentially structural—structuralism, broadly understood, consequently representing a powerful explanans for several fundamentality-related facts. Such assessment requires, first of all, that certain features that are normally attributed to the fundamental architecture of reality, and to the relation of ground itself, be made explicit at the outset. First of all, grounding is typically understood as a relation that gives rise to partial orders, i.e., ‘vertical’ hierarchies in which no element grounds itself (irreflexivity), if a grounds b then b does not ground a (asymmetry), and if a grounds b and b grounds c then a grounds c (transitivity)⁴. In most cases, well-foundedness is also assumed, meaning that grounding chains have an ultimate, ungrounded basis. While there is intense discussion surrounding each one of these features, the resulting foundationalist view of reality is no doubt (still) the dominant one. However, it is not absurd to conceive of universes that lack a foundation, i.e., an ‘ultimate’, ‘basic’, or ‘fundamental’ level. Similarly, it can be argued that grounding relations do not necessarily give rise to ‘vertical’ hierarchies. As we will see later on, this is directly relevant in the context of a discussion of the metaphysical import of contemporary physics, and of the status of physics-based structuralism in particular.

. Ontic Structural (Real)ism As mentioned, structural realism attempts to re-establish the connection between the success of scientific theories and their probable truth, threatened by considerations having to do with the patent discontinuities that characterise the history of science— such as, for instance, the shift from Newtonian mechanics to relativistic physics. It does so by pointing out the structural continuity that exists between (some parts of some) subsequent theories across theory-change. In the terminology of the earliest recent advocate of the view (Worrall ), there may well be radical discontinuity at the level of the (putative) ‘natures’ of things—i.e., individual objects with essential, intrinsic properties. However, very often—if not always—one finds instead a lot of continuity at the level of ‘structure’: that is, at the level of the behaviour of things, corresponding to their extrinsic properties and relations with other things, as mirrored (at least in those domains in which this is possible) by the mathematical ³ Indeed, grounding can be conceived of as an objective, factive relation, i.e., one that exists in the world independently of human practices. An alternative, ‘sentential’, or ‘operational’, view of grounding (Fine ) is available, however, according to which grounding is an operator that connects sentential expressions. This latter view allows one to remain neutral on ontological questions. ⁴ It is possible, perhaps necessary, to work with a many-one conception of ground. If one does this, transitivity must be replaced by a cut rule, and irreflexivity and transitivity require the introduction of partial grounds and appropriate rules for moving from full to partial ground. Also, grounding is normally assumed to be hyperintensional (necessarily co-referring terms cannot be substituted one for the other salva veritate) and non-monotonic (it is possible that a grounds b, but a and c do not ground b).

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   formalism of the relevant scientific theory. Worrall’s well-known example has to do with the shift from Fresnel’s ether theory of light to Maxwell’s electromagnetism: in spite of the radical change in ontology (from a vibration in an elastic solid medium to a wave in a field), the equations describing the behaviour of light remained exactly the same. Typically, the equations of the old theory are not preserved without modifications in the new theory, but ‘only’ as limiting cases of the new equations. However, the basic idea remains the same: the realist can regard preserved formal structure as the (approximately) true part of the relevant theories, and the part responsible for their success. Epistemic structural realists (such as Worrall himself) conceive this as an epistemological position, to the effect that we can be realist about the relations that are described by the (preserved) formal structure of our theories. Ladyman () introduced instead the ‘ontic’ variant of structural realism (OSR), whereby not only is structure all we can be realist about, but also all there is at the fundamental level. Ontic structural realists contend that this move allows one to fill an existing gap between epistemology and metaphysics, and to put indications coming from physical theory and from the history of science together in a virtuous consilience (see also Ladyman and Ross ). In more recent years, supporters of OSR, while still unambiguously presenting their view as a form of scientific realism, have changed their focus. Rather than the history of science and the problem of theory-change, the central issue is now that of identifying the most plausible ontology for our best current science, and in particular for our best current physics. In view of this and of the focus of the present paper, we will set the realism/antirealism issue aside from now on, and focus on the metaphysical content of the thesis, i.e., on ‘ontic structuralism’ (OS) as it is alleged to follow from contemporary physics. What does such content amount to, exactly? The immediate answer is, of course, in terms of structures being ontologically fundamental. Let us make this a bit more precise before proceeding. A key fact OSists refer to is that the theories that are currently regarded as fundamental by the physics community are ‘gauge theories’, i.e., theories containing more variables than the actual number of degrees of freedom of the relevant physical systems, from which the physically meaningful degrees of freedom are ‘selected’ as invariants under certain transformations. The relevant transformations are, in particular, those captured by mathematical entities known as ‘groups’. Based on this, for instance, quantum objects are reconceptualised by ontic structuralists “as representations of symmetry groups, where the symmetries reflect spatiotemporal, i.e. external, and internal degrees of freedom as well as permutation invariance” (Lyre , p. ). In particular, OSists take particles to correspond to the invariant group representations whose states are not decomposable into sub-states which are equally invariant under the relevant transformations—so called irreducible representations. Mathematical groups thus carry direct ontological weight, and in a way that tilts the metaphysical scales towards structuralism. For, groups essentially encode relations between sets of entities, and the former play a more fundamental role than the latter in group transformations. Hence, there is reason, for the realist at least, for giving the status of fundamental entities to relations rather than objects.⁵ ⁵ Of course, the story is slightly different in other domains. For instance, talk of ‘particles’ or ‘objects’ may not be appropriate in the context of quantum field theory (see Glick ), or in the context of spacetime. The key idea remains, however, that relations play a fundamental role, and their relata are derivative,

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   



The same conclusion, according to OSists, follows if, rather than considering general formal features of physical theories, one looks at the peculiar descriptions that such theories provide of their specific domains of application. For a typical example, consider again quantum mechanics, and the phenomenon known as entanglement. As is well-known, at least some quantum composite systems are entangled, which means, roughly, that their physical state encodes information which is irreducible to that contained in the separate component states. For instance, the so-called singlet state tells us that the system composed of particles  and  is such that its components are anti-correlated and, consequently, will necessarily turn out to have opposite spin values in the given spatial direction upon measurement. Yet, taken separately,  and  both have the same probability of having spin up and spin down, which would entail that same-spin outcomes for joint measurements should instead be possible. This is explained by OSists by arguing that the anti-correlations (in our example, ‘ . . . has opposite spin to . . . ’) exhibited by entangled systems correspond to ‘ontologically thick’ relations, not reducible to the monadic properties of their relata and, as a matter of fact, more ontologically basic than particles.⁶ Analogously, a structuralist reading of space-time can be endorsed (see, e.g., Muller ), according to which neither substantivalism (space-time points are fundamental substances) nor relationism (space-time relations are derivative on material objects) is correct, and it is instead the case that spatio-temporal relations—not points or material objects—are primitive and ontologically basic. In the spatio-temporal scenario, as a matter of fact, some (see Pooley ; Stachel ) argue that the structuralist case is even stronger. For, according to these authors, far from there being the sort of metaphysical underdetermination that seems to exist for quantum entities (an underdetermination that, in any case, has been used in support of structuralism—see, in particular, French , ), the evidence is unambiguously in favour of space-time being best interpreted in fully structural terms. A key question concerns the exact ontological nature of these physical relational structures. If they are to be regarded as fundamental, how is this claim to be intended exactly? Answering these questions requires one to use distinctively metaphysical notions and concepts, and in particular those related to ontological dependence and grounding. This invites reflection on a particular question of fundamentality (let’s call it QF): the question concerning which entities are fundamental (and how this is to be translated in precise ontological terms). Notice, however, that there is also another question of fundamentality (henceforth, QF) that deserves attention: the question concerning what view(s) of the ‘architecture of reality’ emerge(s) from a structuralist view of the fundamental items. As we will see, even if it has not been done much in the literature, examining QF alongside QF is quite important in the context of an assessment of structuralism as a metaphysical thesis. Indeed, tackling QF and QF together is not only possible but also advisable, and leads to the identification of several different options, which may shed light on the debate and constitute a useful guide for future discussions of structuralism. if not altogether eliminable. My focus in the chapter will be mostly on quantum entities, hence material objects broadly understood, but I will also consider, more briefly, space-time. ⁶ Entanglement seems to be an even more radical phenomenon in the context of quantum field theory. For discussions, see Clifton and Halvorson () and Lam ().

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  

. Structuralist Ontologies for Contemporary Physical Theories Starting from QF, it must be pointed out, first of all, that OS is in fact a family of philosophical views, sharing the abovementioned assumption that relational structure is (epistemically and) metaphysically fundamental. One possible classification is as follows: (a) Strong OS: there are only relations at the fundamental level, and objects can be dispensed with in discussions of fundamentality, as they are entirely derivative with respect to relational structure; (b) Moderate OS: objects and relations are on a par, as all properties are, at root, relational in nature, but objects also exist as (nothing but) “placeholders” in the relevant structures; (c) Contextualist OS: some properties of objects might be monadic and nonstructurally-reducible, but objects are dependent on structure nevertheless, because identity facts are always contextual, i.e., extrinsically (relationally) determined.⁷ Strong OS is sometimes presented as an eliminative metaphysics, in the sense that the slogan “structure is all there is” should be taken literally (see, e.g., French , ). Of course, though, no OSist denies that we experience objects, whence the need to elaborate on the notion of elimination.⁸ Since this is definitely not an easy task, and the eliminative claim appears in any case stronger than is required by the relevant physics, the recent literature has rather focused on strong OS intended as a ‘priority-based’ claim, one that does not entail elimination/reduction. It is in this latter sense that the position will be intended from now on. French () cashes it out in terms of essential ontological dependence: objects only exist if the relevant structures exists; and the dependence is such that there is nothing to objects— intrinsic properties, identity, constitution, whatever—that is not reducible to structure. McKenzie () suggests that OSists who endorse this strong thesis should rather employ the notion of ontological determination, i.e., grounding: there are no facts about objects and their properties which are not fully analysable in terms of facts about relational structure, since the former obtain entirely in virtue of the latter. The essential idea is clear: QF is answered by strong OSists by excluding objects from the inventory of ontologically fundamental items. How does this relate to QF? To begin with, a recurrent complaint towards strong OS consists in the so-called ‘no relations without relata’ objection. According to it, a metaphysics with only relations at the fundamental level is unworkable, because not ⁷ This use of the notion of contextuality should not be conflated with others. For example, it has nothing to do with contextual dependence in quantum mechanics, which is dependence from the measurement setup. More generally, it is not to be intended as introducing an element of observer-dependence or subjectivity. Rather, it is the claim that (some of the) properties of physical systems—in particular, their identities—are fundamentally extrinsic, and may vary depending on what other physical systems they are related to. ⁸ One option here (Ladyman and Ross ) is to argue that objects are pragmatic devices used by human agents to orient themselves in regions of space-time. In this sense, objects could even be regarded as cognitive epiphenomena.

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   



only are relations defined in terms of objects standing into them, but they ontologically depend on the latter. A possible reaction is to urge a change in the relevant conceptualformal apparatus, so that—regardless of the fact that objects ‘come first’ at the level of commonsense intuition and established linguistic practices—relations truly are regarded as fundamental, our customary ways of thinking and speaking having no direct ontological import. This is the approach recommended, for instance, by French (see, e.g., French , p. , which usefully refers to the work of Mertz ). While this idea is certainly fascinating, it does require a large amount of conceptual revision, and is still work in progress. For this reason, OSists may prefer, and have in fact preferred in many cases, a ‘bite the bullet’ strategy. This may lead to the idea that relations do have relata, but these are always analysable in terms of further relations. The emerging infinite chain of grounding/dependence appears at least implicitly accepted by some authors (e.g., Ladyman and Ross , p. ), thus pointing to an anti-foundationalist variety of (strong) OS, whereby structuralist analyses are in principle always possible. The underlying argument, in more detail, can be reconstructed as follows: () () () ()

There are no relations without relata Every relatum is structurally analysable Hence, there are neither fundamental relations, nor fundamental relata Hence, there is no foundation.

Conclusion () above indicates that strong OS as an answer to QF may be part of the same package with what one may call relational infinitism as an answer to QF. Relational infinitism is the view that, indeed, it is relations all the way down, and the structure of reality does not have a fundamental basis. Although relations are no longer fundamental in the sense that there is a fundamental level and it is occupied by relations only, the package still qualifies as structuralist—for it has it that at each level relata are always dependent on relations (for a defence of metaphysical infinitism, see Morganti ). An analogous line of reasoning is taken by others to point to a form of monistic foundationalism. The underlying idea is simple (see, e.g., Schaffer ): since (the assumption is) there must be a starting point, a ‘source of being’, we should reject conclusion () above, and accept instead a conclusion that preserves foundationalism, although not in its most familiar version. This clearly requires one to modify the premises above. Again putting it in the form of an explicit argument: () () () (*)

There are no relations without relata Every relatum is structurally analysable Hence, there are neither fundamental relations, nor fundamental relata Hence, there is no foundation in the direction in which structural analysis normally goes, i.e., ‘downwards’ () Hence, the foundation must correspond to the whole comprising all structures into a unique, all-encompassing Structure.⁹ ⁹ It is important to point out that this does not mean that the foundation is to be encountered after a finite number of steps. The intuition that there must be a foundation must be kept apart from the intuition that we should be able (at least ideally) to actually get there. For the same reason, infinitism is not the claim that the foundation is infinitely far away: it is the claim that there is no such thing.

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   Again, the fact that at every level relations turn out to be ontologically prior to relata, makes this a structuralist metaphysics. Monism can, however, be formulated in different ways. In particular, French (, p. ) looks favourably at a view of ontic structuralism as existence monism, the view being that there is one fundamental structure with no proper parts, identical to the universe as a whole. This picture of reality surely meshes well with an eliminative attitude towards objects. However, it also requires some sort of error theory accounting for our mistaken impression that there are in fact parts in the all-encompassing One. Alternatively, following Schaffer () and Ismael and Schaffer (forthcoming), one may propose that ontic structuralism be understood as priority monism, the view according to which the one Structure is ontologically prior to its components, but the latter exist nonetheless. While Schaffer’s earliest arguments for priority monism stressed the alleged unacceptability of endless chains of dependence (granting the move from (*) to () above), it must be noted, Ismael and Schaffer focus instead on the peculiar features of entangled systems and, in particular, regard the whole as a common ground for the space-like separated, yet correlated, events occurring in typical EPRBell scenarios (so lending more direct support to () above). In connection to this, it is also possible (Schaffer b) to apply the structuralist claim to space-time, for instance in a supersubstantivalist context whereby objects are identified with (properties instantiated at) space-time regions. In terms of fundamentality, then, once one answers both QF and QF above four possibilities emerge from the strong OSist view that only relational structure is fundamental: () Foundationalist OS: relations are analysable in terms of relata, and there are only relations at the fundamental level; () Relational infinitist OS: relations are analysable in terms of relata, but these always reduce to further relations; () OS as existence monism: relations may be analysable in terms of further relations ad infinitum and this, together with the ‘interconnectedness’ of things of which entanglement is paradigmatic, invites one to invert the intuitive direction of ontological priority and claim that the universe as a unitary Structure is the fundamental entity, with no actual but merely epiphenomenal components; () OS as priority monism: same as existence monism, but without the reduction of the components of the Structure to mere epiphenomena. Each one of these options can be applied either globally or only to specific domains, such as space-time points, or quantum particles, or relativistic fields etc.¹⁰

¹⁰ Something else must be pointed out concerning the dialectics between infinitism and monism. First, (as argued by Bohn  and Morganti  among others) it is far from obvious that, while there may be an infinity of layers on ‘one side’, there is certainly an endpoint on the other, and in particular in the direction of the Whole. Indeed, one may well conjecture that every structure is part of a larger structure. Therefore, () and () actually comprise two different scenarios—one in which grounding/dependence goes ‘downward’ and never bottoms out, and one in which it goes ‘upwards’ and never ‘culminates’, as it were. In fact, the combination of the two is possible as well, with no ultimate step either way, from which a stronger form of infinitism ensues. Further ramifications emerge if one decouples priority/dependence and

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   



Having thus started to see the complexity of the interplay between the debate about OS and the debate concerning fundamentality, especially when one distinguishes clearly QF from QF and seeks answers to both, we can now move on to the other two forms of physical structuralism identified above. Moderate and contextualist OS do not eliminate objects from the list of the fundamentalia. Instead, they give fundamental status to both objects and relations. The moderate position was originally proposed (by Esfeld and Lam ) with a view to accounting for the nature of space-time in structuralist-yet-non-eliminativist terms, so steering clear of the no relations without relata objection: space-time points exist, the claim was, but they are nothing but the “bare occupants” of the relevant places in the spatiotemporal structure. Later, the view was turned into the more general thesis according to which objects cannot be eliminated, yet have no intrinsic features whatsoever. Setting aside worries with symmetric dependence (between objects and relations)¹¹ and the idea of a bare particular with no characterisation whatsoever,¹² it is clear that the force of this view is directly proportional to that which one attributes to the no relations without relata objection (and, of course, inversely proportional to the infinitist and monist answers to it). On a slightly different note, according to some (see Stachel  and Ladyman ), it is really just identity facts that require—or, perhaps, allow for—a structuralist treatment. That identity is contextual, i.e., entirely extrinsic, is argued for essentially on the basis of the following reasoning: (a) If identity were primitive and intrinsic the truth of haecceitism (the doctrine that there can be differences between what distinct possible worlds say about certain individuals inhabiting them that do not correspond to overall qualitative differences between those worlds) would follow; but (b) Haecceitism is contradicted by contemporary science: in particular, our best theories of both material particles and space-time are permutation-invariant, i.e. assign no physical significance to exchanges of exactly similar particles or points (quantum statistics and the diffeomorphisms involved in the so-called ‘hole argument’ in the context of general relativity (see Norton ) are relevant here); therefore, (c) The identity of the basic physical constituents of reality is not primitive and intrinsic to them. In terms of fundamentality, then, as far as QF is concerned moderate OS is the claim that: Symmetric dependence OS: both objects and relations are fundamental—the former being essential to the existence of the latter, and the latter essentially determining the properties and/or identities of the former. part/whole relations, and allows for the two to run in different directions. But we don’t need to get into these details here. ¹¹ We will get back to symmetric dependence later on. ¹² Not even identity, notice: according to their traditional metaphysical characterisation, bare particulars unify and individuate bundles of properties, hence are provided with intrinsic identity—but now identity becomes itself structural.

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   While the no relations without relata objection has no bite in this case, it is important to note that, when it comes to QF, the possibility of reality being infinitely layered has to be taken into account nonetheless. For, focusing for simplicity on part–whole relations, it could be that there are some fundamental simples, i.e., mereological atoms that do not depend on more basic objects, the relations having those objects as relata qualifying as equally fundamental. But it could also be that objects are infinitely analysable in terms of proper parts, leading to an analogous multiplication of relations in spite of the fact that the latter always have relata. Consequently, as in the case of strong OS, one could again claim either that there is no fundamental level (infinitist symmetric dependence OS), or that the only fundamental object—and the only entity that is not symmetrically dependent on anything—is the universal object (monistic symmetric dependence OS, intended as either existence monism or priority monism). As for the contextualist option, the idea that identity facts are extrinsic can be understood in the sense that only objects are fundamental, although they obtain their identity from their being connected to all the other objects that exist in the relevant domain—in which case one doesn’t really depart from the traditional object-based ontology. Alternatively, it can be intended as the claim that objects are fundamental (they do not depend on relations at least for their intrinsic properties) but relations also are (as they do not depend on objects but rather determine the identities of the latter), so collapsing onto symmetric dependence OS as defined above. Either way, it looks as though no metaphysical options additional to those already identified needs to be taken into account as far as contextualist OS is concerned.¹³ In view of the foregoing, we can conclude that the following interpretations of OS as a fundamentality thesis—in the broad sense, i.e., intended as an answer to both QF and QF—can be endorsed: ‘traditional’, foundationalist OS (with a fundamental level inhabited by either relations only or by both objects and relations); infinitist OS (with objects only, or with both objects and relations; going upwards, downwards or both), existence monism OS (with objects only, or with both objects and relations), and priority monism OS (with objects only, or with both objects and relations). The question now is: how is one to evaluate these views, both from the a priori point of view of metaphysical analysis and from the perspective of current science, and possibly choose among them?

. Structures, Objects, and Fundamentalia in Physics: An Assessment Again starting from strong OS, let us now assume that it is at least possible that relations are fundamental and relata are entirely derivative (and that conceptual/linguistic resources to express this thesis can be found). Even so, important difficulties remain. In particular, a seemingly lethal threat emerges from the empirical side. The problem ¹³ In addition to this, scepticism towards the idea that a specific contextualist ontology is worth examining might stem from doubts concerning the strength of the view from a naturalistic perspective. The reason being that contextualist OS makes a claim exclusively about identity, while the initial intuition, and apparent strength, of OS was that many—if not all—properties of physical systems appear to be structural—permutation invariance being only one of the relevant physical facts.

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   



is that, while, for obvious reasons, strong OSists must show that there is nothing else in the essence of objects beyond what is reducible to structure, this appears not to be the case: as argued by McKenzie (), in particular, at least some of the essential intrinsic features of the particles that populate the actual world are not fully analysable in group-theoretic terms.¹⁴ This is not an in principle impossibility, and things may change in the future as the philosophical analysis of physics develops. Until then, however, there is reason for scepticism towards strong OS, at least to the extent that the latter is expected to apply to the fundamental constituents of material objects as described by our best current scientific theories. This, notice, holds regardless of the prospects for a radical revision at the logical and metaphysical level, making relations truly independent of objects for their definition, individuation, and essential nature. Importantly, the story could be different for space-time points, as it seems more plausible that the nature of the latter is indeed wholly determined by extrinsic relations, and is therefore expressible entirely in terms of the metrical features of the spatio-temporal field. This is the story that is told, for instance, by Muller (). Muller claims that if one looks at the General Theory of Relativity and the properties and relations identified as invariants under automorphisms of the four-dimensional space-time manifold, then space-time points are differentiated (merely) by spatiotemporal irreflexive and symmetric relations.¹⁵ Unlike in the case of material objects, it is not absurd to think that there is nothing to be added to these relations, that is, that space-time points have no intrinsic features whatsoever, and are indeed nothing but placeholders in the spatio-temporal structure. We thus see that a fundamentality claim might require a differentiated treatment and application depending on the relevant domain—something that, incidentally, should not come as a surprise from a naturalistic viewpoint. In particular, the sort of ‘no-go’ result holding for the structuralist reduction of the state-independent properties of quantum particles (see above) has no counterpart in the space-time case. On the other hand, here too further work is required. For, irrespective of the status of the relevant relations, and of the ‘weak’ identities that they are alleged to provide spacetime points with, there is more to be taken into account. In particular: . It is far from clear that the structuralist reduction—even granting that it is possible—is recommended¹⁶ given the input coming from physics—at least to the extent that such input consists in the permutation invariance of space-time points, whereby switching two entities with each other, unlike in the classical domain, does not give rise to a new physical state. This, as illustrated above, is often taken to rule out intrinsic natures, especially identities. However, it is in

¹⁴ For, in order to tell apart different particle kinds one has to add something to the analysis in terms of irreducible representations—the latter leaving matters underdetermined. ¹⁵ In particular, Muller employs the light-cone relation, that is, the relation that relates two points if and only if there is some point inside a light-cone of one of them but outside a light-cone of the other. This sort of weak discernibility is invoked, by Muller and others, also in the case of quantum particles, which are (alleged to be) differentiated by the sort of irreducible relations mentioned earlier when discussing entanglement, i.e., relations such as ‘ . . . has opposite spin to . . . ’. ¹⁶ Let alone required: all sensible naturalists agree that no metaphysics can ever follow deductively from the relevant physics.

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   fact not the case that primitive intrinsic identity entails haecceitistic differences. For, what is true of distinct worlds is not univocally determined by the nature of the identity of the objects that inhabit them. A counterpart-theoretic treatment of possible worlds, for instance, may allow one to assume primitive intra-world identities together with anti-haecceitism; and the same would be the case for some sort of ‘Leibnizian super-essentialism’. Additionally, and perhaps more to the point, permutation-invariance does not in fact entail the falsity of haecceitism: for, even if haecceitistic permutations are genuinely possible, this need not have empirically relevant consequences.¹⁷ . The philosophical discussion on the metaphysical nature of space-time cannot but take into account the most recent theoretical developments are in the field of quantum gravity, which attempts to put together in a coherent framework general relativity and quantum theory, and should therefore be regarded as a more fundamental theory. In many of its variants, however, quantum gravity leads to the conclusion that space-time is in fact an emergent entity, hence cannot possibly be fundamental—not even when one focuses on relations rather than objects.¹⁸ So far we have discussed strong OS in connection to QF. What about QF? For reasons analogous to those just mentioned, infinitist versions of strong OS appear weak from the naturalistic viewpoint. For, on the one hand, infinitist strong OS legislates that the world must be such that infinite series of structural analyses can in principle be carried out, hence must itself be infinitely analysable (be it because it is gunky, that is, such that everything has proper parts, or because some other, non-mereological relation occurs at every successive level without ever bottoming out, or culminating, into something fundamental hence unanalysable). However, this is established on merely a priori grounds, i.e., based on purely philosophical arguments against the possibility of relations without relata. If, on the other hand, the view is argued for on empirical grounds, it doesn’t seem to be particularly wellsupported by current physics. For, quantum mechanics and quantum gravity appear to point, if anything, in the direction of the discrete, hence ultimately non-analysable; while general relativity and, in particular, quantum gravity definitely do not lend support to the view that space-time is an infinitely complex basic entity. Something similar holds for the anti-monistic idea that there is in fact no ‘universal structure’ and everything is a proper part of something. For, while cosmological models with infinite series of universes (either in the temporal ¹⁷ Consider for example, the account of quantum statistics suggested in Morganti (), according to which all state-dependent properties of quantum particles in many-particle systems are holistic, or ‘collective’, properties that can only be consistently attributed to the system as a whole. Clearly, an exchange of particles would not make a statistically relevant difference in such a scenario. Analogous considerations may lead one to reject the widespread idea that the ‘hole argument’ is best dealt with by dropping the intrinsic identities of space-time points. ¹⁸ Which is, of course, compatible with the claim that space-time is fundamentally structural. Clearly, this is also relevant in connection to the sort of supersubstantivalist structuralism argued for by Schaffer () (see above), with respect to which it must similarly be stressed that a priori arguments are not particularly effective in the present context, and it is essential to look carefully at all the relevant, most up-to-date physics. For an attempt in this direction in connection to supersubstantivalism, see Lehmkuhl ().

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   



sense of a cyclic universe without beginning or in the spatial and temporal sense of a multiverse in which every universe is a by-product of another (Smith ; Poplawski ; Smolin ) have been formulated and surely deserve attention, at present they definitely qualify as minority views. Regardless of the interest that both physicists and philosophers may have in these infinitist models, at least at the abstract theoretical level, this is no doubt a fact to be taken into account by naturalistically inclined philosophers. What about strong OS as a form of monism, then? Here, there is a crucial question as to how one is to establish whether physics really urges a conceptual revision to the extent that the direction of priority must be inverted and the whole cosmos is fundamental. True, given what we said so far this may appear to be a reasonable option from the point of view of philosophical analysis, and one in favour of which good empirical arguments can be provided.¹⁹ However, for any argument in favour of monism based on the available evidence and a non-empirical foundationalist/structuralist assumption, an equally compelling argument based on the same evidence and the non-empirical assumption that the direction of dependence goes ‘downwards’ can be formulated, leading to some form of non-monistic holism (more on which in a moment). In other words, one might no doubt try to claim that monistic OS fares better than the alternatives when it comes to explaining certain aspects of reality. Yet, while there is unquestionable evidence that at least some parts of reality are mutually interconnected, this need not be understood in terms of monism—nor is monism clearly at an advantage with respect to the alternatives from the explanatory viewpoint. Additionally, surely an empirical reason for choosing existence monism rather than priority monism, or for doing the opposite, will be hard to find.²⁰ The foregoing, notice, also holds for moderate OS in the form of monistic symmetric dependence OS. At this point, however, supporters of OS might argue as follows: since (i) there are good reasons for being structuralists and adopting priority monism and (ii) moderate structuralism minimises the costs of such a move,²¹ then, in spite of the doubts that have just been raised it is nonetheless reasonable to endorse monistic ¹⁹ Schaffer, for instance, does formulate several such arguments (in addition to his argument from the possibility of gunk, which appears quite weak in light of the considerations reported in the main text). One concerns space-time, and we have already looked at it. Another concerns quantum mechanics, and was mentioned earlier—it will be discussed it a bit more in the next section. Yet another one is based on the idea that the cosmos is the only entity to which the laws of nature properly apply (Schaffer ). ²⁰ In connection to this, it is worth pointing out that the very divide between empirical and nonempirical considerations is not straightforwardly identified. For instance, Schaffer takes as a reason for regarding the whole as fundamental the fact that there is a clear asymmetry between emergence (the whole may have properties that the parts taken together lack) and submergence (the parts may have properties that the whole lacks). And he seems to regard this as something like an a priori truth. However, while prima facie this appears unquestionable, empirical considerations (broadly understood) may in fact show that what we intuitively take to be the case is not obviously so. Calosi (), for instance, takes modal interpretations of quantum mechanics to provide (possibly actual) examples of submergence. ²¹ As it avoids the ‘no relations without relata’ objection and the corresponding revision at the linguistic/conceptual level, it does not require the elimination of objects from the domain of fundamentalia, and accounts for the relevant physics without attributing a problematic central metaphysical role to the infinite.

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   symmetric dependence OS—which would thus emerge as a ‘winner by elimination’, as it were. Is this so? Before answering, let us make a crucial point explicit. It has by now emerged clearly that metaphysics is doubly underdetermined—by the evidence, and by the reconstruction of it provided by physical theory. In view of this, in evaluating and selecting our metaphysical hypotheses we should (also) consider extra-empirical, or ‘theoretical’ factors²²—which, incidentally are (or should be) well-known to philosophers of science. However, the assessment of hypotheses on the basis of extra-empirical factors is notoriously problematic, and no mechanical computation of what hypotheses are preferable all things considered is available (and perhaps none is likely ever to be forthcoming). Thus, one should be careful before claiming that a particular metaphysical view is the best option all things considered. In our case, is monistic symmetric dependence OS really the best fit with the physics, once everything has been taken into account?²³ While a conclusive answer to this question would require nothing less than a complete theory of theoretical virtues and hypothesis selection in metaphysics—hence, cannot be provided here—I will, more modestly, close the chapter with a few considerations that may encourage a negative answer and the need to consider further options. The underlying question, in particular, will be: what if we were to drop the assumption that reality is vertically structured into layers or levels?

. Metaphysical Coherentism? In view of the considerations made in the previous sections, to repeat, the structuralist may argue that—methodological complications notwithstanding—it can in any case be agreed that monistic symmetric dependence OS is the only viable metaphysics that remains after a critical scrutiny; and that, a fortiori, structuralism is the best way to go for the metaphysician of science trying to say something about the fundamental architecture of reality. I don’t think this is the case.²⁴ Speaking of extra-empirical virtues, one, often neglected, thing we know is that— regardless of how much physics forced us to revise our beliefs in the past and will probably force us to think in new ways in the future—one of the leading principles of learning from experience and belief revision is certainly what Quine called the ‘minimal mutilation’ of established beliefs. That is, *explanatory power and fit with the empirical data being equal*, we should, and in fact do, always prefer the hypotheses that imply the least revision in the web of beliefs that we currently entertain. As we will see in what follows, there is in fact a metaphysical view additional to those considered so far that not only constitutes an alternative to OS, ²² I will set aside here the issue of whether extra-empirical factors are truth-tracking or instead have merely pragmatic value. ²³ On the other hand, can one establish a best fit between metaphysics and physics without making any assumption going beyond physical theory? ²⁴ The claims and arguments to follow echo those in Morganti (, ). Thompson () also develops a view of reality that allows for ‘metaphysical interdependence’, which she develops based on an analogy with coherentism about justification in much the same way in which we will do in a moment. However, Thompson’s are more general arguments in favour of the non-symmetry of grounding relations, while here we are interested, more specifically, in alternatives to physical structuralism based on symmetric relations of grounding/dependence among objects.

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   



but is also arguably superior to all the various formulations of it once one considers the balance between explanatory power, preservation of established beliefs and fit with the available evidence. The view essentially builds upon a couple of elements that have already been mentioned in passing. To start pinning it down, let us consider first a neighbouring philosophical issue. In the course of a discussion of mathematical structuralism, Linnebo (; –) considers the following view of mathematical objects, which he calls ‘objects depend on objects’ (ODO) mathematical structuralism: (ODO) Each object in the domain of a mathematical structure D depends on every other object in D. That we should take seriously, and indeed endorse, something like a (meta)physical version of ODO (independently of whether or not, in some form, it can be turned into a full-blown version of physical structuralism²⁵) is what I will argue in the rest of the chapter. To see how this may work, and in particular how an ODO-like answer to QF can be formulated and bear on one’s answer to QF, let me now introduce what I will call ‘metaphysical coherentism’. As with coherentist solutions to the problem of justification in epistemology, metaphysical coherentism abandons the idea of a pyramidal structure of directed, asymmetric relations with an ultimate foundation. As in Quinean webs of beliefs, the entities that make up reality are instead mutually related in structures that are at least partly composed by ‘cycles’ or ‘loops’—exactly what ODO recommends and what, recall, we already encountered in the course of the discussion of moderate (i.e., symmetric dependence) OS. An obvious worry, as mentioned in passing in the context of that discussion, is that explanatory relations cannot be symmetric, on pain of vicious circularity and consequent loss of explanatory power—symmetry and transitivity leading to a violation of irreflexivity, hence to things ontologically depending on themselves, and consequently acting as trivial explanantia of their own existence/identity/nature. In reply to this,²⁶ one can argue, first, that the idea that genuine metaphysical explanations must be based on irreflexive relations can be questioned (see, e.g., Fine ; Jenkins ; Correia ). And, secondly, that it is in fact not true that cycles entail the reflexivity of the relevant relations. For, one may reject transitivity and substitute it with something weaker: for instance, quasi-transitivity as understood in social choice theory (allowing for chains in which the relevant relations do not apply at all steps); or quasi-reflexivity (allowing for something to be related to itself if and only if it is related to something other than itself ). Both notions could be used to express the idea that any entity in a given domain may fully ground (i.e., wholly determine) any other entity except itself, and is at most in a relation of partial ground (i.e., partially determines) with itself. Slightly differently, in analogy with sophisticated forms of epistemological coherentism, it could be said that, say, a particle or a space-time point depends on itself at most partially, as it, together with a set of other particles or space-time points, determines the holistic structure that, ²⁵ Something that, for instance, Wolff () is skeptical about. ²⁶ Bear in mind that the alleged problem with symmetric grounding/dependence also affects moderate OS. Therefore, in this respect at least, OS and coherentism stand or fall together.

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   in turn, acts as the ground for some of its features (in particular, features that are not in turn part of the ground for the whole). In this framework, the role played by the whole appears to be irreducible to the mere existence of symmetric cycles of dependence, and sufficient for avoiding triviality charges. Talk of holism leads us back to monism, and in particular to the form of OS as priority monism proposed by authors such as Ismael and Schaffer. Focusing primarily on entanglement, as we have seen, Ismael and Schaffer establish an analogy between common cause explanations and what they call ‘common ground’ explanations. On the basis of it, they conclude that the typical non-separability exhibited by entangled systems is best explained in terms of the entangled particles having the entire composite system as their ontological ground, determining the properties they may be detected to have, and the way in which the latter are interrelated. From this, Ismael and Schaffer move on to suggesting that the whole cosmos is an entangled whole and, as such, turns out to act as a common ground— holism thus entailing, or at least strongly suggesting, the ontological priority of the whole on its parts. Importantly, metaphysical coherentism doesn’t do this, as it does not require the idea that wholes are (or may be) prior to their parts (as a matter of fact, it doesn’t even require the existence of wholes), and simply posits more or less ordinary objects. This entails not only that coherentism is different from monism, but also that it is arguably at an advantage in terms of amount of conceptual revision requested.²⁷ Notice that metaphysical coherentism also steers clear of the claim that all properties are extrinsically determined, hence of the shortcomings of strong, eliminative OS. And while it shares this advantage with moderate forms of OS, analogously sidestepping the no relations without relata objection, at the same time it sticks to a more traditional view whereby objects are truly fundamental and relations ‘merely’ determine their particular ways of being, without ascending to the status of fundamental entities. This is an important point: metaphysical coherentism is similar to moderate OS (and, in particular, to monistic symmetric dependence OS) in that it posits symmetric dependence relations; however, it is at the same time importantly dissimilar, as the symmetrically dependent entities are not objects and relations, but objects only.²⁸ Lastly, unlike infinitist varieties of OS, metaphysical coherentism avoids postulating infinite ‘vertical’ chains of structural dependence (be it upwards, downwards, or both) for merely a priori reasons.²⁹ Summing up, once one accepts the minimisation of revision (together, of course, with the ability to account for the relevant facts at least as well as the alternatives) as a

²⁷ Which obviously entails that coherentism is better than existence monism, as the latter, as explained earlier, adds to priority monism the further cost of requiring an error theory with respect to our experience of a seeming plurality of objects, properties, facts,etc. In the context of discussions of OS, this translates into the possibility of avoiding the elimination of objects and/or the claim that they are mere epiphenomena. ²⁸ That is, while moderate OSists include physical objects and physical relations in the ‘ontological inventory’, coherentism only posits physical objects possibly related by symmetric dependence (nonphysical) relations. This is a gain both in terms of simplicity/parsimony, and closeness to common sense/established beliefs. ²⁹ Notice, however, that the view does not need to rule out such chains: ‘horizontal’coherentist networks may coexist with ‘vertical’, say, mereological structures.

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   



guiding criterion for theory-choice, then there is reason for regarding metaphysical coherentism as preferable to the various forms of OS discussed earlier in this chapter. If, on this basis, one indeed endorses metaphysical coherentism, differentiated ‘local’ applications may follow: for instance, it may turn out that the quantum domain is best accounted for in terms of objects that are (or may be) mutually related—for instance, when they are entangled, or when it comes to the general features of particle-types, captured by group theory; but it may at the same time turn out that coherentism collapses onto moderate OS when it comes to space-time points (if not on strong OS, provided one has a story to tell concerning the purely relational nature of space-time).³⁰ Further details, however, need not and cannot be given here, and for the time being the foregoing sketch of metaphysical coherentism will have to suffice. Whatever one makes of metaphysical coherentism (and structuralism, for that matter), the more general fact emerges that, no matter how ‘close’ to science one develops his/her metaphysics, the under-determination of metaphysical hypotheses by the empirical data is an unavoidable fact that must always be taken into account, lest one be misled in thinking that his/her favourite metaphysics somehow ‘flows naturally’ from science. As we have seen, this entails that pragmatic factors are necessarily relevant when it comes to defining one’s metaphysical interpretation of science—in the present case, one’s understanding of the physical world in terms of fundamentality. Since this is true also in the case of theory-choice in science, this might be a starting point for an argument to the effect that the difference between scientific and metaphysical hypotheses is only one of degree, not of kind; and that, consequently, the only coherent way of avoiding metaphysical questions related to science altogether would be by being a strict instrumentalist about scientific theories. For, only the latter attitude seems to entitle one to reject inference to the best explanation at the level of philosophical analysis, and to consequently accept whatever scientists say without expecting to be able to add anything to that. (Another, much less appealing, alternative is, of course, scepticism.) If one doesn’t go down this road, and sticks to a minimal amount of realism, however, then there is nothing like the natural philosophical understanding of a scientific theory. And even identifying the metaphysics that best fits our best science, or is most successful in taking it at face value, is by no means a task that can be carried out easily.³¹ These considerations, it seems to me, are hardly confinable to a specific case. Rather, it looks like the sort of dialectics partly elucidated in this chapter is ³⁰ In connection to this it is interesting to notice that Bigaj (Chapter , this volume) puts forward his radical structural essentialism as a tool for the substantivalist, whereby the metrical-topological structure defines all the features of space-time points, but the latter are nevertheless genuine existents and not mere epiphenomena (or so I have understood). If the structure is itself considered substantial, then Bigaj effectively endorses moderate OS. If, on the other hand, the only genuine existents are the points— perhaps not allowed to ‘freely recombine’ in Humean fashion and instead subject to specific constraints of reciprocal dependence—then one gets metaphysical coherentism. ³¹ To be clear, I am not claiming that one can do a better or worse job at taking into account the input coming from science when doing metaphysics, and at defining metaphysical hypotheses that fit with contemporary physics. What I am suggesting is that there is no metaphysics that can be uncontroversially be said to take the theory at face value, or fit the latter obviously better than the others. For, given underdetermination and the ineliminable role of non-empirical presuppositions and preferences, establishing the relevant degree of fit cannot be done on entirely uncontroversial, objective grounds.

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   an instance of much more general facts about the nature of metaphysical hypotheses and the relationship between science and metaphysics, with respect to which a sophisticated and moderate naturalistic approach (see, e.g., Morganti and Tahko ) appears highly advisable.

. Conclusions OS as a fundamentality thesis can assume many different forms, which is not only possible but advisable for philosophers to distinguish precisely from one another and carefully assess. To do so, it is important, first of all, to see that OS involves both a discussion of the relation between objects and structures and a discussion of the way in which (physical) structures (and perhaps objects) give rise to the (metaphysical) structure of reality—two related but importantly different (sets of ) questions that I dubbed QF and QF in this chapter. Although no final verdict can be formulated, based on the relevant philosophical notions (essentially, grounding and ontological dependence) and the relevant physical data (essentially, the indications coming from quantum theories, general relativity, and quantum gravity), some claims can be made concerning what answers can be given to QF and QF, which one should be preferred, and on what basis this would be done. First, OS comes either in a strong variety or in a moderate variety. In the former case, as things stand now, it fails in providing a fully structuralist analysis of objects. At best, it is viable in the case of space-time. Moreover, unless a new logical and metaphysical framework is provided in which relations come to be the basic building blocks in a foundationalist context, strong OS adds purely metaphysical and essentially empirically under-determined elements to the evidence coming from physics—in the form of an infinitist or a monistic metaphysics. Moderate OS, instead, might be a more plausible view, establishing a symmetric dependence between two fundamental categories but keeping objects in the picture. However, especially when it comes to QF, it too must be evaluated at least partly on the basis of non-empirical factors and, once this is done, does not appear fully satisfactory. In view of all this, another view may be regarded as a serious contender, if not as the best available option (at least unless one wishes to make the implausible claim that the minimization ceteris paribus of the revision of established beliefs is not an important desideratum in our search for knowledge). This option is what I called ‘metaphysical coherentism’ (a more detailed development of which is provided elsewhere). At least from the viewpoint of a moderate form of naturalism, it could be suggested that, in spite of the several stronger claims made by supporters of OS, what physics invites us to take seriously is not a radically novel view of fundamentality per se, but rather the interconnectedness of things in the holistic sense—where, crucially, it is still the things that qualify as fundamental. Be this as it may, careful reflection on ontic structuralism and metaphysical fundamentality is certainly advisable, if only because it may lead us to realise that certain assumptions that are by and large taken for granted in the literature might have to be questioned, and perhaps even ultimately abandoned. This applies, most notably, to the assumption that reality is structured into vertically arranged layers - an assumption that, vague claims to the contrary notwithstanding, physical structuralists have shared with their opponents so far.

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4 Fundamental and Derived Quantities J. E. Wolff

. Introduction It is fairly standard in contemporary metaphysics to distinguish between fundamental and non-fundamental properties. As I will be using these terms here, this distinction is meant to capture David Lewis’ idea that some properties are perfectly natural, and that these elite properties are what make for objective similarity among objects, as well as doing all kinds of other metaphysical heavy lifting. In order to perform all these metaphysical duties, fundamental properties are usually said to be intrinsic and undefined; they are the properties in terms of which less natural properties are defined, but not vice versa (Lewis ). In keeping with the aim of naturalistic metaphysics, the expectation is that science will reveal to us which properties are fundamental. Standard candidates for fundamental properties on this naturalistic conception are properties like mass and electric charge, which feature prominently in physical theories. A noteworthy feature of these standard candidates for fundamental properties is that they are quantities. Does this mean that all quantities are plausible candidates for fundamental properties, or should we instead expect there to be a distinction between fundamental and non-fundamental quantities, such that only fundamental quantities are candidates for being among the elite properties? In this chapter, I argue that the distinction between fundamental and nonfundamental quantities does not arise from considerations internal to physics or metrology. My discussion is limited to physical quantities, by which I mean attributes like mass, electric charge, temperature,and others, which are often put forward as candidates for perfectly natural properties, and which occur in laws of physics. Quantities are abundant in physics, and there are good reasons to question whether all of them are fundamental properties, as we shall see in Section .. I begin with a recent argument by Hicks and Schaffer (), which observes that laws contain both fundamental and non-fundamental properties. This observation undermines a common assumption in naturalistic metaphysics, namely that only fundamental properties occur in (fundamental) laws of nature. While I disagree with some of the details of Hicks’ and Schaffer’s analysis, I concur with their conclusion that not all J. E. Wolff, Fundamental and Derived Quantities In: The Foundation of Reality: Fundamentality, Space, and Time. Edited by: David Glick, George Darby, and Anna Marmodoro, Oxford University Press (2020). © J. E. Wolff. DOI: 10.1093/oso/9780198831501.003.0005

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 . .  properties that occur in laws of nature are plausible candidates for elite properties. This raises the question of how we might be able to distinguish, from within science, between fundamental physical quantities and non-fundamental physical quantities. Systems of units, which traditionally distinguish between base quantities and derived quantities, are a natural place for such a distinction. After looking carefully at what goes into setting up a system of units, I present three challenges to the idea that the distinction between base and derived quantities can be used to establish a metaphysically robust distinction between fundamental and non-fundamental quantities: (a) different systems of units are possible; (b) the distinction between base quantities and derived quantities is explicitly conventional; (c) the independence of base quantities from each other is explicitly conventional. Finally, I consider whether these challenges can be met by so-called ‘natural systems of units’. Systems of natural units provide something of a response to worries (a) and (c), but do not really help to resolve the problem that the distinction is conventional. I conclude that the distinction between fundamental and non-fundamental properties, at least as far as physical quantities are concerned, is less naturalistic than previously thought.

. Fundamental Properties and Laws It is commonly assumed that only fundamental properties may occur in fundamental laws of nature (Armstrong ; Lewis ). Since the properties occurring in laws of physics tend to be quantities, a close link between fundamental properties and physical laws suggests that some quantities are (candidates for) fundamental properties in the metaphysician’s sense. A common guiding thought has been that quantities that occur in fundamental laws of nature are good candidates for fundamental quantities, and indeed, for being fundamental properties. This thought has often been turned into a requirement, namely that “only fundamental properties can be invoked in fundamental laws”.¹ Accordingly, one might expect that any quantity that occurs in a (fundamental) law is a fundamental property. It turns out, however, that there are reasons to suspect that not all quantities that occur in fundamental laws are in fact fundamental properties. It has recently been argued (Hicks and Schaffer ), that this requirement may be too strong of a connection between laws and fundamental properties, since many laws invoke not only potential candidates for fundamental properties, but also apparently derivative quantities. For example, F = ma, a plausible historical candidate for a fundamental law of physics,² at least prima facie invokes force, mass, and acceleration. Of these three quantities, however, only mass has a claim to be a candidate for a fundamental quantity in the sense intended by metaphysics, argue ¹ This is the formulation offered by Hicks and Schaffer (, p. ), who then go on to criticise this link between fundamental laws and fundamental properties. ² One epistemological concern about the link between fundamental properties and fundamental laws is that it seems to require that we can independently assess the fundamental status of laws and the fundamental status of properties, at least if the link between them is to be a substantive requirement. That is to say, we need to be able to identify candidates for fundamental laws independently of whether they invoke fundamental properties, and we need to be able to say something about whether a property has a claim to being fundamental independently of whether it occurs in a fundamental law.

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   



Hicks and Schaffer. For the force invoked in the law is the net force acting on a physical body, which is in turn the result of any number of contributing forces. If we are to think of force as a fundamental quantity at all, the contributing forces have a better claim to being fundamental than the net force, even though the latter occurs in a fundamental law. Acceleration, on the other hand is defined as the change in velocity over time, with velocity in turn defined as change of position over time. Since a standard expectation for fundamental properties is that other properties are defined in terms of them, but not vice-versa, it seems that acceleration is not a plausible candidate for a fundamental quantity either (Hicks and Schaffer , p.). That fundamental properties are undefined, while some quantities in fundamental laws seem to be defined, hence plays a key role in Hicks’ and Schaffer’s criticism of the link between fundamental laws and fundamental properties. To assess this criticism, we need to get a clearer sense of what it means for a quantity to be irreversibly defined. Hicks and Schaffer argue that force and acceleration are not fundamental quantities, because they are defined in terms of other quantities. The definitions in d2 x question are given by further equations: in the case of acceleration, a= dv dt = dt 2 . The intended reading is that acceleration is defined as change of velocity over time, not just that the value of acceleration changes as the value of the derivative of velocity over the derivative of time. The relation needed for this reading would be that there is a definitional dependency of acceleration on velocity and perhaps on position, but not vice versa. The equation itself does not really yield this definitional dependency, since we can calculate the value of any one quantity given values for the other two, so we need to appeal to other considerations. Hicks and Schaffer point to the role played by position in Newtonian mechanics (Hicks and Schaffer , p. ). Newton’s conception of motion is tied to his conception of absolute space and absolute position, yet absolute positions cannot be recovered from accelerations alone. It is on these grounds that Hicks and Schaffer conclude that the definition of acceleration in terms of position and time is irreversible. They similarly argue for the irreversible dependence of resultant forces on component forces by arguing that the latter cannot be recovered from the former alone. Whether the values of a quantity can be recovered from the values of the others without additional input serves as a test for whether a quantity is irreversibly defined. A quantity whose values can be thus recovered from the values of other quantities, but which is not itself sufficient to recover the values of those quantities, is irreversibly defined. They contrast these cases with cases of reversible definitional dependence, as exhibited by ρ ¼ m=V. Density is defined as mass over volume, yet mass can in turn be defined as density times volume. This definition, according to Hicks and Schaffer (, p. , n. ) is reversible. While no explicit argument is given, Hicks and Schaffer might argue in the following way: from any two of density, mass, and volume, the remaining third can be calculated without additional information. This seems to contrast with acceleration and net forces, where additional information is needed to recover the absolute position of a body, or the component forces acting on a body. Merely knowing the net force or the acceleration is not enough to find out the component forces or absolute positions.

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 . .  Upon inspection, however, the contrast becomes less clear. Volume is usually not taken to be an undefined primitive. Instead, length is taken to be a primitive and volume is defined in terms of it.³ Since the volume formula depends on the shape of the body, the relevant lengths cannot be recovered from mass and density alone, even though volume can. Using the kind of recovery principle employed by Hicks and Schaffer, then, even ρ ¼ m=V begins to look irreversible. Where before we might have worried that quantities are symmetrically interdefined, it now looks as though the recovery test might be too stringent, ruling out apparently reversible definitions as well as irreversible ones. Given that ρ ¼ m=V has been used both to define density (as done in the International System of Units, for example), and to define mass (as historically done by Newton in the Principia),⁴ denying that the definition of density and mass is reversible would seem to go against scientific practice. A second concern for Hicks and Schaffer’s test for the non-fundamentality of physical quantities is that it requires particular interpretations of physical theories. These interpretations are themselves often controversial and may change over time even as the laws continue to be fundamental. Acceleration appears to be irreversibly defined if we accept absolute motion as the quantity with respect to which velocity and acceleration are defined. Yet this is not the understanding of motion and acceleration in neo-Newtonian theories, which might seem more relevant for evaluation given that Newtonian absolute space is no longer considered a live part of physical theorising.⁵ The concern is not that F=ma is not a plausible candidate for a fundamental law of nature, which I’m happy to grant. But the irreversibility of the definition of acceleration requires the Newtonian interpretation of acceleration as a change in motion, and of motion as motion with respect to absolute space. But as the further development of physical theorising shows, these interpretations are optional, and arguably not recommended in light of more recent theories. Once we take into account further interpretations of quantities, however, we should note that other interpretations of these equations are possible. Whether we should interpret an unrecoverable quantity as more fundamental is an open interpretative question. In some cases, like kinetic energy, the appeal of the newly introduced quantity is precisely that it allows us to do calculations in the absence of detailed information about the mass and velocity of the particles composing the system under investigation. Net forces might be thought of in a similar way: they are epistemically more accessible quantities that suffice for many purposes. In both cases one might arguably conclude that the ‘true causes’ can be found in the masses, velocities, and contributing forces at work in a given situation, and that the latter are hence ‘more fundamental’.⁶

³ As we shall see below, once we distinguish a quantity and its dimensions, the sense in which volume is defined in terms of length becomes clearer. ⁴ “Quantity of matter is a measure of matter that arises from its density and volume jointly” (Newton [] , Definition ). ⁵ Hicks and Schaffer acknowledge this point, but set it aside as not central to their argument. ⁶ Such a conclusion is by no means mandatory, though. Cartwright (, ) famously argues that only resultant forces exist, whereas Creary () defends the reality of component forces. For recent discussions see Wilson () and Rowbottom ().

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   



The relationship between (absolute) position and acceleration is quite different, by contrast. Accelerations are not quantities introduced to describe the behaviour of a system where we might not know the nitty gritty details of the motions of component particles. Instead, accelerations are empirically observable difference makers, whereas absolute positions are not. The ultimate conclusion we’ve drawn is that there are no absolute positions. A similar example is provided by the relationship between electric field strength and electric potential. The strength of an electric field at a point, mathematically, is the negative gradient of the potential, and the exact value(s) of the potential cannot be recovered from the field strength alone. Yet the electric field is typically regarded as a bona fide physical quantity, whereas the potential is regarded as a mere mathematical quantity. The reason is that absolute values for potentials make no observational difference; only differences between potentials make an empirical difference.⁷ The case of acceleration compared to absolute position, and to some extent even compared to velocity, is quite similar. While we can of course locate bodies at relative positions and describe their relative motions, any absolute position, and indeed, following relativity theory, any absolute motion, remain elusive. Our inability to recover particular values for quantities like absolute position or absolute potentials from the relevant laws does not indicate that these quantities are more fundamental, then. On the contrary, it might indicate that the quantities are not physically significant, because they fail to make an empirical difference. Hicks’ and Schaffer’s observation that defined or derivative quantities occur in laws of nature is correct, but their alternative proposal for distinguishing fundamental and non-fundamental quantities heavily depends on the interpretation of particular theories. The argument relies on importing the idea that the less fundamental is defined in terms of the more fundamental from the metaphysics discussion to the situation in physics. As we’ve just seen, however, it is quite difficult to say, exactly when physical quantities are defined in terms of one another, and even where such definitions are offered, it is not clear that the ostensibly defining quantities qualify as more fundamental, all things considered. Indeed, as the examples of acceleration and electric field strength suggest, quantities that make an empirical difference may count as more fundamental even when they were mathematically defined in terms of quantities that do not. This suggests that definitional relationships between quantities are difficult to establish from laws alone, and that considerations from physics might speak against taking apparently defined quantities to be less fundamental than their definientia. Instead of using Hicks’ and Schaffer’s criterion of irreversible definition, the application of which requires a commitment to particular interpretations of physical theories, we might turn to metrology for help. At first glance it might be surprising to turn to metrology, since systems of units are a paradigmatic example of conventional stipulations. The reason systems of units are relevant to the question of fundamental quantities is that systems of units explicitly distinguish between base and derived quantities. Moreover, this distinction is then used to describe the relationship not just between quantities but between the dimensions of quantities. If we suspect, as

⁷ As was vividly explained by Maxwell in his elementary treatise on electricity (Maxwell , pp. –).

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 . .  Bradford Skow () has recently argued, that the dimensions of a quantity reveal something about the nature of that quantity, the relationship between the dimensions of quantities might be more indicative of the fundamental or non-fundamental status of a quantity than interpretations of the status of the quantity within particular theories. If primary or base quantities in the metrological sense are fundamental in the metaphysician’s sense, this would offer another way of reading off fundamental properties from scientific theories. Indeed, prima facie it would also offer a way of finding fundamental properties without having to detour through laws at all, thereby circumventing the problem of having to determine which laws ought to count as fundamental. To assess the viability of this strategy, we need to take a closer look at the way in which metrologists draw the distinction between base quantities and derivative quantities.

. Systems of Units and the Distinction of Quantities into Base and Derived Quantities Metrologists traditionally introduce a distinction between base and derivative quantities in the context of setting up a system of units of measurements. The most widely used system of this kind is the International System of Units (SI), with the base quantities length, time, mass, thermodynamic temperature, electric current, amount of substance, and luminous intensity (BIPM ). In the SI, these base quantities are given with the following base units: the metre, the second, the kilogram, the Kelvin, the Ampere, the mole, and the candela. From these, all other quantities and their units are defined. Base quantities act as undefined primitives in a system of units, and are hence prima facie plausible candidates for fundamental quantities.⁸ To set up a system of units, “it is necessary first to establish a system of quantities, including a set of equations defining the relations between those quantities” (BIPM , p. ). The relevant equations are drawn from standard physics and engineering textbooks. As examples the SI offers F=ma, relating force, mass, and acceleration, or T=/ mv², which relates the kinetic energy of a particle to its mass and velocity. Within this system of quantities, a distinction is then drawn between base quantities and derived quantities, and it’s worth noting that metrologists tend to emphasise that this distinction is conventional: “From a scientific point of view, the division of quantities into base quantities and derived quantities is a matter of convention, and is not essential to the physics of the subject” (BIPM , p. ). The small set of base quantities is then associated with corresponding base units; while the base units are defined in the SI, the base quantities are typically left ⁸ To avoid confusion, it is perhaps worth pointing out that all units, including base units, are defined in a system of units. The definition of base units is quite different in character from the sense in which derived quantities are defined. Defining a base unit for a quantity might be understood as something like reference fixing. Defining a quantity in terms of other quantities, by contrast, is a kind of derivation of the defined quantity from the base quantities using the equations assumed as part of the ‘system of quantities’. To say of a quantity that it is defined suggests an asymmetrical dependence relation of that quantity on some other quantity or quantities, which seems to be exactly what we are looking for when trying to distinguish fundamental from non-fundamental quantities.

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   



undefined. The equations relating the different quantities are needed to guide the introduction of derived units for the non-base quantities in the system of units. The units for derived quantities are typically arrived at by replacing the quantity equations with “unit-equations”, in which the familiar quantity symbols are replaced by the standard units of the base quantities. For example, T=/ mv² becomes Joule = kg m²/s².⁹ Systems of units associate each base quantity with a base dimension as well as a base unit. The base dimensions of the SI are as follows: length L, mass M, time or duration T, electric current I, thermodynamic temperature Θ, amount of substance N, and luminous intensity, J. These base dimensions can then be used to define the dimensions of derived quantities in terms of the base dimensions. “In general the dimension of any quantity Q is written in the form of a dimensional product, dim Q = Lα MβTγIδΘεNζJη where the exponents α, β, γ, δ, ε, ζ, and η, which are generally small integers which can be positive, negative or zero, are called the dimensional exponents” (BIPM , p. ). This general fact makes it possible to keep the set of dimensions small. The units of derived quantities are found by looking at their dimensions, and their dimensions in turn are found by looking at the relations in which they stand to other quantities.¹⁰ Quantities and units are abundant, while the set of dimensions remains small. As far as metrology is concerned, a quantity is defined in terms of another quantity (or quantities), if its dimensions definitionally depend on the dimensions of these quantities. This helps to shed light on some of the defined quantities discussed in earlier sections of the chapter. The dimension of volume is L³, that of velocity is LT1, that of acceleration is LT2. Unlike the equations among quantities, the relations among their dimensions can be understood as relations of asymmetrical dependence or irreversible definition. For whereas we can easily reverse equations to calculate the value of whichever quantity we are most interested in from given values for the other quantities, the situation is quite different for dimensional equations. These equations do not relate changeable values, which suggests a symmetric supervenience relationship for equations among quantities. Instead, they seem to make explicit the dependence of the dimensions of derived quantities on the base quantities, by showing how the dimensions of derived quantities can be defined in terms of the dimensions of base quantities. The requirements for setting up a system of units make it clear very quickly that looking to metrology does not avoid reference to laws altogether. After all, the equations relating quantities will typically be laws. What is interesting, however, is that the laws are not there to identify the base quantities, since those are determined by convention, but rather to relate the derivate units to the base units in a coherent fashion. This makes it clear that if metaphysicians wish to draw on the base quantities

⁹ More precisely, each base quantity has an associated base dimension and an associated base unit. The relations between base units and derived units are strictly speaking relations between dimensions, not units. ¹⁰ Interestingly, this epistemic direction can be turned around. Sometimes we might not know the exact form of the equation relating several quantities, but we do know what the units should be. Dimensional analysis can then help determine what the equation must look like.

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 . .  of systems of units as candidates for fundamental quantities, they need to accept that laws of nature relate fundamental and non-fundamental quantities, instead of involving only fundamental quantities. Hicks’ and Schaffer’s observation that fundamental laws sometimes contain derivative quantities begins to look like a standard feature of laws, if we assume that the relevant sense of derivative quantity is given by metrology. We still need to appeal to an external criterion to distinguish fundamental and non-fundamental quantities, but the explicit distinction into base and derived quantities drawn by systems of units might seem to provide such a criterion. Systems of units, then, draw a distinction between base elements and derived elements: we have base quantities, dimensions, and units and we have derived quantities and their units, where the latter are determined by finding the right combination of base dimensions. For the purposes of finding a distinction between fundamental and non-fundamental quantities, the distinction between base and derived quantities might seem promising, but as we shall see next, there are problems with appealing to this distinction for the purpose of drawing a metaphysical distinction between fundamental and non-fundamental quantities.

. Problems with Taking the Base Quantities to Be Fundamental Properties There are three obvious problems with naively taking the base quantities of the SI to be fundamental quantities. The first is that the SI is only one among many different systems of units. There are two respects in which systems of units can differ from one another. A system of units might use the same base quantities and dimensions as the SI, but use different units as base units, for example by using the inch as base unit for length and Fahrenheit as base unit for temperature. As long as the base dimensions and quantities are the same as those of the SI, this alternative system of units and the SI will belong to the same class of systems of units.¹¹ By contrast, different classes of systems of units employ different quantities as base quantities. For example, while the SI recognises seven base quantities, the Gaussian system of units only recognises three base quantities—length, time, and mass—and standardly uses centimetre, second, and gram as their base units. Units for other quantities, including electromagnetic quantities, are then defined in terms of these three base units. While conversion from Gaussian to SI units is straightforward for mass, length, and time, which occur as base quantities in both systems, the conversion of units treated as derived in the Gaussian system to units treated as base units in the SI is more complicated. The Gaussian system and the SI belong to two different classes of systems of units, because they differ in their base quantities. Systems of units within the same class of systems of units do not make a difference for the question whether we can take the base quantities of our systems of units to be fundamental properties. Since all systems of units within the same class agree on

¹¹ Skow calls systems of units systems of “scales” and similarly distinguishes classes of systems of scales with different base dimensions from systems of scales within the same class (Skow , p. ).

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   



the base quantities, the plurality of systems within the same class is compatible with the view that their base quantities are (uniquely) fundamental. That the choice of unit is conventional is compatible with the view that the choice of base quantities reflects which quantities are fundamental. The existence of multiple different classes of systems of units, by contrast, presents a potential problem for the view that the base quantities of our systems of units are fundamental quantities. Since different classes of systems of units use different base quantities, we face a plurality of candidate sets for fundamental quantities. This raises the question whether there is a preferred class of systems of units. Skow suggests that to find the relevant base quantities, we should look to systems of units used by scientists when solely concerned with the formulations of laws (Skow , p. ). The contrast is meant to be with situations in which computational ease determines the choice of units, and perhaps also with cases where quantities are chosen as base quantities because their units are easier to realise as measurement standards (as in the case of electric current). It’s not clear, though, that the purpose of being concerned with the formulation of laws is specific enough to single out a unique preferred system of units. As we’ve seen, some laws are presupposed when setting up the system of units to relate quantities to one another. This is vital for constructing derived units from base units in a systematic fashion. Choosing a system of units does not proceed independently of what the intended system of laws is supposed to be. More importantly, since the laws relate base quantities and derivative quantities, even if scientists are only concerned with the formulation of the laws, the distinction into base quantities and derivative quantities can be made even after the laws have been agreed upon. The same set of laws can give rise to different systems of units using different quantities as base quantities. Perhaps what Skow is getting at is that we should take seriously base quantities in systems of units that somehow reflect their status in physical theory, not their computational or operational convenience. Doing so might explain why many physicists (and philosophers) would prefer to use electric charge as a base quantity, not electric current, even though the latter, but not the former is a base quantity in the SI. But this means using yet another way of determining which quantities qualify as fundamental, different from either the laws or the system of units in question. We shall return to the question of a preferred system of units in Section .. The second problem is that the SI itself emphasises that the distinction between base quantities and derived quantities is conventional. Not only do different systems of units employ different base quantities, but even that there is a distinction between base quantities and derived quantities is regarded as a matter of convention. This is important to note, because otherwise one might propose that the different systems of units have the status of different theories, with at most one of them getting it right. More specifically, one might suggest that some class of systems of units gets it right in that it picks the right base quantities and dimensions. The choice of base quantities and dimensions does not uniquely determine the choice of units and one might well hold that the choice among different systems of units in the same class of systems of units is entirely conventional, while holding that the choice of class of system of units is not. This seems to be Skow’s proposal, since he suggests that there is a distinction

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 . .  between base (primary) and derivative (secondary) quantities that holds not just relative to a system of units (Skow , p.).¹² A third concern is that while the SI “conventionally” stipulates the base quantities to be mutually independent, there are interdependencies in the definitions of their units. For example, the definition of the mole—the unit of amount of substance— uses the kilogram, and the definition of the metre involves the speed of light and thereby the second. While these interdependencies hold, in the first instance, among the definitions of the base units, they reflect relationships among the base quantities as expressed in equations relating them. This suggests that we should not take the independence of the base quantities from one another in the SI to be indicative of their status in physical theorising—physics may very well regard some of them as heavily dependent on others. All of this should give us pause when pursuing the strategy of determining which quantities are fundamental by starting from the distinction between base quantities and derived quantities. A possible solution to these concerns might be to find a preferred set of units, or at least a preferred class of systems of units. If such a class of systems of units could be found, its base quantities would make for plausible candidates of fundamental quantities. So called ‘natural’ systems of units might seem like a plausible candidate for preferred systems of units. Unlike conventional systems of units, natural systems appeal to physical constants to define units. In the next section we shall see how such systems cope with the challenges outlined here.

. Natural Units and Universal Constants The idea of using physical constants, such as the speed of light in vacuum in the definition of units goes back to the nineteenth century, but has recently become quite central to metrology. With the  reform of the International System of Units, all SI base quantities will be defined in terms of physical constants. What do such definitions look like, and can systems based on physical constants provide a response to the worries raised in the previous section? Physical constants are an appealing choice for defining units, because their values are presumed not to change, yet this stability is a matter of nature, so to speak, not of human intervention or stipulation, as might be the case with a material prototype. We first have to distinguish between dimensional and dimensionless constants. A dimensionless constant is simply a number, such as the fine structure constant α /. The constants used to define units, by contrast, are dimensional. The dimensions of physical constants are determined by the laws in which they occur together with the requirement that these laws are dimensionally homogeneous. The dimensions of physical constants ensure the dimensional homogeneity of equations relating quantities with different dimensions. The dimensions of a physical constant hence reflect the form of the law relating the different quantities. For example, Planck’s constant h occurs in the equation E ¼ hυ, where E is the energy and υ the ¹² Skow explicitly connects drawing this distinction in a non-relativized manner to the notion of fundamental properties, but admits that he has no independent argument for thinking that the distinction holds independent of a particular class of systems of units (Skow , p.).

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   



frequency of a photon. As a result, Planck’s constant has dimensions [energy][time], although as we shall see, this is not the only way to think about the dimensions of Planck’s constant. There are two different ways in which constants can be used in the definition of units: we can either ‘fix’ the numerical value of the constant in particular units, or we can set the constant itself to . The first strategy is used in the new SI, the second strategy is used by so-called natural systems of units. Let’s look at the two strategies in turn to see what the consequences of either strategy might be for our problem of finding fundamental quantities. Even prior to the current SI reform, the metre was defined in terms of the speed of light. This was done by fixing the value of the speed of light in vacuum at ,, metre/second. By fixing the exact value of c in certain units, namely metre and second, the metre can then be defined as the distance light travels in /,, seconds. By fixing the value of a natural constant (in certain units), this approach to defining units avoids reliance on artefacts or particular physical systems. Previously the metre had been defined, first, as the length of a particular material prototype, and later as a fixed multiple of the wavelength of orange light of krypton-. In the new SI, all units will be defined by fixing the numerical value of constants, using the following constants: hyperfine structure transition frequency of Caesium ΔvCs to define the second, the speed of light in vacuum c to define the metre, Planck’s constant h to define the kilogram, elementary charge e to define the Ampere, the Boltzmann constant kB to define the Kelvin, the Avogadro constant, NA to define the mole, and luminous efficacy, Kcd to define the candela (BIPM ). Each of the base units thereby receives a defining constant with a fixed numerical value. How do the three concerns regarding the conventionality of the choice of base quantities play out in the new SI? In the first instance, the new SI marks a change to the definitions of the base units, so it is not obvious that this change should affect the question of whether the SI base quantities are fundamental. An interesting consequence of the reform, however, is that the distinction into base units and derived units is only maintained for the sake of tradition. “Defining the SI by fixing the numerical values of seven defining constants has the effect that this distinction [into base units and derived units] is, in principle, not needed, since all units, base as well as derived units, may be constructed directly from the defining constants” (BIPM , p. ). The reason it is possible to construct derived units directly is that dimensional constants can be expressed in different SI units. For example, Planck’s constant, h is usually expressed in units J s (Joule times second). The Joule is not an SI base unit, but if we fix the numerical value of h, we can ‘construct’ the Joule directly, without detour through the kilogram. The new SI retains the distinction into base and derived units mainly for tradition. If the distinction into derived and base units was ‘conventional’ before, it has now become obsolete as far as metrology is concerned. If the distinction between base units and derived units is obsolete, how can we determine which quantities in the new SI might be candidates for fundamental quantities? Before the reform, the base quantities were the only obvious candidates, but since the reform, different quantities also have a claim to being considered fundamental from the perspective of the SI. The quantities related by Planck’s

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 . .  constant are energy and frequency, neither of which is among the traditional base quantities of the SI. Their status as derived quantities means that their dimensions are expressed in terms of the base dimensions of the SI: ML²T2 in the case of energy, and T1 for frequency. When we try to find out which quantities are fundamental from the new SI, then, we are confronted with a bit of a puzzle. Not only does the new SI state that we no longer need a distinction between base and derived units, it also seems that by using constants to define units, more quantities are ‘brought in’, so to speak, because the laws in which these constants occur relate quantities that are themselves not among the base quantities of the SI. In this respect the new SI differs from the template on which the reform is based, the redefinition of the metre. The defining constant for the metre, the speed of light in vacuum, relates two quantities that are already base quantities in the SI: time and length, with base dimensions T and L respectively. For the speed of light, then, there was no question of which dimensions the constant has, since it was already expressed in terms of base dimensions of the SI. By contrast, h, but also elementary charge, e, have dimensions that are not base dimensions of the SI and need to be expressed in terms of base dimensions of the SI before they can be used to define ‘base’ units of the SI. Two further observations confirm that base and derived quantities in the new SI are not helpful for determining fundamental quantities. Since each ‘base’ unit receives its own defining constant in the new SI, the SI retains its non-minimal set of base units. Other systems of units, with different sets of base units hence remain possible. Moreover, which constants to use as defining constants is (to a certain extent) a matter of choice. For instance, whether Avogadro’s number or the molar mass of carbon- is fixed seems to be a matter of choice. Similarly, the reason to use Planck’s constant, rather than the gravitational constant, G, is that the former can be measured with much greater precision than the latter. While this is a compelling reason for metrological purposes, from a philosophical perspective this would be an excellent epistemological reason to prefer Planck’s constant, but by itself not an indication that the gravitational constant is insufficiently fundamental. The new SI is an odd hybrid, in that it uses dimensional constants to define the units, while retaining features of the old SI, including the now obsolete distinction into base and derived units and the same seven quantities as the basis for the system of units. For the purposes of metrology this might be fine, but if we are looking to recover something like the distinction between fundamental and non-fundamental quantities, the new SI is not promising. In using physical constants to define units, the SI is moving in the direction of what is sometimes called a natural system of units. Natural systems of units avoid definitions of units in terms of artefacts or other anthropocentric elements. Instead, definitions of units are based on universal physical constants. The new SI is moving in the direction of such a system, but stops short of becoming a full-fledged system of natural units in two respects. First, it retains the same seven base units as before. A full-fledged system of natural units sets the defining constants equal to unity, that is, it turns the constants into the (base) units of the system. The new SI, by contrast, explicitly fixes the values of the defining constants when expressed in current SI units, in order to then express the old SI units as multiples (or fractions) of the fixed values of the constants. The constants are hence used to define the units,

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   



but are not themselves the units. Second, precisely because it does not set the defining constants to unity, it can retain seven distinct defining constants, whereas that is not possible in full-fledged systems of natural units. Natural systems of units, by contrast, will choose a limited number of constants to be set to unity and this choice is constrained by interdependencies among the constants.¹³ How do the three concerns regarding the conventionality of the choice of base quantities play out in a fullfledged system of natural units? Planck’s system of natural units, which is perhaps the best known system of natural units, uses the following five defining constants: the speed of light, c, the h the Boltzmann constant gravitational constant G, the reduced Planck constant ℏ ¼ 2π 1 kB and the Coulomb constant ke= 4πε0 , where ε₀ is the permittivity of free space. In Planck units c ¼ G ¼ ℏ ¼ kB ¼ ke ¼. The resulting system of units is commonly used in theories of quantum gravity.¹⁴ Planck units do better than conventional systems of units when it comes to the interdependencies among defining constants. Because the defining constants in a fully natural set of units are set to unity, the relations between them have to be taken into account (see note  above). The claim that c, G, and ℏ can all be set to  in a coherent system of units is not conventional, but is constrained by the relationships between these constants. Planck units are not the only possible system of natural units: other constants can be chosen as the constants to be set to unity. A historically important example are Stoney units, which set c ¼ G ¼ e ¼ ke ¼, where e is the elementary charge. Planck units might nonetheless seem particularly natural even among natural systems of units, because the defining constants are ‘universal’ in that they do not depend on the properties of any particular system or entity. By contrast, Stoney units, which use the charge of the electron, e, as a defining constant invokes a particular type of entity. Furthermore, on some accounts of quantum gravity, Planck length is a ‘minimal length’, either in the sense of being the shortest length we can experimentally probe even in principle, or in the sense of there being no shorter length at all, perhaps because more fundamental structures are not spatio-temporal.¹⁵ If so, Planck units might seem to hold special significance even among natural systems of units and can lay claim to being a naturally preferred set of units. Does this mean we finally have a distinction between fundamental and non-fundamental quantities? Planck base quantities are usually understood to be (Planck) length, mass, and time, units for which can be found through combinations of the defining constants. qffiffiffiffiffi qffiffiffiffiffi ℏG , Planck time is t ¼ For example, Planck length is lP ¼ ℏG 3 P c c5 , and Planck mass is ¹³ A good example of these interdependencies is provided by the fine structure constant, α, which is a dimensionless constant approximately equal to /. Since α is dimensionless, it cannot be set equal to . 2 ee . As a result, a coherent system of units Several important dimensioned constants are related to α by α ¼ kℏc cannot set all of them equal to . ¹⁴ Sometimes Planck units are regarded not merely as a (mathematically) convenient set of units to use, but as characteristic of the scale at which a theory of quantum gravity can be found (Planck scale). This stronger interpretation of the role of Planck units is more controversial, however (see Meschini  for discussion). ¹⁵ Wüthrich (Chapter , this volume) argues that the non-fundamentality of spacetime has farreaching consequences for metaphysics. Different scenarios for minimal length in the context of quantum gravity are explored in Hossenfelder ().

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 mP ¼

. .  qffiffiffiffi

ℏc G.

This choice of base quantities, and hence base dimensions, is not forced

upon us by choosing the defining constants of the Planck units as our defining constants. In the first instance, c is a velocity, so one might expect that we should treat velocity as a base quantity and length and time as derived quantities. Similarly, ℏ has units of mechanical action (energy times time), and so we might conclude that mechanical action is more appropriately considered a base quantity than energy, let alone mass.¹⁶ Both in Planck units and in the new SI, base dimensions of length, time, mass are presupposed and extracted from the defining constants. But the defining constants themselves do not determine which quantities are to be treated as base quantities. The dimensions of a physical constant do not by themselves reveal which dimensions, if any, are base dimensions, just like the laws do not reveal which quantities, if any, are fundamental quantities. In fact, a general concern about systems of natural units is that the dimensions of the defining constants become invisible once the constants are set equal to . Whether the dimensions of c are LT1 or whether [velocity] is a base dimension is not clear in such a system. If there is a deep distinction between base quantities and derived quantities, full-fledged natural systems of units fail to express such a distinction. Even if we take Planck units as a preferred set of units, we do not thereby arrive at a unique set of base quantities that could be candidates for fundamental quantities. Natural systems of units certainly provide a less arbitrary basis for unit definitions. But this does not help with the original problem of distinguishing between more and less fundamental quantities. The main problem is that natural systems of units can be built from a set of dimensional constants, but that this leaves it open which quantities or dimensions should count as base quantities or dimensions in the new system of units. Indeed, such systems of units arguably render the distinction between base quantities and derived quantities obsolete. This undermines, rather than strengthens, the claim that the distinction between fundamental and non-fundamental quantities can be found in physics or metrology.

. Conclusion I’ve argued that the distinction between base quantities and derived quantities as found in systems of units is unsuitable as a basis on which to distinguish between quantities that are fundamental properties and those that are not. Perhaps this result won’t seem surprising, since systems of units are expected to be highly conventional. Systems of units are nonetheless relevant to the question of fundamental quantities, because the choice of base quantities determines the base dimensions. Since the definitional relationships between quantities are best understood to hold between their dimensions, rather than between the quantities directly, we need to detour through systems of units to ensure that we’ve selected a coherent set of base dimensions. Both natural and traditional systems of units show that we do not need to treat all quantities in fundamental laws as fundamental, since we can derive the dimensions ¹⁶ Compare van Remortel () for reasoning along these lines.

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   



of some of them from a small set of base dimensions. At the same time, the very distinction into base dimensions and derived dimensions becomes obsolete in the context of a natural system of units. The development of natural systems of units, which depends on selecting a set of physical constants as defining constants, seems to indicate that what is fundamental are not particular quantities, but the laws that hold between them, since the dimensions of the physical constants depend on the laws in which they occur. The discussion in the first part of the chapter showed, however, that laws by themselves do not single out particular quantities as fundamental. This leaves us with a puzzle about how to capture the idea that some quantities are more fundamental than others in a naturalistic fashion.

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5 Privileged-Perspective Realism in the Quantum Multiverse Nora Berenstain

. Metaphysics and Perspectival Facts “Consider a world made up of pointlike particles moving around in threedimensional space. In four-dimensional spacetime—the bird perspective—these particle trajectories resemble a tangle of spaghetti. If the frog sees a particle moving with constant velocity, the bird sees a straight strand of uncooked spaghetti. If the frog sees a pair of orbiting particles, the bird sees two spaghetti strands intertwined like a double helix. To the frog, the world is described by Newton’s laws of motion and gravitation. To the bird, it is described by the geometry of the pasta – a mathematical structure. The frog itself is merely a thick bundle of pasta, whose highly complex intertwining corresponds to a cluster of particles that store and process information.” (Hut, Alford, and Tegmark , p. )

“I am a Tralfamadorian, seeing all time as you might see a stretch of the Rocky Mountains. All time is all time. It does not change. It does not lend itself to warnings or explanations. It simply is. Take it moment by moment, and you will find that we are all, as I’ve said before, bugs in amber.” (Kurt Vonnegut Jr. )

Metaphors of the bird perspective and the frog perspective allow us to visualize certain aspects of the metaphysical pictures that result from our efforts to make sense of various physical systems, structures, and phenomena. Alternatively called the God’s-eye view and the ant’s-eye view (Silberstein, Stuckey, and McDevitt ), these two viewpoints delimit the perspective of an observer viewing or modeling a system from an external vantage point and an observer experiencing that system from within, respectively. When a physicist studies the equation for Newtown’s second law of motion, they theorize the laws from the external, mathematical bird perspective. When they watch someone throw a Frisbee, they observe its motion from the internal frog perspective. As sentient observers, we are capable of making inferences and observations from either or both of these perspectives. We form our quotidian picture of the world by combining inferences and knowledge obtained Nora Berenstain, Privileged-Perspective Realism in the Quantum Multiverse In: The Foundation of Reality: Fundamentality, Space, and Time. Edited by: David Glick, George Darby, and Anna Marmodoro, Oxford University Press (2020). © Nora Berenstain. DOI: 10.1093/oso/9780198831501.003.0006

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- 



from both. Often, there are tensions between these two pictures. When such tensions arise then, which picture ought to be ascribed metaphysical priority? In this chapter, I address two applications of the question of whether the bird view or the frog view is metaphysically fundamental. I raise several concerns about approaches to metaphysics of science that prioritize the fundamentality of the frog view over the bird view. Specifically, I discuss shortcomings of this approach with attention to two families of metaphysical views: () presentist interpretations of special relativity and () non-multiverse interpretations of quantum mechanics. I locate these approaches in the intellectual tradition of a priori metaphysics that aims to preserve ‘common-sense’ intuitions and argue that they are not compatible with a naturalistic methodology that is attentive to the history of modern science. I also suggest that the fundamentality they ascribe to the content of our intuitions and phenomenal experiences is in tension with an evolutionary understanding of the development of our cognitive faculties and what Silberstein, Stuckey and McDevitt () have called our resulting dynamical bias. Consider the distinction between the bird view and the frog view in terms of the metaphysics of perspectival facts. A perspectival fact is a fact that obtains from within the frog view. Here, the term “perspectival fact” is used to refer to any fact expressed by a proposition whose truth is relative to the perspective(s) or location(s) within the world of some (possible) observer(s). First-personal facts (e.g., “I am a professor”), spaced facts (e.g., “The Air & Space Museum is three miles away from here”), and tensed facts (e.g., “It is snowing in Cedar Rapids”) are examples. Contrast these with non-perspectival facts, such as “The author of ‘Against Privileged-Perspective Realism in the Quantum Multiverse’ is a professor,” “The Air & Space Museum is three miles away from Dupont Circle,” and “It is snowing in Cedar Rapids at : pm EST on January , .” We may then ask, are the perspectival natures of the first group of facts features of fundamental metaphysical reality or merely features of the statements by which they are described? As Fine (, p. ) puts it, “Is reality itself somehow tensed, or spatiocentric, or first-personal, or is it merely that we describe a tenseless or spatially uncentered or impersonal reality from a tensed or spatiocentric or first-personal point of view?”¹ The question is whether some statements of irreducibly perspectival facts express metaphysical truths or whether metaphysical truths are only non-perspectival. Privileged-perspective realism (PPR) accepts that certain irreducibly perspectival facts are constitutive of reality and asserts that there is a single metaphysically privileged standpoint from which these perspectival facts obtain. It is possible to hold a PPR view about any sort of perspectival fact. If I accept PPR about firstpersonal facts, I believe that certain first-personal facts are constitutive of reality, and that there is only one first-personal standpoint (presumably, my own) from which such facts obtain. Thus, the only first-personal facts that are constitutive of reality are the ones concerning me. While this particular version of PPR may seem odd, some forms of PPR are widely defended. Presentism, for instance, is PPR about tensed

¹ For Fine, metaphysical truths are irreducible, so the question becomes whether there are perspectival facts that are not reducible to non-perspectival facts.

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

 

facts. Many of its defenders find it to be self-evidently true given our experience of time. Presentists believe that reality itself is tensed and that the irreducibly tensed facts that are constitutive of reality are the ones that obtain in relation to the present. Of course, this view is not without significant problems.

. The Argument from Special Relativity against PPR about Tensed Facts One of the strongest challenges to PPR about tensed facts arises from the argument that it is incompatible with special relativity. Putnam () presented an early version of this argument. Stein () and Saunders () have both offered critiques and defended reformulated versions of it. Fine () explicates how mathematical and theoretical considerations create difficulties for the intuitively appealing metaphysical view of PPR about tensed facts. I discuss Fine’s introduction of four metaphysical principles that provide conceptual scaffolding for characterizing the differences between privileged-perspective realism and anti-realism about perspectival facts. In his discussion of McTaggart’s classic () argument for the unreality of time, Fine demarcates the four conceptual principles that a view about time can accept: realism, absolutism, neutrality, and coherence. He defines each principle as follows (, ): . Realism—Reality is constituted (at least in part) by tensed facts. . Neutrality—No time is privileged, the tensed facts that constitute reality (if any do) are not oriented towards one time as opposed to another. . Absolutism—The composition of reality is an absolute matter, i.e. not relative to a time or other form of temporal standpoint. . Coherence—Reality is not contradictory, it is not constituted by facts with incompatible content. While the four principles are together incompatible, the combination of any three results in a consistent metaphysical view. Different combinations of these principles produce the conceptually possible views of tensed facts. Anti-realist views about tensed facts such as eternalism, for example, maintain Absolutism, Coherence, and Neutrality. Eternalism gives up Realism about tensed facts, as it accepts that reality contains no tensed facts that cannot be adequately expressed by or reduced to tenseless facts. Thus, rather than “It is snowing in Cedar Rapids” uttered at : pm EST on January ,  expressing a fact that is part of metaphysical reality, “It is snowing in Cedar Rapids at : pm EST on January, , ” uttered anywhere would express such a fact. No feature of reality is left out by including the fact expressed by the latter statement in the set of metaphysical facts but not that expressed by the former. That all metaphysical facts expressing the temporal location of an event or the temporal distance between events hold independently of temporal location or spatio-temporal frame of reference for the eternalist reflects the view’s Neutrality about tensed facts. PPR views, on the other hand, such as pre-relativistic presentism, maintain Realism, Absolutism, and Coherence. The view is realist because it takes reality to

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

contain tensed facts that are not reducible to or expressible by any tenseless facts. It is absolutist because these facts hold absolutely rather than relative to a specific perspective. The view is coherent because it does not countenance inconsistent facts as part of reality. Lastly, the view gives up Neutrality because it privileges a specific perspective or stand point—namely, the present one—from which the irreducibly tensed facts that are constitutive of reality hold. In his discussion of Realism about tensed facts, Fine () distinguishes between what he calls ontic presentism and factive presentism. Ontic presentism is the ontological view that only present things exist, while factive presentism is the view that reality is partly constituted by irreducibly tensed facts. Ontic presentism thus presupposes factive presentism. When philosophers consider special relativity’s compatibility with presentism, they tend to address the ontological question of what the presentist should take to be real. But Fine introduces the following as a more basic question that the presentist must answer: Since the presentist believes in tensed facts, what facts should the presentist take to be tensed in light of special relativity? The obvious but pre-theoretic answer would be just those facts that obtain with respect to the present moment. But special relativity’s preclusion of an absolute notion of simultaneity poses a problem for the pre-relativistic conception of a tensed fact as one that obtains relative to a time. Consider the following: If e and f are events, then the propositions that they are occurring now are tensed. If it makes sense to say that the proposition that e is occurring now and the proposition that f is occurring now are true at any given time, then we can ask if they are true at the same time. For the propositions to be true at the same time is for the events to be simultaneous. The pre-relativistic concept of tense presupposes absolute simultaneity, but since there is no absolute simultaneity under special relativity the tensetheoretic realist must replace the temporal moment as that with respect to which a tensed proposition obtains. There are two options for what the realist about tensed facts may take to that with respect to which a tensed proposition holds: a space-time point, or an inertial frame of reference plus a time. If we evaluate a tensed proposition with respect to a spacetime point, then a tensed proposition would be one that declares that a given event holds (or is earlier or later than) here-now (Stein ). Harrington () defends a version of this view. Alternatively, since each frame of reference gives rise to a framework of times, a tensed proposition could be evaluated with respect to a frame plus one of its times. Tensed propositions would then be those that say a given event is now or that a given thing is now-at-rest, relative to the frame of reference. Fine concludes that neither of these two options is feasible. The first option, what Fine calls the locational account, faces a significant problem. Because it involves the merging of temporal and spatial relativity, it fails to preserve one of the primary motivations for presentism, namely the sense that there is an important metaphysical distinction between time and space. The presentist accepts that there is a metaphysically privileged point (or, more precisely, hyperplane) of time but no corresponding metaphysically privileged area of space. Tensed facts are partially constitutive of reality, while facts that are true relative to a spatial location— spatio-centric facts—are not. Thus, if the presentist accepts the collapse of temporal

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

 

and spatial relativity into spatio-temporal relativity, they must relinquish one of the primary motivations for their view. Aside from the locational account, the other option for the presentist is to evaluate tensed facts with respect to an inertial frame of reference plus a time. But both accounts face the problem of arbitrariness. If only one inertial frame + time is to be constitutive of reality, which one is it? Fine suggests that the only plausible answer seems to be “the one that I occupy.” But there can be no good reason to prefer my inertial frame to any other as the metaphysically privileged perspective with respect to which tensed facts hold. If I accept the locational account, then the metaphysically privileged location from which to answer a tensed question will be the space-time point from which I ask it. If you ask the question from a space-time point that is within my lightcone (i.e.—in my absolute past or future), I may have a presentist reason to exclude your frame of reference from reality, since you pose the question from a different time than my present. But suppose you instead ask the question from a point that is only space-like separated from me. If the only difference between our positions is a space-like separation, I have no reason to accept my frame of reference as constitutive of reality while excluding yours. This suggests that the primary motivation for presentism (and, I argue, for other PPR views), the thought that there is something irreducibly special and important about one’s own viewpoint such that it must be foundational to metaphysical reality, is arbitrary and unmotivated. Consider the same parity objection applied to the inertial-frame option. Under this picture, when I ask a question, the relevant standpoint from which to evaluate it is the frame at which I am at rest when I ask the question and the time within that frame at which I ask it. If you are within my light-cone at the moment I ask the question, the fact that you are in my absolute past or future may give me presentist reason to exclude your standpoint. If, however, we are in relative motion and coincide at the location from which I ask the question, then the only difference between us is our relative motion (Fine , p. ). But in that case, it is unclear how the difference in our relative motion could give me reason to exclude your frame of reference from reality while admitting mine. The presentist may respond that the answer lies in an appeal to indexicality. A presentist may reject another observer’s standpoint from reality because she is not here-now or is not now-at-rest (from the presentist’s viewpoint). While this appeal to indexicality might be plausible for the pre-relativistic presentist, it does not have the same impact for the presentist revising her position in light of special relativity. While it may be plausible that a past observer does not occupy a standpoint of reality because she is not present, the same cannot be said for the postrelativistic presentist’s appeal to indexicality. This, as Fine puts it, simply collapses to, “You do not occupy the standpoint of reality since you are not me” (, p. ). The presentist’s attempt to maintain their intuition in light of special relativity again leads to arbitrary and unmotivated privileging of their own standpoint, resulting in a surprising apparent dependence of the tensed on the first-personal. Of course, a natural objection is that, since we know that special relativity is not a final theory of physics and is thus likely to be replaced by a future theory, there is no need to ensure that our other theories are compatible with it. However, the case is a bit more complicated. On the one hand, followers of Kuhn () hold that some scientific paradigm shifts, such as the transition from the classical to the quantum or

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

the Newtonian to the Einsteinian constitute such radical, revolutionary changes, that the theories are incommensurable and thus there can be no theoretical content preserved in the transition. Structural realists such as French and Ladyman () and Ladyman and Ross (), however, have shown that there is often more continuity across theory change than is recognized. Consider the transition from Newtonian gravitation to General Relativity. This shift exemplifies a radical change in ontology across the two theories but one that preserves a great deal of structural continuity. Under Newton’s law of universal gravitation, gravity was thought to be an attractive force between massive objects that acted on bodies at a distance. Einstein’s General Relativity dispensed with this ontological commitment, understanding gravitation to be an attribute of curved spacetime rather than a force propagated between bodies. Despite the difference in the theories’ underlying ontologies, there is structural continuity between them, as Newtonian Gravity obtains in the limit of weak gravitational fields in General Relativity (Silberstein, Stuckey, & McDevitt , p. ). There are myriad examples of structural continuity across theory change. Quantum mechanical models mathematically reduce to classical models in the limit of large numbers of particles or the limit of Planck’s constant becoming arbitrarily small (Ladyman and Ross ). The Correspondence Principle shows that the behavior of systems described by QM reproduces classical physics in the limit of large quantum numbers. Structural continuities between classical mechanics and special relativity also exist. While the coordinate systems that arise from inertial frames of reference of stationary bodies are related by Lorentz transformations in special relativity rather than by Galilean transformations as they are in classical mechanics, as the value of c—the velocity of light in a vacuum—goes to infinity, the mathematical structure of Lorentz transformations increasingly approximates that of Galilean transformations (Ladyman and Ross ). The transition from Fresnel’s theory of optics to Maxwell’s theory also shows retention of structure across theory change (Worrall ; Ladyman ). While the two theories differ in the interpretations and ontologies ascribed to their structural contents (e.g. light travels through an elastic solid ether for Fresnel but through an electromagnetic field for Maxwell), the differential equations remain constant during the transition from the former to the latter. While structural realists present this argument partly in response to Laudan’s () pessimistic meta-induction, one need not accept structural realism in order to take the basic point of the argument: when a theory successfully makes a novel prediction, as Newtonian gravitation and Fresnel’s optics did, we should expect some aspect of the theory’s structural content to be retained in whatever theory replaces it. Applying this insight to the case of special relativity yields the expectation that some of its structural content, namely those features of the theory that are responsible for its successful novel predictions, will be retained in whatever theory follows it. This would include the assumption that c has a non-zero value and thus that there is an upper-limit on the speed of information transfer within the universe.² ² Many physicists and philosophers take this claim to have already been straightforwardly falsified by Bell’s Theorem. But again, the case is not so simple. What Bell’s inequalities showed was that no local hidden variable theory can accurately reproduce the observed probabilities of his experiment. What the experiments

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

 

Special relativity has extraordinary predictive power and empirical confirmation. It also works best against a backdrop of four-dimensional space-time. While space and time are sometimes pre-theoretically taken to be separate, special relativity indicates that they are different dimensions of single manifold. The presentist’s effort to maintain the intuition that there is a metaphysically significant distinction between space and time illustrates the strained results of prioritizing the metaphysical fundamentality of bird-view facts over frog-view facts. Because our movement through time is more constrained than our movement through space, the presentist infers that there must be some fundamental metaphysical asymmetry between space and time. They assume that the way the world appears to us at the very limited spatio-temporal scales to which we have direct experiential access must be the way the world is metaphysically. To reject PPR about tensed facts, on the other hand, is to recognize that phenomenology of time should not be considered more metaphysically fundamental than the picture of reality indicated by the structural content of our most predictive scientific theories. One approach to scientific metaphysics that takes the bird view as prior is Silberstein, Stuckey, and McDevitt’s () Relational Blockworld (RBW). Like Wharton, Miller, and Price (), these authors are motivated by seemingly insurmountable challenges and incompatibilities in modern physics to jettison the dynamical worldview in favor of a Langrangian-first schema. They argue that the Newtonian approach to physics that takes motion and change in time to be the primary objects of physical explanation has largely failed. In its place they propose a metaphysics that accepts the Lagrangian schema, whereby the explanatory value of the least-action principle combines with both the initial conditions of a system as well as its future states to explain the system’s present state. While the Langrangian-first approach is very much in line with taking the primacy of the bird view over the frog view, Silberstein, Stuckey, and McDevitt don’t quite go far enough with their own approach, as they rule out a multiverse view on the seemingly shaky grounds that it is unfalsifiable.³ In the next section, I argue that interpretations of quantum mechanics that rule out the quantum multiverse can be understood as endorsing a form of PPR, namely realism about what I call ‘world-indexed’ facts. I consider parallels between realism about irreducibly world-indexed facts and realism about irreducibly tensed facts. I argue that the realism about irreducibly world-indexed facts is motivated by the same overvaluing of phenomenology and intuition that is responsible for attempts to preserve presentism in light of special relativity. Thus, non-multiverse interpretations of quantum mechanics fall prey to the same methodological objections straightforwardly show is that a number of plausible assumptions such as locality, separability, and independence are incompatible. At least one must be relinquished, but it is not clear which one. It is not straightforward that locality is violated by Bell’s inequalities, as one possibility is interpreting quantum entanglement to be something that occurs outside of spacetime in Hilbert space. This is one way that the Everettian can go. Thus, one of the arguments in favor of the Everettian interpretation is that it is able to retain locality (by positing entanglement as a process that occurs in Hilbert space rather than within spacetime) and thus compatibility with special relativity’s most empirically predictive foundational assumption. ³ As Carroll () argues, the cosmological multiverse is confirmed the same way conventional science is, via abduction, Bayesian inference, and successful empirical prediction.

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that undermine the non-naturalistic motivations of PPR more generally. Using the Copenhagen interpretation as a case study, I consider some of the ways this form of PPR is vulnerable to the same objections as presentism.⁴ Combining a rejection of PPR about tensed facts with a rejection of PPR about world-indexed facts results in a picture that has been called the quantum block universe (Saunders ) and described as “a multiverse of discrete, parallel, block universes which are identical to each other up to certain points in the MWI ‘trunk’ before they diverge according to the MWI branching” (Mckenzie ). In what follows, I discuss paths that lead there.

. The Metaphysics of Quantum Mechanics Interpretations of quantum mechanics differ in how they respond to the measurement problem. The Schrodinger equation describes the evolution of quantum systems, and its dynamics are unitary and deterministic. Before measurement, a quantum system can be in a state of a superposition. If a quantum system is in a state of superposition at one time, the dynamics predict that it will be in a state of superposition at a later time. But we never seem to measure or observe quantum systems in superposed states, nor do we have much idea of what such a state would look like. When a measurement is taken, the wave function appears to instantaneously collapse, leaving a definite outcome where before there was only probability. The different interpretations have different views on the reality of the wave function and different explanations of its apparent collapse. One way to think about the measurement problem is in terms of three compelling yet conflicting principles. Ladyman and Ross (, p. ) offer the following characterization of the measurement problem in terms of three intuitive principles that cannot all be true: () All measurements have unique outcomes. () The quantum mechanical description of reality is complete. () All time evolution of quantum systems is in accordance with the Schrodinger equation. Since these principles cannot all be true for a quantum system that is initially in a superposition of the property that is being measured, which one should be relinquished? How an interpretation of quantum mechanics solves the measurement problem plays a large role in determining its metaphysics. Since a comprehensive survey of QM interpretations is beyond the scope of this chapter, I look only at the Everett and the Copenhagen interpretations to illustrate the metaphysical issues at stake. The Everett interpretation rejects (), which results in a multiverse or ‘many ⁴ Copenhagen is not the only interpretation that rules out a quantum multiverse, though it is the one I focus on here as a primary example of a collapse theory that produces a PPR metaphysics. While other collapse theories avoid some of the worst problems with Copenhagen, such as those surrounding the question of when physical measurement occurs) they are still susceptible to the objections I consider here against PPR views. For instance, GRW theory is vulnerable to the same objections regarding reliance on adding ad hoc postulates to the theory in order to avoid the accepting a quantum multiverse.

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worlds.’ The Copenhagen interpretation and other collapse theories reject (), which entails that quantum mechanics does not give a complete account of the dynamical properties of quantum systems.

.. The Copenhagen Interpretation and PPR When Bohr () first introduced the Copenhagen interpretation, he did so by way of the principle of complementarity. The principle states that an experiment may show matter to behave as a particle or as a wave, but never both at once. The Copenhagen interpretation explains the determinate nature of a property after observation via the presumed mechanism of wave-function collapse. Because the Copenhagen interpretation gives up (), an additional postulate must be added to the theory to specify the conditions under which collapse occurs. The theory that collapse occurs in response to a measurement of a system being taken faces a number of well-known problems, such as the fact that measurement is not a well-defined notion and cannot be precisely identified and applied at the quantum scale.⁵ Another downside of the view is that it violates the principle of locality. The principle of locality states that an object can only causally influence and be influenced by objects in its immediate surroundings, not by distant objects. Thus, locality prohibits action at a distance. Compatibility with locality (conceived as compatibility with Lorentz-invariant space-time) is what allows quantum mechanics to be unified with special relativity. Since violations of locality involve something (namely information) traveling faster than the speed of light, as would be the case if a change in some entity were to directly influence a distant object instantaneously, they render a theory incompatibility with special relativity’s upper-limit on the speed of information transfer. Thus, a major cost of the Copenhagen interpretation is that its acceptance of non-locality precludes its unification with special relativity. The Copenhagen interpretation can be understood as promoting a kind of privileged-perspective realism about world-indexed facts. A world-indexed fact is the quantum-world analog of a tensed fact. It is a fact that obtains only with respect to some world or branch within the quantum multiverse.⁶ While the existence of world-indexed facts follows from the Everett interpretation, those who do not subscribe to Everett can think of world-indexed facts using quantum worlds as theoretical constructs much in the same way that Lewis’s (a) opponents treat the objects of possible-worlds semantics. Consider: A physicist in our world learns the potential locations of an electron before measurement by identifying the electron’s wave function. Since the electron’s wave function can be obtained by anyone using the same equations, regardless of which branch of quantum reality they inhabit, this reflects that the wave function represents a bird-view fact that is not dependent on one’s location in a given quantum world. The physicist then takes a measurement and determines the electron’s location to be at space-time point m₁. However, in other possible measurements, the electron is determined to be at locations other than m₁. When the ⁵ I will not address this significant worry, however, as the primary concern here is with issues that this interpretation shares with other collapse theories. ⁶ Issues related to the terminology of ‘worlds’ and ‘branching’ are addressed in the following section.

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physicist utters, “The electron is located at m₁,” they are uttering a proposition whose truth is evaluated with respect to the branch or world at which it is uttered.⁷ This proposition therefore expresses a world-indexed fact. World-indexed facts are contrasted with those that are not indexed to any particular world. Since the proposition, “In quantum-world q₅, electron z is at m₁” obtains whether or not it is uttered in q₅, it does not express a world-indexed fact. Call the non-perspectival fact it does express a ‘world-less’ fact. The four principles that Fine identifies can be used to categorize views in many areas of metaphysics. Generally, PPR views retain Realism, Absolutism, and Coherence while relinquishing Neutrality; and anti-realist views about perspectival facts give up Realism while retaining Neutrality, Absolutism, and Coherence. This outline of metaphysical commitments can help to illuminate some of the differences between various interpretations of quantum mechanics, specifically between collapse theories and no-collapse theories. In the next section I suggest that implicit in collapse theories (and specifically in non-multiverse interpretations of quantum mechanics) is a form of PPR—specifically, about world-indexed facts. If one were to extend Fine’s four principles to possible views about world-indexed facts, they might look like this: . Realism—Reality is constituted at least in part by irreducibly world-indexed facts. . Neutrality—No world is metaphysically privileged over any other. . Absolutism—The composition of reality is an absolute matter, i.e. not relative to a world. . Coherence—Reality is not contradictory, it is not constituted by facts with incompatible content. In terms of Fine’s principles, the Everett interpretation denies Realism about world-indexed facts and retains Absolutism, Coherence, and Neutrality. In this way, it is analogous to eternalism, which denies Realism about world-indexed facts while retaining Absolutism, Coherence, and Neutrality. The Copenhagen interpretation, on the other hand, retains Realism about world-indexed facts along with Absolutism and Coherence and denies Neutrality. In this way it is similar to presentism, which also maintains Realism, Absolutism, and Coherence but denies Neutrality. The theoretical parallels among PPR about world-indexed facts and other privileged-perspective realisms are striking. PPR about world-indexed facts accepts Realism (about perspectival facts), Absolutism, and Coherence, while rejecting Neutrality. It accepts that there is a metaphysically privileged standpoint of reality, and that standpoint is this quantum world. The fact that PPR about world-indexed facts rejects Neutrality reflects its acceptance of frog-view facts as fundamental constituents of reality. By assuming that consistency with the frog perspective takes precedence over theoretical considerations, the Copenhagen interpretation limits the scope of metaphysical reality to include only what is observable in this quantum world. Just as the ontic presentist limits their ontology to what exists from their own

⁷ See Wilson () for an account of the formal semantics that this picture involves.

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temporal perspective, the Copenhagen theorist limits their metaphysical reality to what obtains from their location in the set of quantum worlds. The view that metaphysical reality contains irreducibly world-indexed facts that cannot be expressed by any world-less facts is PPR about world-indexed facts. This view forms part of the metaphysical picture of the Copenhagen interpretation. The Copenhagen interpretation posits a set of postulates indicating that the true description of metaphysical reality obtains from a privileged perspective. The Copenhagen interpretation’s claim that only the experimentally observable constitutes reality is equivalent to the claim that only what is observable from within this quantum world constitutes reality. The Copenhagen interpretation takes the reality that is apparent to us as the inhabitants of a single quantum world to be the whole of metaphysical reality. This view denies that physical reality could extend beyond what we experience and have practical access to.⁸ Contrast this with the Everettian ascription of fundamentality to the bird view, which takes the wave function to be a fundamental component of metaphysical reality. What the Everettian takes to follow from this—that all outcomes denoted by the wave function are equally real—is just what the Copenhagen theorist rejects in order to maintain the privileged perspective of this world over those of other quantum worlds. The Copenhagen interpretation faces some of the same challenges that threaten factive presentism. Just as special relativity implies that there can be no physical reason to privilege any spatio-temporal perspective as more metaphysically real than any other, nothing in the quantum wave function or in quantum theory itself indicates that there is anything more metaphysically real about the outcome we observe than the other outcomes reflected by the probability amplitude of the wave function.Both presentism and the Copenhagen interpretation posit the existence of a metaphysically privileged perspective that does not come directly out of the theory but must be added in by hand in order to ‘save’ our experience. A tenet of scientific realism is that we have reason to believe in the physical reality of mathematical structures posited by scientific theories when they successfully produce novel predictions and explanations. Berenstain () demonstrates how, for instance, the mathematical formalism of quantum incompatibility predicts that whenever a physical system is in an eigenstate of one of a pair of quantum properties it will always be in a superposition of the other.⁹ The higher-order property of incompatibility of two such properties is defined in terms of the non-commutativity of their corresponding operators. The mathematical structure that describes the relations of these properties is responsible for the measurable prediction that when two observables are incompatible they will not both be simultaneously instantiated with determinate values by a single system. The quantum wave function and Schrodinger dynamics are also mathematical structures that play a central role in producing the enormous empirical success of quantum theory. As Deutsch emphasizes, ⁸ Tegmark (, p. ) refers to the rejection of this possibility as the “Omnivision assumption: physical reality must be such that at least one observer can in principle observe all of it.” ⁹ The linear algebra predicts that it will not be possible to simultaneously measure definite values for two incompatible observables in a quantum system. For instance, since the operators corresponding to spin in the x-direction and spin in the y-direction do not commute, they represent incompatible observables.

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“The formalism, of quantum theory did not come out of nowhere. It is the solution of a scientific problem and, as always in science, the problem was not primarily what mathematical formula best predicted the outcomes of experiments. It was what mathematical structures correspond best to reality” (, p. ). The Copenhagen does not ascribe physical reality to these mathematical structures that have effectively produced both prediction and explanation of observable phenomena. Instead it treats facts determined through observation to be the fundamental constituents of reality, seemingly renouncing scientific realism about some of quantum theories most empirically successful entities. Like other interpretations of scientific theories driven by a need to maintain a version of PPR, the Copenhagen interpretation elevates the way the world appears to be to the internally located observers above the way that our best theories suggest it is. The interpretation treats the scope of information obtained through observation as metaphysically universal while limiting the scope of information derived from the mathematical structure of the wave function. The view privileges facts that obtain from the frog perspective as more fundamental than those that obtain from the bird perspective. Like presentism, then, the Copenhagen interpretation seems to undervalue the role that the mathematical structure of an empirically successful theory can play in accessing the modal, physical, and metaphysical nature of the universe.¹⁰

.. The Everett Interpretation: Neutrality and the Multiverse Contrast the Copenhagen interpretation and its resulting PPR metaphysics with the metaphysics of the Everett interpretation. Named after Hugh Everett, who first introduced what was known as the “Relative-State formulation” of quantum mechanics (Everett ), the view was later defended and renamed the “Many-Worlds” interpretation by DeWitt and Graham (). Carroll and Singh () take the Everettian interpretation to offer the “most pure, minimal” quantum ontology, which includes only the stripped-down elements of a vector in Hilbert space and a Hamiltonian, from which everything else emerges. Deutsch (, ) objects to the fact that it is even called an interpretation rather than simply being acknowledged as the theory of quantum mechanics itself—no interpretation necessary. While other interpretations require the addition of ad hoc postulates to quantum theory in order to redeem intuitions about the uniqueness of the universe, the Everett Interpretation, as Wallace (, p. ) describes it, “is a pure interpretation of quantum mechanics. It leaves the quantum formalism, dynamics and state space alike, completely alone.” In other words, it is the view that “quantum mechanics is everything, and it is right” (Wallace , p. ). Nothing more needs to be added to the quantum formalism, which has already produced an enormously empirically successful theory, to get a complete story of quantum phenomena. A central feature of the Everett interpretation is that QM is a unitary theory, as all isolated systems evolve according to the Schrodinger equation. The Everett interpretation thus maintains the determinism of quantum mechanics. When a system in ¹⁰ See Berenstain () for a discussion of the relationship between mathematical structure and the modal structure of the physical world.

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a state of superposition is measured, there is no assumption of a unique definite outcome and there is no presumption of wave-function collapse (Tegmark ). The wave function is considered to be a genuine physical entity. The result of our measured observation is just one part of the unitary quantum state represented by the wave function. Though all of the probabilistic results occur in different branches of the quantum multiverse, only one outcome is manifest in each branch and thus only one is observable in each world. The wave function appears to collapse to the observer by way of quantum decoherence while in fact retaining its physical reality. While the observer sees only one experimental result, all of the possible outcomes denoted by the wave function occur in different worlds that are likely practically inaccessible to the observer in this world because of the rate at which they are expanding away from one another.¹¹ While different possible outcomes for a physical event or experiment occur in other branches of the physical universe, as Tegmark () argues, the assumption that the view entails some sort of “splitting” or “branching” of the universe as an event occurring within time is a misrepresentation of the theory. This sentiment echoes Saunders’s claim in his early description of what he refers to as the “quantum block universe.” Saunders writes, “The notion of ‘splitting’ is particularly inapposite . . . (no such process is defined in the block universe); nor is our world subject to “splitting,” no more than “now” contains different times.” (Saunders , p. ). While “branching” language is commonly used in discussions of the Everett interpretation, nothing in the theory mandates an assumption that this occurs as an event within a background of time rather than the assumption that the multiverse has a branched structure that is already determined within something like a background eternalist framework. The version of the Everett interpretation that is most compatible with the bird view is one that takes the multiverse to be made up of various world-states combining to produce world-lines in a structured graph network that instantiates a branched structure, producing what can then be referred to as the quantum block multiverse. Fundamentally then, on this picture, there is just one thing in our quantum ontology—the universal quantum state. In a block multiverse picture, nothing changes and nothing branches; worlds are simply ordered. Some world-states are accessible from other world-states and others are not. I follow Wilson’s () suggestion that the Everettian employ the language of the multiverse in order to pick out relevant parts of the universal quantum structure in metaphysical theorizing. He writes, At the fundamental level, the ontology is monistic—there is just one single highly-structured object, the universal quantum state. Call the language which describes reality in these terms the Universal State language. By use of decoherence theory to pick out privileged structure from the universal state, we can construct a language (call it the Pluriverse language) in which we quantify over structural features of the universal state: branches, branch segments, and so on.

¹¹ For arguments that the quantum multiverse and the cosmological multiverse may well be one and the same, see Boddy, Carroll, and Pollack () and Tegmark ().

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The Pluriverse language can be thought of as the working language of metaphysical theorizing for Everettians. (, p. )

The language of ‘worlds’ and ‘branches’ is used interchangeably to identify locations within the multiverse, and the multiverse can be taken to refer to the universal quantum state. Quantum worlds sometimes have overlapping parts with other quantum worlds, just as branches on a tree sometimes share parts with other branches on the same tree. Smaller branch chunks may be referred to as ‘branch segments,’ etc. This quantum block version of the Everett interpretation avoids the metaphysical problems associated with interpretations of quantum mechanics theoretically motivated by frog-perspective considerations like many collapse interpretations. The view maintains the metaphysical priority of the bird-view over the frog-view, thus avoiding the objections that PPR interpretations face. I now turn to two common objections to the Everett interpretation and suggest that reconsidering them within a framework that privileges the bird perspective can resolve the apparent issues.

...     ’   A tempting objection to multiverse metaphysics is that it posits an extraordinarily high number of unnecessary worlds in order to make sense of quantum theory. This is a clear violation of Ockham’s razor, as the objection goes. However, it is not clear at all that Ockham’s razor tells against, rather than in favor of, a multiverse picture. On some readings of Ockham’s razor, the Everett interpretation comes out ahead. As Baker () notes, a distinction is often made between syntactic simplicity, which captures the number and complexity of hypotheses or axioms in a theory, and ontological simplicity, which is the number and complexity of things posited. Baker refers to minimizing the former as elegance and the latter as parsimony. On the measure of syntactic simplicity, the Everett interpretation is hard to beat. It takes the formalism of QM at face value and avoids adding ad hoc fundamental postulates, as so many collapse theories must. GRW, for instance, must introduce a new fundamental constant, which is a serious cost to the interpretation’s syntactic simplicity. Those who object to the quantum multiverse on the basis of Ockham’s razor prioritize ontological simplicity over syntactic simplicity, parsimony over elegance. This choice of theoretical values can be understood as stemming from a privileging of the frog perspective over the bird perspective. An example from computation reveals that it is often far simpler, i.e. it takes fewer bits of information, to produce all the entities in an infinitely large set than to produce only a single member of a set. In computation, the algorithmic content of a number is defined as the length in bits of the shortest computer program that can produce that number as output. However, the set of all integers can be generated by a trivial computer program (Tegmark , p. ). The algorithmic content of the entire set is much smaller than that of even a single element of the set. The standard of ontological parsimony applies to that which results from a theory, while syntactic simplicity applies to the algorithmrequired to produce that ontology. In this case, the theory that posits the existence of an infinitely large set of objects is less computationally complex than one that posits the existence of only a single member. These two theoretical virtues come apart and often pull in

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opposite directions. Applying simplicity to the output of an algorithm—which is what ontological parsimony measures—reflects the assumption that the frogperspective has metaphysical primacy. Whereas applying simplicity to the axioms required to produce the output—what syntactic simplicity measures—reflects taking the bird view to be the perspective that is metaphysically fundamental. So parsimony and elegance are often in conflict, and the Everett interpretation has an advantage over non-multiverse interpretations of QM on syntactic simplicity. But is a quantum multiverse picture actually worse off on measures of parsimony than a nonmultiverse picture? One common line of reasoning is that ontological parsimony only applies to types of entities rather than to tokens. While some, such as Nolan (b) and Cowling (), argue for a further distinction between qualitative and quantitative parsimony, the latter’s status as a theoretical virtue is much more contentious than the former. Lewis (, p. ), for instance accepted the theoretical virtue of qualitative parsimony but recognized no such presumption in favor of quantitative parsimony. On a view like his, a theory that posits the existence of numbers as a new type of entity is ceteris paribus less ontologically parsimonious than one that does not. But a theory that posits only even numbers is not more ontologically parsimonious than one that posits all the natural numbers. By this measure of parsimony, the Everett interpretation is no worse-off than non-multiverse pictures, since it isn’t positing any new type of entity. We already have a world, and the theory just posits more of the same. It does not commit us to any new sort of thing or unique substance, so it is not worse off than non-multiverse theories with respect to qualitative parsimony. If quantitative parsimony is a consideration, it is presumably one that tells against a multiverse picture. On the other hand, qualitative parsimony may actually tell in favor of an Everettian multiverse. The multiverse that follows from the Everett interpretation would be more qualitatively parsimonious than the single-universe picture, for instance, it allows us to do away with the qualitative distinction between the physically possible and the physically actual or even between the physical and the mathematical as Tegmark () has argued. It is thus by no means obvious that considerations related to Ockham’s Razor constitute an objection to a multiverse metaphysics motivated by the Everett interpretation of QM.

...      Another family of objections to the Everettian multiverse turns on a series of seeming misunderstandings about the metaphysical implications of subjectivity, first-personal facts, and self-locating uncertainty. Consider, for instance, Albert and Loewer’s () explanation of their objection to the Everett interpretation based on issues of subjective conscious minds, introspection, mental states, and personal identity. The heart of the problem is that the way we conceive of mental states, beliefs, memories, etc., it simply makes no sense to speak of such states or of a mind as being in a superposition. When we introspect following an x-spin measurement we never, as apparently predicted by the theory, find ourselves in a superposition of thinking that spin is up and thinking that spin is down. If introspection is to be trusted, and it seems part of our very concept of mental states that it is trustworthy at least to this extent, then we are never in such superpositions. (p. )

Albert and Loewer construct an argument that any many-worlds interpretation has to be committed to non-physicalism because mental states won’t supervene on brain

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states. Their argument turns on the assumption that for any observer in a multitude of brain states across the multiverse, there will only be one mental state that they can be in. They take this view to be “especially pernicious” since “it entails that mental states do not even supervene on brain states . . . since one cannot tell from the state of a brain what its single mind believes” (). This understanding assumes that there is a further question that the Everett interpretation gives rise to but doesn’t have the tools to answer: which of these physical observers is the subjective mind? In many ways, these issues seem to boil down to the same questions and confusions that arise for Parfit () about personal identity in any context, not just within a branching quantum multiverse. The problem of fission for personal identity captures the same concern that Albert points to in the Everettian context: which one of these people/minds/counterparts is really me? Indeed, Papineau () suggests that the Everettian can straightforwardly invoke Parfit’s account of personal survival as opposed to personal identity. His work demonstrates “how to talk coherently” about a conception of self in terms of branching experiential world-lines made of overlapping pasts that eventually diverge in physically indistinguishable copies. Given that Parfit has demonstrated the coherence of this hypothesis, Papineau writes, “I don’t see what else argues against it, except its unfamiliarity” (p. ). Carroll and Sebens () warn against what they take to be the naïve reaction that self-locating uncertainty poses a crisis for multiverse accounts. They take self-locating uncertainty to be a generic and expected feature of human cognition in a multiverse in which there is a temporal lag, however brief, between branching and post-measurement updated knowledge states. They argue that the Born rule can be derived from quantum mechanics given assumptions about the uniquely rational way for an observer to apportion credences in the time between wave function branching and the observer’s knowledge updating based on registering the measurement outcome. They explain, “In our approach, the question is not about which observer you will end up as; it is how the various future selves into which you will evolve should apportion their credences.” Albert and Loewer’s worry about personal identity seems to arise from an assumption of PPR about world-indexed facts. Specifically, the view commits to PPR about world-indexed facts that locate a specific observer-copy and index their specific conscious experience to a single universe or location within the quantum multiverse. This move works as an explication of their view but not as an objection to the Everett interpretation, as it simply begs the question against the anti-realist about irreducibly world-indexed facts. The anti-realist in both cases suggests that such a question can only be asked from within a certain standpoint, either from within a single branch or universe within the quantum multiverse, as is the case for world-indexed facts, or from within a specific frame of reference plus a time, as is the case for tensed facts. Beyond that, for the anti-realist, there is no further substantive question to be asked.¹² Deutsch (, p. ) explains the ambiguities of ordinary language that give rise to the confusion that Albert and Loewer demonstrate by suggesting that two

¹² That Albert and Loewer take there to be a further question of which person or physically indistinguishable quantum counterpart is really them further draws out the analogy between PPR about worldindexed facts and PPR about tensed facts. The presentist looks at the description of the world that arises from a complete set of tenseless facts and takes there to be a further unanswered metaphysical question: which one of these times is really now?

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apparently conflicting reports of what an observer is seeing—one report of seeing tea, the other of coffee, both of which are true—are to be understood as merely a special case of a general multiplicity in physical reality writ large. He writes, I have just said that I cannot see the coffee, and that I am having the perception of seeing the coffee. This is no contradiction, merely two different uses of the word ‘I’. The problem here is that ordinary language implicitly makes the false assumption that our experiences (and observable events in general) have a single-valued history. To help resolve the ambiguities created by this assumption, Lockwood introduces the term Mind to denote the multiple entity that is having all the (‘maximal’) experiences that I am in reality having, and reserves the term mind for an entity that is having any one of those experiences. So I (the Mind) am both seeing tea and seeing coffee, and am simultaneously reporting both experiences, but I (the mind), who am writing ‘tea’ am seeing only tea. Similarly, we call multi-valued physical reality as a whole the multiverse, to distinguish it from the universe of classical physics in which observables can take only one value at a time. (p. )

Deutsch embraces Lockwood’s () use of capitalization to distinguish between the two under-specified uses of ‘I’ and ‘mind.’ Deutsch and Lockwood’s use of ‘Mind’ is evaluated from the bird perspective, as it quantifies over the domain of the entire multiverse, while ‘mind’ is evaluated at and indexed to a single quantum world. Given this clarification, it seems that the only way to maintain Albert and Loewer’s objection is by claiming that there can simply be no physically indistinguishable copies of subjective observers. Tegmark () discusses a dubious assumption on which objections to the quantum multiverse are often based which further brings out the intuition that motivates Albert and Loewer’s objection. Tegmark identifies what he refers to as the “no-copy assumption,” that “no physical process can copy observers or create subjectively indistinguishable observers” (). This possibility is unacceptable for Albert and Loewer because they have already concluded that it would leave a well-defined question unanswerable. Albert and Loewer’sconfidence that the question of which physically and subjectively indistinguishable quantum counterpart is really the observer can be diagnosed as the same sort of certainty in our special and unique nature that has long motivated resistance to paradigm-shifting changes in scientific worldviews.

. Privileged Perspectives and Induction over the History of Science The history of modern science can be viewed as a series of discoveries that have continually dethroned humankind from a presumed special and unique place in the universe. The transition away from Ptolemy’s geocentric model and the corresponding Aristotelian worldview which posited that the sun and the planets revolved around Earth in circular orbits exemplifies this. Ancient Greek and Roman astronomers believed the circle to be the most perfect shape and thus assumed that celestial bodies must move in a circular motion as the heavens were perfect and unchanging. When observations such as Mercury’s retrograde motion created challenges for this theory, astronomers postulated the existence of a complex and arbitrary system of epicycles and deferents to account for differences in orbit in order to retain the

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Ptolemaic picture. Astronomers posited ad hoc mechanisms to save what they thought must be the correct theory and resist the transition to the Copernican heliocentric theory. The Copernican model removed Earth and thus humanity from its presumptive place at the center of the cosmos and cast it into the realm of ordinary planetary bodies orbiting the sun. Cosmology is not the only area of science in which our intuitions fall short of grasping the nature of reality, however. Biology is another. The transition from The Great Chain of Being picture to Darwin’s theory of evolution by natural selection showed that humans could no longer take themselves to have been divinely created separately from the biological processes that produced other lifeforms. Darwin’s theory unified the processes that produced the origins of humanity with those that created other animal species. Humanity was again removed from its presumed pedestal in nature. In modern physics, Einstein overturned the accepted cosmological assumption that Newtonian mechanics was the ultimate physical theory that applied to the entirety of the universe. Whereas human experience of space, time, and length gave rise to the presumption that these quantities were invariant and absolute, Einstein’s theory of special relativity showed them to be dependent on the observer’s frame of reference. While the spatio-temporal interval remained invariant in his theory, the other quantities became relative to an inertial frame. The reach of phenomenological intuition was once again shown to exceed its grasp. The presumed metaphysical uniqueness of the continuously moving now could no longer be retained. The moment in time that we inhabit was shown to be nothing more than one moment among many, all equally real within the space-time manifold that structures them. PPR about tensed facts became untenable. Now once again, as the Everett interpretation suggests that we live in a quantum multiverse filled with indistinguishable copies of ourselves, many are affronted by the sense that quantum multiverse has removed the specialness and perceived ineffability of our individual subjectivity. These are indeed psychologically challenging scientific discoveries, but the fact that they are difficult for us to reconcile is not an argument against them. Deutsch (, p. ) suggests this psychological difficulty underlies many instrumentalist interpretations of wildly successful scientific theories, from the Catholic Church’s permission to use heliocentric theory “purely as a means of predicting astronomical observations, but not if it was interpreted as a factual theory of where and what the planets and the Earth are” to interpretations of quantum mechanics that refuse to acknowledge the quantum multiverse. What “these miscellaneous revisionist views of scientific theories have in common,” Deutsch charges, “is a loss of philosophical nerve.” Each of these times that humans have prioritized the frog view over the bird view, we have held on to failing worldviews at the expense of new developments in science. We have significant inductive evidence that PPR fails. Each of these scientific paradigm shifts has demonstrated that taking the frog perspective to have metaphysical priority is an obstacle to achieving greater scientific understanding of the world we live in. Science has progressively shown that our position in the world is not special. We do not live at the center of the universe. We do not live under special terrestrial laws that differ from the laws that govern the celestial sphere. We did not

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come into being by processes that differ from those that produced the rest of the animal kingdom. Relativity shows that no time is special and that all times exist. ‘Now’ is merely indexical like ‘here.’ Given the success of quantum mechanics and the fact that the Everett interpretation is the theory in its purest form, perhaps we should accept that the apparent difference between the physically actual and the physically possible is merely indexical as well.¹³

. Evolution and Intuition, Scope and Scale The motivation for many PPR views, including PPR about tensed and world-indexed facts, is the intuition that the way the world appears to be from within the frog perspective reflects fundamental metaphysical reality. The advocate of PPR in both the tensed and world-indexed cases insists not just that metaphysical reality makes sense of our experiences of mid-level macroscopic phenomena, but that reality must be the way it is portrayed by our mid-level macro experiences at all other scales as well. Rather than taking the content of our experiences to apply only to the limited domains and scales that we are actually capable of experiencing, the PPR advocate takes the content of our experiences to be unrestricted in scope and universal in scale. The mistake of the privileged-perspective realist is to weight the picture of reality painted by our intuitions so heavily that empirical and theoretical investigations must be brought in line with that picture rather than vice versa. Since our direct experiences say nothing about the way the universe is at quantum or cosmological scales, it is a mistake to project the content of our mid-level macro experiences onto the world at those scales. Similarly, our experiences of mid-level macroscopic phenomena are silent on whether a quantum multiverse exists; nothing in our experience is incompatible with the existence of a multiverse. To deny the quantum multiverse on the basis of experience is also to project the content of experience beyond its domain and presume that it is universal in scope. There is a naturalistic explanation for the inductive failure of our intuitions when it comes to discovering the metaphysical nature of reality. Ladyman and Ross () have noted that our intuitions about the physical world are well-developed to understand the mid-level macro scale of physical phenomena that made up our ancestral environments and that continue to constitute the scale at which humans experience the universe. But we do not have any naturalistic reason to think that our pre-theoretic intuitions about the quantum and cosmological track metaphysical reality at those scales. People are probably also relatively reliable barometers of the behavioural patterns of animals they get to spend time observing, at making navigational inferences in certain sorts of environments (but not in others), and at anticipating aspects of the trajectories of

¹³ Lewis (b) argues for this claim in his On the Plurality of Worlds, but he gets there via a route of apparent IBE from semantic theory rather than from quantum theory. His final picture is also radically different from the quantum multiverse, as it is characterized by concrete possible worlds generated from the Principle of Recombination of properties limited only by logical possibility, which bear no spatiotemporal relations to one another.

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medium-sized objects moving at medium speeds. However, proficiency in inferring the largescale and small-scale structure of our immediate environment, or any features distant from the parts of the universe distant from our ancestral stomping grounds, was of no relevance to our ancestors’ reproductive fitness. Hence, there is no reason to imagine that our habitual intuitions and inferential responses are well-designed for science or for metaphysics. (Ladyman and Ross , p. )

Metaphysics that aims to make sense of our best physical theories should not be expected to save ‘common-sense’ intuitions. The frog view is pragmatically and instrumentally valuable for navigating the physical world, but we should not mistake our limited inside view of the universe for its metaphysical nature. We have betterdeveloped ways of learning about the universe at the quantum and cosmological scales than relying on intuitions that stem from our phenomenology of the comparatively tiny range of scales which human experience is capable of capturing.¹⁴ Tegmark (, p. ) echoes Ladyman and Ross’s point when he writes, “Evolution endowed us with intuition only for those aspects of physics that had survival value for our distant ancestors, such as the parabolic trajectories of flying rocks. Darwin’s theory thus makes the testable prediction that whenever we look beyond the human scale, our evolved intuition should break down.” The history of science is filled with an overwhelming number of cases that demonstrate the limits of our capacities to intuitively discover the nature of physical reality. We don’t just know that our intuitions have failed, evolutionary biology offers an explanation of why we ought to expect them to fail in such cases. Tegmark () critiques what he calls the “pedagogical reality assumption,” which is the assumption that “physical reality must be such that all reasonably informed human observers feel they intuitively understand it” (p. ). Rejection of the multiverse interpretation of QM is in the lineage of other metaphysical theories that attempt to force science to fit with our pre-theoretic, ‘common-sense’ intuitions about the way reality should be. The metaphysical presumption of wave function collapse as physical reality is motivated by the desire to make the facts that obtain from the frog-view part of the ultimate description of reality. The Everett interpretation can maintain the apparent reality of wave function collapse while reducing frog-view facts to world-less facts about the wave function. When there are clear tensions between the picture of reality suggested by our best physical theories and the picture suggested by our intuitions, the latter ought to be the one relinquished.

¹⁴ Ladyman and Ross (, p. ) emphasize how severely limited the range of scales of human experience is compared to the range of scales at which the universe is structured. They write, “We occupy a very restricted domain of space and time. We experience events that last from around a tenth of a second to years. Collective historical memory may expand that to centuries, but no longer. Similarly, spatial scales of a millimetre to a few thousand miles are all that have concerned us until recently. Yet science has made us aware of how limited our natural perspective is. Protons, for example, have an effective diameter of around  15m, while the diameter of the visible universe is more than ¹⁹ times the radius of the Earth. The age of the universe is supposed to be of the order of  billion years. Even more homely sciences such as geology require us to adopt time scales that make all of human history seem like a vanishingly brief event.”

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. Conclusion I have considered a number of problems with the practice of offering metaphysical interpretations of scientific theories that are grounded in an ascription of metaphysical fundamentality to the frog perspective. I have argued, using the examples of special relativity and quantum mechanics, that there are structural similarities in PPR metaphysical interpretations of scientific theories across domains of theoretical physics. Conceived in terms of which combination of Fine’s principles a metaphysical view accepts, the structural similarities among PPR views reflect an overvaluing of the first-personal and subjective and create vulnerability to objections from arbitrariness due to their rejection of Neutrality. I have suggested that much of the motivation for PPR metaphysics is simply psychological, and I have demonstrated that metaphysical assumptions that are motivated by psychological tendencies are difficult to make compatible with induction over the history of science. Such assumptions should therefore not be driving views in the metaphysics of science, especially, as I have argued, in either quantum or cosmological domains.

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SECTION 2

Quantum Mechanics and Fundamentality

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6 Super-Humeanism: The Canberra Plan for Physics Michael Esfeld

. The Canberra Plan Consider how Jackson (, p. ) describes the task of metaphysics: Metaphysics, we said, is about what there is and what it is like. But of course it is concerned not with any old shopping list of what there is and what it is like. Metaphysicians seek a comprehensive account of some subject matter – the mind, the semantic, or, most ambitiously, everything – in terms of a limited number of more or less basic notions. In doing this they are following the good example of physicists. The methodology is not that of letting a thousand flowers bloom but rather that of making do with as meagre a diet as possible. . . . But if metaphysics seeks comprehension in terms of limited ingredients, it is continually going to be faced with the problem of location. Because the ingredients are limited, some putative features of the world are not going to appear explicitly in the story. The question then will be whether they, nevertheless, figure implicitly in the story. Serious metaphysics is simultaneously discriminatory and putatively complete, and the combination of these two facts means that there is bound to be a whole range of putative features of our world up for either elimination or location.

This quotation can be considered as a locus classicus of what is known as the Canberra plan: metaphysics is ontology, answering the question of what there is in such a way that something is admitted as fundamental and it is then shown how everything else that exists is included in what is fundamental (location). This implies that the propositions that describe what is fundamental entail all the other true propositions about the world. This chapter sets out a proposal how to implement a Canberra plan for the physical domain. In doing so, the proposal made here deviates in three main aspects from positions that are connected with the Canberra plan. () In the first place, as far as the methodology is concerned, this chapter is an essay in metaphysics of science or naturalized metaphysics rather than conceptual analysis and a priori metaphysics (“armchair metaphysics”). When it comes to a concrete proposal about what is fundamental in the world, the benchmark is not only to take science, notably physics, into consideration, but to show in precise terms how the proposal in question matches our central physical theories. This benchmark amounts to a naturalization Michael Esfeld, Super-Humeanism: The Canberra Plan for Physics In: The Foundation of Reality: Fundamentality, Space, and Time. Edited by: David Glick, George Darby, and Anna Marmodoro, Oxford University Press (2020). © Michael Esfeld. DOI: 10.1093/oso/9780198831501.003.0007

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of metaphysics. However, this does not mean returning to positivist metaphysics. There is no question of reading the proposal for what is fundamental off from the mathematical structure of physical theories (e.g. by translating this structure into first order logic and taking the propositions with existential quantifiers thus obtained to settle the ontological commitments, as in Quine’s vision of metaphysics, see Quine ). Formulating such a proposal is genuine philosophical work, the guideline being “making do with as meagre a diet as possible”, as Jackson puts it. That is, the criterion is parsimony, together with coherence and empirical adequacy. In other words, the question that naturalized metaphysics, thus conceived, seeks to answer is this one: What is a minimal set of entities that form an ontology of the natural world, given what science tells us about the natural world? In a nutshell, the criterion is minimal sufficiency, neither necessity, nor a priori knowledge and conceptual analysis. () Moreover, taking recent developments in metaphysics into account that have moved away from the notion of supervenience, “location” is in the following conceived in terms of identity. The task hence is to spell out what there is in terms of a minimal set of entities and then to show how everything else is located in that set in being identical with something in that set. The advantage of working with the notion of identity is that it is clear and simple: if A and B exist, then B is either identical with A or not identical with A. If B is identical with A, then endorsing B requires no ontological commitment that goes beyond A—given A, B comes for free. If B exists and is not identical with A, then B is something in addition to A. Hence, an ontological commitment going beyond A is necessary. Thus, if A is the set of entities originally admitted as minimal, that set then has to be enlarged to include B. Working with identity instead of supervenience is no deviation from the Canberra plan as set out in Jackson or in Lewis’s metaphysics of Humean supervenience, since for them, in brief, supervenience is a vehicle to establish identity and ontological reduction of everything else to the supervenience basis (e.g. Lewis a, introduction, and Jackson , ch. ). The fact that identity is symmetrical does not infringe upon the ontological reductionism of locating everything in a minimal set of entities. Suppose that the configuration of point particles of the universe is that minimal set of entities—that is, all the point particles of the universe conceived as permanent with their relative positions (i.e. the distances among these particles) and their motion (i.e. the change of their relative positions, that is, the change of their distances). The basic notions then are “point particle”, “distance”, and “motion” (i.e. “change of distance”). The problem of location then is to show how everything else is identical with some specific sub-configuration of the universal configuration of particles and its motion. Suppose that organisms, including minds, exist. The task then is to spell out by using only the basic notions of point particles, distances, and change of distances which specific configurations of particles and their motions are organisms and which ones are minds, given certain normal or standard conditions in the environment. If this cannot be done, more than particles standing in distance relations and the change of these relations has to be admitted in the ontology. Hence, the reductionism consists in establishing that the configuration of particles of the universe is everything that

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exists, and this is done through the symmetry of the identity relation by showing that some sub-configurations of particles of the universe are organisms so that organisms are certain specific particle configurations, some further sub-configurations are minds so that minds are certain specific particle configurations, etc. () In the third place, ontological reductionism thus spelled out in terms of identity commits one to the view that the propositions describing the minimal set of entities entail all the true propositions about the world. However, there is no commitment to a priori entailment endorsed in this paper. The core idea of the Canberra plan is that given the basic notions that describe the minimal set of entities, all further notions are introduced through a functional definition that describes their causal or functional role for the behavior of the entities in that set (see e.g. Lewis , ). But it is not mandatory to consider these functional definitions as being a priori, that is, as being the business of conceptual analysis only; they may rather be a matter of scientific investigation. In any case, given these functional definitions, the propositions describing the minimal set of entities then entail all the other true propositions about the world. Thus, supposing that the minimal set of entities is the particle configuration of the universe, if the notions describing the mind can be defined in terms of their causal role for particle motion, then the description of the particle configuration of the universe and its change entails all the true propositions describing minds. The chapter sketches out an ontology of physics in this vein (Section .), then goes into the dynamical structure of physical theories (Section .) and finally considers possible limitations of this naturalistic ontology when it comes to the mind (Section .).

. Physics: Ontology Naturalized metaphysics turns to physics when it comes to determining the minimal set of entities on the basis of which everything in the domain of the natural sciences can be understood. Consider what Feynman says in the introduction to the Feynman Lectures on Physics: If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generations of creatures, what statement would contain the most information in the fewest words? I believe it is the atomic hypothesis (or the atomic fact, or whatever you wish to call it) that all things are made of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another. In that one sentence, you will see, there is an enormous amount of information about the world, if just a little imagination and thinking are applied. (Feynman et al. , vol. , ch. , section )

This is the old and venerable hypothesis of atomism, going back to the Presocratics Leucippos and Democritos and being turned into a precise physical theory by Newton. The idea is that matter is composed of point particles and that everything in the physical domain is identical with configurations of these point particles. Consequently, all the variation and change in the physical domain can be accounted

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for in terms of differences in the spatial arrangement of these point particles and the change of their spatial arrangement. The philosophical task then is to elaborate on the atomic hypothesis in such a way that it is clearly spelled out what the commitment to a minimal set of entities is, what the basic notions are that express that commitment and how all the further notions come in as being defined in terms of their function for that minimal set of entities. In that respect, as is clear since Leibniz’ criticism of Newton in the famous LeibnizClarke correspondence, the ontological commitments to an absolute space in which the atoms are embedded and an absolute time in which their motion takes place amount to endorsing a surplus structure, which shows up, in the case of Newtonian physics, in the guise of a commitment to ontological differences that do not make a physical difference (see notably Leibniz’ third letter, §§ –, and fourth letter, § , in Leibniz , pp. –, –; English translation Leibniz and Clarke ). Consequently, on a minimalist ontology, the commitment should only be to spatial relations (i.e. distances) among the point particles as the relations that define, in Leibnizian terms, the order of coexistence of the point particles, and the change of these relations, with time being the measure of that change (order of succession, in Leibnizian terms; see notably third letter, § , and fourth letter, § , in Leibniz , pp. , ). By the same token, it amounts to a commitment to a surplus structure to conceive the point particles as being equipped with some intrinsic properties—in other words, to conceive physical parameters such as mass and charge as referring to intrinsic properties of the particles. As, for instance, Mach (, p. ) stresses when commenting on Newton’s Principia, “The true definition of mass can be deduced only from the dynamical relations of bodies.” That is to say, both inertial and gravitational mass are introduced in Newtonian mechanics through their dynamical role, namely as dynamical parameters that couple the motions of the particles to one another. In general, even if attributed to the particles taken individually, mass, charge, etc. express a dynamical relation between the particles instead of indicating intrinsic properties of the basic objects. Hence, mass, charge, etc. are parameters that can be defined in terms of what they do for the basis objects, namely in terms of their causal or functional role for the motion of the basic objects. The commitment to surplus structure lies in the fact that if one conceives mass, charge, etc. as intrinsic properties of the particles that are dispositions with the acceleration of the particles being their manifestation, there are situations (including entire possible worlds) conceivable in which these dispositions do not manifest themselves, or situations in which they cancel each other out—such as a situation in which the attractive motion resulting from the gravitational mass of two particles is cancelled out by the repulsive motion resulting from these two particles having the same charge so that there is no relative motion of the two particles. If one conceives mass, charge, etc. as intrinsic, categorical properties of the particles so that the functional role that these properties exercise for the motion of the particles is contingent, the surplus structure then is that one is committed to quidditism, namely to a purely qualitative essence (quiddity) of these properties that is not accessible to physics (humility) (see e.g. Lewis ).

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Leaving absolute space and time as well as intrinsic properties aside for these reasons, we can formulate atomism as ontology of the natural world in terms of the following two axioms: () There are distance relations that individuate simple objects, namely matter points. () The matter points are permanent, with the distances between them changing. Consequently, the specific notions endorsed as primitive are the ones of matter points, distances and change of the distances (see Esfeld and Deckert , ch. ., for details). The reason for singling out the distance relation is that it is the first and foremost candidate for the world-making relation, at least insofar as the physical world is concerned. All and only those objects that stand in a distance to each other make up a world. No other relation has been proposed in the literature that could do the same for the physical world (Lewis’s hypothetical basic relations of like-chargedness and opposite-chargedness, for instance, would not do, since, as Lewis (a, p. ) notes himself, these relations fail to distinguish the objects that stand in them as soon as there are at least three objects). The world-making character of the distance relation, at least with respect to the physical world, is the reason why Leibniz considers space as the order of coexistence. The distance relation defines extension, which means that the objects that stand in this relation are not extended themselves. They thus are point objects, more precisely matter points, taking up the Cartesian definition of matter as res extensa. In other words, the distance relation is such that it makes it that the objects that stand in this relation are matter points. The distance relation is irreflexive: nothing can stand in a distance to itself. It is symmetric: if matter point i is at a certain distance to matter point j, then j is at the same distance to i. It is connex, meaning that any two objects in a configuration stand in a distance relation to each other. It fulfils the triangle inequality—that is, for any three matter points i, j, k, the sum of the distances between i and j and j and k is greater than or equal to the distance between iand k. What is important are the ratios between the distances—that is, not how far is i from j in absolute terms, but how far is i from j in comparison to how far is i from k, and k from j. To carry out such comparisons and to represent the distances, one needs numbers, more precisely real numbers. But this does not smuggle in any infinity. If there are N matter points (at a minimum ), there are finitely many distance relations, namely /N(N–) distance relations. This holds also if N is the number of point particles in the universe. Obviously, there is more to the concrete distance relations that exist in the universe than just fulfilling the triangle equality (to that extent, the criticism of Lazarovici  is well taken). Nonetheless, all there is to them can be expressed in terms of scale invariant quantities. Thus, the most elaborate proposal how to understand physics by being committed only to relational quantities is the shape dynamics developed by Julian Barbour and collaborators since the s (see Barbour ; Mercati  for an overview). All that there is to the particles is contained in the shape of the particle configuration, which requires endorsing distances and angles that define the shape as primitive, but which is scale invariant.

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Unless one takes the radical ontological structural realism proposed by French and Ladyman to be intelligible (see e.g. Ladyman and Ross , chs.  and ; French , chs. –)—or, to put it more mildly, maintains that the formulation of ontology calls for a change in first order logic—relations require objects as that what is related by the relations in question (see Morganti in this volume for a discussion of the different forms of ontic structural realism). Consequently, both the relations and the objects that stand in the relations in question are ontologically basic. It is not clear to say the least how relations could constitute objects, or how objects could emerge from relations, in the sense that objects are somehow derived from relations. Objects may be nothing more than the nodes in a network of relations—in the sense that all there is to the objects is that they are points standing in distance relations, they do not have any intrinsic properties, or a haecceity—but, still, the objects are ontologically distinct from the relations: the points are ontologically distinct from the distances that obtain between them. By the same token, it is, pace Heil (, ch. ) and Lowe (), unclear to say the least how there could be only objects making up a world, but no world-making relations that bind these objects together so that they make up a world. No one has put forward a proposal how to derive relations from objects. Hence, both relations and objects are basic; but one gets the individuation of objects for free from the relations. In other words, given that one has to recognize relations anyway, one can employ the relations endorsed as basic as being that what individuates the objects. One thereby obtains a structural individuation of objects, by contrast to an individuation through an intrinsic essence, a bare substratum, or a primitive thisness (haecceity). Thus, the distance relation can individuate the matter points provided that it fulfills the following requirement: if matter point i is not identical with matter point j, then the two sets that list all the distance relations in which these points stand with respect to all the other points in a configuration must differ in at least one such relation. It is such differences in the way in which i and j relate with the other points in the configuration that make it that i and j are different points. Consequently, by imposing this requirement, one avoids having to accept the plurality of matter points as a primitive fact, which would imply that the matter points are bare particulars or bare substrata. The distance relation also accounts for the impenetrability of matter without having to invoke a notion of mass: for any two matter points to overlap it would have to be the case that there is no distance between them. This requirement implies that any model or possible world of this ontology has to include at least three matter points that are individuated by the distance relations among them. Consequently, symmetrical configurations are ruled out, but also, for instance, the configuration of an isosceles triangle. Nonetheless, this is no objectionable restriction: having empirical adequacy in mind, there is no need to admit worlds with only one or two objects or entirely symmetrical worlds (see Hacking  and Belot  for also ruling such worlds out as being metaphysically possible).

. Physics: Representation If the ontology is exhausted by matter points standing in distance relations and the change of these relations so that “matter points”, “distances”, and “change of distances” are the primitive notions, these notions are not sufficient to formulate a

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physical law. The reason is that a configuration of matter points described in terms of these notions does not contain any information about its evolution. Of course, these notions are sufficient to describe all the change that occurs in the configuration of matter of the universe. But that would just be a description that lists all the change that actually happens by contrast to a law that simplifies—that is, a law that is such that given some initial configuration of matter, the law tells us something about the evolution of the configuration (in the case of a deterministic law, it contains the information about the whole past and future evolution of the configuration). In order to achieve such a law, we have to embed the configuration of matter in a geometry and a dynamics: we have to conceive the configuration of matter as embedded in a geometrical space (such as Euclidean space) and we have to attribute parameters to the configuration of matter that are introduced in terms of their functional role for the change in the distance relations among the matter points. These can be parameters that are attributed to the matter points individually (such as mass, momentum, charge), to their configuration (such as total energy, or an entangled wave function), or constants of nature (such as the gravitational constant). They can always remain the same (such as mass and charge) or vary as the distance relations among the matter points change (such as momentum, fields, a wave function). Let us call these parameters and the geometry the “dynamical structure” of a physical theory, and let us use the term “primitive ontology” for the ontology that is defined in terms of the basic, primitive notions on the basis of which the dynamical structure then is introduced in terms of its functional role for that ontology. Consider how Hall (, § .) describes the task of a physical theory: the primary aim of physics – its first order business, as it were – is to account for motions, or more generally for change of spatial configurations of things over time. Put another way, there is one Fundamental Why-Question for physics: Why are things located where they are, when they are? In trying to answer this question, physics can of course introduce new physical magnitudes – and when it does, new why-questions will come with them.

That is to say: the new parameters—that is, the parameters that go beyond the primitive parameters of relative positions of the basic objects and the change of these positions—come in through the functional role they play for the motion of the basic objects. The benchmark for these parameters then is that they enable the formulation of laws that simplify the representation of the change that occurs in the configuration of matter—by contrast to merely dressing a list of that change— without losing the information about the change that actually happens. The task hence is to specify a dynamical structure such that, for any configuration of matter given as initial condition, the law fixes (or at least puts a constraint on) how the universe would evolve if that configuration were the actual one. The dynamical structure thereby goes beyond the actual configuration of matter: it fixes for any possible configuration of matter what the evolution of the universe would be like if that configuration were actual. It thereby supports counterfactual propositions. The ontological parsimony of this proposal then shows up in refusing to extend the ontological commitment beyond the primitive ontology—such as the primitive ontology defined by matter points individuated by distance relations and the change of these relations. In other words, the dynamical structure of a physical theory,

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insofar as it can be construed in terms of its functional role for the primitive ontology, does not call for an additional ontological commitment. Of course, physics explains the motions of bodies by using a geometry and dynamical parameters that figure in laws. However, the argument for an ontological commitment to the geometrical space and the dynamical parameters employed in physical theories cannot simply be that they figure in our best physical theories. Reading the ontology off from the mathematical structure of physical theories would be begging the question of an argument for ontological commitments that go beyond what is minimally sufficient to account for the phenomena in a scientific realist vein, namely the commitment to a primitive ontology as given by the two axioms of distance relations individuating matter points and the change in these relations. In a metaphysics based on science, the argument can only be that by subscribing to ontological commitments that go beyond that minimum, one achieves a gain in explanation. However, our scientific understanding of the world comes to an end once the salient patterns in the change of the distances among the basic objects are reached, such as e.g. attractive particle motion. In whatever way one spells out in philosophy of science what explanation means—covering law, causal explanation, unification, just to name the most prominent accounts—explanation is achieved when one shows how a phenomenon that calls for an explanation falls into a common pattern (such as e.g. the pattern of attractive motion). To vindicate scientific explanations, no ontological commitment going beyond the minimal commitments spelled out in the previous section is called for (see Loewer ). Subscribing to an ontological commitment with respect to the parameters that are introduced in terms of their function for the particle motion does not amount to a gain in explanation. Quite to the contrary, it entails serious drawbacks that come with ontological surplus structure. The argument for this claim is the one illustrated in Molière’s piece Le maladeimaginaire: one does not explain why people fall asleep after the consumption of opium by subscribing to an ontological commitment to a dormitive virtue of opium, because that dormitive virtue is defined in terms of its functional role to make people fall asleep after the consumption of opium. By the same token, one does not obtain a gain in explaining attractive particle motion by subscribing to an ontological commitment to gravitational mass as a property of the particles, because mass is defined in terms of its functional role of making objects attract one another as described by the law of gravitation. Of course, mass, charge and the like are fundamental and universal physical parameters, by contrast to the dormitive virtue of opium. But the point is that they are defined in terms of the functional role that they exert for the particle motion. Why do objects move as they do? Because they have properties whose function it is to make them move as they do. An ontological commitment to such properties does not yield a gain in explanation. The same holds for forces, fields, wave functions, an ontic structure of entanglement in quantum physics, laws conceived as primitive, etc. It also applies to geometry: it is no gain in explanation to trace the characteristic features of the distance relation back to the geometry of an absolute space, because that geometry is defined such that it allows for the conception of distances in that space. Furthermore, as mentioned above, a formulation of physics, including relativistic physics and quantum physics, that employs only relational quantities is available in the guise of shape dynamics.

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It is true that by tracing the distance relations back to an absolute space, or the change in the distance relations back to properties of the particles that are dispositions for that very change, the characteristic features of the distance relations as well as those of the patterns of the change in them come out as necessary instead of contingent. However, shifting the status of something from contingent to necessary does not amount to a gain in explanation. Quite to the contrary, one only faces drawbacks that come with the commitment to a surplus structure in the ontology in the guise of an absolute space, fundamental dispositional properties of the particles, ontic dynamical structures of entanglement, etc.: differences with respect to absolute space that do not make a difference in the configuration of matter, questions such as how an object can influence the motion of other objects across space in virtue of properties that are intrinsic to it, how a wave function defined on configuration space can pilot the motion of matter in physical space, etc. (see Esfeld and Deckert , ch. ., for a detailed argument). The stance that results from these arguments can be dubbed Super-Humeanism: on standard Humean metaphysics (e.g. Lewis a, introduction), the ontology— the Humean mosaic—consists in spatio-temporal relations among points and intrinsic properties instantiated at those points. The Super-Humean deletes the commitment to a substantival space-time and to intrinsic properties, thus avoiding the surplus structure in the ontology that comes with these commitments. Hence, the Super-Humean maintains that there are only sparse points that then are matter points with distance relations individuating these points and change in the distance relations. The laws of physics are our means to represent the salient patterns in that change in a system that strikes the best balance between being simple and being informative about that change. In order to do so, dynamical parameters that enable the formulation of laws have to be introduced in terms of the functional role that they play for the change in the distance relations among the matter points. However, introducing these parameters does as such not lead to an improvement in simplicity. Values for them have to be fixed as initial conditions along with the relative particle positions. Doing so may turn out to be as complicated and messy as specifying initial positions of the matter points (consider e.g. the case of the wave function in quantum physics). The gain in simplicity is that by figuring out these further parameters over and above the relative particle positions in an initial configuration, one determines an initial condition that can be inserted as input into a law so that one obtains as output (in the deterministic case) a description of the whole past and future evolution of the particle configuration (cf. the distinction between two types of physical state in Bhogal and Perry —one type that consists only in the relative particle positions, and a further, broader type that includes what has to be specified as initial condition in addition to that). The argument for not subscribing to an additional ontological commitment to these parameters—beyond the commitment to relative particle positions and their change—is that one can introduce these parameters in terms of their functional role for the change in the relative particle positions (so that they do not have to be endorsed as basic, but are located in the particle motion) and that in refraining from an additional ontological commitment to them, one avoids a commitment to surplus structure.

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There is a well-known problem for Humeanism about laws of nature: the standards for what is simple, what is informative and what is the optimal balance between these two criteria may not be unique. They may even be subjective. Lewis’s strategy to counter this objection is, in brief, to lay stress on natural properties that carve nature at its joints (such as, for instance, mass and charge) (see notably Lewis ). However, all the cognitive access that we have to these allegedly natural properties is the functional role that they perform for the particle motion (see e.g. Jackson , p. ). Hence, their intrinsic essence is beyond our cognitive access (see e.g. Lewis ). Consequently, the standard Humean stance that relies on such natural properties is in trouble already for purely metaphysical reasons, before it comes to the objection from entanglement in quantum mechanics. In any case, the strategy that relies on natural properties cannot be applied to Super-Humeanism, since there are no properties in the minimalist primitive ontology (see the objection that Matarese  builds on this fact). But there are relations in the minimalist ontology, namely exactly one type of natural relation that is the world-making relation (i.e. the distance relation). The predicates that define this relation are not sufficient to formulate laws about the change in the distance relations. Further predicates have to be introduced in terms of their functional role for that change, that is, in exactly the manner in which the standard Humean introduces the predicates that are supposed to refer to natural properties. However, for the role that these predicates play in the formulation of laws of nature it is immaterial whether one takes them to be dynamical variables that are located in the particle motion through the described procedure, or whether one regards them as referring to natural intrinsic properties (to which we have no cognitive access anyway). That is to say: in both standard Humeanism and in Super-Humeanism, the procedure how to obtain laws of nature is the same. This procedure relies only on there being stable patterns or regularities in the particle motion. The patterns in the behavior of the physical objects are what carves nature at its joints. These patterns enable the introduction of predicates in terms of a functional role for the particle motion. Laws of nature are then formulated in terms of these predicates. On the one hand, the metaphysics of matter points individuated by distance relations and the change of these relations is a robust scientific realism: it remains stable from ancient Greek atomism to today’s physics. It is not hit by the stock arguments against scientific realism from pessimistic meta-induction and underdetermination of theory by evidence, since these arguments concern the change in and the underdetermination of the dynamical structure of physical theories, but not the scientific realist ontology. On the other hand, that ontology, although based on science, may appear pretty much like an a priori, armchair metaphysics: all the experimental evidence that we always had and will ever have are relative positions of discrete objects and their change (including dots on a screen, pointer positions, detector clicks, etc.). Consequently, it will always be possible to adopt the stance that whatever dynamics a future physical theory introduces is a dynamics for the change in the relative positions of basic, discrete objects. In other words, it will always be possible to maintain that whatever new parameters a future physical theory conceives, they come in through the functional role that they play for the change in the relative positions of basic discrete objects.

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However, this is not the way in which the ontology defined by the two axioms of matter points individuated by distance relations and change of these relations is conceived as a science based, naturalized metaphysics. The ambition of this metaphysics is to stand out as the best account of the ontological commitments of our physical theories. This has to be established for each physical theory separately, and it may fail in the case of future physical theories. Thus, one has to vindicate Leibnizian relationalism as the overall best proposal for the ontological commitments of classical mechanics (see Huggett ), show how this proposal applies to relativistic physics as well (see Vassallo and Esfeld ; Vassallo et al. ). Furthermore, one has to argue that an ontology of permanent particles being characterized only by their relative positions and the change of these positions makes out the best solution to the measurement problem in quantum physics, both for quantum mechanics (see Miller ; Esfeld ; Callender ; Bhogal and Perry ), and for quantum field theory (see Colin and Struyve ; Deckert et al. ) (see the book Esfeld and Deckert  for setting out this case for both classical and quantum physics).

. Beyond Physics The Canberra plan provides a clear roadmap for both ontology and epistemology. As far as ontology is concerned, the task is to specify a basic set of entities such that everything else can be located in that set and hence be vindicated as being identical with something in that set. Thus, on the naturalized metaphysics sketched out in the two preceding sections, everything in the physical world is identical with a configuration of matter points that is characterized only by the relative distances among the matter points and the change in these distances. The task then is to find out which configurations of matter points are, for instance, water molecules, stones, trees, tigers, tables, etc. In detecting this, the crucial issue is not a configuration at a time, but the salient patterns of relative change in the distance relations in the configuration in question. These patterns provide for the stability of the macroscopic objects with which we are familiar. For instance, an ephemeral table-shaped configuration of matter points would not be a table; only a stable such configuration is a table. As argued in the two preceding sections, to obtain such stable configurations, all that is needed are certain patterns in the particle motion, but not an ontological commitment to intrinsic properties of the basic objects (see also Dickson ). As regards epistemology, all further notions apart from the primitive ones of matter points, distance relations and the change of these relations come in through a definition in terms of their functional role for the change in the distance relations. Given these functional definitions, all the true propositions that employ terms that are thus defined—e.g. all the true propositions employing terms such as “mass” and “charge”—then are entailed by the propositions that describe the universal configuration of matter and its change in terms of the primitive notions. The naturalized metaphysics set out in this chapter does not maintain that these definitions are an a priori affair of conceptual analysis. They are rather a matter of scientific investigation of salient patterns of motion in the world. Consequently, the entailment is not a priori. But the crucial point is that there is entailment given the description of the world in terms of the basic notions and the functional definitions of all the further

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notions. The multiple realizability of functional roles does not infringe upon these entailment relations: the issue is sufficient physical conditions, defined in terms of the basic notions, for these roles to be realized, never necessary and sufficient conditions and thus never biconditionals (see Chalmers , pp. –, on reductionist explanations as well as Esfeld and Sachse ). The Canberra plan can of course be applied beyond the domain of the natural sciences. The crucial issue is the functional definition of the relevant concepts in terms of their functional role for, in the last resort, particle motion. Consider mental concepts: there is no question any more today of behaviorism, that is, of defining mental concepts directly in terms of the effects on the bodily motions of persons. Nonetheless, functionalism in the philosophy of mind is the successor of behaviorism (see e.g. Lewis , section III). The functional definition of each single mental concept can include other mental concepts; but in the end, the functional definition of the whole cluster of mental concepts is one in terms of their causal role for the behavior of the person, that is, for the change in the relative positions of the particles making up the person’s body and its environment. This causal role functionalism is to be contrasted with a normative functionalism according to which the functional definition of mental concepts is an affair of indicating their role in a normative network of justifications (giving and asking for reasons) with that normative network constituting a realm of its own (such a normative functionalism can be traced back to Sellars ; for a contemporary elaboration, see notably Brandom ). Once one has functional definitions in terms of causal roles for in the last resort particle motion at one’s disposal, the true propositions employing the concepts defined in that way are entailed by the description of the particle configuration of the universe and its change. Consequently, the referents of those definitions are identical with certain sub-configurations of particles and their change, insofar as that change manifests certain stable regularities—although it may be quite a complicated (if not desperate) affair of empirical research to detect these sub-configurations. The crucial issue hence is whether all the concepts employed in the description of what there is admit of a definition in terms of their causal role for a basic ontology defined exclusively in terms of primitive physical notions. Functional definitions of this kind are undisputed in the natural sciences: it would be odd, for instance, to postulate a heat stuff to account for thermodynamical phenomena, since these can be defined functionally in terms of changes in molecular motion. By the same token, it would be odd to postulate an élan vital to account for organisms and their reproduction. Since the advent of molecular biology, the evolution of organisms and their reproduction can be accounted for in terms of molecular biology so that functional definitions in terms of causal roles for, in the last resort, particle motion are vindicated. There is no explanatory gap here. However, when it comes to consciousness as well as rationality and the normativity and freedom that are linked to rationality, one may maintain that there is an explanatory gap in the sense that functional definitions in terms of, in the last resort, changes in the configuration of matter do not capture what is characteristic of mental phenomena (see Levine ). Once one has understood the science, it is obvious how a functional definition of, for instance, water in terms of the effects on the interaction of H₂O molecules captures and explains the characteristic phenomenal

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features of water and how a functional definition of organisms captures and explains their reproduction, including the link from genotypes to phenotypes. However, it is not obvious—at least not obvious in the sense of these paradigmatic examples—what the qualitative character of conscious experience, or the normativity that comes with rationality have to do with molecular motion in the brain. Going beyond physics, the Canberra plan is not tied to a physicalist ontology. Its rationale rather is the one of setting a clear standard for ontology and thereby making evident the consequences that follow from endorsing a physicalist ontology as well as the consequences that one has to accept in any ontology that goes beyond physicalist commitments. The point is that any such further commitments come in as new primitives. For instance, in case the mental cannot be functionally defined on the basis of a primitive ontology of matter in motion, then an ontological commitment to the mental is called for over and above the ontological commitment to a primitive physical ontology. Such an ontological commitment then is as fundamental as the commitment to a primitive physical ontology, although the mental may only exist in certain systems in the universe and only at a certain period of time (cf. the Canberra plan of Chalmers  in which conscious experience figures as a further primitive beyond the physical ones). In general, whatever does not come in as being located in the basic set of entities—i.e. in a physicalist ontology as being identical with a specific sub-configuration of particles and their characteristic motion—and hence not as being entailed by the description of the particle configuration of the universe and its change in terms of the basic, primitive notions is itself a further primitive ingredient of the ontology. Of course, the methodology of parsimony advocated by Jackson () in the quotation at the beginning of this paper would be spoiled if the world turned out to be such that the reduction of everything to a few primitives (which do not have to be all physical) were not possible. Is there an alternative to this rigour of the Canberra plan? The most widespread alternative is the attempt to introduce notions that are weaker than identity with the intention of thereby eschewing the commitment to further ontological primitives beyond the physicalist ones. Emergence is the most popular such notion. There is a trivial sense of emergence in that new things come up in the evolution of the universe. However, if what emerges can be functionally defined on the basis of the ontology that is endorsed as primitive, then there is no emergence in any philosophically relevant sense, namely in the sense of something that ontologically goes beyond what was recognized as primitive. If what emerges cannot be thus defined, then we have a situation in which something that is included in the ontology originally admitted as primitive is correlated with something of which it is maintained that it cannot be defined in terms of its causal role for changes in something that is included in that ontology. Consider as a pertinent example a sub-configuration of particles that is a brain and conscious experience conceptualized in terms of qualia. However, we then have a primitive correlation. Saying that conscious experience emerges from neuronal particle configurations does not go beyond stating that there is a (systematic) correlation between certain particle configurations and certain phenomena that do not consist in exercising a certain causal role for changes in the particle configurations in question. Thus, when it comes to ontology, one is committed to more than what was originally admitted as primitive (i.e. matter in motion). Consequently,

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there then are further primitives that hence have the same ontological status as the original primitives, and the notion of emergence is of no use when it comes to ontology. The same goes for the notion of grounding. According to its proponents (e.g. Schaffer a; Fine ), grounding is not identity. Nevertheless, if A grounds B, then B is supposed to come in for free—that is to say, recognizing the existence of B requires no ontological commitment that goes beyond the commitment to A. However, it is not evident to say the least how this can be so. Again, what we have is a (systematic) correlation between two types of phenomena, such as, for instance, particle configurations that are brains and conscious experiences conceptualized in terms of qualia. It is not clear what work the notion of grounding does to explain this correlation by turning it into a priority of A (the brain) over B (the qualia)—unless in the trivial sense that there are particle configurations everywhere and all the time and that only some particle configurations in the universe, which moreover occur only at some times in the evolution of the universe, are correlated with qualia. Nonetheless, the correlation then remains primitive. Hence, again, if A is not identical with B, then there is something that makes B different from A, something which is not included in A, so that an ontological commitment going beyond A is necessary, and that commitment has to be added to the commitments that were originally endorsed as primitive. In sum, the merit of the Canberra plan is to set clear standards for metaphysics. These standards can be illustrated—and the Canberra plan be carried out in concrete terms—for the domain of the natural sciences. Going beyond that domain, the Canberra plan points out what commitments have to be undertaken—namely either to make do with the ontology admitted as primitive for the physical domain or to enlarge that ontology by further primitives that then have the same status as the original primitives—and what the consequences are in each of these two scenarios.

Acknowledgements I am grateful to the participants of the conference “Foundation of Reality: Fundamentality, Space and Time” and two anonymous referees for helpful criticism.

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7 What Entanglement Might Be Telling Us: Space, Quantum Mechanics, and Bohm’s Fish Tank Jenann Ismael

There are complex pressures from many sources in physics for questioning whether space-time is a fundamental structure. Much of the impetus has come from considerations of cosmology and quantum gravity. Many theories of quantum gravity describe fundamental structures that are significantly non-spatiotemporal. In that setting, the motivation comes from high-level theoretical concerns that are quite far removed from experience. Much of the discussion is highly technical and has not penetrated into the philosophical literature.¹ In philosophy, discussion of the possibility that space is not fundamental has centered on a particular proposal for the interpretation of quantum mechanics: David Albert’s wave-function realism (Albert , ). There, the discussion is conducted in the context of standard, non-relativistic quantum mechanics and is quite narrowly focused on whether the virtues of wave-function realism outweigh those of, say, GRW or Oxford Everett. The status of space on Albert’s proposal has mostly been treated as a problem, and one that we would rather do without. There hasn’t been, either in the physics literature or in the philosophical literature, a clear articulation of the general impetus provided by quantum phenomena for the move to an ontology in which space is recovered as an emergent structure. That is what I will try to provide. I will suggest that there are signs in the most basic and familiar features of quantum phenomena that space (-time) is not fundamental. I will present simple low-dimensional examples that reproduce aspects of entanglement and complementarity and explore the difference between two kinds of explanation:one that looks for causal processes passing through the space in which the correlated events are situated, and one that derives them as lower dimensional projections of a higher dimensional reality.

¹ That said, there is a significant, and growing, literature in the philosophy of quantum gravity that considers the reasons (in the context of quantum gravity generally, and within particular approaches) for treating spacetime as non-fundamental: e.g., Castellani & De Haro (this volume), Crowther (), Oriti (), Wüthrich (). Jenann Ismael, What Entanglement Might Be Telling Us: Space, Quantum Mechanics, and Bohm’s Fish Tank In: The Foundation of Reality: Fundamentality, Space, and Time. Edited by: David Glick, George Darby, and Anna Marmodoro, Oxford University Press (2020). © Jenann Ismael. DOI: 10.1093/oso/9780198831501.003.0008

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Figure . The set-up.

This chapter is meant to display in a simple setting a form of explanation that connects quantum phenomena to questions about the status of space. For those who aren’t soaked in the details of quantum mechanics, it is not easy to develop physical intuition for what entanglement and complementarity are. The examples can serve as toy models that exhibit the connection between quantum phenomena and the (possible) non-fundamentality of space in a vivid way.² They also allow us to move discussion in the foundational literature in philosophy away from wave-function realism, and also avoid some of the contested vocabulary. The primary goal is to sharpen our understanding of what is at issue here. The division between those who advocate a non-(fundamentally) spatial account and those who insist that space has to be a fundamental structure touches on a deep, and increasingly pronounced divide between two approaches to metaphysics.

. Bohm’s Fishtank and the Kaleidoscope The first example comes from Bohm (with some embellishment) (see Figure .).³, ⁴ ² Wave function realism (in the form in which it was originally proposed) is a first pass at this form of explanation. It was a bold initiative, and a radical departure from previous ways of thinking. But the case for treating space (time) as emergent should not be judged solely on its merits. ³ The discussion here is non-relativistic and restricted to space. This is partly for visualizability and intuitive naturalness, but for other reasons as well. In a relativistic setting, it is more natural to treat spacetime as the emergent structure, but time raises new questions, and I want to leave those aside. ⁴ I’ve embellished the example and developed it more fully. Bohm presents it as a model of entanglement in Bohm ().

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Imagine that there is a fish tank containing several multi-colored fish being filmed by one camera from the front and another camera from the side. It’s an ordinary three-dimensional tank. But suppose that the cameras project side-by-side images on a flat screen in an adjacent room, integrated seamlessly so that the screen displays just a two-dimensional expanse of color and moving shapes. Imagine someone whose experience was confined wholly to the shapes moving across the two-dimensional screen, who knew nothing directly about the three-dimensional space in which it (and he) was embedded, and who was keeping track of the on screen movements. Such a person would notice correlations between the images appearing on the two sides of the screen. For example, if on the right side of the screen he sees an image of a lionfish from the side, on the left side he will see another such image this time from the front. The two images will generally look different from one another. And we will suppose that if he focuses on either image by itself, he will not be able to predict its changes or movements from one moment to the next, but there will be interesting correlations between them. A flick of the tail or turn through an angle on the left will be mirrored by a flick of the tail and corresponding movement on the right. These correlations will seem to the viewer to be instantaneous. Someone whose experience was entirely confined to the two-dimensional images on screen would naturally try to fathom the mechanism by which the distant events communicate or influence one another. He might look for signals passing between them or search for causes in their common past. We know that the search would be misguided because the correlations aren’t the product of signals or causal influences passing through the space of the screen. They are products of redundancy in the space in which the images of the fish appear, which I’ll call hereafter the image-space. The space in which the images of the fish appear is a lower dimensional projection of the space in which the fish themselves are contained. Where there is one fish in the tank, the viewer of the screen sees two. Where there is one tail flick he sees two. Objects and events located at a single place in the tank produce multiple copies at different places on the screen. Another, more obvious example of correlations produced by redundancy in an image space can be obtained from a kaleidoscope. A kaleidoscope is an optical toy in a tube; it produces symmetrical patterns as mirrors reflect bits of colored glass. It consists of mirrors that run the full-length of the inside of the scope. There is a fixed or detachable object case at the end of the mirror tube that gives the scope its images. In many cases, the scope has a casing that contains glass beads that move freely and independently of one another inside the confines of the case. Changes in the configuration of beads produces changes in the pattern of light and color seen by someone looking through the eyepiece. The viewer doesn’t directly see the threedimensional beads or pieces of colored glass located in the casing. He sees those pieces reflected and refracted through the mirrors to produce a two-dimensional image in which each piece of glass is redundantly represented. Facts about the way the scope is put together determine the invariant features of the image; its boundaries, symmetries, and topology.⁵ The number and angles of the ⁵ Invariant, that is, under transformations that don’t disassemble the scope. These will appear as kinematic constraints on the space of possible images generated by a particular scope. The dynamics of the image space will simply describe its trajectory through the space of possible images.

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mirrors will determine the number of reflections viewed.⁶ Someone looking through the lens wouldn’t be able to predict which image will show up next, but he will notice correlations that would let him use what he observes in one part of the image space to get information about others. He will know that a blue triangle in one place will be matched and mirrored by counterparts in others. The symmetries are different in every case, but every case possesses them. The symmetries of a kaleidoscopic image space are more obvious than those of the fish tank. Viewers of the fish-screen focused on small-ish volumes of the space would see fish-images moving into and out of view and behaving like effectively autonomous two-dimensional realities. But for someone with a mathematical eye and a full view of the two screens, the symmetries of the fish image-space would be equally striking. What is interesting about these examples is that we have a one–many correspondence between events in three-space and events in a non-fundamental image-space that express themselves as instantaneous dependencies between events at different image-space locations. Someone whose view was confined to the image space and who knew nothing of the higher dimensional reality in which they were embedded would be puzzled by the apparently coordinated randomness. He might posit causal mechanisms or signals passing between the two sides of the screen. But of course, as we know, no attempt to explain the correlations dynamically in the lower dimensional space will be correct. It’s not that there is no explanation of the correlations. It is that there is no dynamical explanation in the image space. In the kaleidoscope example, the image that the observer sees when he looks through the eyeglass is twodimensional, but the observer himself, the eyeglass, the bits of glass that generate the image, the placement of the mirrors and the process by which the bits of glass generate the image all live in three space. In the fish-tank example, the surface on which the redundant images of the fish project themselves is two-dimensional but the tank, the fish, the cameras, and the process by which the images on the screen are generated from the movements in the fish-tank, are three-dimensional. One may have an interest in producing a descriptive history of image space, but the physics is given in three dimensions. We wouldn’t expect the screen to have an autonomous, well-behaved dynamics. And no attempt to explain the correlations dynamically in the lower dimensional space will be correct. The fish tank and the kaleidoscope provide low-dimensional examples of a way of explaining phenomena with suggestive similarities to the behavior of entangled particles quantum mechanics. In (standard, non-relativistic) quantum mechanics, the principles for constructing the state-spaces for complex systems—a pair of particles, an object system, and a measuring apparatus, an observer and her physical environment—generate states for the whole that cannot be reduced to states for the components. The result is that the quantum state of a complex system does not in general permit decomposition into ontologically distinct components.⁷ Schrödinger

⁶ For a two-mirror system, a ten-degree angle, divided into , gives  reflections (or an -point star, since  of the reflections will be reversed from the original). A -degree angle divided into  degrees gives  reflections (or a -point star). ⁷ Or at least not in a way that preserves the logic of part and whole, that is to say that it doesn’t permit decomposition into spatially localized components whose intrinsic states can be characterized independently of one another and then pieced together to obtain a complete description of the whole.

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regarded entanglement as the defining trait of quantum theory, and it is the source of some of the central mysteries of quantum mechanics. The components of a system in an entangled state behave in ways that are individually unpredictable, but jointly constrained so that it is possible to forecast with certainty how one component will behave, given information about the measurements carried out on the other even though it is impossible to predict how they will behave individually and even though they are not interacting. The correlations borne by the spatially separated components of a system in an entangled state are well verified and there is no difficulty understanding how to derive them from the formalism. Difficulties surrounding entanglement have to do with trying to arrive at a physical understanding of how entangled particles manage to exhibit the coordinated randomness that the formalism predicts. It is in this connection that we get stories about tachyons, superluminal influence, or even cosmic conspiracy. The fishtank and kaleidoscope present clearly visualizable models of cases in which observers see correlations that are the product of redundancy in the space in which the phenomena are arrayed rather than any real dynamical interaction. There are no signals or causal influences passing between the images, no dynamical interaction defined in image space that explains the correlations. And they suggest a way of seeing what is creating the difficulties in understanding the quantum analogues. Instead of seeing entangled systems as distinct existences interacting in a three-dimensional space (or events arranged in a four-dimensional space-time), we can see them as redundant glimpses of a deeper structure, refracted and reflected to provide multiple representations in the manifest space of everyday sense.

. Complementarity If Schrodinger regarded entanglement as the central mystery of quantum mechanics, others have seen complementarity as more fundamental. Questions of what the fundamental objects are, are of course bound up with questions of what the fundamental quantities are. Here I want to point to the relationship between ‘observables’ in the image space and beables in three-space.⁸ Focus on the changing image of some particular fish as that fish turns through an angle; the aspect that was presented on the screen will disappear and be replaced by another. We can say that the view of a fish from one angle occludes another if they can appear on the screen individually but never together. A full frontal view completely occludes a view from the back; a left side view occludes the right, and so on (see Figure .). These relations of mutual occlusion provide constraints on simultaneous observation of particular pairs of images cast by fishes on the screen, and it would be natural to think of views that excluded one another as complementary. We can see what the back of a fish looks like and what the front looks like by turning the fish ⁸ A beable is just a physical quantity. The term was introduced by John Bell to contrast with the term ‘observable’ so that we can speak of what exists and not merely what is observed. Entanglement by itself doesn’t suggest that the more fundamental space has to be higher-dimensional. It is complementarity (the commutation relations between quantum observables) that pushes us to a higher dimensional space. There are kaleidoscopes, for example, that don’t have glass beads, but generate an image when pointed at a flat, colored surface.

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Figure . Front and side views of a fish.

through an angle, but we can’t simultaneously observe what the back and the front of the fish looks like. Those two images of the same fish never appear together on screen. There are also relations of partial occlusion. The frontal view obscures the side, so that you can’t get a clear view of the side, when looking at the front, but maybe we can attach informative probabilities, given a frontal view of a fish, to what will be seen if it turns through an angle to expose its side. If these relations of full and partial occlusion were systematized, we would get a quite complex network of relations between two-dimensional fish-images. A full three-dimensional portrait could be pieced together from a collection of complementary, two-dimensional views if we understood how that collection was structured in three dimensions, something that we could read off of the relations of mutual occlusion. And conversely, if we had such a portrait, we would know everything there was to know about the images that the fish would project in image-space if it were turned through different angles. Let us distinguish categorical from modal properties. Categorical properties are properties that involve no admixture of possibility. They describe what is the case, but not what could, might, or would be the case under some possible circumstance. The mass of a particle is usually conceived as a categorical property. Its disposition to decay under certain conditions is a modal property. The categorical description of a fish is properly given in three-dimensions, but the three-dimensional description embodies a good deal of complex, modal information about the images the fish would cast when viewed from different angles in image space. Observations in image space correspond to image-space observables. The image a fish presents when viewed from angle a corresponds to the value it has for the observable ‘view from a’. We know that the relations of occlusion and partial

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occlusion among image-space observables derive from the three-dimensional structure of the fish, but someone who didn’t know that and who tried to interpret image-space observables as abiding properties of a two-dimensional object would run into trouble. In visual terms, he would end up with a Picasso-esque construction of two-dimensional images from different angles superimposed on top of one another in a geometrically impossible configuration. In mathematical terms, he would find that there is no reduced, categorical two-dimensional description that captures the invariant content of the information embodied in a three-dimensional description. He could express all of the information contained in a three-dimensional description in a wave-function-like mathematical object which embodied a lot of complex, modal information about the images the fish would cast in image space, i.e., about the ‘values’ of different ‘image-space observables’. But it would be impossible to ground all of the dispositions that a fish has to appear thus and so when viewed from different angles in a two-dimensional description of how it is. The categorical description is irremediably three-dimensional. And this is suggestive of the commutation relations among quantum observables. In classical mechanics, the space of possible states for an n-particle system is a n-dimensional space, usually parameterized by the positions and momenta of constituent particles. Any function of these basic variables is an observable and every system always has a full set of values for all observables. The theory is defined by the axioms governing the behavior of the basic observables—Newton’s equations for the positions or Hamilton’s for positions and momenta. In (standard, nonrelativistic) quantum mechanics, there is more structure on the set of quantities. The space of possible states is a Hilbert space. States are represented by vectors. Physical properties are represented by Hermitian operators on that space.⁹ The dynamics is given by Schrodinger’s equation, which describes the evolution of the state vector if undisturbed. The Eigenstate–Eigenvalue link tells us that we observe the value for observable A iff the vector representing its state is an eigenstate of the A-operator. Now, we know that for any Hermitian operator on a Hilbert space, there are others on the same space with which it doesn’t share a full set of eigenvectors, and indeed some with which it has no eigenvectors in common. It follows that we can never observe simultaneous values for all observables and indeed that there are pairs of quantities whose values we never observe simultaneously. Quantities represented by operators that have no eigenstates in common are canonically conjugate. The most familiar example of such a pair are the position x and momentum px of a point particle in one dimension. These relations of occlusion and partial occlusion are summarized by the Uncertainty relations given for the example just mentioned by, [x,px] = iħ, where [x, px] = xpx px, x is the commutator of x and px, i is the imaginary unit and ħ is the reduced Planck’s constant h/π. As in the case of two dimensional fish-images, the relations take their most extreme form for canonically conjugate quantities, but if we catalogued them, we ⁹ And the Hilbert space associated with a complex system is, of course, the tensor product of those associated with its components. The rule for constructing the state-spaces of complex systems is what gives rise to entangled states.

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would find an interesting network of derivative relations of partial occlusion that would be preserved by free evolution or any attempt at measurement. It is very natural to think that the uncertainty relations simply place epistemic constraints on simultaneous knowledge of a system’s properties but that there must be some more categorical properties that characterize the intrinsic state of quantum systems and that ground the probabilistic dispositions attributed by the quantum state in the way that positions and momenta ground the dynamical dispositions of classical systems, and in the way that the categorical properties of any physical system ought to ground its law-governed behavioral dispositions. But we know a lot about the limits of this way of thinking in quantum mechanics. There has been a century of no-hidden variable results of varying strengths, placing restrictions on attempts to derive those law-governed dispositions from hidden, categorical properties of systems that live in three-space. Most people believe those restrictions are too strong to be met with empirical plausibility.¹⁰ The claim is not that there has to be an intrinsic fact about a particle that determines how it will behave on any given occasion; we are trying to introduce hidden variables that will allow us to say that there is an intrinsic difference between particles that have different law-governed dispositions to show a particular result in a measurement, or an intrinsic difference in a particular particle when it goes from having a chance of ½ to a chance of  of showing some result. We would expect analogous results if we tried to ground the information embodied in the -d description of a fish in categorical properties in two dimensions. The suggestion here is that quantum observables do not behave like categorical properties of a three-dimensional system. If we can derive the commutation relations among quantum observables from categorical properties of objects that aren’t ultimately localized in space, in a manner that mirrors the derivation of the algebra of twodimensional aspects from the three-dimensional description of a fish, that would be an explanation of a quantum effect that should appeal to your sense of naturalness. Qualitatively and intuitively, one way of understanding the pathologies of quantum mechanics is that in quantum mechanics there is nowhere in three-space to house and ground the dispositions attributed by the quantum state. We have lacked an understanding of quantum systems that would explain in categorical terms the measurable dispositions they have to affect ourselves and our measuring instruments in ways that we can predict with statistical accuracy. Treating space-time as a ¹⁰ There are well-known loopholes. Noncontextual hidden variables theories assigning simultaneous values to all quantum mechanical observables are ruled out by Gleason and Kochen-Specker, contextual hidden variables theories in which a complete state assigns values to physical quantities only relative to contexts are left open. See Shimony () for an especially illuminating discussion. We get an analogue of contextuality in the fish-tank example that is suggestive of what might be going on in the quantum case. To ground the dispositions embodied in the three-dimensional description of the fish in categorical properties in two dimensions, we have to relativize image-space observables to camera angles and positions (we say that a fish looks thus and so from this or that angle, but not simpliciter). This technique allows us to smuggle information about the three-dimensional configuration of the fish into two dimensions, but remains a purely formal option so long as camera positions and angles can’t be specified in two-dimensions. Put another way, there’s a failure of supervenience of measurable dispositions on (non-contextual) d observables defined over the image-space in the fish tank example because we lose information about relations between observables embodied in the three-dimensional structure of the fish and allowing contextuality is just a way of smuggling in -d information.

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derivative structure give us somewhere in the actual world to house and ground those dispositions. The examples, of course, should not be over-interpreted. They provide toy models of specific phenomena, and were intended to give only a qualitative feel for how to connect those phenomena to the (non)-fundamentality of space. They don’t substitute for a proper theory. They do, however, allow us to address some of the ambiguities that have hindered discussion of the quantum case.

. Two Kinds of Causal Notions We can distinguish two kinds of causal notions, both of which are familiar from the philosophical literature on causation. There are the causal relationships represented by directed, acyclic graphs (DAGs) and analyzed in terms of intervention counterfactuals (see Figure .). These are introduced to distinguish causal structure from mere probabilistic correlation. One says that there is a causal relationship (as opposed to a mere probabilistic correlation) between X and Y just in case in an intervention on X

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would produce a change in Y (intuitively, when you wiggle X, Y wiggles). Then there are causal processes. These are chains of events called causal interactions, each of which involves the exchange of some conserved quantity. These are sometimes treated as competing accounts of causation, but we don’t need to choose between them. They are both useful, and in classical physics we find both. We have causal processes at the fundamental level, and manipulable causal relationships of the kind captured by DAGs all over the place. The latter are undiscriminating about levels, and neutral about underlying processes. They represent often local, scaffolded relationships among variables that can be investigated experimentally, frequently without knowledge of the underlying processes. They are part of what we might think of the surface phenomena of the world. Causal processes, by contrast, are part of the explanatory substructure. They relate fundamental parameters. They are local and continuous. The early history of quantum mechanics interpretation was (in part) a search for causal processes to explain entanglement, complementarity, and a suite of other effects that were slowly teased out of the quantum formalism and put on display. The quantum formalism as it stood was recognized as an elegant, compact, and precise embodiment of the regular, law-governed effects of quantum systems on macroscopic, spatially localized ‘measuring’ instruments. But people were looking for an understanding of the causal processes in space-time that produced quantum effects. They were hankering after (to use a nice phrase from Bell) an account of what goes on in ‘the limbo between one observation and another’ (Bell ; Butterfield ). As the attempts to explain quantum phenomena got increasingly strained and unnatural, and as the quantum formalism frustrated them systematically, and as no-go theorems of increasing strength made them seem more and more like perversions of a formalism that had its own inner logic, people started to deny the need for a causal process explanation (Hughes ; Bub ; Healey a; Spekkens ; Fuchs, Mermin, and Schack ). Alongside approaches to quantum mechanics that renounced the search for a quantum ontology, there were those who continued the search, but made radical departures from classical ontology.¹¹ All the while, of course, there were still people who were doing the more traditional kind of quantum ontology (GRW, Bohmian Mechanics).¹² The people I’m talking about here were all people who were reacting to the frustrated attempts to fill in the limbo between experiments with continuous trajectories and causal processes. The landscape of possibilities here has become increasingly developed and refined. Against this background, what the toy models suggest is a way of hanging onto all of the causal relationships that we find in space-time, but instead of looking for microscopic causal processes propagating through space-time, we look for a substructure of connections behind the scenes, threading connections through degrees of freedom that aren’t themselves localized in space.

¹¹ Albert (, ) and the Everettians, well represented in Wallace (). ¹² See Lewis () for an overview.

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. Two Notions of Space Just as we can distinguish two notions of causation, we can discern two notions of space. Let’s use Our-space to refer to the manifest space of everyday sense and Urspace to refer to the space in which the fundamental particulars are housed or, if you like this way of speaking: the space that acts as the ground of individuation for the fundamental objects, or perhaps the space in which the world decomposes into separable parts. Each of these ways of speaking evokes a role that Our-space plays in classical physics.¹³ Then question is, then, whether we should see the kinds of connections that the quantum formalism predicts among local beables in Our-space as suggesting the existence of a more fundamental ordering, an Ur-space containing beables whose connections to space-time observables screens off the internal connections among the latter. The symmetry of the connection between the particles in a Bell experiment (or the parts of an entangled system), the absence of evident mediating processes, the complex relations of full and partial occlusion among space-time observables, the fact that the quantum state contains more information than can ever be directly accessed in measurements in space-time all suddenly make sense if Our-space has the status of an image space. They all snap into place, as exactly what is to be expected. The difficulties that we find in attempts to fill in the story about what goes on in the limbo between experiments with continuous trajectories and causal processes in space-time have analogues in the examples for observers in the examples trying to do their physics in the image spaces and make those attempts look misguided.

. The Rationale for Moving to Ur-Space In the contemporary literature surrounding the discussion of wave function realism, there are two sorts of questions that have been distilled out of the controversy, both pressed here by Wayne Myrvold, in a symposium on Albert’s After Physics (Albert ): The general precept at work here seems to be . . . Faced with a physical theory that, taken at face value, seems to violate the condition of separability [i.e., the requirement that there should be no connections between systems located in different parts of space that aren’t mediated by causal processes], we are to find (or construct) another space, such that states of the theory can be represented as an assignment of local quantities to points in that space, and to take some space of this sort as the fundamental space of the theory. If this is the precept in operation, then I have two sorts of question about it. One is: in what sense is a space of this sort . . . more fundamental than the space in which we live and move . . . [and what is the] argument in favour of separability as a requirement on an acceptable theory of fundamental physics. (Myrvold)

The toy models can help us address Myrvold’s questions. First they give us concrete, low-dimensional examples of the sort of relationship that is being proposed to hold between Our-space and an Ur-space. The philosophical discussion about what it ¹³ The reference here is to Gareth Evans’ () discussion of the role of space as the fundamental ground of individuation of objects. Evans is talking about the role that Our-space plays in the conceptual scheme of common sense.

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means to say that space is (or is not) fundamental has been rather cloudy. It turns inward to reflection on what ‘fundamental’ means, then to analysis of the various roles that space plays in our experience, on the one hand, and in physics, on the other. Worries are voiced that on this proposal space would turn to be ‘unreal’ or ‘illusory’, and we begin to worry what ‘real’ means. This is useful in helping us refine our concepts, but it hasn’t helped to clarify the physical proposal.¹⁴ The toy models can at once make the proposal concrete without using that vocabulary, and serve as the basis for refining it. And they make the rationale explicit by highlighting the sorts of clues to which people are responding when they look to a non-spatiotemporal substructure. Kerry Mckenzie in a recent review discussing Albert’s wave-function realism puts the demand for the rationale more spiritedly. She writes that Albert’s view: Seems to be embraced by those who do so . . . only in order that we not be ‘saddled’ with the ‘old-fashioned and unwelcome quantum mechanical weirdness of non-separability’ . . . it seems deeply incongruous to me that respectable philosophers of physics are so sanguine about letting quasi-aesthetic predilections like this do so much ontological work. . . . There is no reflection, not so much as a momentary expression of regret, over the idea that at some point we may have no choice but to retreat to largely individual preferences regarding virtues to support our world view. (McKenzie )

I am suggesting that the case for this form of explanation can be made independently of wave-function realism and that it is not a mere ‘intuition’, expectation derived from classical habits of thought, or ‘quasi-aesthetic’ distaste for non-separability that motivate this form of explanation but clues in the phenomena of a kind that are our best guide to the deep structure of the world. The rationale for suspecting that there might be a structure behind (or underneath) space-time that explains the deepest quantum effects is not a perverse predilection for revisionary metaphysics, or a ‘quasi-aesthetic’ distaste for nonseparability. The complex connections between the events in different parts of space revealed by the quantum formalism are the same kinds of clues that guide us in other kinds of inferences when we look for an explanatory structure. That is why they appeal to our sense of explanatory naturalness. It would be nice here, if we could invoke a formal normative framework for scientific inference so that we wouldn’t have to rely on our sense of explanatory naturalness, but we don’t have a generally accepted framework. Our sense of explanatory naturalness plays an ineliminable role both in science and in everyday inference to the best explanation. Once the ambiguities in the central notions are resolved, it begins to emerge that what separates the two sides of the debate is a very deep division between two ways of doing metaphysics. On the one hand, there is the purist (represented here by David ¹⁴ This is one of the most important reasons for getting away from abstract, verbal presentations. Words like ‘fundamental’ and ‘emergent’ don’t have a clearly defined meaning in current philosophical usage. In physics, and as I will use them here, ‘fundamental’ typically means ‘basic in the ontological ordering’, and ‘emergent’ means ‘not fundamental’. But as soon as one moves outside those circles the terms are used in ways that are complex and contested. Even without a well-defined meaning, the terms are laden with misleading philosophical connotations. On the question of whether space is ‘illusory’ if not fundamental, see Lewis (, ).

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Albert, and elsewhere by people like David Wallace () and myself) whose standards for ontology are mathematical clues that guide physics at the fundamental level. On the other hand, there is the a priori metaphysician whose standards are closeness to common sense. The standards for choosing between fundamental ontologies in physics are the kinds of formal clues that I have highlighted here, and that push in the direction of an Ur-space: one that plays the same role in individuation in fundamental ontology that Our-space plays in individuation of the ontology of common sense, but which is not three-dimensional.

. Reichenbach’s Cube I want to add here, one last example. This one drawn from Reichenbach. It comes from Experience and Prediction.¹⁵ It is not a quantum example. Reichenbach asks us to imagine: . . . a world in which the whole of mankind is imprisoned in a huge cube, the walls on which are made of sheets of white cloth, translucent as the screen of a cinema but not permeable by direct light rays. Outside this cube there live birds, the shadows of which are projected on the ceiling of the cube by the sunrays; on account of the translucent character of this screen, the shadowfigures of the birds can be seen by the men within the cube. The birds themselves cannot be seen, and their singing cannot be heard. To introduce the second set of shadow-figures on the vertical plane, we image a system of mirrors outside the cube which a friendly ghost has constructed in such a way that a second system of light rays running horizontally projects shadow-figures of the birds on one of the vertical walls of the cube . . . this invisible friend of mankind . . . leaves [those inside the cube] to their own observations and waits to see whether they will discover the birds outside. (pp. –)

Penetration through the walls is impossible, so all that they have to go on is correlations in the movements of the shadows, and as Reichenbach observes: If the shade a wags its tail, then the shade a also wags its tail at the same moment. Sometimes there are fights among the shades; then, if a is in a fight with b, a is always simultaneously in a fight with b. (p. )

The story has a hero—unsurprisingly called ‘Copernicus’—who proposes a radical and suggestive theory. He will maintain that the strange correspondence between the two shadows of one pair cannot be a matter of chance but that these two shadows are nothing but effects caused by one individual thing situated outside the cube within free space. He calls these things ‘birds’ and says that these are animals flying outside the cube, different from the shadow-figures, having an existence of their own, and that the black spots are nothing but shadows. (p. )

The example has the same form as Bohm’s fish tank. It occurs in the context of a discussion of the inference from cross modal patterns in sensory phenomena to the existence of an external world. And that analogy is apt for the fish tank and the kaleidoscope, and for the suggestion about the explanation of quantum phenomena ¹⁵ Reichenbach (). There is a very nice formal discussion by Elliot Sober () that connects it to common cause reasoning.

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being made here as well. In each of these cases, we trace correlations to a common source.¹⁶ We have links between ‘images’ or ‘shadows’, which have a secondary existence, connections among which are not explained by causal processes that pass through the space in they live, but common links to something outside the space in which the images are projected. The suggestion is that these same inference patterns are pointing towards treating Our-space as something less than fundamental. These are the kinds of signs that usually lead to a search for degrees of freedom that screen off the connection. What is special about the quantum case is that those degrees of freedom can’t be assigned to volumes of space in the natural and familiar way (viz., as representing non-contextual, intrinsic properties whose effect on other regions is mediated by local influences).¹⁷ The way to think of the epistemic position of the theorist is that she is solving simultaneous equations for the degrees of freedom controlling a range of observable effects, where the nature of those degrees of freedom is itself up for grabs.¹⁸ This is what is going on in the example that Reichenbach was pointing to: the inference from correlations across sensory streams to an external world.¹⁹ In microphysics, the data streams are instrumentally mediated sources of information about a world whose ultimate structure is inferred, and the hope is to approach understanding of the basic elements of nature at least up to the level to which our instruments are able to probe. We form an increasingly articulate understanding of the unobservable substructure of the world as we learn to distinguish separable components, isolate their individual ranges of motion and see how their joint movements produce the visible behavior. The clues that we use in these cases are the same ones to which we are responding in these everyday inferences. We

¹⁶ It might seem surprising that Reichenbach does not try to justify this as an instance of his Common Cause Principle. That Principle received its most explicit defense in his (posthumously published)  book, The Direction of Time (Reichenbach ). The discussion in Experience and Prediction was published in . It would be interesting to know more about the trajectory of his thought in this period. ¹⁷ The idea that there are degrees of freedom that screen off the connection between particles in entangled states is not new. Many people have suggested that we should just think of the quantum state as containing information about degrees of freedom that don’t have a spatiotemporal location, or can’t be localized in any volume of space. In formal terms, that is a natural thing to say. The wave-function contains information about degrees of freedom that can’t be localized. The problem is that if we allow causal agents outside of space, we break the connection between space and causal structure that we have in a classical setting. The suggestion here is that Ur-space just is the space that makes the causal structure explicit. The role played in a classical setting by Our-space is played in this setting by Ur-space. ¹⁸ Consider a doctor trying to explain an array of co-presenting symptoms. She doesn’t start out with a clear and distinct idea of the source (is it microbe? a tumor? an unknown disease?), but there is a default assumption that if the symptoms always present together they probably have a common source. In that case, she can open up the body and check her hypothesis. In physics, this kind of direct verification is not an option. We are like fisherman trying to figure out what is producing ripples on the surface of an opaque body of water, we can send down lines and probes, but we can’t go down there and check our hypotheses. We rely unavoidably on heavily mediated information streams, and it is our theories that individuate their source. ¹⁹ We don’t (of course) make that inference consciously. It is made (effectively) by the brain in processing sensory clues for spatially structured perceptual presentations. Reichenbach is seeking here simply to make the structure of the inference explicit.

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look at mutual constraints on independent variation as symptoms of redundancy in our information and use them to triangulate to a common source.²⁰

. Shifting the Intuitive Weight There are two ontological attitudes one can take in approaching the interpretation of quantum mechanics. One assumes a broadly classical ontology of three-dimensional objects (particles, configurations of particles, or macroscopic measuring devices, as the case may be) and supposes that the quantum laws describe how the categorical properties of these objects change over time. The other attitude turns that story on its head. On this view, what we see in three-space, are mere appearances: i.e., partial and perhaps redundant manifestations of a reality whose intrinsic structure is unknown. Correlations trace to underlying identities, dispositions are grounded in categorical properties, the three-dimensional space of classical physics (or four-dimensional space-time of relativistic physics) is recovered as a derivative structure. In practical terms, making the gestalt shift from the first to the second attitude means abandoning the idea that there should be a well-behaved story about what happens in three-space, calling off the search for a physical process in space-time by which the components of an entangled system influence one another, relinquishing the call for continuous trajectories and rejecting any attempt to describe what is happening in three-space in the gaps between measurements. It has a rather different explanatory task viz.: recovering what we see in three-space from a deeper structure. The two styles of explanation are very different. One asks ‘what are the processes in space-time that explain the correlations?’ The other asks ‘how do correlations in space-time emerge from the structure of the underlying reality?’ Many parts of physics have made this shift already. In cosmology, and quantum gravity, as I said, it is routine to treat space (and perhaps space-time) as a derivative structure. The reasons are complex, and more highly theoretical.²¹ The physics, as always, proceeds under its own steam. But the philosophical imagination is lagging behind. That is in part because the philosophical imagination tends to be guided by common sense, and not by the kinds of formal clues that I have been pointing to here. My colleague Richard Healey is fond of saying that if we want to know what physics is telling us about the world, we should be looking at our best and most fundamental theories: quantum field theory, or quantum gravity, certainly standard, non-relativistic quantum mechanics. He’s right about that. But it is still worth looking back and seeing whether we can see suggestions of a non-spatiotemporal order even in the most familiar and elementary quantum phenomena, the ones that are at the heart of the difference between the classical and quantum world. Whether an approach like this works will remain to be seen. Ultimately, these are physical questions. They won’t be settled by philosophical argument, but by calculation and experiment, and the detailed development of theory. ²⁰ If criteria of identity for underlying beables are given independently, the situation is different. So it is really only when we have no independent criteria of identity for the underlying beables that these kinds of constraints guide individuation, favoring redundancy over coincidence. ²¹ See Wüthrich and Huggett () as well as the references in note .

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8 Wave Function Realism in a Relativistic Setting Alyssa Ney

The purpose of the present chapter is to respond to a thread of recent criticism against one candidate framework for interpreting quantum theories, a framework introduced and defended by David Albert and Barry Loewer: wave function realism (Albert , , ; Loewer ). According to this framework, we should take quantum theories to show us that contrary to what perceptual appearances suggest, the world we inhabit is not fundamentally a world of discrete objects located in a three-dimensional space or even of matter fields in a four-dimensional spacetime. Instead, the world is fundamentally constituted by a less familiar field, the quantum wave function, which inhabits a much different and higher-dimensional space, one adequate to capturing the full range of pure quantum states. The criticisms with which I will be concerned here gain traction based on the fact that wave function realism has until now been formulated and defended solely within the context of nonrelativistic quantum theories. Although nonrelativistic quantum theories are useful as approximations, teaching us important lessons about a range of real world phenomena, it seems that if one is going to learn about our world from a consideration of quantum theories, these are going to have to be lessons that remain even when we move to a relativistic framework. After all, we know our world to be a relativistic world. With this in mind, Wayne Myrvold, Chris Timpson, and David Wallace have all argued that wave function realism yields an interpretation reliant on features of quantum theories that do not carry over to the relativistic setting (Wallace and Timpson ; Myrvold ; Wallace ). Therefore, wave function realism does not adequately capture aspects of the fundamental nature of our world according to quantum theories. It is an interesting question whether wave function realism must, to be viable as an interpretation of quantum theories, have application beyond the domain of nonrelativistic quantum theories. Must an interpretation of a quantum theory be workable as an interpretation for all quantum theories? For the purposes of the present chapter, I’ll bracket this question and address the issue of what good wave function realism may be for interpreting relativistic quantum theories. Myrvold, Wallace, and Timpson raise several specific concerns based on issues that come into play when we move to a relativistic setting. After a brief primer on how wave function realism looks Alyssa Ney, Wave Function Realism in a Relativistic Setting In: The Foundation of Reality: Fundamentality, Space, and Time. Edited by: David Glick, George Darby, and Anna Marmodoro, Oxford University Press (2020). © Alyssa Ney. DOI: 10.1093/oso/9780198831501.003.0009

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in the nonrelativistic context, I’ll recap these objections. I’ll argue that none of these objections give conclusive reason to reject wave function realism;however, some do present a challenge to the wave function realist to be clearer about what her view looks like when extended especially to quantum field theories. And thus I provide a sketch of what the view looks like in relativistic contexts, one that is extendable from the simple case I describe in detail to more complicated ones. This will establish the main point of this chapter, namely that the primary motivation for wave function realism is one that rests in no way on details peculiar to nonrelativistic theories and so there is a prima facie case for a suitable extension to the relativistic domain whatever that may in the future look like.

. Wave Function Realism for Nonrelativistic Quantum Mechanics Wave function realists explore the possibility that what quantum mechanics is trying to tell us about the world is that what appear to be numerically distinct objects located in separate regions of space are in fact manifestations of a more fundamental field, the quantum wave function, living on a less familiar space, one quite different from that presented in ordinary perceptual experience. Practically all discussions of wave function realism have until now been carried out in the context of nonrelativistic quantum mechanics, where one starts the project of interpretation from data suggesting the world is constituted by a determinate number of particles that may be in more or less determinate states (superpositions) of spin, energy, position, and so on, states which may be correlated with one another by entanglement relations. The wave function realist then argues, appealing to the virtues of metaphysical frameworks that are separable and local, arguments that will be fleshed out in more detail below, that the more fundamental reality underlying the appearance of a world of many particles in three-dimensional space is one of a single entity, the wave function, characterized by an assignment of numbers (amplitude and phase values) or spinors (for particles with spin) to a space with the structure of a classical configuration space, one for which every point in the space corresponds to a state in which all particles have determinate positions. Since particles in the quantum description do not all have determinate positions, the wave function at any given time will be a field smeared out over many locations in its space. This field is then said to be the fundamental reality with the initial particle description ontically derivative. A variety of questions are typically raised about this proposal. For example, if the fundamental reality is really a field on a space with the structure of a configuration space (which would mean the fundamental space is extremely-high dimensional, given the number (~⁸⁰) of particles that appear in the observable universe), then why does it seem we inhabit a world of discrete objects in a much lower-dimensional space? Wave function realists have addressed this question, often by arguing that the contingent dynamical behavior of the wave function is such as to generate the existence of what may accurately be described as (non-fundamental) threedimensional objects that may have at least moderately precise three-dimensional locations. Wave function realists and their supporters debate the best way to fill in

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this account. David Albert, Barry Loewer, Jill North, and I have all made proposals (Albert ; Albert and Loewer ; North ; Ney ). The critics with whom I engage in this chapter do not question whether this project may be successfully carried through in the nonrelativistic case.

. Critiques I count six different critiques of wave function realism in the literature citing the difficulties of extending wave function realism to a relativistic setting. I will describe them in chronological order of publication. A. The first critique (Wallace and Timpson ) is that there is no good account of the sort of space the wave function is meant to be defined on when wave function realism is considered as an interpretation of relativistic quantum field theories. In the nonrelativistic case, as has already been mentioned, the wave function is defined on a space that has the structure of a classical configuration space. This is a space with 
N dimensions, where N is the number of particles in the universe, particles which are claimed to be ontically derivative. For the wave function realist, this provides a “top-down” description of the fundamental space, since particles are not fundamental; but one that nonetheless singles out a primitive structure. For a nonrelativistic quantum theory, the character of the configuration space is straightforward, since particle number is assumed to be conserved. However, in quantum field theories, particle number is not conserved; moreover, systems may evolve into states describable as superpositions of particle number. This undermines the possibility of understanding the world as simply constituted by a relativistic wave function as a field on some unique, determinate configuration space. Wallace and Timpson consider an alternative possibility, namely that for quantum field theories, the wave function realist instead postulate an infinite number of (non-normalized) wave functions: a single-particle wave function living on a three-dimensional space, a two-particle wave function living on a six-dimensional space, and so on. However, they (rightly) assume the wave function realist will not prefer to adopt such an ontically profligate metaphysics. B. A second objection is that wave function realism “obscures the role of spacetime in quantum theories” (Wallace and Timpson , p. ). Here Wallace and Timpson are especially concerned with quantum field theories. Standard presentations of quantum field theories work by assigning field operators to localized spacetime regions. (This is true as well in algebraic quantum field theories, in which algebras are associated with spacetime regions.) Systems are not described as in the Schrödinger wave-dynamical representation of nonrelativistic quantum theories in terms of the evolution of a wave function in configuration space. And so the representations on which wave function realism is based simply are not standard representations in relativistic quantum theories. C. A third critique relates to the demand of relativistic covariance. To get a theory of the evolution of the wave function that is relativistically covariant, one sacrifices what David Albert has called ‘narratability’ (Albert ). That is, there will be no unique account of how the wave function evolves from one time to the next. Wallace and Timpson () press this failure of narratability not as leading to a decisive

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refutation of wave function realism. Rather they argue that this failure of narratability is obscure and surprising in the context of wave function realism where one is supposed to be achieving a separable metaphysics. It is, by contrast, unsurprising in the context of rival interpretations of quantum theories, including the one they prefer, spacetime state realism. D. A fourth critique aims to show that in the context of relativistic theories especially, the existence of a wave function is derivative on the antecedent existence of structures defined on ordinary spacetime. Myrvold shows how wave functions, to the extent that they may be constructed in quantum field theories, may be defined in terms of the global quantum state, vacuum state, and field operators associated with points of spacetime (Myrvold ). So, he concludes, the wave function cannot be more fundamental than a spatiotemporal ontology, as the wave function realist believes. E. Myrvold also argues that in the case of relativistic quantum theories, the wave function cannot be thought of as a field in the way the wave function realist presupposes (Myrvold ). This would require its specification by an assignment of local values to points or subregions of the space it occupies. Myrvold argues that facts about the state of the wave function at arbitrary subregions of the wave function’s space are not local to those regions. He argues as follows. Consider two states. S is a state in which a single-particle wave function takes on a nonzero value at some point x. S is a state just like the first near x, but differs in that there is with certainty a particle confined to a spacetime region R, far from x. It follows that in S, the probability that an array of detectors spread through space will report a particle detection at x and nowhere else is zero. From the Born rule we may infer that the single-particle wave function for S must take the value zero at x. And so, nonzero value of a single-particle wave function at x is incompatible with there being a particle definitely located in region R, no matter how far away R is. Myvold’s conclusion is that the assignment of the nonzero value to some point near x in S is not a local fact about that point. This, he then claims, means that wave functions are not like traditional fields, contra the wave function realist.¹ F. Finally, David Wallace argues that the privileging of the position basis is problematic in the context of quantum field theories, for which quantum states and observables are more typically defined in terms of a momentum basis (Wallace ). We can see that at least in the nonrelativistic case, the wave function realist does privilege the position basis as she defines the wave function as an assignment of values to points in a space with the structure of a classical configuration space as ¹ Simon Saunders (p.c.) has questioned whether this argument of Myrvold’s ought to be classified as an objection to wave function realism that concerns its extension to relativistic quantum theories, claiming there is nothing particularly relativistic about the example. (Though the other examples Myvold provides to illustrate the premises of the argument are relativistic, the arguments based on them use the same basic reasoning as that presented above.) This is fair enough, but all variants of the argument involve situations in which particle number is not conserved. Whether one wants to think of the failure of particle conservation as essentially a relativistic phenomenon or not, Myrvold’s argument at least offers a concern for the extension of wave function realism to examples of more realistic quantum theories than those in which particle number must be conserved. For this reason, I address it in this chapter.

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opposed to some other kind of state space. One might think it is unproblematic that quantum field theories generally characterize states in terms of a momentum basis, as one can simply Fourier transform to achieve a position representation. But this is not straightforward in the case of quantum field theories. There it turns out that even if representations of exact states of position can be achieved, these have the consequence of violating Lorentz covariance (cf. Teller , pp. –). And so if the privileging of position is an essential feature of wave function realism, not merely confined to the interpretation of nonrelativistic quantum mechanics, wave function realism cannot be generalized into an interpretation of relativistic quantum theories. These are the objections. What all of these critics of wave function realism prefer is an interpretational framework according to which quantum theories describe an assignment of objects (operators, algebras) to spacetime regions rather than to the high-dimensional space preferred by the wave function realist. This is a view that Wallace and Timpson label spacetime state realism. Adopting such a framework avoids the privileging of the position basis and the dependence on a fixed number of particles that the framing of an ontology in terms of a configuration space representation requires. However, before giving up the high-dimensional metaphysics favored by the wave function realist and moving to spacetime state realism, it is worth recalling why the wave function realist favored such a picture in the first place. I will argue that the wave function realist has the same reason to reject spacetime state realism as she had for rejecting a fundamental metaphysics of particles arranged in three-dimensional space. The motivations for the high-dimensional picture carry over to the relativistic case, even if the applicability of a high-dimensional configuration space representation does not. Shrugging off the constraints of the wavefunction-in-configuration-space view, we can begin to see what a more plausible relativistic extension of wave function realism may look like and how one may respond to the six criticisms above.

. Wave Function Realism in Relativistic Quantum Theories As was briefly mentioned, the advantage wave function realism has over rival interpretations of quantum theories one finds today, keeping in mind these discussions are generally held in the context of nonrelativistic quantum mechanics, is that it provides a metaphysics that is fundamentally both separable and local. In the nonrelativistic case, this discussion begins by considering what appear to be spatially separated particles of a determinate number. However at least prima facie, the case for wave function realism may be extended to consider any type of events occurring at distinct spacetime locations for which there appear to be correlations induced by entanglement. The wave function realist will take these correlations to suggest that what appear to be distinct events occurring at distant spacetime locations are manifestations of a more fundamental ontology in a higher-dimensional framework. The question of how to extend wave function realism to a relativistic context is complicated by the fact that the basic mathematical structure of relativistic quantum theories and quantum field theories, especially when interactions are factored in, is

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not entirely clear. Disputants seem to agree however that whatever the best way to express the mathematical structure of these theories, they ultimately involve an assignment of values of some kind (operators or algebras) to regions of spacetime. From this, the critics of wave function realism infer that the ontology of these theories must involve features instantiated at spacetime regions. For example, consider Wallace and Timpson’s spacetime state realism. They argue: Suppose one were to assume that the universe could be divided into subsystems. Assign to each subsystem a density operator. We then have a large number of bearers of properties – the subsystems [they take these to be spacetime regions] – and the density operator assigned to each [spacetime region] represents the intrinsic properties that each subsystem instantiates, just as the field value assigned to each spacetime point in electromagnetism, or the complex number assigned to each point in wave-function realism, represented intrinsic properties. (Wallace and Timpson , p.)

Wallace and Timpson also allow that one can also think of the ontology in terms of an assignment of algebras of operators to each spacetime region rather than operators of some kind (Wallace and Timpson , p.). But either way, we have a fundamental ontology of intrinsic properties attributed to spacetime regions. This is what the wave function realist wishes to call into question. Wallace and Timpson both acknowledge and embrace the fact that although relativistic quantum systems can be characterized in terms of an assignment of algebras or density operators to spacetime regions, interpreting this literally leads to a nonseparable metaphysics. Facts about the assignment of such features to the subregions of a spacetime region R do not determine the assignment of features to R. We may debate the importance of having a metaphysics that is separable. Wallace and Timpson question its importance, while I in earlier work attempted to lay out the case for separability and locality in some detail (Wallace and Timpson ; Ney ). Wave function realists are motivated by the desire to have a separable metaphysics, and there does not seem to be any prima facie argument that the desire for a relativistic theory conflicts with the desire for a theory that is separable and local.² And so spacetime state realism, although it may have the virtue of producing a metaphysics that is close to the mathematical structure of quantum field theories (insofar as that is a virtue), will only be a stopping point for the wave function realist on the way to a fundamentally separable metaphysics. The simplest way for the wave function realist to proceed in developing an interpretation of relativistic quantum theories would be in many respects analogous to the way she achieved the high-dimensional representation in the nonrelativistic case. Starting from an apparent ontology of localized subsystems in threedimensional space with nonseparable features, the wave function realist posited a more fundamental, higher-dimensional space in which each point corresponded to an entire three-dimensional configuration of subsystems. In the relativistic case, let’s consider as a starting point a version of spacetime state realism in which operators are assigned to subregions of the total spacetime manifold. As mentioned, for entangled systems at a total region R, the features of two nonoverlapping but together ² If anything, relativity seems to favor local interpretations. See Ney ().

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exhaustive subregions of R, R and R, will not in general determine the features of R. So, for example, let’s start by considering a Fock space representation in which â†(k), â(k), and N(k) denote raising, lowering, and number operators for quanta (here, we’ll consider bosons) of momentum k and their assignments to regions of spacetime.³ Suppose there is an assignment of N(k) to a total spacetime region R with expectation values that would lead one to predict a total of either two or three quanta with determinate momentum k in R.⁴ The system is in a superposition of a state with three quanta in R and zero quanta in R and a state with zero quanta in R and two quanta in R. This may be described in terms of the total number of quanta of momentum k: j3>R1  j0>R2 þ j0>R1  j2>R2 ; or it may be described in terms of raising operators applied to the vacuum state: ak † ak † ak † j0>R1  j0>R2 þ j0>R1  ak † ak † j0>R2 :5 Confined to the low-dimensional image, there is nonseparability. The facts about the quantum state at R are not wholly determined by the states at its subregions. What things are like at R is inextricably correlated with the situation at R, but this fact is left out if one just looks at what things are like at R and R individually. To achieve a separable ontology, the wave function realist can postulate a higher-dimensional space in which each point corresponds to an assignment of operators to the subregions of R. In this case, she will postulate a field with nonzero amplitude at a point corresponding to three quanta at R and zero at R, and another point corresponding to zero quanta at R and two at R. Generalizing from this simple case in which we are just considering two regions R and R, and assuming the spacetime representation from which we begin is continuous, the higher-dimensional space will be continuously-infinite-dimensional with each point corresponding to an assignment of operators to all spacetime points or smallest regions in the low-dimensional representation. One thus recovers a separable fundamental metaphysics.

. Interpretations and Interpretational Frameworks At this stage, we may note that we are no longer considering wave functions on a space with the structure of a configuration space as the central elements in the wave function realist’s basic ontology. What we have instead is a field defined on another ³ One may think of a Fock space representation as involving a weighted sum of a single particle representation, a two-particle representation, a three-particle representation, and so on. ⁴ A word about the use of basis states that involve the assignment of particles to definite spacetime regions. Strictly speaking, such states are precluded by quantum theory. Nonetheless in practice physicists do represent states in this way. Such practice may be justified by the fact that we can understand talk of a particle being definitely located in a given region as communicating that the expectation value for its being located anywhere outside of that region approximates the vacuum. Similarly, the representations used here may seem to falsely suggest that vacuum states are separable rather than entangled states. However this representation of a vacuum state at a localized region again speaks to the expectation values for locating particles inside the region approximating null values. Thanks to David Wallace for discussion. ⁵ Note that normalization coefficients have been elided for presentation.

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

kind of high-dimensional space, one for which locations are correlated with assignments of values to regions in a four-dimensional ontology. This is a simple consequence of the fact that the low-dimensional, nonseparable representations from which the separability argument departs are different in the relativistic case than they were in the nonrelativistic case. The point I want to make here—and this is the central point of the present chapter—is that there is a deeper interpretative ideology underlying wave function realism: that which guides one to a metaphysics for quantum theories lacking fundamental nonseparability and nonlocality. This ideology or interpretative framework—call it Localism if you like—leads one to adopt a picture of a wave function on a space with the structure of a classical configuration space in the context of nonrelativistic quantum theories. But it will lead one to adopt a different metaphysical interpretation for relativistic quantum mechanics and quantum field theories.

. Response to Objections Once we see that the wave function in configuration space view is only an instance of a broader strategy for interpretation applied to the special case of nonrelativistic quantum mechanics, and that relativistic implementations of this strategy will not rely on configuration space representations, some of the objections to wave function realism canvased in Section . may be quickly dispensed with. A. Recall the first objection (from Wallace and Timpson) was that wave function realism, as applied to relativistic quantum theories in which particle number is not conserved, led to an ontically profligate picture of an infinite sequence of configuration spaces. Viewing the wave function in configuration space picture as an implementation of an interpretative strategy that applies only to nonrelativistic theories, this objection does not get off the ground. The wave function realist need not and should not offer the wave function in configuration space picture as an interpretation of relativistic quantum theories. There would be something to this objection were the wave function realist not able to suggest an alternative picture applicable in the case of relativistic quantum theories. But, as we saw in Section ., these concerns are unfounded. In the case of relativistic theories, the wave function realist can start from the same place as the spacetime state realist, with a nonseparable metaphysics of operator assignments to spacetime regions, and reason from there to a more fundamental, separable metaphysics in a high-dimensional space in which each point corresponds to a total assignment of operators to local spacetime points or regions. Wallace argues that: [W]avefunction realism seems to rely on features of toy NRQM which, far from being universal features of any realistic quantum theory, drop away as soon as we generalize. (Wallace )

The response is that wave function realism applied to toy nonrelativistic quantum mechanics relies on features of toy nonrelativistic quantum mechanics. However, the wave function realist achieves an interpretation of more realistic quantum theories by

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

 

relying on the features of these more realistic theories, those same features on which the spacetime state realist bases his interpretation. B. The second objection Wallace and Timpson () raised for wave function realism is that the theory “obscures the role of spacetime in quantum theories.” What seems to be underlying this objection is a kind of normative constraint that metaphysical interpretations of physical theories should themselves remain close to the mathematical structure of those theories. Wallace and Timpson defend their own spacetime state realism for the fact that “it adds no additional interpretational structure (given that the compositional structure of the system is, ex hypothesi, already contained within the formalism); and it gives an appropriately central role to spacetime” (Wallace and Timpson , p.). Myrvold presents similar concerns as well, noting that characterizing the ontology of quantum field theories in terms other than an assignment of values to spacetime regions “is not what is done in the usual presentations” (Myrvold , p. ). He also notes that “the empirical content of [QFTs] remains tied to observables that are associated with regions of spacetime” (p. ). The first thing to note is that the wave function realist does not reject the truth or empirical adequacy of the spacetime representations that are characteristic of usual presentations of both conventional and algebraic quantum field theories. Her claim is not that these representations should be rejected, but rather that they should be seen as metaphysically explained or grounded in terms of a more fundamental representation of a field in a high-dimensional space. This is a metaphysical claim motivated, as we have seen, by a commitment to fundamental separability and the thought that nonlocal correlations should be explained in terms of a more fundamental picture lacking them.⁶ Because the wave function realist takes as a given the truth of spacetime representations, only questioning their fundamentality, there is no problem with her using them to understand the empirical content of quantum field theories or even their role in standard expositions of quantum field theories. How closely a metaphysical interpretation of a physical theory should adhere to the picture immediately suggested by the formalism is a vexed issue. Insofar as this is a desideratum, then we can see that wave function realism when applied to nonrelativistic quantum mechanics does better at achieving it than does the strategy when applied to relativistic quantum theories, since textbook discussions of nonrelativistic quantum mechanics often rely on the wave function in configuration space picture. However, in the relativistic case, the wave function realist does have a justification for moving beyond the sort of picture suggested by the relativistic quantum formalism. And this is that what she is trying to do is seek out what the strange observations constituting our evidence for quantum theories are telling us about the fundamental structure of our world. Of course it makes sense, when we construct quantum theories that do justice to our evidence and are relativistically covariant, that we will start with spacetime representations. However, persistent correlations that we observe between spatiotemporally distant events suggest that ⁶ Although as I’ve noted, the kind of spacetime state realism Wallace and Timpson prefer appears to achieve locality as well. The wave function realist appeals to locality rather to distinguish her metaphysics from other interpretational frameworks, such as that of Bohmian mechanics.

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

there is something more basic underneath this spacetime picture. Or maybe not, at this stage at least, it is not possible to know whether there is or there isn’t. But the aim of wave function realism is to spell out what this more basic picture looks like if nonlocal correlations do possess a more fundamental explanation. And it does not seem possible to achieve a simple explanation of them without moving beyond a spacetime framework. C. The third critique Wallace and Timpson raised for wave function realism, recall, was that relativistically covariant quantum theories violate narratability—the history of events cannot be told as a single story evolving over time, but will vary depending on the way one slices up the manifold into times—and this violation is surprising from the point of view of a separable account like wave function realism but not from a nonseparable account like spacetime state realism. To see how narratability fails, consider one of the simple cases presented by Albert (). Albert describes a system S consisting of four spin-/ particles. From the perspective of a frame of reference K, particles  and  are permanently located at their respective spatial positions, and particles  and  move with uniform velocity along parallel trajectories intersecting the paths of particles  and  at spacetime locations P and Q respectively. P and Q are simultaneous from the perspective of K. The spin states of the particles may be described as jφ>₁₂jφ>₃₄, where: pffiffiffi pffiffiffi jφ>AB ¼ 1= 2 j " >A j # >B  1 2j # >A j " >B Albert notes that from the perspective of K, there is no difference between the following two Hamiltonian descriptions of S’s evolution. The effect of the two Hamiltonians may be summarized simply as: H₁: H₂:

S evolves freely Particles exchange spin upon contact.

Interestingly, although reference frame K recognizes no difference between evolution according to H₁ and H₂, since from the perspective of K, the state of S is always jφ>₁₂jφ>, this is not so in any other frame of reference K’. And so the story or “narrative” of the evolution of S over time is dependent on one’s reference frame. Nor can one transform between the description of K and other frames by a Lorentz transformation. No such transformation is possible since it would require mapping identical states in K to distinct states in K’. And so specifying a system’s state at all times in any one frame of reference is not sufficient to specify all facts about that system. Wallace and Timpson argue that if one adopts spacetime state realism, narratability failure becomes natural given the interpretation’s postulation of nonseparability: [S]uppose that we have any spacetime theory which (i) is non-separable, so that there can be simultaneous spacetime regions A and B such that the state of A[B is not determined by the states of A and B separately, and (ii) is also covariant. Covariance entails that there can also be non-simultaneous spacetime regions whose joint state is not fixed by their separate states. This opens up the possibility of failure of narratability: specification of global states on elements of one foliation on their own will not in general fix the joint states of non-simultaneous regions. (Wallace and Timpson , p.)

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 

Let’s see in more detail how this is supposed to work. Start () by assuming nonseparability so that there are regions A and B that are simultaneous according to some reference frame K and the facts about A[B are not wholly determined by the intrinsic facts of A and B taken separately. Then, () assuming Lorentz covariance, there is another frame K’ according to which A and B are not simultaneous. Suppose () that according to K’, B is in the future of A. The facts about A and B individually do not determine the facts about A[B. So in particular, () the facts about A do not determine the facts about A[B. The failure of narratability is a situation in which () the fact of whether some event x follows another y, or the chances of y’s following x, is not absolute, but depends on one’s reference frame. This could be, for example, the facts about A not determining whether B follows or with what chances it is likely to follow from A. So () is interpretable as an example of (). Narratibility failure is thus something one might expect given the assumptions of nonseparability and covariance. At least this is what Wallace and Timpson appear to have in mind. The key issue though, if narratability failure is to provide a case for spacetime state realism over wave function realism, is whether the implication works in the other direction, i.e. does the failure of narratability suggest a failure of separability? And does it do so in a way so as to suggest the failing of wave function realism? It is not clear why it should. In demonstrating the failure of narratability, one implicitly relies on spacetime representations in which one can make sense of the notion of Lorentz covariance. One might argue that the fact that there is no absolute distinction between simultaneous and non-simultaneous space time regions means one can argue from narratibility failure (and Lorentz covariance) to failure of separability just as we showed above one can argue from the failure of separability (and Lorentz covariance) to the failure of narratability. But this would only get us to nonseparability in spacetime. As we have seen, the wave function realist believes there is a more fundamental metaphysics underlying spacetime representations. And her view is that what manifests itself as nonseparability in ontically derivative spacetime representations is a more fundamental metaphysics that is separable. So there is no direct argument from narratability failure to fundamental nonseparability of the kind that would undermine wave function realism. Even if narratability failure implies spacetime nonseparability, spacetime nonseparability is compatible with both fundamental nonseparability (spacetime state realism) and fundamental separability (wave function realism). D. Myrvold objected to wave function realism that in quantum field theories, wave functions are constructed from structures defined on spacetime and so they cannot be more fundamental than spacetime structures. He argues that: the nonrelativistic theory of systems of a fixed, finite number of degrees of freedom . . . should, rather, be regarded as a low-energy, nonrelativistic approximation to a relativistic quantum field theory (which itself might be a low-energy approximation to something else). (Myrvold , pp. –)

He demonstrates the derivation of wave functions in the context of both nonrelativistic and relativistic quantum field theories to show how the wave function

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

metaphysics is, in the context of actual physical theories, an effective metaphysics merely, not fundamental. We need not get into the details of these derivations though to see that this criticism of wave function realism is misplaced. First, the fact that wave function representations may be mathematically derived from spacetime representations does not show anything about the direction of ontic priority. The wave function realist (at least in the nonrelativistic context) only argues that wave functions are ontically prior to spacetime structures. Myrvold’s demonstrations do not show that spacetime representations cannot similarly be derived from wave function representations, or the kinds of representations I have argued the wave function realist should take to be tracking fundamental structure in the case of relativistic quantum theories. And second, as I’ve tried to argue above, the wave function realist need not deny that the metaphysics of wave functions in configuration space is an effective metaphysics. She may concede that this picture arises only when one subjects what is a more fundamental metaphysics to nonrelativistic idealization. Indeed it is the one of the main claims in the chapter that this is precisely the case. I have also made suggestions for how the wave function realist may take steps toward articulating what a more fundamental metaphysics for quantum theory looks like (Section .). My disagreement with Myrvold is only that the wave function in configuration space picture is not an effective metaphysics for a more fundamental spacetime metaphysics. Rather the wave function in configuration space picture is an effective metaphysics for a more fundamental state space picture. The nature of the more fundamental state space will vary depending on the details of the quantum field theory under consideration. As there is no unique quantum field theory, there will be no unique more fundamental space according to the wave function realist. Regardless, viewing the wave function in configuration space picture as an effective metaphysics need not require invoking spacetime as a fundamental background. E. Myrvold’s other criticism relates to the wave function realist’s claim that her fundamental metaphysics consists of a field in a high-dimensional space. As we saw above, Myrvold presents an example intended to show that the central object in the wave function realist’s fundamental metaphysics is not a field, as it isn’t something defined by local assignments of values to regions of the space the object occupies. There are several issues with Myrvold’s argument, the first being its reliance on an interpretation of relativistic quantum theories I have already noted the wave function realist should reject—one according to which what fundamentally exists is a plurality of wave functions of increasing dimension. But set that issue aside. Recall that Myrvold asks us to consider two states of a single particle wave function: S and S. In S, the wave function takes on a nonzero value only at some point x. S is one in which there is with certainty a particle located at some region R at some distance away from x. Myrvold’s argument may be summarized in the following way: . That a single-particle wave function takes on a nonzero value at a point x is incompatible with there being a particle located in region R that does not overlap with x, no matter how far R is from x. . So, facts about the assignment of values of wave functions to regions of wave function space depend on circumstances in distant regions.

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 

. So, wave functions are not defined by an assignment of local quantities to regions of the space they inhabit. . So, wave functions are not fields in the usual sense. I wish to argue that depending on what one means by ‘assignment of local quantities’, the wave function realist has good reason to reject either the inference from () to () or from () to (). Before getting to this, it is worth noting why Myrvold thinks this critique is so powerful against the wave function realist. This is because, as Myrvold recognizes, one of the central motivations for wave function realism is to provide a separable metaphysics for quantum theories. But if the state of wave functions do not depend on an assignment of local features, then the wave function realist’s metaphysics will not be separable. So, it seems the key point is really what I have stated as subconclusion () above. A natural understanding of what is meant by an assignment of local quantities is an assignment of features that is not determined by what takes place at distant locations. Assuming this understanding, it is easy to see that () does not follow from Myrvold’s earlier claims. What we find in S and S are two distinct wave functions, where the state of each wave function is determined by the assignment of amplitudes (and phase, though this is ignored in the example) to each point in its space. The amplitude of the single-particle wave function at x in S is not equal to the amplitude of the single-particle wave function at x in S and this is what determines the fact that these are distinct wave functions. Now Myrvold notes that we can infer that the single-particle wave function for S differs from that of S simply by knowing in S there is a particle definitely located at a distant region R. This is due to a modal dependence between the fact about the particle and the fact about the wave function. But, that there is a particle definitely located at R is not what ontically determines the amplitude of the single-particle wave function at x for S according to the wave function realist. The facts about the amplitude of a wave function at different points in its space are brute. According to the wave function realist, facts about the locations of particles are ontically derived from facts about the wave function (or wave functions). The fact about dependence stated in () obtains due to this fact of ontic priority. So, () simply does not follow from (). This part of Myrvold’s argument simply begs the question against the wave function realist. It assumes facts about the locations of particles determine facts about the wave function rather than vice versa. Now Myrvold would likely reject this understanding of what makes for an assignment of local quantities. Myrvold himself makes the following terminological remarks helpful to interpreting his argument: We will say that a physical quantity is intrinsic to a spacetime region if the fact the quantity has the value that it has carries no implications about states of affairs outside the region. A local beable, as we understand it, is one that can be regarded as an intrinsic property of a bounded spacetime region, and will be said to be local to that region. (Myrvold, , p. )

So, let us evaluate the argument with this understanding in mind. What is going on in this case, we can see, is that in S, since we are told that a particle is definitely located at R, we must infer that the single-particle wave function must take on value one at

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

location R. If single-particle wave functions were normalized, this would mean that it must take on a value of zero at every other location. As a rule, these individual (single-particle, two-particle, etc.) wave functions won’t be normalized. Nonetheless, we can still infer that the amplitude of the single-particle wave function at x is zero, since it is still reasonable to require that the amplitude-squared of all of the wave functions at all of the points in their respective spaces must add exactly to one. Now one might read Myrvold as saying that it is this fact that demonstrates that the assignment of amplitudes to the wave function is not an assignment of local quantities. The facts about the assignment of values at R carries implications for the assignment of values at other locations. Under this understanding of ‘assignment of local quantities’ then, () follows. But why should this make the wave function un-field-like? Consider paradigmatic examples of entities representable as fields with amplitude and phase: bodies of water or waves on a string. Since actual water waves or waves on a string are always made of a finite amount of material, there necessarily will be implications regarding what amplitude the wave takes on at one location and what it takes on at others. So if one likes, in a sense one can say that waves of these kinds are not defined by an assignment of local features; there are dependence relations obtaining between the values the wave takes on at distinct locations. But noting that the wave function too has this feature does not make it in any way un-field-like unless all of these things are un-field-like as well. And so even if the move to premise () is valid, the further move to () is not. And note that even if () is conceded in this way, this is not relevant to the matter of separability. Whether the wave function metaphysics is separable is a matter of whether facts about features at composite regions are determined by facts about features at the regions making up those composites. In other words, it is only the first interpretation of ‘assignment of local quantities’ that is relevant to the issue of separability. For reference, Don Howard defines separability in the following way: It is a fundamental ontological principle governing the individuation of physical systems and their associated states, a principle implicit in many classical physical theories. It asserts that the contents of any two regions of space-time separated by a nonvanishing spatiotemporal interval constitute separable physical systems, in the sense that () each possesses its own, distinct physical state, and () the joint state of the two systems is wholly determined by these separate states. (Howard )

Separability is a matter of determination not dependence. Separability may obtain even if there is a dependence in values taken on by the wave function at distant regions. This latter fact has no bearing on whether what happens at a composite region is determined by what happens at the regions out of which it is composes. F. The final objection one finds in the literature challenges the wave function realist’s privileging of the position representation, noting that such representations fail to be straightforward in relativistic quantum theories (Wallace ). My response to this objection is similar to the response I have argued the wave function realist should make to the first objection. This is that the objection mistakenly assumes that all features of the interpretation the wave function realist gives for nonrelativistic quantum mechanics will carry over to interpretations of relativistic

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theories. As illustrated in Section ., just as the wave function realist bases her interpretation of NRQM on standard presentations of NRQM, the wave function realist will base her interpretation of QFTs on standard presentations of QFTs. In the absence of a Lorentz covariant position representation of a QFT, the wave function realist will construct her higher-dimensional representation using a different kind of basis. The example in this chapter used the momentum basis. Importantly, the wave function realist may start from the same formal framework as the spacetime state realist.

. Conclusion We may see that at least some of the main objections raised to wave function realism regarding the putative difficulties of extending the view to relativistic quantum theories rest on the false assumption that the resulting interpretation of relativistic quantum theories must postulate a field in configuration space. Others rest on the failure to see that the wave function realist allows the truth and/or legitimacy of spacetime representations, claiming only that these are determined by the higherdimensional facts she posits. So, it seems to me there are no clear barriers to extending wave function realism to the relativistic domain. Wave function realism should still be on the table as one among many interpretations of quantum theories. Although the wave function in configuration space view is really only plausible as an interpretation of nonrelativistic quantum mechanics, it is a particular case of a broader framework for interpretation applicable to relativistic quantum theories as well. Insofar as it provides the interpreter with a separable and local fundamental metaphysics, it may benefit from an air of comprehensibility that does not similarly apply to alternative approaches. This itself makes it worth keeping on the table.

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9 In Defense of the Metaphysics of Entanglement David Glick and George Darby

. Introduction Anyone who accepts quantum theory must allow that entanglement exists, in some sense. But it’s important to distinguish several related claims: . Non-locality: The existence of certain non-local phenomena, such as measurement results in an EPR experiment. . Non-separability: The existence of composite systems assigned quantum states that cannot be recovered from the quantum states of the constituents of the system. . Entanglement realism: The existence of some entity or entities in our fundamental ontology constitutive of entanglement. The first two theses are relatively uncontroversial, but as we’ll see, the lattermost is a point of contention among interpreters of quantum theory. Our focus here will be with (), but let’s briefly consider () and () before setting them aside.

.. Non-locality Quantum theory predicts correlations in measurement outcomes incapable of locally causal explanation. This follows from Bell’s theorem, which shows that any locally causal theory must conform to Bell’s inequality, which is violated by the predictions of quantum theory. After the experiments of Aspect in  and those following him, we may now say that “[v]iolations of Bell’s inequality show that the world is nonlocal” (Maudlin b, p., original emphasis). Non-locality, then, is hard to deny. It’s an observable fact about our world that such correlations exist. Even those views that deny the reality or fundamentality of measurement outcomes must accept non-locality in some form at some level of description. On the Everett interpretation, for instance, the experience of an agent will almost certainly include evidence of non-local correlations in measurement outcomes. Non-locality is part of our evidence for quantum theory, and

David Glick and George Darby, In Defense of the Metaphysics of Entanglement In: The Foundation of Reality: Fundamentality, Space, and Time. Edited by: David Glick, George Darby, and Anna Marmodoro, Oxford University Press (2020). © David Glick and George Darby. DOI: 10.1093/oso/9780198831501.003.0010

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hence, it must be recovered by any interpretation lest it be empirically incoherent (Barrett ).¹

.. Non-separability The term “non-separable” is sometimes taken to have metaphysical implications. Our use here has no such implications. The claim is simply that the application of quantum theory involves assigning quantum states to composite systems that cannot be factored into quantum states of their constituents. It is a further claim—which some deny—that these states represent particular physical properties or entities. The standard example of non-separability is a two-particle system assigned a pure state ψ₁₂ that doesn’t factor into pure states of the individual particles: ψ₁₂ 6¼ ψ₁ψ₂ where ψ₁₂2H₁₂, ψ₁2H₁, ψ₂2H₂ and H₁₂ = H₁H₂. One can also define nonseparability for an arbitrary quantum state in terms of density operators. Here the quantum ρ₁₂, is non-separable just in case it violates: P P state of the joint system, ρ₁₂ = iαiρ₁iρ₂i, with αi> o, iαi = . Non-separability is a widely accepted feature of the standard quantum formalism and its application. There are difficult questions about the correct formal structure for quantum theory in general, and how to characterize non-separability in particular, but it remains a core feature of any quantum theory that some states will have this character. Indeed, in introducing entanglement, Schrödinger () called non-separability “not one but rather the characteristic trait of quantum mechanics” (p., original emphasis). Thus, any view that accepts the quantum formalism in anything like its standard form will allow for the non-separability of quantum states.

.. Entanglement Realism Entanglement realism requires more than merely recognizing the existence of nonlocal phenomena and non-separable quantum states. In particular, it requires that non-separable states represent genuine entanglement in the world. The details of how entanglement is understood may vary, but the realist regards entanglement as part of the structure of our world rather than a mere appearance. Consider an analogy with retrograde motion, the apparent “backwards” motion of certain planets through the sky relative to the background of stars. On the Ptolemaic astronomical model, planets travel along circular orbits around the earth and smaller epicycles around a point on the main orbit. The combination of the motion along the epicycle and the main orbit allows a planet to travel backwards for brief periods of time, allowing for their retrograde motion. On the Copernican model, however, planets travel along orbits around the sun and apparent retrograde motion occurs when a planet overtakes, or is overtaken by, the earth. Hence, retrograde motion is real on the Ptolemaic view, but ¹ Wallace () claims that Bell’s theorem simply doesn’t apply to the Everett interpretation given the latter’s denial of unique measurement outcomes. And non-ontic views such as QBism and Healey’s pragmatism claim to be fully local. However, the point remains that when Alice and Bob compare measurement results from an EPR experiment they (almost certainly) find violations of Bell inequalities. One may deny the inference from these correlations in reports to correlations in reality, but even so, there will remain an analog to non-locality in reports—once Alice finds out Bob’s measurement results, she finds violations of Bell inequalities. Denying the veridicality of such reports brings one perilously close to solipsism (Norsen ).

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merely apparent for the Copernican. It seems natural to say that the Ptolemaic position is realist about retrograde motion while the Copernican position is antirealist. Analogously, the entanglement realist embraces entanglement as a genuine feature of reality that is not merely apparent. This licenses the metaphysics of entanglement, which seeks to understand the nature of this feature of our world. Although the term may not be in wide usage, one can find several authors gesturing at entanglement realism. For instance, Wallace (), in describing entanglement on the Everett interpretation says that “picturesquely, we can think of entanglement between states as a string connecting those states, representing the nonlocal relation between them” (p.). Healey () rejects entanglement realism when he argues against the claim that “ascribing an entangled state to quantum systems is a way of representing some new, non-classical, physical relation between them” (p.). And Esfeld () suggests a version of entanglement realism when he claims that “[w]hatever entanglement may exactly be, it is a relation among quantum systems. ‘Being entangled with’ is a property that is predicated of at least two quantum systems; it is thus a relational property” (p.). There are several ways of denying entanglement realism. Most straightforwardly, views that deny the quantum state a representational role do not find entanglement in the world. QBism, for instance, takes quantum states to encode an agent’s beliefs as they relate to certain experimental procedures and future experiences (Fuchs et al. ). Healey’s pragmatist view of quantum theory differs significantly from QBism, and gives quantum states a more objective status (Healey b). But, Healey agrees that the primary function of an entangled state is to tell an agent to expect Bell-type correlations when comparing the outcomes of certain experiments. On such views, entanglement, as a relation between quantum states, concerns a particular kind of belief an agent might have—one which would lead them to expect non-local phenomena.² Such non-representational views aren’t the only way to be an antirealist about entanglement. An interpretation may grant the quantum state a representational role, but still deny that entangled states describe worldly entanglement. Indeed, below we claim that two “realist” approaches to the metaphysics of quantum theory are committed to entanglement antirealism. On these views, quantum states do sometimes describe features of the world, but not in a manner that countenances entities, properties, or relations constitutive of entanglement. Wavefunction realism and Super-Humeanism (or “Bohumeanism”) are often motivated by a desire to avoid what are perceived as problematic metaphysical implications of entanglement, but we argue that these problems aren’t so great as to motivate eliminating entanglement in the manner they suggest. Before closing this introduction, it’s worth noting that, other things being equal, realism has certain benefits. First, it accords with a flat-footed understanding of the metaphysical implications of physics. While it’s clear that one cannot read-off metaphysics from physics, it is nevertheless an aim of naturalistic metaphysics

² Or reports of apparently non-local phenomena. See note .

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to let physics guide much of our metaphysical theorizing. The realist attitude recommends taking theories at face-value and committing to the ontology they describe.³ Second, realism has certain explanatory power. The realist is able to provide explanations involving the nature of reality itself rather than merely being limited to the predictive features of our scientific theories. In the case of entanglement, this suggests that the realist will be able to offer explanations of non-local phenomena unavailable to the anti-realist—viz. those based in genuine physical entanglement.⁴

. Humeanism and Entanglement One motivation that might be given for abandoning realism about entanglement is the perceived threat to the doctrine of Humean supervenience, which lies at the heart of Lewis’s neo-Humean approach to metaphysics. Humean supervenience, in Lewis (a), says that everything supervenes on the monadic properties of point-like things plus spatiotemporal relations. But entanglement seems to show that there are things that don’t supervene on that basis, and hence that Humean supervenience is false. To put it in the terms of the present volume, the charge against Lewis is that he assumes the wrong things to be fundamental. The response that we favor is that this problem for Humean supervenience is nowhere near as serious as is often made out. This is because, for all that has been shown so far, it is perfectly compatible with Humean supervenience to simply include in the supervenience basis whatever troublesome relations are supposed to be required by entanglement. Obviously one cannot do that for every threat to Humean supervenience, or the thesis would be trivial. But so long as the relations to be added are not powers, or lawmaking relations between universals, or otherwise obviously unHumean, it seems that admitting, say, an “opposite-spin” relation into the ontology will do no harm at all—the natural thought would be that such relations can be analyzed as “external” in Lewis’s sense.⁵ And, surely, the burden of proof is at least partly on the anti-Humeans to show that the relations in question cannot be thus analyzed—and nothing like that has been shown so far. In other terms, Lewis might be wrong about what is physically fundamental, but that’s okay: what matters is what is metaphysically fundamental. Since the context of much of this discussion is in the movement towards “naturalistic” metaphysics—the demand for metaphysicians to take notice of natural science—the proper response for Humeans is to accept the extra ontology and await a demonstration that it conflicts with their ideology. An alternative response, which we follow Esfeld and Deckert () in calling “Super-Humeanism”—though as will become apparent it is not clear that there is ³ Of course, it’s sometimes impossible to take a theory at face-value, and hence some amount of (re-)interpretation is required. The point here is that realism requires an attitude or methodology of beginning from a flat-footed literal reading of a physical theory and going from there. ⁴ We will return to explanatory power of entanglement realism in Section . below. ⁵ External relations are those that supervene not on the intrinsic qualities of their relata, as internal relations do, but on the intrinsic qualities of the composite of the relata—spatiotemporal relations being the paradigm case.

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anything super about it—is advocated in various forms by Esfeld and Deckert (), Miller () (under the heading “Bohumeanism”), and Bhogal and Perry (). Our claim in this section is that Humeans should reject Super-Humeanism and should take the more neutral stance, holding that whatever the fundamental ontology turns out to be, it will be analyzable in terms of some fundamental things standing in natural external relations. The reasons for this are best seen in light of the alleged advantages of SuperHumeanism; we will focus on two: . The claim that Super-Humeanism is somehow more faithful to the true nature of entanglement, and . The claim that it avoids quidditism. We’ll argue that both claims are problematic.

.. Super-Humeanism and the Nature of Entanglement Under the heading of “Quiddistic Entanglement,” Bhogal and Perry consider something like the Humean response that we prefer. They say that one way out of the conflict between HumeanSupervenience and QM: is to deny that entangled states imply anything about any other states. This view accepts that there is a world where two particles are in a Singlet state but yield matching outcomes to properly calibrated x-spin measurements. This is an option for the Humean. Its major drawback is that it involves a unintuitively quiddistic conception of entanglement—it violates the intuition that entanglement implies something substantive about the particles so entangled. Perhaps such a quidditism isn’t so bad for the Humean, after all if the Humean is to use recombination she will require a quiddistic conception of the properties that make up the mosaic. But our view does not have a quiddistic conception of entanglement; it retains a more intuitive understanding of the phenomena. (Bhogal and Perry , p.)

The idea that entanglement is captured by relations in the supervenience basis leads to quidditism because it divorces entanglement from any necessary connection to lawlike behaviour. Once the relations are in the supervenience basis they may be recombined in any combinations whatsoever, including those that give the same measurement results to particles in the singlet state, as Bhogal and Perry point out. But the option rejected here seems to us to be exactly the right one to consider: Humeans are already committed to properties understood as quiddities, so why not have a few more? This parallels the preferred response above: Lewis is already committed to a class of natural external relations, so why not accept some more to account for entanglement. We accept, of course, that entangled states imply something about their substates; what we deny is that they imply anything with metaphysical necessity. The intuition that entanglement is something substantive is satisfied by its special status in requiring natural external relations beyond the spatiotemporal. Anything more involved than that is unHumean. The claim that entanglement “implies something substantive about the particles so entangled” may mean a number of things, of course. Take a view on which the entangled particles somehow cease to become numerically distinct, or are “nonindividuals,” or something similar. This kind of view does indeed appear to imply something very substantive about the particles. Or suppose that one thought that the correspondence between the part–whole relation and a relation of ontological dependence somehow became inverted in entanglement: whereas usually the whole

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depends on the parts, now the parts depend on the whole; one can see how entanglement would then imply something strikingly substantive about the particles.⁶ In a sense, we want to say that the entanglement-as-external-relations view also implies something substantive about the entangled particles of our world—viz. that they instantiate a certain natural relation. On the other hand, there is nothing implied about the particles considered individually—indeed, that is the whole point. The Super-Humean view also, of course, implies nothing substantive about the particles considered individually—in some variants they are just featureless blobs. Nor is there anything even as substantive as the instantiation of a natural relation—the whole point of a Humean treatment is, of course, to make things unsubstantive—Humeans being suspicious of substantive things like powers and necessary connections as being ultimately unintelligible. So, in a sense, the SuperHumean view can’t imply anything substantive about entanglement, because Super-Humeanism doesn’t imply anything substantive about anything. What the Super-Humean view does imply, and presumably the advantage that Bhogal and Perry have in mind, is that there is something about entangled particles that is necessarily true about entangled particles: necessarily, if x and y are entangled then p, where p is a fact involving laws and thus ultimately—on the best-system view of laws—about the whole mosaic. But what is unclear now is why Humeans should particularly value that. Maybe we should understand “substantive” as something like observable. Bhogal and Perry need not mean actually observable, but perhaps reflected somehow or other in the pattern exhibited by the Humean mosaic. And it does do this, of course: to be entangled is just a matter of things being thus and so in the Humean mosaic. So obviously being entangled implies—strictly implies—something about the mosaic.⁷ But, as Bhogal and Perry acknowledge, Humeans, especially those following Lewis, are already in the business of denying that many things hold of metaphysical necessity. In a different context, but illustrating perfectly the general attitude, it famously seems that on Lewis’s theory of transworld identity, Lewis should happily accept that he could have been a fried egg. Nothing particularly substantive, in this sense, is implied about anything, and the behavior of relations—whether spatiotemporal, or those capturing entanglement—are no exception.

.. Humeanism and Quidditism Esfeld and Deckert, and Bhogal and Perry, all suggest that avoiding quidditism is a desirable feature of the Super-Humean strategy over regular Humeanism. But is it really? The first question is what exactly is supposed to be wrong with quidditism anyway. Esfeld and Deckert () say that Lewis’s quidditism is “occult” (p.) and ⁶ See Section . below for a discussion of this approach to the metaphysics of entanglement. ⁷ It implies something either directly, in terms of the outcomes of measurements actually performed, or indirectly, in terms of the outcomes of measurements that are not, but could be, performed—such counterfactual outcomes also being a matter of the mosaic (everything supervenes on the mosaic, including counterfactuals).

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“baroque” (p.). But it’s not like quidditism is an extra bit of ideology, like powers again—it’s just a thesis about the distribution of stuff over the plurality, or about the extent of the plurality. It’s hard to see what exactly is “baroque” about that. The question of whether quidditism is occult raises an interesting parallel with what exactly Humeanism is supposed to be, and why we should try to defend it in the apparent conflict with quantum mechanics. On one way of thinking about it, Humeanism is a claim about the extent of the Lewisian plurality of worlds. If there is in some world a pair of objects, mereologically separate enough to qualify as “distinct existences”, but in no world is there one without the other, then this is a “necessary connection between distinct existences”; Humeanism then is the claim that this never happens. Likewise, on a similar way of thinking about it, quidditism is a claim about the extent of the Lewisian plurality of worlds, this time requiring no constraint on the pattern of instantiation of fundamental properties and relations. There is a different way of thinking about what Humeanism is: the rejection of “occult” metaphysics such as powers—that is, put in terms of ideology rather than ontology. In a similar way, one can think of quidditism not in terms of the way the plurality is, but in terms of the way in which Lewis understands natural properties: are there universals, for example, to correspond to natural properties? If so, then quidditism again easily follows, because properties just correspond to universals, so of course universal F can play the same role in one world as universal G plays in another—this just requires the right pattern of instantiation of F and G. But again, there is nothing “occult” about this—it’s just the way of thinking about natural properties that is central to Lewis’s metaphysics, and is only occult if natural properties are (Lewis , p.). Esfeld and Deckert advertise the avoidance of quidditism as an advantage of Super-Humeanism because (a) quidditism, and quidditistic humility, are inessential features of true Humeanism, and, moreover, are undesirable features, and (b) their metaphysics avoids quidditism but is in other respects like Lewis’s Humeanism. But for the reasons above, (a) is unconvincing. As for (b), does the Super-Humean strategy really avoid quidditism? In a sense, of course, they have avoided quidditism for monadic properties by not having any monadic properties. But, they do still have relations—the fundamental distance relations—so why are they not still committed to quidditism about the relations? And if they are, then doesn’t this undermine the whole argument? Quidditism about relations is still quidditism! Now of course there may be reasons why entanglement relations must be different in kind from spatiotemporal relations, such that one could see spatiotemporal relations—but not entanglement—in a non-quidditistic way, but that has to be spelled out, and that has not been done. Moreover, it seems clear to us that it cannot be done, because quidditism—about monadic properties, relations, whatever—is not dependent on the precise properties or relations in question—it falls straight out of the general metaphysical framework. Lewis himself seems clearly to accept this: In Lewis (), not only does he not accord relations different status to monadic properties so far as quidditism is concerned, he positively treats them the same, and proposes to “speak of ‘fundamental properties’ for short, but they fall into several categories. There are all-or-nothing monadic properties. There are all-or-nothing

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n-adic relations, at least for smallish n. There are properties that admit of degree . . .” (p.). And, of course, there are relations that admit of degree, assuming that spatiotemporal relations are treated as such. His subsequent discussion of quidditism takes this equivalence for granted. Thus, it seems clear that quidditism applies to relations just as much as to monadic properties. And so nothing is gained, from this point of view, by moving to an ontology that includes only relations among the fundamental properties. Thus there is nothing super about “Super-Humeanism” over bog-standard Humean supervenience apart, of course, from the fact that Super-Humeanism is apparently compatible with quantum mechanics whereas standard Humean supervenience is not. But then, of course, the obvious move of acknowledging the lessons of entanglement rather than evading them, and making the appropriate additions to the supervenience basis, works just as well to achieve the required reconciliation.

. Holism and Entanglement Another strand of entanglement antirealism is motivated by the suggestion that entanglement involves a kind of ontological holism. Some forms of holism clearly endorse entanglement realism. For instance, the relational holism of Teller () maintains that entanglement is manifested in relations between entangled particles that fail to supervene on their intrinsic properties. Relational holism is closelyaligned with the approach we favor in the Humean context, and shares many of its virtues.⁸ However, other forms of holism have the effect of eliminating entanglement from fundamental reality. In particular, wavefunction realism maintains that the fundamental ontology of physical reality is located in a high-dimensional space free from non-local influences and entanglement relations. “Wavefunction realism” is a misleading term. A number of views claim that the wavefunction (or quantum state) is real and reserve a place for it in their fundamental ontology. The version of wavefunction realism we have in mind here, is the one defended in, e.g., Albert (, ); Loewer (); Ney (, , ); North (). On this view, the wavefunction is understood as a physical field defined on a high-dimensional space—in the case of ordinary non-relativistic quantum mechanics, a n-dimensional configuration space, where n is the number of particles that exist. Because the wavefunction is part of the most fundamental description of the world, and is defined on configuration space, it is this n-dimensional space rather than ordinary -dimensional space that is fundamental. Far from denying the reality of entanglement, defenders of wavefunction realism often point to entanglement and non-locality as sources of motivation. There are at least two arguments used to motivate wavefunction realism on the basis of entanglement. The first is formal in nature. It begins with the observation that one cannot adequately characterize an entangled system with wavefunctions ⁸ Unlike the Humean approach discussed above, Teller maintains that spacetime is substance, and hence, that spatiotemporal relations do supervene on the intrinsic properties of their relata (i.e., are internal rather than external). Thus, he takes the relational holism implied by entanglement to be a novel feature of the quantum world.

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defined on -dimensional space. One cannot distinguish an entangled pure state from a similar mixed state using only individual position wavefunctions in -space. “To capture the difference between [pure and mixed] states, we must use the additional spatial degrees of freedom we have in configuration space. Entanglement is a completely pervasive feature of a quantum world, and what is needed to capture states that are entanglements of position . . . is a configuration space representation” (Ney , p.). This point is correct as far as it goes, however, the fact that individual position wavefunctions in -dimensional space are incapable of capturing entanglement needn’t recommend moving to configuration space. One may opt instead to (a) adopt a different understanding of the quantum state than one based on individual position wavefunctions (e.g., the spacetime state realism of Wallace and Timpson ) or (b) recognize more than individual position wavefunctions in one’s fundamental ontology (e.g., physical relations constitutive of entanglement). Thus, this motivation for moving to configuration space will only appeal to those who already accept a fundamental ontology free from entanglement. Another argument from entanglement to wavefunction realism is explanatory in nature. In light of Bell’s theorem, one cannot explain non-locality in familiar causal terms without invoking action at a distance.⁹ However, Ismael and Schaffer () argue that a non-causal explanation of non-locality can be given in terms of “common ground.” The core idea is that if entangled particles are fundamentally nondistinct, then correlations in measurement outcomes involving them may be accounted for by appeal to grounding relations—properties of the individual particles are determined by properties of their common ground. There are a variety of ways of developing a common ground account, some of which are eliminative and some of which aren’t. Our contention here is that the non-eliminative versions do a better job of explaining what’s special about entanglement. Wavefunction realism may be considered to be a version of the common ground approach. For the advocate of wavefunction realism, the manifestation of nonlocality in -space is a reflection of a more fundamental state of affairs in n-space. In particular, the coordinated behavior of distinct particles in -space is explained by appeal to the fact that, fundamentally, they have a common ontological basis in the fundamental n-space. Moreover, the fundamental states of affairs involve no non-locality; happenings at one location in n-space are independent of happenings at “distant” locations.¹⁰ Thus, wavefunction realism seems to offer the prospect of a satisfying ontic explanation of non-locality that avoids action at a distance, or fundamental entanglement in the world. The problem is that this implementation of the common ground strategy is global insofar as everything in -space has a common basis in the ontology of n-space. The particles involved in an EPR experiment are fundamentally non-distinct, for

⁹ We are taking “familiar causal terms” here to refer to either a direct causal connection or a common cause explanation capable of “screening-off” the correlata in the sense of Reichenbach (). Retrocausal explanations are also being set aside. ¹⁰ Note that the “locality” of points in configuration space is only analogous to locality in ordinary space. The notion of “distance” involved requires a metric on n-space, which would be analogous but not identical to the familiar notion of spatial distance.

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example, but so are any pair of particles anywhere in the world. Arguably, this results in a kind of (priority) monism, as defended by Schaffer (). But part of what is required in an explanation of non-locality is an account of what is special about the systems that manifest it, and on monism, having a common ground is a perfectly generic feature of everything found in -space. Schaffer () argues that entanglement (“non-separability” in our terminology) is itself perfectly generic in quantum theory, and hence, that monism is well supported. This suggests that wavefunction realism may provide a satisfying common ground explanation after all. However, there are several issues with this reply. First, it’s not at all clear that “the cosmos forms one vast entangled system” (p.). While it is true that some interpretations of quantum theory may recognize a universal wavefunction in which the entire universe is described by a massively-entangled quantum state, other interpretations reject either: (a) that there is a universal wavefunction or (b) that such a state would involve widespread entanglement. Second, even if there is an entangled universal wavefunction, there is a distinction to be drawn between subsystems that display non-local behavior and those that don’t. Careful preparation is required to display non-local phenomena capable of violating a Bell inequality, and the vast majority of observable phenomena do not display nonlocality in this sense. Believers in an entangled universal wavefunction typically appeal to decoherence to establish the required distinction (e.g., Wallace ), but then it seems to be decoherence rather than common ground doing the explanatory work. In sum, an appeal to common ground cannot explain non-locality in the sense of saying what’s special about systems that display it if every physical system shares in that common ground. According to the wavefunction realist, the appearance of entanglement in -space is evidence that there is something wrong in our choice of fundamental space. They advocate moving to n-space where problematic non-locality is no longer present. In moving to a higher-dimensional space, the wavefunction realist has eliminated entanglement rather than embraced it. The wavefunction realist advertises this as an explanation of entanglement, but it more resembles an error theory. It can explain why there appears to be entanglement, but in the end, it is only a feature of the redundant representation of a reality free from entanglement (cf., Ismael, Chapter , this volume). Thus, for the wavefunction realist, entanglement is not a part of the fundamental structure of the world. Fortunately, one can take up the common ground strategy without eliminating entanglement. The key is to locate entanglement in the same space where we find non-locality, that is, ordinary -dimensional space.¹¹ The relations of ontological dependence (or grounding) that link entangled physical systems can obtain without invoking a more fundamental non-spatiotemporal level of reality. For example, consider a case of ordinary physical composition, say, hydrogen and oxygen atoms composing a water molecule. One may regard the composite systems—the water

¹¹ Or, more accurately, -dimensional spacetime. We will continue to speak in terms of + dimensional space and time as this is the setting in which ordinary, non-relativistic quantum mechanics is often situated. That said, none of the arguments given depend on this being the fundamental space rather than relativistic spacetime.

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      



molecule—as dependent on (or grounded in) the atoms that compose it. As a result, features of the composite may be explained by features of its constituent parts. For instance, the overall electrical neutrality of the (non-ionized, liquid state) water molecule may be explained by the respective charges of the hydrogen and oxygen atoms (ions). It is often supposed that ontic explanations require relations of dependence (causal or non-causal), so that there must be a real-world dependence relation underlying explanations like this one (Koslicki , p.; Schaffer , pp. –). In this case, the dependence relation is that of a whole on its parts. But, given that ontological dependence and composition are distinct notions, it’s at least conceptually possible that parts could depend on wholes. Thus, one can maintain that entanglement involves cases of part-on-whole dependence. Two electrons in the spin singlet state depend on the two-electron whole they compose. This dependence relation, in turn, can support an ontic explanation of the anti-correlation of their spins described by quantum theory: the two-electron system has net spin = , hence, if one electron is spin +½ along a given axis, the other electron must be spin ½ along that axis. This has the same structure as the ordinary case, only with the direction of part-whole dependence reversed. Unlike wavefunction realism, this version of the common ground story needn’t be global; part-on-whole dependence is a unique feature possessed by all and only entangled systems. Hence, the account recognizes entanglement in the form of characteristic relations of ontological dependence in ordinary -space. This style of explanation also admits of an alternative reading. It could be that part-on-whole dependence indicates we have individuated systems in too finegrained a manner. For instance, if we try to treat a system as having parts that it lacks, then we should expect to find this non-standard pattern of ontological dependence. Thus, when we attempt to ascribe quantum states to entangled particles—e.g., by taking the partial trace of the density operator for the composite system—we are carving up reality into smaller bits than it really contains. On this reading, entanglement doesn’t consist in a unique form of ontological dependence, but rather the ontological holism of entangled composites (Healey a). This brings us close to the motivation for wavefunction realism discussed above, but there remain two crucial differences: (a) the wholes in question are located in -dimensional space, and (b) all and only systems of mutually entangled particles are wholes. Thus, only if everything is truly entangled would this result in monism, and even then, entanglement—in the form of ontological holism—would remain a genuine feature of our world. Bell () claimed that non-local correlations “cry out for explanation.” By saying what’s special about entangled physical systems, the present account goes further in answering this demand. Of course, if one adopts an interpretation which features a universal wavefunction with widespread entanglement, then we may be drawn to a version of monism on this view as well. However, there still may be a level of description at which it’s appropriate to deploy effective wavefunctions which will be more restrictive in ascribing non-separable quantum states. If so, this approach may be deployed at that level: (all and only) the effectively entangled particles ontologically depend on the wholes they compose. This sort of non-global and effective application of the common ground explanation is unavailable on wavefunction realism, which is a global claim about fundamental reality.

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

    

The non-global common ground explanation of non-locality sketched here is incomplete in several respects. Most significantly, substantial work is required to fully integrate it into the extant interpretations of quantum theory. That said, it is prima facie open to any interpretation that posits systems of particles in nonseparable quantum states to regard particles in such systems as ontologically dependent on the wholes they compose. The details of how such entangled states come about and are dissolved can be treated as a separate matter. Thus, providing a common ground explanation of non-locality doesn’t require wavefunction realism. One can instead retain entanglement as a genuine feature of our world by recognizing part-on-whole dependence. This alternative does a better job of explaining what’s special about systems that display non-local behavior.

. Implications for Spacetime Fundamentality Our primary aim in the foregoing discussion has been to recommend entanglement realism over views that we see as eliminative. The discussion also has some important implications for the role of spacetime in our thinking about quantum theory. A common feature of both of the approaches criticized above—Super-Humeanism and wavefunction realism—is the fundamental place they give to spatiotemporal (or quasi-spatiotemporal) relations. First, consider the Super-Humean. Esfeld and Deckert maintain that the only features of matter points are their spatiotemporal relations. This elevates background spacetime to an almost analytic status; the view cannot be articulated in a world in which fundamental spacetime or spacetime relations cannot be defined. Second, consider wavefunction realism. On this view, ordinary -dimensional space has been replaced by a fundamental n-dimensional configuration space. But, the analogous “quasi-spatial” relations between points in configuration space are given fundamental status. Thus, both views eschew putative entanglement relations in favor of an ontology of only (quasi-)spatiotemporal relations. However, this simplified ontology has certain costs. We’ve already discussed the potential explanatory cost in eliminating entanglement, but there is another important cost as well: the commitment to only (quasi-)spatiotemporal relations leaves hostages to fortune. In the case of wavefunction realism, it is an open question whether the approach can be extended beyond ordinary non-relativistic quantum mechanics (Wallace and Timpson ; although see Ney, Chapter , this volume, for a response). In the case of Super-Humeanism, the need to adopt something like a “primitive ontology” interpretation of quantum mechanics both leaves hostages to relativistic fortune and offends against the naturalistic metaphysical spirit by allowing background metaphysical preconceptions to dictate the choice of an idiosyncratic approach to a physical theory. Moreover, if spacetime is emergent as some approaches to quantum gravity seem to suggest (Huggett and Wüthrich ), then clearly approaches that seek to eliminate relations of entanglement in favor of fundamental (quasi-)spatiotemporal relations will be untenable. This again recommends a common sense realism about entanglement relations over eliminativism.

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SECTION 3

Spacetime Theories and Fundamentality

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10 On the Independent Emergence of Space-time Richard Healey

. Introduction Space and time have been fundamental to metaphysics and physics since the pre-Socratics. Their union remained fundamental after Minkowski’s pronouncement that special relativity doomed each separately to fade away as a mere shadow, and Einstein’s declaration that space-time itself exists only as a structural quality of the gravitational field described by general relativity. But difficulties in meshing general relativity with quantum theory have prompted a widespread conviction that space-time is not fundamental but emergent. This raises a difficult ontological question: What is there more fundamental than space-time from which it might emerge? Rather than try to answer this question I suggest we reject its presupposition that emergence demands a precisely characterized ontological basis. Jeremy Butterfield (a,b) took emergence to mean properties or behavior of a system which are novel and robust relative to some appropriate comparison class. Karen Crowther (, p.) added a dependence requirement based on Bedau’s (, p.) condition that emergent phenomena are dependent on, constituted by, or generated by underlying processes. Together, these criteria would make spacetime emergent just in case spatiotemporal phenomena display novel and robust behavior relative to some comparison class of underlying non-spatiotemporal processes on which they depend. This chapter explores the possibility that space-time emerges in the domain of a quantum theory of gravity in a way that does not so depend on objects, properties or processes represented by that theory. The possibility is suggested by two relational approaches to quantum theory: Carlo Rovelli’s () relational quantum mechanics and my () pragmatist view, according to which any assignment of a quantum state is relative to the physical situation of an actual or hypothetical agent. By contrast, Rovelli takes the assignment of a state to a system in quantum theory always to be relative to some other physical system—a view he deploys (Rovelli and Vidotto ) to secure the conceptual foundations of a theory of loop quantum gravity from which (general-relativistic) space-time emerges only as a classical limit. But can a quantum theory of gravity help itself to talk of physical systems or situations without representing them in Richard Healey, On the Independent Emergence of Space-time In: The Foundation of Reality: Fundamentality, Space, and Time. Edited by: David Glick, George Darby, and Anna Marmodoro, Oxford University Press (2020). © Richard Healey. DOI: 10.1093/oso/9780198831501.003.0011

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

 

space-time? In the rest of this chapter I show how it can in a pragmatist view of quantum theory (Healey , a): I consider Rovelli’s relational approach elsewhere (Healey ). The following section offers a preliminary analysis of the relation between fundamentality and emergence. This exposes two problems in evaluating the suggestion that space-time is not fundamental but rather emerges within a quantum theory of gravity. Leaving the actual construction of such a theory in the capable hands of physicists, I introduce a strategy for attacking the second, conceptual, problem. Section . illustrates this strategy by applying it to theories of the free electromagnetic field, for which we have a successful quantum theory from which a corresponding classical theory emerges—Maxwell’s theory of electromagnetic radiation. I argue that this provides an example of the emergence of a phenomenon within a quantum theory that describes no underlying physical processes. The argument depends on the pragmatist view that a quantum state is not what Bell () called a beable, even when a statement assigning that state is objectively true. So while it is not the function of |ψ> to represent a physical object or magnitude (“an element of physical reality”), some quantum states are objectively real. Section . explains why this view does not conflict with recent results such as those of Pusey, Barrett, and Rudolph (PBR ) and Colbeck and Renner (). Applying the pragmatist view to the states of a conjectured quantum theory of gravity, Section . presents a resolution of the conceptual problem diagnosed in Section .. This shows how space-time might emerge within a quantum theory of gravity that describes no more fundamental physical structure. Section . says why this emergence would be independent. Finally I draw some morals of this resolution for the relation between metaphysics and fundamental physics. Even if there is such a thing as ultimate reality, fundamental physics need not describe it and much that is real (even in physics) is not grounded in it.

. Fundamentality and Emergence ‘Fundamentality’, an ugly word used almost exclusively by philosophers, purports to designate a monadic property of a fundamental object. But the adjective ‘fundamental’ is implicitly relational: It is preferable to use the relational term ‘fundamental to’ from which it derives. This reframes discussion of fundamentality as a question of whether x is fundamental to y, for suitable instantiations of the variables x, y. Their prominence in everyday, scientific and philosophical thought about the world suggests replacing x by space and by time. Three salient instances for y then yield the theses: Space, time are fundamental to us. UsS,T RealityS,T Space, time are fundamental to reality. PhysicsS,T Space, time are fundamental to physics. These theses are by no means mutually exclusive: indeed, some may be taken to imply others. But UsS,T does not imply PhysicsS,T. Though in some sense fundamental to us, life, air, water, food, clothing, housing, each other, language, sanitation,

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

antibiotics, government, law, colors, causation, probability, atoms, the force of gravity are not (all) fundamental to physics. There are naturalisticallyinclined philosophers who would deny that any of these are fundamental to reality. Some physicalist philosophers may endorse this conditional statement: If x is not fundamental to physics, then x is not fundamental to reality Insofar as life, air, water, food, clothing, housing, other people, language, sanitation, antibiotics, government, law, colors, causation, probability, atoms, the force of gravity are not fundamental to physics they may conclude that none of these are fundamental to reality. I’ll raise doubts about this conditional statement later (see Section .). But note that anyone interested in the foundation of reality who accepts the conditional is then led to ask whether space and time are fundamental to physics. That question is easily answered. Space and time have not been fundamental to physics at least since Einstein () published his special theory of relativity. As Minkowski () put it three years later: Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.

Indeed, even before relativity theory there was a strong scientific case that space-time is more fundamental to physics than space and time individually. So space and time are not fundamental to physics insofar as space-time is fundamental to physics, but space and time are not. This motivates revising PhysicsS,T to read: PhysicsS-T

Space-time is fundamental to physics.

Acceptance of Minkowski space-time as the basic geometric setting for special relativity seemed to close the case; and general relativity (the theory that succeeded it ten years later) merely introduced a host of alternative structures this space-time might take, depending on how matter and energy are distributed in it. Many philosophers have come to believe that space-time is fundamental to physics, even though space and time are not. But this was not Einstein’s own view. As he put it (Einstein, ): Space-time does not claim existence on its own but only as a structural quality of the [gravitational] field. (p.)

He came to think of the gravitational field, (represented by the four-dimensional metric tensor gμν), as more fundamental than space-time, so that If we imagine the gravitational field, i.e. the functions gμν to be removed, there does not remain a space, but absolutely nothing, and also no “topological space”. . . . . There is no such thing as an empty space, i.e. a space without a field. (p.)

Physics has moved on since Einstein. Even without an agreed, successful theory of quantum gravity, we have reason to believe that neither space-time nor a (classical) gravitational field will be fundamental to physics. The following quote (Rovelli and Vidotto, ) expresses a common (though not universal) attitude amongst contemporary physicists.

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

 

G[eneral] R[elativity] is not just a theory of gravity. It is a modification of our understanding of the nature of space and time. Einstein’s discovery is that space-time and the gravitational field are the same physical entity. Space-time is a manifestation of a physical field. All fields we know exhibit quantum properties at some scale, therefore we believe space and time to have quantum properties as well. (p.)

The idea is to follow Einstein’s lead into the quantum realm by considering space-time to arise as a manifestation of a quantum gravitational field. This idea could be made to work if we could find something underlying space-time—some entity x that is fundamental to space-time to instantiate the expression ‘x is fundamental to y’, with ‘y’ instantiated by ‘space-time’. But how might we instantiate ‘x’? We don’t know! But the progress of physics over the past half century suggests at least the form of an answer to this question. Einstein thought space-time is the same physical entity as a classical gravitational field: But today we should expect it to arise as a manifestation of a quantum gravitational field. If so, from the perspective of a theory of quantum gravity, space-time will appear as an emergent phenomenon, (a structural quality of ) a classical gravitational field that emerges in some appropriate limit of that theory. Indeed, philosophers of physics often speak of the emergence of space-time in a quantum theory of gravity. Like fundamentality, emergence is best thought of as a relation: x emerges from y. One might assume this is simply the converse of the relation x is fundamental to y. But while metaphysicians typically regard is fundamental to as an ontological relation, philosophers of science typically take emerges from to relate theories rather than things. So there are ample opportunities for confusion here! In a pair of recent papers, Jeremy Butterfield (a,b) has taken emergence to mean properties or behavior of a system which are novel and robust relative to some appropriate comparison class. He investigates such emergence by analyzing possible relations (e.g. reduction, supervenience, limiting case) between a theory T describing such properties or behavior and a second theory T taken to be applicable to the same system. On this approach, space-time would emerge in a quantum theory of gravity if and only if theoretically-described spatio-temporal phenomena count as novel and robust relative to a class of (presumably) non-spatiotemporal phenomena described by a quantum theory of gravity. An emergence relation between theories generally involves an inverse relation between their ontologies. The emergence of geometric optics from wave optics depends on the identification of a ray with an element of a propagating wave-front. Butterfield (a) gives the example of the emergence of the theory of thermodynamics (for an isolated gas in equilibrium) from the theory of statistical mechanics, which depends on the identification of a gas with a vast number of particles. In such cases the emergence of T from T requires the ontology of T to be fundamental to that of T. Two problems arise if we try to take this approach to the emergence of space-time within a quantum theory of gravity: . We have no agreed quantum theory of gravity to serve as theory T. . We don’t know how any such theory could be applied to a system without assuming that system is in some sense spatio-temporal.

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Spacetime

Conventional quantum field theorist



Boundary spacetime

     -

Post-Maldacena string theorist

Genuine quantum-gravity physicist

Figure . Pre-general-relativistic physics is conceived on spacetime. The recent developments of string theory, with bulk physics described in terms of a boundary theory, are a step towards the same direction. Genuine full quantum gravity requires no spacetime at all. Source: Rovelli and Vidotto (), p. .

These problems are nicely illustrated in Figure ., copied from (Rovelli and Vidotto ), p.. The figure sketches three approaches toward the hoped-for theory of quantum gravity. The left side depicts a conventional attempt to craft a theory of gravity as a quantum field theory on an assumed space-time background (possibly with more than three spatial dimensions). The center depicts an attempt to create a quantum theory of gravity in the bulk as dual to such a conventional quantum field theory defined on its boundary, where a background space-time with multiple spatial dimensions is assumed for both. The right side depicts a prospective quantum gravity theory that does not presuppose any background space-time. None of these approaches has yet proved wholly successful, so the figure illustrates the fact that we still lack a satisfactory theory of quantum gravity (problem ). While the fanciful figure on the right shows how hard it is even to imagine what a system described by such a theory could be like if it is in no sense spatio-temporal (problem ). Half a century of work by many of our best physicists has not yielded an agreed solution to the first problem. I can merely encourage them in their continuing efforts! But my recent understanding of quantum theory has suggested a largely independent strategy for tackling the second problem. In the next section I’ll illustrate the strategy by applying it to an analogous case in which we do have a well-developed quantum theory from which classical behavior emerges: the quantum theory of the electromagnetic field.

. How Light Emerges within a Quantum Field Theory The Quantum theory of free Electro-Magnetism (QEM) is traditionally arrived at by canonical quantization of a classical field theory—Maxwell’s equations in a vacuum. This involves replacing magnitudes such as electric and magnetic fields by mathematical operators that act on a space of quantum state vectors (Fock space). Unlike classical electric or magnetic field magnitudes, these operators do not have the function of representing physical magnitudes. So one can ask what this quantum theory represents—in J.S. Bell’s terminology, ‘what are its beables?’ (i.e. “elements of

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

 

physical reality”, if the theory were true). This is one case of the more general question “What are quantum field theories about?”, or in fancier language “What is the ontology of a quantum field theory?” Two main candidates have been proposed for the ontology of a quantum field theory: particles, and (classical) fields. Unfortunately, philosophers have formulated strong objections to both proposals (Fraser ; Baker ). Physicists typically finesse such objections by regarding particles and fields as not among the theory’s beables, but rather as emergent entities. As one recent text (Lancaster and Blundell ) puts it: Every particle and every wave in the Universe is simply an excitation of a quantum field that is defined over all space and time. (p.)

The idea is that one kind of excitation may be treated as if it consisted of particles, while a different kind of excitation may be treated as if it consisted of classical waves. Here ‘being treated as’ does not mean ‘is composed of/identical to’ since the treatment is successful within limits, and only for certain purposes. But how is this ontological emergence supposed to work? Metaphysicians enamored with mereology may be uncomfortable with the idea of an emergent entity—something with no well-defined set of constituents. But our world is full of them. They include epidemics, hurricanes, water waves, Jupiter’s red spot, phonons, and many other varieties of quasi-particles in condensed matter physics. The excitations from which photons and light waves emerge are quantum states of QEM. This suggests that quantum states may be the real beables of this theory, while more familiar entities like particles and waves emerge from particular kinds of excited states in certain circumstances. However, the ontological status of quantum states (“wave functions”) is currently a topic of intense controversy among physicists and philosophers working on conceptual foundations of quantum theory. In my view, quantum states are not beables of any quantum theory. In Section . I’ll defend this view against objections based on some influential recent arguments. But for now I’ll treat it as an assumption to show how it can help us solve problem (as specified in Section .). For once granted, I think we can see how a theory of quantum gravity might be applied to a system without assuming that system is in some sense spatio-temporal. A solution to that problem would provide an additional argument for the assumption. It follows that quantum states are not beables of QEM: QEM has no beables! While ocean waves emerge from excited states of water, photons and (classical) light-waves do not emerge from anything described by QEM. Objection: That’s crazy: everyone agrees that QEM describes the quantum/quantized electro-magnetic field! Reply: Everyone agrees that mathematical objects (electro-magnetic field operators) appear in models of QEM that have been successfully applied to physical systems. But these models are not used to represent or describe the electro-magnetic field (classical or quantum). An application presupposes the existence of the system to which it is applied, and we can call this “the quantum/ quantized electro-magnetic field”. But QEM is applied to it without in any way

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

describing its features: successful application does not entail description. The quantum electro-magnetic field remains purely schematic in QEM. It is generally acknowledged that CEM (the Classical theory of free ElectroMagnetism) emerges from QEM in a suitable limit (“of coherent states with large mean photon number”). But one can acknowledge such theoretical emergence while denying that QEM or its application describes any beables more fundamental than the (emergent) classical fields described by CEM. Here we have an example of two theories where the emergence of T from T does not require the ontology of T to be fundamental to that of T.

. The Quantum State Is Not a Beable Pusey, Barrett, and Rudolph () proved a result that might be taken to show that the quantum state is a beable. The proof is valid, but like all proofs its premises may be questioned. It does not show that quantum states are beables. The argument depends on few assumptions. One is that a system has a ‘real physical state’ not necessarily completely described by quantum theory, but objective and independent of the observer. This assumption only needs to hold for systems that are isolated, and not entangled with other systems. Nonetheless, this assumption, or some part of it, would be denied by instrumentalist approaches to quantum theory, wherein the quantum state is merely a calculational tool for making predictions concerning macroscopic measurement outcomes. The other main assumption is that systems that are prepared independently have independent physical states. (p.)

As the argument proceeds, it becomes clear that the physical states referred to in the last sentence are taken to be real, so “the other main assumption” depends on the main premise stated in the second quoted sentence. But what is a ‘real physical state’? To clarify the notion of a physical state, the authors consider the classical mechanics of a point particle moving in one dimension. They take its (instantaneous) physical state to be completely specified by its phase space point with coordinates (x,p), functions of which constitute its physical properties. By taking these to include constant functions such as mass and charge, they apparently take the physical state of a classical particle to determine the values of all magnitudes pertaining to it at a time. So the physical state of a single classical particle moving in one dimension would be real if it correctly specified the particle’s instantaneous physical properties. By analogy, the physical state of a quantum system would be real only if it specified the system’s instantaneous physical properties, and complete if it specified all of these. Pusey et al. () take the analogy with classical statistical mechanics to motivate imposition of a second necessary condition on the reality of a physical state λ of a system in quantum theory: Now consider a quantum system. The hypothesis is that the quantum state is a state of knowledge, representing uncertainty about the real physical state of the system. Hence assume some theory or model, perhaps undiscovered, that associates a physical state λ with the system.

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

 

If a measurement is made, the probabilities for different outcomes are determined by λ. If a quantum system is prepared in a particular way, then quantum theory associates a quantum state (assume for simplicity that it is a pure state) |ψ>, but the physical state need not be fixed uniquely by the preparation—rather, the preparation results in a physical state according to some probability distribution μψ(λ).

So Pusey et al. () count a physical state λ as real only if λ both specifies a system’s instantaneous physical properties and determines the probabilities for different outcomes if a measurement is made on it. In Leifer’s () terminology, such a state λ is an element of an ontological model. PBR clearly view their proof as directed against the conjunctive hypothesis that a quantum system has a real physical state λ and that this does not uniquely determine its quantum state. Given the second (independence) assumption, their proof shows that this conjunctive hypothesis is false. So if a quantum system has a real physical state, then this uniquely determines its quantum state. Now one thing that trivially uniquely determines that quantum state is the quantum state itself. So their result is consistent with the assumption that the quantum state is physically real. But it is equally consistent with the assumption that the quantum state is not real. A system’s quantum state does determine the probabilities for different outcomes if a measurement is made on it. But PBR count it as real only if it also specifies the system’s instantaneous physical properties. The result that the quantum state is real follows only if that is indeed a necessary condition for its reality and the condition is in fact satisfied. PBR maintain that this result indeed does follow. They take the fact that the quantum state can be inferred uniquely from λ to show that the quantum state is itself a physical property of the system (p.). But this follows only if λ does specify a system’s instantaneous physical properties. However, that is a consequence of its reality only given what they take to be required for a physical state to be objectively real and independent of the observer. This is essentially the requirement that such an objectively real state be what Bell called a beable—a physical magnitude or property, at least as represented by some physical theory.¹ In the pragmatist view of quantum theory I am assuming here, the quantum state of a system is not a beable. But it is still objectively real, no matter what any observer may take it to be. Indeed, anyone who accepts quantum theory should believe that systems would have had quantum states even if there had been no observers. The quantum state is not merely a calculational tool for making predictions concerning macroscopic measurement outcomes. A state vector |ψ> (or other mathematical representative of a quantum state) does have a distinctive non-representational role within quantum theory. A statement in which |ψ> is used to assign a quantum state to a particular system at a time is objectively true or false, from which it trivially follows that |ψ> represents something objectively real. According to Healey (a), this is not a (monadic) property of the system, since a quantum state is assigned relative to the physical situation of an actual

¹ Here is a more precise statement of their requirement. A theory represents a real physical state only if that state uniquely determines the value(s) of one or more magnitudes within the theory.

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     -



or hypothetical agent. So the objective reality represented by |ψ> involves a relation to a physical situation, relative to which it is assigned. But the function of that statement within the theory is not to describe physical properties of the system at that time. Instead, a quantum state functions as a source of advice on the significance of statements about the values of magnitudes and on how credible are those that are meaningful enough to be entertained. The Born rule is key to both these functions of the quantum state. Applied to a system’s quantum state, it directly assigns a probability to each significant statement about the value of a magnitude on that system, not about the outcome of its measurement. But not every statement about the value of a magnitude is significant in a given physical situation, and the Born rule may never be legitimately applied to all such statements at once. The relevant feature of the physical situation is the system’s environment. So assessing the legitimacy of an application of the Born rule to a particular statement assigning a value to magnitude on the system requires information about the environment and its interaction with the system. Such information may be provided independently of quantum theory. But it does not have to be, since quantum theory itself may be used to model the system’s interaction with its environment. This is the role of models of environmental decoherence. In a model a quantum state may be assigned to a larger system that includes both the original system and its environment, and their interaction modeled by an interaction Hamiltonian. If the reduced quantum state of the system would evolve into a form that is robustly diagonal in some orthonormal basis, then each statement assigning a value to a magnitude equal to an eigenvalue of the corresponding operator is certainly significant enough to be assigned a probability by the Born rule applied to the quantum state originally assigned to the system alone. But the Born rule is not legitimately applied to a statement assigning a value to a different magnitude corresponding to an operator none of whose eigenvectors is even close to an element of that orthonormal basis. It is by restricting legitimate applications of the Born rule in this way that one secures consistency with so-called no-go theorems such as those of Kochen and Specker () and Bell (, ).² The Born probability assigned to a significant statement about the value of a magnitude on a system has the same status as the quantum state used to assign it. It, too, is objective and relative to the same physical situation as the quantum state, though it is not a beable. But it is not objective in Maudlin’s (a) sense. there could be probabilities that arise from fundamental physics, probabilities that attach to actual or possible events in virtue solely of their physical description and independent of the existence of cognizers. These are what I mean by objective probabilities. (p.)

It is important to note that the reason Born probabilities aren’t objective in Maudlin’s sense is not that they are dependent on the existence of cognizers. Born probabilities, like quantum states, would exist in a world without cognizers. ² The use of models of decoherence for magnitudes represented by operators with continuous spectrum is analogous but involves additional complexities though no essentially new concepts. In no case does this use of a model depend on perfect decoherence in a fixed orthonormal basis.

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

 

Maudlin’s reason for not counting them as objective would be that they are relational: they do not attach to actual or possible events in virtue solely of the physical description of those events. Since a quantum state is not a beable, it is not even a candidate for the job of real physical state λ as this appears in an ontological model. As Leifer () puts it in his comprehensive review article: The idea of an ontological model . . . is that there is some set of ontic states that give a complete specification of the properties of the physical system as they exist in reality. (p.)

PBR prove that in any ontological model of (a certain fragment of ) quantum theory, the quantum state is uniquely determined by the state λ of the model. But while the quantum state is objectively real, it is not a beable, and so is not determined by the ‘real physical state’ of any ontological model. Quantum theory has no ontological model even though the quantum state is objectively real. Leifer () analyzes similar proofs by Colbeck and Renner () and by Hardy (). Since these all assume the ontological models framework it is not necessary to consider them further here. No so-called ψ-ontology theorem can show that a quantum state is a beable.

. How Space-time Might Emerge in Quantum Gravity The emergence of the classical electro-magnetic field from QEM may provide us with a good conceptual model for the emergence of space-time in a quantum theory of gravity. The first step toward understanding the emergence of space-time would be to follow Einstein in identifying space-time with the classical field gμν of the general theory of relativity (GR). We seek a quantum theory of gravity (QG) from which GR will emerge as a suitable limiting case, much as CEM emerges from QEM in the limit of coherent states with large mean photon number. As a quantum theory, QG need have no beables of its own. Suppose we had a satisfactory quantum field theory of the gravitational field QG. Such a theory would be clearly formulated and mathematically consistent. It would be predictively successful and explanatorily powerful: it might even permit us to control natural phenomena we cannot now control. But, as a quantum theory, it would not describe or represent the properties of a quantum gravitational field. Instead, it would contain mathematical models for application through the assignment of quantum states to a quantum gravitational field. But that field would not be a beable of the theory, and it would figure in no ontological model within or underlying that theory. The emergence of GR (and hence space-time) need require no ontological dependence of gμν on anything more fundamental described or represented by QG. We could say that the quantum gravitational field is fundamental to space-time. But a successful QG need not describe this field at all, and so neither as spatiotemporal nor as spatiotemporal in some extended sense (cf. string theory). Scientists could justifiably celebrate such an extraordinary extension of human knowledge. But metaphysicians might feel dissatisfied by such an ineffable fundamental ontology.

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     -



. What Makes Such Emergence Independent If space-time were to emerge within a quantum theory of a gravitational field whose properties and behavior it does not represent, then its emergence would undermine the claim that general relativity (or some other classical space-time theory) can be reduced to that quantum theory. This is a consequence of Butterfield’s (a,b) understanding of reduction as a relation of deduction or determination between theories. While seeking to distance the concept of emergence from that of reduction, Crowther () still requires emergent phenomena to be dependent on, constituted by, or generated by underlying processes. That requirement could not hold either, if space-time were to emerge in the way suggested in the previous section. The emergence of space-time within a quantum theory of gravity would count as independent, whether dependence is understood as a relation between theories or a relation between processes described by them. But the relational nature of quantum states provides an additional reason for taking the emergence of classical space-time within a quantum theory of gravity to be independent. In my () pragmatist view, any assignment of a quantum state is relative to the physical situation of a hypothetical agent. So to apply any quantum theory (including a hypothetical, successful QG) it must be possible to contemplate an agent’s occupying such a situation—to place oneself in that agent’s position, if only in thought. A key aspect of a hypothetical agent’s physical situation is its space-time location, as becomes clear from a consideration of the different quantum states correctly assigned to the same quantum system by spatially separated agents Alice and Bob in Bell tests on entangled pairs (see Healey b). If this is right, then any application of any quantum theory implicitly assumes the existence of space-time sufficient to permit contemplation of a hypothetical agent’s space-time location. A successful QG need not itself describe an agent, its physical situation or spacetime location. But the theory could not be successful if it were impossible to apply it, and so the success of a QG theory implicitly assumes the existence of space-time enough to permit its application. If a classical space-time does indeed emerge from a successful QG, then that assumption is justified, at least for any conceivable application. So we can certainly contemplate applying such a QG to extreme situations including events we typically describe as occurring within the event horizon of a black hole, or in what we think of as the extremely early universe. We can do this because the “Archimedean platform” from which we contemplate applying the theory is our own, relatively benign, physical situation: it is not that of the target of the application, and it is an adequacy condition on any successful QG that a classical space-time should emerge in such benign situations.

. Fundamentality: Physical and Metaphysical Metaphysicians sometimes talk of ultimate reality: this would presumably be something more fundamental than space and time, if anything is. The obscure metaphysical formulations: x is fundamental to y  y is grounded in x x is an element of ultimate reality  nothing (else) is fundamental to x

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

 

suggest an obscure metaphysical criterion of reality: y is real  y is grounded in ultimate reality and a (slightly less?) obscure suggestion: y is real  y is grounded in ultimate physical reality These formulations inherit their obscurity from that of the grounding relation, a currently fashionable item in the metaphysician’s toolkit. But their obscurity is not the only thing wrong with them, as we can see by reflecting on this examination of the status of space, time and the quantum state in fundamental physics. Is the quantum state real? Quantum states play a key role in our best fundamental physics, according to which a statement (correctly) assigning a quantum state to a system is objectively true. So we should believe that the quantum state is real. To note that the quantum state plays this role in physics rather than in some other discipline, one might choose to say that quantum states are physically real. But a quantum state is not a beable. So accepting quantum theory involves accepting quantum states as real even though they are neither elements of nor grounded in an ultimate (physical) reality to which that theory is applied. So, quantum states provide counter-examples to the obscure metaphysical criterion of reality, and to the (slightly less?) obscure suggestion. Their primary role is to serve as a source of the Born probabilities we derive from them. So quantum probabilities are also real, though not grounded in ultimate (physical) reality. I think that probability is a source of counter-examples to the obscure metaphysical suggestion even outside of its application in quantum theory, as also are color, causation, laws, modality, etc. But I won’t try to argue that here! Are space and time real? Each remains fundamental to us in our daily lives, and to most of science. Scientific theories involving these concepts emerge from more fundamental theories of (classical) space-time, especially general relativity. In classical physics, the relation between pre-relativistic and relativistic theories and what they represent can be made to fit conventional models of reduction (Butterfield), and emergence (Crowther). Here space and time emerge from something more fundamental on which they depend. I leave it to metaphysicians to decide whether this means that space and time are grounded in space-time. We currently have no completely successful quantum theory of gravity. This makes it premature to render a final verdict on the status of space-time. I have argued that space-time may emerge from a successful quantum theory of the gravitational field that represents nothing more fundamental. If that turns out to be right, then space-time emerges independently within that theory. The only sense I can make of the suggestion that space-time would then turn out not to be fundamental is that it would depend for its existence on something, we know not what. That might satisfy a Kantian, but I think it would greatly disappoint a contemporary, naturalistically inclined metaphysician.

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11 Duality, Fundamentality, and Emergence Elena Castellani and Sebastian De Haro

. Introduction The idea of duality has been at the centre of many important developments in theoretical physics in the last  years. Dualities are interesting for a variety of reasons: they are powerful tools in theory construction, they allow physicists to do calculations in regimes that are otherwise inaccessible to existing theories, and they give new insights into the physics of a problem. Thus, from a philosophical point of view, they allow for interesting test-cases of various forms of empirical and physical equivalence. They also offer insights for emergence, especially the emergence of spacetime and fundamentality (see Rickles ; Dieks et al. ; De Haro ; Wüthrich ). In this contribution, we aim to illustrate their import for some aspects of emergence and fundamentality that have not yet received much attention in the philosophical literature. In general, the recent discussions of duality in the philosophical literature have focussed on three main questions: namely, what is the best formulation of dualities, how do dualities bear on theory individuation, and when can dual theories be said to be physically equivalent, i.e. to describe the same physical situations. More recently, also other aspects of dualities have started to be addressed in the philosophical literature: such as their heuristic uses in theory construction (De Haro ) and their empirical consequences (Dardashti et al. ). Our motivation in bringing together duality, fundamentality, and emergence rests on two points. First, the fact that dualities can render a theory more tractable than its dual, depending on the context in which it is used—something that is not evident if one would regard a duality as a mere translation between two theories, like translating a text from English into French. By illustrating a notion of epistemic emergence in examples of dualities, we can better understand why physicists regard dualities as surprising and as useful for theory development: the point is precisely that dualities are not mere “automatic translation devices”. Second, we wish to explicate the sense in which it is sometimes claimed that fundamentality in physics is not absolute but relative, because the relation of fundamentality can change depending on the energy

Elena Castellani and Sebastian De Haro, Duality, Fundamentality, and Emergence In: The Foundation of Reality: Fundamentality, Space, and Time. Edited by: David Glick, George Darby, and Anna Marmodoro, Oxford University Press (2020). © Elena Castellani and Sebastian De Haro. DOI: 10.1093/oso/9780198831501.003.0012

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

     

or the distance scales involved (see for example Susskind , p. , and some of the contributions in Castellani and Rickles ). Of course, a discussion of such philosophical notions as fundamentality and emergence requires us to be explicit about how they are intended. Therefore, let us specify, at this point, the sense in which we will use the notions of duality, fundamentality, and emergence. A duality in physics is, roughly speaking, a relation of formal equivalence between different theories. More specifically, it is an isomorphism between theories (for more details, see Section .).¹ In the cases we will consider, this relation will be between theories whose energy scales (or other significant parameters in the theory, such as a length scale or the strength of an interaction) are inverted, one with respect to the other: so that quantities in a high-energy regime are mapped to dual quantities in a low energy regime in the other theory. This is precisely the reason why duality, as a formal relation with significant interpretative consequences, offers new perspectives the interconnected notions of fundamentality and emergence. Regarding ‘emergence’, let us start with the general notion which can be found in O’Connor and Wong (), where emergent entities are characterised as those that “arise out of more fundamental entities and yet are ‘novel’ or ‘irreducible’ with respect to them”. Since our purposes in this chapter are epistemic—we are interested in what can be derived, predicted, and sometimes also explained, by duality—we will take emergence to be a matter of irreducibility, as above; but we will consider theories, rather than entities in the theory’s domain of application. Thus we take emergence, in general terms, to be the ‘lack of derivability from an appropriate comparison class’.² This characterisation is general enough that it allows us to treat the cases we wish to deal with. Furthermore, lack of derivability may lead to novelty, so all our cases of emergence will contain novelty as well. This conception is close, for example, to Bedau’s () epistemic emergence, as we will discuss in Section ..³ Notice that the above characterisation of emergence involves two sets of theories. We will dub these the ‘top’ and ‘bottom’ theories, with the comparison class being the ‘bottom’ theory, and theories that emerge being the ‘top’ theory.

¹ For a conceptual introduction to dualities, see De Haro et al. (). In this chapter, we will take our notion of duality from De Haro () and De Haro and Butterfield (). See also the contributions to the special issue on dualities, Castellani and Rickles (). ² As we have argued elsewhere (De Haro , section ..), the notion of ‘novelty’ is in some respects more useful than derivability, because it is more general and covers more cases. Cases of lack of derivability are always cases of novelty, while the opposite is not the case. However, in the case of epistemic emergence, ‘lack of derivability’ suffices. ³ An important distinction in the literature is between ontological and epistemic emergence, the distinction being in the kind of novelty or irreducibility that is considered. For example, ontological emergence is often associated to the irreducibility of laws, to the appearance of novel causal powers, and to supervenience; while epistemic emergence is often related to systemic features of complex systems that could not be predicted, or to ‘lawlike generalizations within a special science that is irreducible to fundamental physical theory for conceptual reasons’ (O’Connor and Wang ). As we already mentioned, we are interested in epistemic emergence: which motivates our choice of ‘irreducibility’, as lack of derivability, in our notion of emergence above. For more on this, see section . For discussions of ontological emergence in the context of dualities, see De Haro () and Dieks et al. ().

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, ,  



Turning now to ‘fundamentality’, we will adopt, in our use of the notion, the sense in which, in physics, bottom descriptions are usually thought to be more fundamental than the top ones, since the top theories can, at least partly, be derived from the bottom ones—emergence thus consisting in the fact that this derivation fails. Thus the relation between bottom and top (the ‘comparison’ involved in making a judgment of emergence) involves fundamentality. We will give more details on this point in Section .., just mentioning here some examples in order to illustrate the above sense of fundamentality. In physics, the top and bottom descriptions are often related by the variation of a continuous parameter (typically, an energy scale, length, or strength of an interaction), where the top and bottom theories are the endpoints of the variation. This variation takes us through a sequence of coarse-grainings: from the more fundamental (bottom) theory to the less fundamental (top) theory. The bottom theory is more fine-grained, i.e. it describes the domain in more detail. The top theory is more coarse-grained, it describes the domain in less detail, and so its applicability and its accuracy are reduced, relative to the bottom theory. An example of this is the relation between statistical mechanics (the bottom theory) and thermodynamics (the top theory), regardless of the question of emergence. Another example is the distinction that physicists make between ‘elementary’ vs. ‘composite’ particles, characterising the elementary particles as more ‘fundamental’ because they are the classical particles that are the starting point for a quantum theory. We will join physicists in using this jargon, which indeed reflects their epistemic procedures—distinguishing between what is elementary and what is composite in a given theory is one way to make progress in physics towards finding an appropriate starting point for a new theory. We wish to emphasise the epistemic import of the above notion of fundamentality, as well as of the notions of ‘elementary’ vs. ‘composite’ used here: they should not be understood ontologically in the sense of what is “out there in the world”, since—as we will see—such a characterisation may well end up being problematic, when applied to these two notions. Indeed, as we will discuss with respect to the case studies of Section ., the distinction between what is elementary and what is composite may well end up being untenable, if it were to apply to the theory’s domain of application—it is a feature of the theory’s descriptive apparatus that distinguishes some characteristic patterns or behaviours, without correlating with a literal mereological distinction. Nevertheless, the distinction is of epistemic importance, since it can give a starting point for a new theoretical description. Finally, let us note that when we talk about the ‘description’ given by a theory, we mean the theory’s interpretation or the way it represents approximately a domain of application. However, as we said, this interpretation need not be invariant under the duality map; thus, it does not immediately follow that the interpretation should be taken literally, as part of the theory’s ontology. This is indeed one of the main questions in the recent studies of dualities: whether the ontology of dual theories can be obtained regardless of dualities (in which case the duality typically does not leave the ontology invariant) or whether the theory’s ontology is described only by those parts of the theory that are isomorphic under duality—what is usually called the ‘common core’ between the two theories that is left invariant under the

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

     

duality—so that the duality should be part of one’s starting point in constructing the theory’s ontology. While we cannot go in detail into this discussion here—nor will our chapter depend on it—let us say that our position is a cautious one, which admits a common core interpretation as the most natural interpretation of the duality in some cases, but not in others (the main cases, of sine-Gordon/Thirring and gaugegravity dualities, discussed in this chapter, do admit common-core interpretations). Anyway, we are not specifically interested in ontological aspect here, but rather in analysing in more detail how different theoretical descriptions describe their intended domain, and in the specific advantages of particular descriptions for given domains. For example, different theoretical descriptions of the same domain may emphasise different aspects, which may even be novel with respect to each other, or incompatible with respect to one another—hence the relevance of the notion of emergence. A useful analogy may be with the different coordinate patches of a manifold: different coordinate patches describe different regions of the manifold and ascribe different local properties to it that only need be compatible at the overlaps—outside the overlaps, the properties may be very different. This is all regardless of whether a global, single-patch description exists or not: in some cases, such a global description will exist, and in others it will not. The structure of the chapter is as follows. In Section ., we discuss in some detail the notions of duality, fundamentality, and emergence. In Section ., we illustrate the interplay of these notions by means of two case-studies; in Section ., we discuss to what extent the above analysis can suggest a new way to construe the relation between fundamentality and emergence, and thereby conclude.

. Duality, Fundamentality, and Emergence As we said, in this section we discuss in turn the notions of duality (Section ..), fundamentality (Section ..) and emergence, of the weak, epistemic kind (Section ..).

.. Dualities in Physics Dualities apply to a wide variety of theories, ranging from condensed matter physics to classical and quantum field theory to string theory. Our focus will be with dualities in classical and quantum field theories and string theory. First, there is the dual resonance model of the late sixties, from which early string theory originated. Successively, one of the most important developments of this idea was the generalization, proposed by Claus Montonen and David Olive in , of electromagnetic duality in the framework of quantum field theory. This was later extended to the context of string theory, where dualities have also spawned recent developments in fundamental physics, offering a window into non-perturbative physics, and motivating both the M theory conjecture and gauge-gravity duality (see Section ..). The analogy and contrast between dualities and symmetries can be helpful before we characterise dualities more precisely. While a symmetry, in physics, is a relation within a single theory (typically, an automorphism of the space state and-or the set of quantities of the theory), a duality is a similar kind of relation, but between

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, ,  



different theories (namely, where the notion of ‘automorphism’ is replaced by ‘isomorphism’).⁴ A bit more precisely—and our discussion will not hinge on the details—a duality is a bijective map between the states and quantities of two theoretical descriptions, such that the dynamics and the values of the quantities are preserved (for details, see De Haro  and De Haro and Butterfield ). Indeed, duality turns out to be a matter of isomorphic theoretical descriptions of the same physics. The papers just mentioned illustrate how a number of examples of duality in string theory and in quantum field theory indeed instantiate this notion. A well-known example of a duality is the position-momentum duality in basic quantum mechanics. Although this is an elementary example, and one does not expect it to have the interesting properties of the dualities one finds in string theory and in quantum field theory, it is nevertheless illustrative of the general notion of duality just introduced. Namely, in basic quantum mechanics one starts with an algebra of operators, which is usually the Heisenberg algebra for position, x, and momenta, p: ½x; p ¼ iℏ:

ð11:1Þ

Notice that Equation (.) underlies Heisenberg’s uncertainty principle for position and momentum. The above algebra of operators, Equation (.), can be represented in two different ways, depending on whether one takes the position, or the momentum, to have a well-defined value: which is in agreement with Heisenberg’s principle. Accordingly, the wave-function that one constructs can be either a function of the position, or a function of the momentum. The transformation that relates these two representations of the wave-functions is the Fourier transformation.⁵ Leaving the problem of measurement aside: one can show that all the quantities, i.e. all the matrix elements of all the quantities constructed from x and p, can be calculated either as functions of the position basis or as functions of the momentum. The Fourier transformation relates the two, for all the matrix elements. Thus the Fourier transformation is a duality in the above sense, albeit a very simple one. Although the position-momentum duality just discussed is not a case of weak/ strong coupling duality, we can draw a useful analogy with the weak/strong coupling dualities that we will introduce in Section .. In fact, the Fourier transformation has the property of “inverting the uncertainties” of the states, since it maps a welllocalised wave-function to an ill-localised one. Namely, take a wave packet with spread Δx. This means that the position of the particle is known with uncertainty Δx. Let us also assume that the particle is well-localized, i.e. that Δx is very small, compared to a relevant length scale. Then it follows from the Fourier transformation

⁴ For the relation between dualities and symmetries, see De Haro and Butterfield (). ⁵ A Fourier transformation is a mathematical technique widely used for waves, e.g. to decompose sound waves (which are described as the vibrations of air in space, i.e. the oscillations are functions of the position in space) into elementary frequencies (so that the oscillation function is now not a function of the position, but of the frequencies or, equivalently, of the momenta: see also note ). The Fourier transformation relates the two descriptions.

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

     

of the wave-function (alternatively, it follows from Heisenberg’s uncertainty principle) that the uncertainty in the particle’s momentum will at least be of order ℏ=2Δx, which is very large if Δx is small. Thus the Fourier transformation maps a localised particle to a delocalised one. Informally speaking, we can say that ‘having a well-defined position’ is a typical particle property, while ‘having a well-defined momentum’ is a wave-like property.⁶ This can also be seen as a duality between particle and wave-like properties,⁷ and it traditionally goes under the name of ‘wave-particle duality’. Just to give an idea, the analogy between the situation just described and the weak/ strong coupling duality that we will discuss below goes as follows. In the more sophisticated cases of dualities in string theory and quantum field theory, when one theory is weakly coupled, so that the classical approximation holds good (as in the case of the well-localised particle), the other theory is strongly coupled, i.e. it is highly interacting and quantum (as in the case where the momentum is ill-defined, so that the description in terms of a single momentum value is not valid).

.. Emergence and Fundamentality In Section ., we characterised epistemic emergence as the ‘lack of derivability from an appropriate comparison class’, where this lack of derivability implies that there is novelty in the theoretical description. Thus the novelty is, in this case, not to be found in the world but in the way the theory describes the world. At this point, a further refinement of this ‘lack of derivability’ will help in clarifying our case studies: for lack of derivability can be a matter of principle—an intrinsic limitation of a theory (e.g. the theory is not general or precise enough that the emergent theory can be derived from it), or it can be a matter of practice. The latter option—lack of derivability in practice—is considered, for example, by Bedau (), who proposes a weak notion of emergence that allows derivation, but only if it is by simulation. More precisely, Bedau defines a macrostate to be weakly emergent from the microscopic dynamics iff it can be derived but only by simulation. Clearly, this lack of derivability is not one of principle; rather, it is a practical one, since a way to derive the behaviour does exist: that is, simulation. But, according to Bedau, “the algorithmic effort for determining the system’s behaviour is roughly proportional to how far into the future the system’s behaviour is derived” (p. ). Thus, despite the in-principle availability of derivation, such a derivation is unreachable in practice for epistemic agents with limited resources (such as limited information about the initial conditions, computing power, etc.). Although in this chapter we are not concerned with simulation, the above example does illustrate well the distinction between weak and strong epistemic emergence that ⁶ A wave is typically characterised by its wavelength, λ. But by de Broglie’s relation between the momentum and the wavelength of the particle, viz. λ ¼ h=p, a wave can also be characterised by its momentum. This is the reason why we say that dependence on the momentum is a typical wave-like property since, against the background of de Broglie’s relation, it is also dependence on the wavelength. ⁷ We are being informal here, since the aim of the example is only to illustrate a duality that is analogous to the case weak/strong coupling duality. The notion of wave–particle duality of course depends not only on Heisenberg’s uncertainty principle, but also on whether the behaviour is “particle-like” or “wave-like”, on detection with a measurement apparatus.

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, ,  



we are interested in (in this respect, see also Guay and Sartenaer ). Strong epistemic emergence is the lack of derivability in principle—the theory simply lacks the resources to derive whatever is emergent from it. Weak epistemic emergence is the lack of derivability in practice—some derivations may be available, but they are difficult to carry out within the theory’s methods or resources, so that the situation is, in practice, as if one was dealing with strong emergence. We agree with much of the philosophical literature in thinking that epistemic emergence is a genuine form of emergence. In fact, we agree with Bedau () that weak emergence is not to be dismissed as merely “subjective” or as referring only to human factors, since it stems from the complexity of the theoretical description, which is an objective feature of the theory. In particular, we assume that epistemic emergence has to do with the limited applicability of a theoretical description, given its own techniques and the description of the world that it gives. Indeed, the cases of weak epistemic emergence that we will consider here will be such that, once a theoretical description ceases to be applicable because it loses its practical predictive power, a novel description emerges through duality that does have that predictive power and applicability. More precisely, the fact that these cases of emergence are weak depends on the assumption that we have exact dualities between the theoretical descriptions: then, the limitations are indeed practical rather than a matter of principle, stemming from the fact that the specific theoretical description can only be (easily) used in a limited domain. But since the duality is exact, the dual description could in principle be derived by using the duality, and then the two theories are equivalent when the full domain is considered: it is only that it is very difficult to do so in practice. Our examples from Section ... are of this type. However, there are cases in which a proof that the duality is exact is not currently known, and even the theoretical descriptions are only approximations covering their own limited domain of phenomena, so that the two descriptions could be only approximately dual. In this case, if it turns out that the duality is approximate— that is, if it turns out that the two descriptions are not exact duals of each other, and there is no better description that renders them dual—then epistemic emergence is strong, since derivability ultimately fails. Thus a verdict whether duality is weak or strong depends on delivering a proof of the duality, and on having theoretical descriptions that are not approximations—a situation that is rarely met in physics.⁸ In fact, we proceed on the plausible assumption that these dualities will one day be proven, so that epistemic emergence is weak. So far we have talked about emergence of theoretical descriptions through duality. This indeed covers all the cases we will discuss (see Sections ..., ..., and ..). However, as we will see, the process of finding a new, dual, description often begins with identifying new, emergent quantities, so that a problem that looked intractable in the original description, becomes the starting point for the new description, where it is tractable. Typically, these new quantities represent a new

⁸ For cases with a proof of duality, see De Haro and Butterfield (). The examples in Section .. are cases of dualities the existence of which is well-supported by the evidence, but which have not been proven. For a discussion of the evidence, see De Haro et al. ().

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

     

class of particles; however, recall that we are not committed to the ontological status of these particles. We now turn to discuss fundamentality. It is useful to keep in mind that weak emergence relates (what in physics jargon is usually called) the top and bottom theories or entities, where the bottom theory or entity is regarded as more fundamental, because (at least in principle) the top theory or entity can be derived from it. This one-way implication—the top theory can be derived from the bottom theory, but not the other way around—means that the bottom theory is more general than the top theory, and covers more cases, so that the bottom theory can make at least some predictions that in practice cannot be made using the top theory. There are various mechanisms for this, of which we here mention three that we will encounter in the examples: () A well-known example is coarse-graining, where the bottom theory is more fine-grained, so that it describes the domain of application in more detail. The top theory is more coarse-grained, it describes the domain in less detail, and so its applicability and its accuracy are reduced, relative to the bottom theory. A well-known example is the relation between statistical mechanics and thermodynamics, where the former describes the fine-grained microdynamics, while the latter can be derived from it (at least in important classes of examples) by using appropriate bridge laws (and this possibility to derive one from the other is of course regardless of possible emergence). () Another example is provided by weak/strong dualities inthe framework of perturbative quantum field theory. The perturbative Feynman diagram techniques of quantum field theory do not work at strong coupling, and there are no general and systematic methods for dealing with strong coupling situations in quantum field theory. This is why the presence of a weak/strong coupling duality—which allows us to transform the strong-coupling problem into a weak-coupling problem—is a significant theoretical advantage, and forms the starting point of a new theoretical description that is more fundamental in that regime, in the sense we have defined in Section ..— it is a theoretical description with predictive power, while the original theoretical description has no predictive power. But this notion of fundamentality is relative, as we will see, i.e. it depends on the domain to which the theory is applied. () In some cases, rather than varying a continuous parameter (or in combination with it) the duality involves a (not necessarily invertible) change of variables, whose definition can contain an infinite number of terms. Again, such infinite redefinitions allow us to change a theoretical description that is practically intractable into a tractable one.

.. Three Options for Emergence In ()–() in the previous section, we considered changing either a single parameter (the scale, or the coupling, in ()–()), or variables (). But these options can be combined. In that case, it is not a priori clear what emerges from what, and so what is the fundamentality relation. Our discussion of (), for example, assumed the

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, ,  



direction of coarse-graining as the emergent one, and our discussion of () assumed that, in the variation of the coupling strength, the theories that we find at the endpoints of the variation are the relevant bottom and top theories. But what happens if we combine options () and (), or combine options () and (), etc., so that we have “different directions”? We distinguish three different options for combining (), (), and (): Option (i). This is the option usually endorsed in the philosophy of science literature in connection with reduction: a finer-grained theoretical description is viewed as more fundamental than a coarser-grained theoretical description. This is the sense according to which, for example, particle physics is considered a more fundamental description (because it is more “fine-grained”) of the physical world than condensed matter physics, condensed matter physics more fundamental than chemistry, chemistry more fundamental than biology, and so on. In particle physics, in particular, the physical description and its degree of fundamentality is interpreted as related to a physical scale. Accordingly, the fact that at physical scale ranges we can have remarkably different physics has found an explicit realization in the so-called effective field theory approach. In general terms, an “effective field theory” (EFT) is a theory which “effectively” captures what is physically relevant in a given domain. More precisely, it is a convenient, appropriate description of the relevant physics in a given region of the parameter space of the physical world.⁹ A key point in the EFT approach is the separation of the physics at the chosen energy scale from the physics at much higher energies: an EFT describes the physics relevant at a given regime and this low-energy description is largely independent of the high-energy theory. In this sense one can say that the low-energy theory is emergent with respect to the high-energy one. In fact, both the decoupling between what happens at a high-energies and what happens at a low-energies, and the emergence of new properties and behaviours at different energy ranges, were used by the Nobel Prize P. W. Anderson, in his  seminal article entitled “More Is Different”, for arguing against the view that highenergy physics is more fundamental than condensed matter physics (i.e. arguing against (i)). Option (ii). This is the option endorsed by those who argue in the following way (with respect to the direction of emergence and its use in defining fundamentality): because of the facts of emergence and decoupling, unlike option (i), fundamentality is not to be based on the position on the energy scale, but it is to be discussed at the same scale, i.e. each and every scale, considered on its own. Option (iii). This option is a bit unusual, in the sense that according to it emergence and fundamentality appear as disconnected. As we will see, it concerns cases in which what emerges in the dual description plays a fundamental role in the new dual description. To illustrate how this is possible, we will discuss two case studies, both involving weak/strong coupling duality. In particular, we will show

⁹ There is a lively philosophical discussion on EFTs: a seminal reference is Georgi (, p. ). On the concept of a parameter space, see Section .. below.

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

     

how this kind of duality can provide interesting examples of weak epistemic emergence.

. Two Case Studies In this section, we introduce the two case studies that we will use to analyse the bearing of duality on fundamentality and emergence: namely, “generalized electricmagnetic duality” (Section ..) and “gauge-gravity duality” (Section ..). Since these case studies are examples of weak/strong coupling duality, we will start with briefly introducing, in Section .., this kind of duality in the framework of perturbation theory in quantum field theory. Let us note, at this point, that in addition to the emergence of theories, we will also consider emergence of “effective entities”: these are not the entities in the theory’s domain of application, but rather variables of a theoretical description that are interpreted as effective entities that appear in the theory’s domain of application when a certain approximation is made. The idea is that these effective entities are features of a theoretical description, with various particle-like or object-like behaviours. Thus, in the rest of the chapter, our ‘effective entities’ should be understood in this way—they are useful features of a theoretical description with various particlelike or field-like properties. Like our use of ‘effective entity’, which is epistemic, also our use of names such as ‘particle’, ‘elementary’, and ‘composite’ is epistemic. Thus our use of these words does not commit us to their existence as entities in the theory’s domain of application. Rather, it reflects the effective entity’s possession of particle-like properties, etc., as described by a specific theory.

.. Weak/Strong Coupling Duality and Perturbation Theory Weak/strong coupling duality has become a basic ingredient in fundamental physics, especially since the s. In general terms, it is a duality such that the weak coupling regime of one theory is mapped to the strong coupling regime of the other theory. The special interest in this form of duality stems from the fact that it is seen as a new tool for getting information on physical quantities in the case of large values of the coupling constant, where the usual perturbative methods fail,¹⁰ by exploiting the results obtained in the weak coupling regime of the dual description. Let us unpack some of the notions used above, especially: ‘couplings’ (or coupling parameters), and ‘perturbation theory’. A coupling is, roughly, a parameter characterising the strength of a force. Thus Newton’s constant, GN , is the coupling parameter of the gravitational force, and the spring constant, k, is the parameter ¹⁰ “Failure of perturbative methods” here means that the expansion in Section .. (below) does not converge, because g is not small (in a weaker sense, it means that one has to take into account an infinite number of terms in this expansion, which in practice is often impossible to do). This makes dualities particularly interesting and useful in the context of quantum field theory and string theory, since we usually know only the perturbative part of a theory, that is its ‘weak coupling’ regime. Dualities thus can be used to relate what is still unknown to what is known.

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, ,  



characterising the strength of Hooke’s law, viz. the coupling of the spring force. In Maxwell’s theory, the electric charge, e, plays the role of the coupling. In this respect, we will also consider another important parameter, Planck’s constant ℏ;¹¹ it is the dimensionful parameter that typically indicates the importance of quantum effects (in other words, quantum effects are large or small compared to this parameter). Although Planck’s constant is strictly speaking not a coupling constant in the way just described (it does not characterise the strength of a force, but rather the importance of quantum effects), we will see that it plays a similar role as the coupling constants do. Let us write the coupling constant as g, for whatever force is present in the problem. Since the coupling constant characterises the strength of the force, an expansion of the physical quantities around the point g ¼ 0 is an expansion subject to the assumption that the force is weak, and so that the interactions are small: QðgÞ ¼ Qð0Þ þ gQ1 þ g 2 Q2 þ . . .

ð11:2Þ

where QðgÞ is the quantity of interest, as a function of the coupling. The above expansion is called the ‘perturbative expansion’ of the theory, i.e. it is an asymptotic expansion for small interactions, or weak coupling.¹² In quantum field theories, where the interactions are of a quantum mechanical nature from the start, the above expansion turns out to coincide with the expansion in ℏ, as we will discuss in Section .... So, the first term is the classical contribution, and the sub-leading terms are quantum corrections.¹³ An important ingredient of quantum field theories is the so-called ‘flow of the couplings’. Namely, unlike ordinary quantum mechanics where the coupling g is a constant, the coupling in quantum field theory is a function of the momentum, k, i.e. g = gðkÞ, where k is like energy. This has to do with the effects of renormalization, that is the basic fact that—due to the infinite number of particles that are assumed to be present in quantum field theory—the self-interactions and mutual interactions of fields give rise to new terms that have to be taken into account in the interactions of the theory.¹⁴ In fact, the coupling constant gðkÞ satisfies an equation, the renormalization group equation, which fixes the dependence of the coupling on k. This equation describes the ‘flow’ of the coupling constants (if there are more than one) in their parameter space. We will get back to this notion in Section ... We now turn to illustrating our two case studies.

¹¹ The constant ℏ = h=2π, where h is Planck’s constant, is called the reduced Planck constant. For simplicity, we will continue to call it Planck’s constant. ¹² We will not enter here into the details of whether this expansion converges. This is obviously an important issue. However, in theories with dualities it is usually a good assumption (modulo technicalities), because the duality ensures that the regimes of both small and large g are under control. In those cases, the difficulty will be not the convergence, but the fact that one needs to take into account an infinite number of terms (see note ). ¹³ This coincides with the celebrated Feynman diagram expansion, which may be familiar to some readers. ¹⁴ For a philosophical review, see Butterfield and Bouatta (); for a brief discussion, close to our second case study where we will discuss renormalization, see Dieks et al. (, p. ).

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

     

.. Generalized Electric-Magnetic Duality Electric-magnetic duality (EM duality) represents the first form of duality to be explicitly applied in twentieth-century fundamental physics. The idea that there is a substantial symmetry between electricity and magnetism is an old one, dating back to the nineteenth century where it played a role in Faraday’s discovery of electromagnetic induction and was first made more precise with Maxwell’s formulation of his famous equations regulating the behaviour of electric and magnetic fields. In its contemporary form, its origin and first developments are due to P.A.M. Dirac’s famous papers (Dirac  and Dirac ) on his “theory of magnetic poles”. In fact, the very idea of weak/strong duality stems from Dirac’s seminal work and its successive generalizations in the context of field and string theory. From the viewpoint of the issue at stake here—namely the significance of duality to the discussion of fundamentality and emergence—EM duality in its generalised form is particularly interesting because of the following novel feature: that is, that the weak/strong coupling nature of the duality manifests itself in the fact that under EM duality it often happens that what is viewed as ‘elementary’ in one description gets mapped to what is viewed as ‘composite’ in the dual. At first sight, this interchange between what is ‘fundamental’ and what is ‘composite’ could be taken to suggest an ontological, relative notion of fundamentality. But this reading is too quick. Actually, what this case seems to best suggest is a form of epistemic relative fundamentality or “representational fundamentality” (as argued in Castellani ). In what follows, we will enter into some details of the generalized EM duality case study, in order to identify those specific features that illustrate the option (iii) (mentioned in Section ..), in the relationship between duality, emergence and fundamentality. We will structure this brief overview of the main features of the EM duality according to its actual historical development. In Section ..., we discuss the classical formulation of EM duality in the context of Maxwell’s electromagnetic theory, turning in Section ... to its extension to the quantum context with Dirac’s “Theory of Magnetic Poles”. Then, in Section ..., we discuss the generalization of EM duality within the framework of quantum field theory.

... -     In Maxwell’s theory, there is an evident similarity in the role of electric and magnetic fields. This similarity is complete in the absence of source terms (electric charges and currents), and this is mathematically expressed by the fact that Maxwell’s equations do not change in form when the roles of the electric field E and the magnetic field B are exchanged in the following way: D : E ! B; B !  E: The transformation D is a duality transformation, which leaves Maxwell’s equations invariant.¹⁵

¹⁵ Since the transformation is on the same theory, this is a case of self-duality, i.e. a symmetry.

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, ,  



When electric source terms are present, however, the Maxwell equations are no longer invariant under the duality transformation, D. In order to restore the duality of the theory in the presence of source terms, one needs to postulate the existence of magnetic charges beside electric charges and, accordingly, to modify Maxwell’s equations. In their new form, these equations are then invariant under the duality 0 transformation D , which at the same time exchanges the roles of the electric and magnetic fields, and of the electric and magnetic sources, as follows: 0

D : E ! B; ðe; je Þ ! ðg; jg Þ;

B! E

ð11:3Þ

ðg; jg Þ ! ðe;  je Þ

Here, ðe; je Þ represents the electric charge and electric current, and ðg; jg Þ the magnetic charge and magnetic current.¹⁶ There is a problem, however: isolated magnetic charges, i.e. the so-called magnetic monopoles, have never been observed. Breaking a magnet bar in two parts, one obtains two smaller magnets but never an isolated North pole or an isolated South pole. Assuming, nevertheless, the existence of magnetic charges in order to save the duality between electricity and magnetism, leaves this question to be addressed. In fact, the extension of EM duality to the quantum context, as we will see below, allowed Dirac to give the following answer: isolated magnetic poles have never been observed because an enormous amount of energy is needed to produce a particle with a single magnetic pole.

...      The extension of EM duality to the quantum context was carried out by Dirac in the two papers (, ) in which he developed his theory of magnetic monopoles. In these papers, Dirac proved that it is possible for a magnetic charge, g, to occur in the presence of an electric charge, e, without disturbing the consistency of the coupling of electromagnetism to quantum mechanics.¹⁷ As Dirac proved, the condition for this to be possible, known as Dirac’s quantization condition, is the following one: eg ¼ 2πnℏcn ¼ 0;  1;  2; . . .

ð11:4Þ

where c is the speed of light. Dirac’s condition thus established the existence of an inverse relation between electric and magnetic charge values, with many relevant consequences.¹⁸ In particular, from the viewpoint of interest here, this condition ¹⁶ For more detail on this and the next subsection, we refer the reader to Castellani (, ). ¹⁷ Turning from the classical to the quantum formulation of electromagnetic theory with magnetic sources posed a consistency problem: the electromagnetic vector potential A, playing a central role in coupling electromagnetism to quantum mechanics, is introduced in standard electromagnetism by taking advantage of the absence of magnetic source terms. ¹⁸ First, as said, it provided an explanation of why isolated magnetic poles had never been observed. Second, it explained the quantization of the electric charge: the mere existence of a magnetic charge, g somewhere in the universe would have implied the quantization of electric charge, since any electric charge should occur in integer multiples of the unit.

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

     

provided the basis for the idea of weak/strong coupling duality. Indeed, by combining Dirac’s condition with the fact that EM duality interchanges the roles of electric and magnetic charges, as above (i.e. combining Equations (.) and (.), we obtain the following inverse relations: e!g¼

2πnℏc e

g ! e¼ 

2πnℏc : g

This means that, if the charge e is small (i.e. weak coupling), the dual charge g is strong (strong coupling), and vice versa: in other words, in a quantum context EM duality relates weak and strong coupling. However, it is only with the generalization of EM duality to the framework of quantum field theory that the idea of weak/strong coupling duality started to acquire its modern meaning and fruitfulness. Thus, we now turn to this decisive step in the history of EM duality, with a particular focus on the related interchanging role of ‘elementary’ and ‘composite’ between the entities in the dual descriptions.

... -/   -  Historically, the seminal contribution for the generalization of EM duality to the quantum field theories of particle physics was the  work by Montonen and Olive, entitled “Magnetic monopoles as gauge particles?”, where they formulated their celebrated EM duality conjecture: that is, in their own words, the conjecture that “there should be two ‘dual equivalent’ field formulations of the same theory in which electric (Noether) and magnetic (topological) quantum numbers exchange roles” (Montonen and Olive , p. ). In order to understand the physical implications of this conjecture, let us take a step back and mention a previous result: namely, the duality between the so-called sine-Gordon theory and massive Thirring model.¹⁹ This duality, which was firmly established by works of S. Coleman and S. Mandelstam in the mid-s, originated from pioneering contributions by T.H.R. Skyrme between the end of the s and the beginning of the s. Let us mention two things in particular: (a) his pioneering idea that a soliton could be interpreted as a quantum particle,²⁰ and that a dual correspondence could be established between this sort of particle—which is extended, and therefore not considered as elementary—and the familiar elementary particles of quantum field theory; (b) his conjecture that the nucleons (spin / fermionic states) could emerge as the soliton states of a purely bosonic field theory. ¹⁹ These are two field theories in one space and one time dimension, describing, respectively, a massless scalar field ψ (with interaction density proportional to cosβψ) and a massive self-coupled fermionic field. See Castellani (, section ..). ²⁰ Solitons are extended solutions of classical non-linear field equations, so called by Zabusky and Kruskal () to indicate humps of energy propagating and interacting without distortion. They were first discovered in nineteenth century hydrodynamics in the form of ‘solitary water waves’, whence their name.

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, ,  



Skyrme’s conjecture was confirmed in  by Coleman and Mandelstam’s work proving the dual equivalence of the sine-Gordon and massive Thirring models in general terms. From the viewpoint of this chapter, we will focus on the following results: (a) The equivalence was proven to be a weak/strong coupling duality: the weak coupling regime of the sine-Gordon fields corresponds to the strong coupling regime of the massive Thirring model, and vice-versa. (b) This duality implies, in particular, a precise correspondence between the soliton states of the quantized sine-Gordon theory and the elementary particle states of the dual massive Thirring model. In other words: by means of the weak/strong coupling duality, the sine-Gordon quantum soliton was proven to be a particle (the “elementary” fermion of the massive Thirring model) in the usual sense of the concept in particle physics.²¹ Thus, in the full quantum theory, particles could appear as solitons or as elementary particles, depending on the way the theory was formulated (whether as the quantum sineGordon model or as the massive Thirring model): their status was equivalent. Coleman (, p. ) famously commented on this fact in terms of a situation of democracy among the particles: “Thus, I am led to conjecture a form of duality, or nuclear democracy in the sense of Chew, for this two-dimensional theory.”²² As we already mentioned in Section .., in discussing the present case study we will take the notion of fundamentality to be based on the ‘elementary vs. composite’ distinction: a particle in a given theoretical description is considered to be more fundamental if it is elementary, and less fundamental if it is composite (or extended). The exact equivalence between the two theories, i.e. result (b) above, is worth stressing. For it means that the fermionic state of the Thirring model is already there in the sine-Gordon theory, and vice-versa: a bosonic state of the sine-Gordon theory is already there in the massive Thirring model.²³ In this sense, there is no ontological emergence, because the two theories describe exactly the same states, quantities, and dynamics (though, as we will argue later, there is epistemic emergence). Sine-Gordon/Thirring duality was the first explicit example of a weak/strong duality with a corresponding dual interchange of elementary particles and solitonic particles in the framework of a quantum field theory. It was therefore natural to try to extend these ideas to the more realistic case of a physical space-time of three space and one time dimensions. This was proposed by Montonen and Olive in their  work, in terms of a generalization of Dirac’s EM duality in the context of a unified quantum field theory of weak and electromagnetic interactions.

²¹ That is, structureless particles arising from the quantization of the wave-like excitations of the fields. ²² On the idea of nuclear democracy in Chew’s S-Matrix approach in the s, according to which no hadron was more fundamental than the other, see in particular Cushing (). Comments on this can be found in Castellani (, section .). ²³ The fermionic state is non-perturbative (i.e. not visible at weak coupling) in the sine-Gordon theory; as is the bosonic state not visible in the weakly coupled Thirring model. But the states are there nevertheless. For more details, see De Haro and Butterfield (, section ..).

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

     

Just as for Dirac’s theory, the duality considered by Montonen and Olive is a case of self-duality: the very same theory has two equivalent dual descriptions. What is of particular interest, here, is the kind of situation that this generalized EM duality represents: a quantum field theory describing both particles with “electric” charge e, and particles with “magnetic” charge g,²⁴ which can have two different classical limits, depending on which coupling (charge)—e or g—is kept fixed while taking the classical limit ℏ ! 0.²⁵ Accordingly, there are two possible scenarios, corresponding to the two dual descriptions of the same quantum theory: () If the “magnetic” coupling g is kept fixed, then, from Equation (.), the classical limit, ℏ ! 0, corresponds to weak electric coupling, viz. e ! 0: in this case, the electrically charged particles play the role of elementary particles, and the magnetically charged particles of solitons. () If the “electric” coupling e is kept fixed, then, from Equation (.), the classical limit, ℏ ! 0, corresponds to weak magnetic coupling, viz. g ! 0: in this case, the magnetically charged particles play the role of elementary particles, and the electrically charged particles of solitons. The particles, whether electrically or magnetically charged, are all present in the complete quantum theory. In this sense, they all are equally “fundamental”, from an ontological point of view. What the duality implies, however, has rather to do with their different modes of appearance when considering the different classical limits of the quantum theory (the dual perspectives). They interchangeably play the role of “elementary” (i.e. “fundamental”) or “solitonic” particles, depending on the perspective under which the theory is considered. At this point, what conclusions can be drawn with respect to fundamentality and emergence in the light of the cases of weak/strong duality in QFT just illustrated? As seen, these are both cases in which a weak/strong coupling duality is accompanied by an interchange of the ‘elementarity’ and ‘compositeness’ of the particles in the quantum theory. In short, we can make the following considerations, at this stage: (A) In the case of sine-Gordon/massive Thirring model duality, we have two different quantum theories—a bosonic field theory vs. a massive fermionic theory—which are related by a weak/strong coupling duality, such that an elementary particle in one theory becomes a soliton state in the other. Regarding fundamentality: as said above, the effective entities of the two theories are ontologically equally fundamental, since all the states and operators (for both

²⁴ See for example Sen (, Section ); Polchinski (, p. ). Electric and magnetic are here to be intended in a generalized sense. For a more detailed treatment, we refer to Castellani (, section ..). ²⁵ Planck’s constant ℏ is of course a dimensionful constant of nature, and we cannot change its value. What we have in mind here is that we consider a sequence of semi-classical solutions of the theory, with successively larger values of the typical action in the solution (in comparison with ℏ) while we keep the couplings (including ℏ) fixed. Taking pffiffiffi the ℏ ! 0 thus involves comparing different physical systems. This is the case also if one takes e.g. e= ℏ as one’s expansion parameter. Also, notice that in the quantum field theory literature, e is not measured in Coulombs, pffiffiffibecause it has been divided by the square root of the vacuum permittivity. This is the reason why e= ℏ can be taken to be a dimensionless parameter, and electric and magnetic charges can be related through Equation (.).

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, ,  



particles and solitons) are already there in the two theories. Therefore, there is no ontological emergence, as already discussed. However, we can also look at the different roles that the bosonic and fermionic particles play in the two dual descriptions: in one description, the bosonic particles are the elementary particles, while the fermions only emerge as solitons in the high-energy limit. In the dual description, it is the reverse. In other words: being elementary or being a soliton is a matter of the convenience of the description, i.e. it depends on the specific fields one is working with, and the relation between the two pictures is like a (admittedly, very complicated) change of variables. Thus, this kind of emergence is only weakly epistemic. If we take the elementary particles to be more fundamental than the composite ones in a given description, then the notion of fundamentality is relative to that description, and—like emergence—fundamentality in this example is relative, from an epistemic point of view. (B) In the case of generalized EM duality (Montonen-Olive duality), the difference with the previous case is that emergence takes place within the same theoretical context, since it is a case of self-duality (i.e. the duality map does not take us out of the theory). Nevertheless, we can reach the same conclusions as in (A). That is, in one description the electric particles are elementary (and the magnetic particles are then solitons or “composite”), while in the other description it is the opposite.²⁶

.. Gauge–Gravity Duality Around , the discovery of string dualities and of D-branes (which are extended, non-perturbative objects in string theory) motivated the idea of the existence of a relation between gravity theories and gauge theories. The microscopic entropy counting of Strominger and Vafa () for extremal black holes, which is seen as one of the successes of string theory, vindicated this relation between gauge theory and gravity: since the entropy of a black hole (the gravitational object par excellence) is calculated by counting microstates in an associated gauge theory. In , Maldacena took this relationship a step further, by relating string theory in anti-de Sitter space, or AdS (i.e. a space with a negative cosmological constant)²⁷ to a quantum field theory (QFT) which is scale invariant, i.e. a gauge theory.²⁸ This is called ‘gauge-gravity duality’.

...  /   Gauge-gravity duality is a case of weak/strong coupling duality similar to the ones described above, since when the coupling of the bulk gravity theory is weak (viz. far away from the centre of the bulk) the gauge theory is strongly coupled. This can be seen as follows: both theories have two parameters in terms of which one can do a perturbative expansion (recall Section ..). In the gravity theory, we have ²⁶ For more detail on this case and the successive extension of the idea of generalized EM duality to the context of string theory, see Castellani (, section ). ²⁷ The space is actually only required to be asymptotically, locally AdS, rather than pure AdS. ²⁸ The QFT does not actually need to be exactly scale invariant. It is sufficient that it has a conformal fixed point.

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

     

Newton’s constant, GN , and the radius of curvature of the AdS space, l (Newton’s constant is proportional to the string length, α0 ).²⁹ On the other hand, in the gauge theory we have the coupling constant, g (which determines the strength of the interactions), and the rank of the gauge group, N (the number of colours in the theory; for Quantum Chronodynamics this would be N ¼). These parameters are related between the two theories as follows (see De Haro et al. , section ..): GN π 3 ¼ 2N 2 ℓ

ð11:5Þ

ℓ4 ¼ g 2 N; α0 2

ð11:6Þ

where the parameters on the left-hand side are those of the gravitational theory, and the parameters on the right-hand side are those of the gauge theory. Gravity is weak if Newton’s constant, GN , is much smaller than the radius of curvature of the AdS space, l, so that Gl3N  1. Equation (.) above then implies that the number of colours has to be large, i.e. N  1. Also, quantum corrections will be suppressed if the radius of curvature of the space is much larger than the string length, l4  1 (so that the effects of the finite string length cannot be seen, and we basically deal α0 2 with a point-particle theory), so that Equation (.) gives g 2 N  1. Now it was argued by ‘t Hooft () that, in a gauge theory with N colours, the natural coupling constant is not g, but rather the combination g 2 N. In other words, when g 2 N  1 perturbation theory in the gauge theory breaks down, because the theory is strongly coupled. This is why the weak-gravity, semi-classical regime of the string theory corresponds to a strongly coupled gauge theory. The converse of this statement is of course also true: if the gauge theory is taken to be weakly coupled, so that g 2 N  1, then the semi-classical (gravity) approximation to the string theory cannot be trusted, because an infinite tower of string corrections will give non-zero contributions. Thus, the way gauge-gravity duality is a weak/coupling duality is similar, to some extent, to the example of electric-magnetic duality, in that the coupling constants of the two theories are inversely proportional to each other. However, gauge–gravity duality brings in a new element: the coupling constants do not have fixed values on the two sides, but can vary according to the details of a specific physical situation.³⁰ Let us just note, here, that the gauge theory coupling is a function of momentum, gðkÞ, while the string theory coupling is a function of the position in the AdS space. The region in which the gravity approximation is valid (i.e. the region where the gravity coupling is weak) is the region far away from the centre of the AdS space (if there is e.g. a black hole at the centre of AdS, the curvature will be strong). So, weak coupling requires large distances, far away from the centre, on the gravity side. But, as

²⁹ These parameters can be written alternatively in string theory language, in terms of the string length (squared), α0 , and the string coupling, gs . The string length determines how a string differs from a point particle, and the string coupling determines the perturbative expansion of the string theory. The expressions given in Equations (.) and (.) are for a five-dimensional gravity theory, which is the original example considered by Maldacena (). ³⁰ For more details, see Dieks et al. (, p. ).

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, ,   Strongly-coupled gravity theory

Duality

Weakly-coupled gauge theory

Renormalization group

Bulk to boundary

String theory



Duality

Gauge theory

Figure . Duality relations vs. renormalization group flow.

we saw above, weak gravity coupling is dual to strong coupling in the gauge theory, which happens at high momenta (and hence high energies). In short: large distances (weak coupling) in the gravity theory correspond to high energies, i.e. small distances (strong coupling) in the dual gauge theory, and viceversa. Thus, this is analogous to, although distinct from, the Fourier transformation example discussed in Section ... One new element of gauge–gravity duality is the fact that the motion from the boundary towards the centre of the space, in the gravity theory, increases the gravitational coupling of the theory because the curvature radius increases. The dual of this inward motion, in the gauge theory, is motion from the UV towards the IR, i.e. towards low energies. This ‘motion’, which is interpreted in terms of spatial variation in the gravity theory but in terms of energy variation in the gauge theory, is called the ‘renormalization group flow’ of a gauge theory.

...     –  Let us now take stock of what we said so far. Recall, from Section .., that the top and bottom theories are distinguished by the value of a parameter. In the gauge theory, the parameter is the momentum scale, which is dual to the radial direction in the gravity theory. Thus, we naturally get the diagram in Figure ., where the vertical direction corresponds to the ‘motion’ discussed above (the RG flow), while the horizontal direction is the duality map. In Figure ., we have a horizontal relation between two theories which are dual, where the duality inverts the values of the couplings, according to Equations (.)–(.). But, in addition, we also have a vertical direction, which corresponds to the ‘motion’ in coupling space: radial motion on the gravity side, vs. the renormalization group flow on the gauge theory side. At each value of the coupling in the vertical direction, we have a pair of theories that are dual to each other. Thus, as illustrated in Figure ., we have two directions in which to consider the emergence and fundamentality issues: viz. the vertical and the horizontal directions. Indeed, the theories on Figure .’s bottom are the exact, non-perturbative theories, and (under the assumption of an exact gauge-gravity duality) they are dual to each other. These are the fundamental theories (assuming fundamentality in the sense specified in Section ..), and the top-row theories are effective theories, and therefore less fundamental than these. Thus fundamentality increases in the

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

      Strongly-coupled quantum gravity theory

Emergence Weakly-coupled gauge theory Fundamentality

Increase coupling / go to smaller distances

Weakly-coupled gravity theory (general relativity)

Increase ‘t Hooft coupling

Emergence Strongly-coupled gauge theory Fundamentality

Figure . Emergence and fundamentality.

downward vertical direction, but not in the horizontal direction. Emergence in the vertical direction is discussed in detail in Dieks et al. (, p. ) and De Haro (, pp. –), with the conclusion that there is ontological emergence in this direction. For example, on the gravity side we have Einstein’s theory of general relativity with specific matter fields emerging in the low-energy limit of the underlying string theory. So far, we have the ordinary situation for effective field theories— that is, option (i). To get option (iii), we need to change the picture slightly, so that the theories on the bottom row are the weakly coupled string theory (i.e. the semi-classical gravity theory, general relativity) and the strongly coupled gauge theory which is dual to it (see Figure .). In that case, the duality relation relates a weakly coupled theory to a strongly coupled theory. For this duality, there is a slight difficulty in identifying what is composite/solitonic and what is elementary, because we lack good descriptions of the strongly coupled theories. Nevertheless, we can still identify the weakly coupled theory as the more fundamental description, in the epistemic sense discussed in Sections . and .., that it is the description in which calculations can be done reliably: and we can identify its strongly coupled dual as the less fundamental one, since that description is out of control, when the coupling is weak in the other theory.³¹ Let us now examine epistemic emergence in the horizontal direction, i.e. along the duality. The question is whether the duality considered here can give rise to emergence. Notice that regarding ontological emergence, Dieks et al. (, p. ) and De Haro (, p. ) concluded that there cannot be any, because the two theories are exactly dual (i.e. equivalent), and therefore there can be no novelty, and so no emergence of one theory from the other.³² But this verdict can be modified when we consider emergence in the epistemic sense, i.e. as irreducibility or novelty of description, and fundamentality not as a ³¹ It is also very likely that there is a story about what is elementary and what is component in each description, as in the cases discussed in Section ..: but we will set this issue aside. ³² This verdict is subject to a specific interpretation, namely a so-called internal interpretation. Notice that from the mere presence of a duality, which is a formal relation, one cannot make a verdict about ontological emergence. To that end, one needs to consider the interpretation of the two theories, which in Dieks et al. () and De Haro () was done for internal interpretations.

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, ,  



property of the full theory, but as a property of the particular theoretical description. In this sense, one can indeed say that the weakly coupled gravity description emerges from the strongly coupled gauge theory. Namely, imagine that one is working within the strongly coupled gauge theory, so that one is in a highly quantum regime where the usual perturbative description breaks down, and is unable to make any predictions. And assume that one then stumbles upon the duality, which points to a useful change of variables (in this case, an exceedingly complicated—presumably infinitely long—change of variables!), that allows one to reformulate the theory as a semiclassical gravity theory. That is, thanks to the duality it becomes possible to identify a set of variables that are more fundamental in this regime since, unlike the old variables, they give a good description, and can be the starting point for arriving at the gravity theory. The interpretation of these variables is not at all in terms of quantum fields, but in terms of semi-classical gravity. In this case, the gravity theory (and the objects within it) is indeed epistemically emergent. Note that emergence here involves not just the change of variables (which is the mechanism () for emergence in Section ..), but also a weak/coupling duality (mechanism ()), and that this change of variables involves coarse-graining (mechanism ()).³³

. Discussion and Conclusions The initial question motivating our contribution was the way in which the notions of fundamentality, emergence and duality can be intertwined, and how this connection can shed new light on fundamentality. More precisely, the novel feature is based on duality: how dualities are applied in contemporary physics, in particular the weak/ string coupling duality, and the implications of this kind of duality for the philosophical discussion of fundamentality. Our starting point was to consider the three ways, (i) to (iii), of considering the relation between fundamentality and emergence. While (i) and (ii) are commonly discussed in the literature on fundamentality in physics, duality suggests (iii) as a new way to construe the relation between emergence and fundamentality. We illustrated this in the two cases taken from quantum field theory (sine-Gordon/massive Thirring duality, and generalized EM duality) and in the case of gauge–gravity duality. The conclusion was similar for all these cases: the physical effective entities and the theories, as they are described, can play a more or less fundamental role, depending on the description chosen. And as such, there is emergence of a more fundamental description (because more elementary, in the sense specified in Sections .. and ..) out of a strongly coupled description.

³³ Emergence through a change of variables has also been considered recently by Franklin and Knox (), who define emergence in terms of ‘novel explanation’. According to Franklin and Knox, changes of variables can give novel explanations of phenomena, because using the appropriate variables allows one to abstract away from irrelevant details and capture the salient explanatory elements. Although epistemic emergence is for us a matter of lack of derivability and therefore of descriptive, rather than explanatory, novelty, the role that changes of variables play in Franklin and Knox’s account is similar to ours (another important difference is that, because of the weak/strong coupling dualities in this chapter, our changes of variables often contain infinitely many terms, as well as coarse-graining).

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

     

Notice that the direction of emergence is here opposite to that of fundamentality. Ordinarily, it is the non-elementary effective entities that emerge at lower energy. Note that this ‘inversion of the direction’ is possible because our notions of fundamentality and emergence are epistemic. What is fundamental in this sense is not fixed once and for all by the ontology, but depends upon the description. Thus a more fundamental description can emerge within a strongly coupled theory. And this is made possible by the fact that quantum field theories can have more than one classical limit. In general, we expect that each classical limit will have its own emergent effective entities. Let us underline that the kind of weak epistemic emergence found here is tightly connected with the notion of perturbation theory discussed in Section ... This is also the reason why we found no emergence in the cases of classical electromagnetism (Section ...) and Dirac’s quantization of it (Section ...), where there are no perturbative expansions or perturbative duality, but only an exact one. It is only in the more sophisticated cases of quantum field theories and string theories that we get sufficient complexity to allow this kind of emergence. As a final remark, let us recall that, in the example of gauge/gravity dualities, a spatial direction and the curvature of space emerge, together with the gravitational force, from a quantum field theory. This is of course an intriguing form of emergence of (at least one dimension of ) space, along with new topological and geometric structures of the spacetime. We hope that regarding this as a case of epistemic emergence can cast light on the emergence of spacetime in theories of quantum gravity more generally: namely, as the emergence of spacetime from a reformulation of a non-spatiotemporal theory.

Acknowledgements We thank Jeremy Butterfield and three referees for their comments. We thank the Vossius Center for History of Humanities and Sciences for its support, as well as the members of the Amsterdam Philosophy and History of Gravity group, for a discussion of the manuscript.

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12 Radical Structural Essentialism for the Spacetime Substantivalist Tomasz Bigaj

. Introduction The controversy regarding the nature of space and time is one of the oldest and most persistent in the history of philosophy. Are spatiotemporal objects, such as points and regions, fundamental entities not reducible to other ontological categories? Or, perhaps, they are mere constructs that owe their existence to more fundamental objects (things, events) and relations among them? In this chapter I will not attempt to survey possible scientific and philosophical arguments in this debate. Instead, I will adopt the broad position of spatiotemporal substantivalism with its central thesis of the fundamental and irreducible character of space-time, and from this perspective I will consider what further amendments to this doctrine are required in view of the well-known developments in modern physics. In particular, I will focus on the modifications that substantivalism has to undergo in the light of the challenge posed by the diffeomorphism-invariance of General Relativity. This challenge is typically presented in the form of the so-called hole argument. I will try to argue that the best response to the hole argument that the substantivalist can offer is to adopt the ontological stance I dub Radical Structural Essentialism (RSE). This doctrine is based on the assumption that fundamental objects by necessity exist as elements of a relational structure that determines their identities and therefore enables us to place these objects in alternative, possible scenarios. The appropriate structure represents essential features of these objects, i.e. features that are necessarily possessed by these objects (whose possession is basically what it means to be a given object). The radical character of the defended doctrine comes from the fact that under this view being part of a given essential structure is both necessary and sufficient for being this very object. No further identifications are required, and indeed permitted. Radical Structural Essentialism can be contrasted with other essentialist approaches to spacetime offered by various authors. I will point out some advantages that my preferred solution possesses in comparison with its direct competitors. Furthermore, I will focus on two important details that perhaps have not received sufficient attention in the literature. One question is whether the metric structure that is commonly accepted as representing the essence of spatiotemporal points and Tomasz Bigaj, Radical Structural Essentialism for the Spacetime Substantivalist In: The Foundation of Reality: Fundamentality, Space, and Time. Edited by: David Glick, George Darby, and Anna Marmodoro, Oxford University Press (2020). © Tomasz Bigaj. DOI: 10.1093/oso/9780198831501.003.0013

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

 

regions is in fact genuinely relational. I will survey the basic mathematical framework in which the metric properties of spacetime are expressed, and I will argue that the relational character of this framework comes from the underlying topological notions upon which the metrical features are founded. The second issue arises in connection with the general problem of what it means to identify a given essential structure in a possible scenario (a possible world). The standard criterion of identity is based on the notion of isomorphism, while I will argue that we need a weaker concept of embedding here. Thus I will develop a conception according to which the necessary and sufficient condition for actual objects to appear in a counterfactual scenario is that the essential structure containing these objects is embeddable in an alternative structure. One of the founding principles of General Relativity is general covariance, which roughly speaking reflects the assumption that the equations of this theory should remain the same under any arbitrary choice of coordinate systems.¹ Thus when we apply any smooth transformation of coordinates to a particular solution of the equations of GR, we will obtain another solution, physically indistinguishable from the original one. Changing the coordinates does not amount to any significant alteration—what we obtain as a result is a different description of the same reality. However, the very same mathematical procedure of transforming coordinates can be interpreted in a more substantial way as involving a change in the metric and physical properties of individual points (this is the essence of the so-called active interpretation of diffeomorphical transformations—see note  for a brief discussion). Here is where the notorious hole argument comes on the scene (see e.g. Norton  and Pooley  for overviews). A hole transformation moves the metric and physical fields inside a particular region, while leaving the outside region unchanged. This results in a peculiar form of indeterminism, as the physical situation outside a given region does not uniquely determine the assignment of properties to points inside this region. The hole argument does not affect views that deny the objective and independent character of spacetime. Relationism with respect to spacetime deals easily with the challenge, as it sees spatiotemporal points as mere ‘reflections’ of physical objects—fields, events etc.—and their spatiotemporal relations (e.g. colocation). However, it is commonly agreed that the hole argument creates a potential difficulty for substantivalism. It has become customary to defend the doctrine of substantivalism by delving deeper into the metaphysical underpinnings of the application of a hole transformation. The transformation itself is interpreted as taking us from the actual physical situation to a possible scenario, and this interpretation opens the door to employing the rich conceptual apparatus of the metaphysics of modality, including such notions as transworld identity, representation de re, and crucially the concept of essential properties. However, this metaphysical strategy to defend (and indeed formulate)

¹ The exact meaning of the requirement of general covariance and its role in the development of GR is a matter of debate (see Norton ; Pooley ). We don’t have to enter this controversial issue, as what we need here is a slightly less general and more precise concept of diffeomorphism invariance. I am grateful to Antonio Vassallo for pressing me on this point.

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

substantivalism has come under severe attack. Shamik Dasgupta () argues that the positions in the substantivalism–relationism debate, as well as the arguments used in the debate, should not be cashed out in modal terms. One reason for this assertion seems to be that wildly diverse metaphysical views on modality which have nothing to do with the central issue of the status of spacetime can nevertheless impact some arguments in the debate and thus muddle the distinction between the key positions there. An example illustrating this point involves the well-known Leibniz shift argument. It is commonly accepted that Newtonian substantivalists are committed to the view that it is metaphysically possible for the world to be located three feet away from its current location, while Leibnizian relationists deny such a possibility. And yet Dasgupta points out that we may remain substantivalists while rejecting the possibility of a Lebnizian shift on the basis of some unrelated choices made in the metaphysics of modality (such as accepting Spinozism which assumes that everything that is actual is also necessary). I agree that it would not be wise to make the ontological positions of substantivalism and relationism hostages to our preferred views on modality. However, in the current analysis I am interested primarily in the question of what modal characteristic of spacetime should be adopted in the light of the challenges to substantivalism such as the hole argument. Thus from my perspective it would be incorrect to approach the problem with already fixed views on modality. My intention instead is to rely on some form of abductive reasoning (or inference to the best explanation) in order to arrive at the best modal characteristic of spacetime for the substantivalist that can deal with the infamous hole argument and related problems.² Leaving this issue aside, I will proceed straight away to a brief characterization of the essentialist approach to the hole argument in GR. The plan of this chapter is as follows. In Section . I will identify and discuss three essentialist responses to the hole argument: Tim Maudlin’s metric essentialism, David Glick’s Minimal Structural Essentialism and my own variant of essentialism that I label Radical Structural Essentialism. I will compare these versions of essentialism by showing how they deal with some challenges arising in the context of space-time theories. Section . will discuss in detail why structures consisting of the metric tensor should be seen as relational. In Section . I will argue that the structural criterion of representation de re (or the counterpart relation) should be based not on the notion of isomorphism between structures, but rather on the weaker notion of embeddability. The chapter ends with some general remarks regarding the ontological interpretation of our best theories of space-time.

² Incidentally, I do not endorse the distinction made by Dasgupta between two ways of dealing with the hole argument: one based on so-called thin substantivalism with no modal component, and the other involving the assumption of thick substantivalism plus some ‘bare’ modal claims (such as essentialism or anti-haecceitism). The reason for my reservation is that I don’t fully understand the notion of grounding in terms of which Dasgupta distinguishes thin from thick versions of substantivalism. In particular, I don’t see how grounding can be defined in non-modal terms (especially, since it is assumed that grounds metaphysically necessitate what they ground), and for that reason I am not convinced that it is possible to separate bare modal claims from specific modal characteristics of spatiotemporal points and regions.

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

 

. Three Variants of Essentialism .. Maudlin’s Metric Essentialism The standards for an essentialist solution to the hole argument have been set by Tim Maudlin (Maudlin , ). His trailblazing approach to the metaphysical problem that the hole argument presents us with is based on the assumption that spatiotemporal points possess their metric properties (encompassed chiefly in the metric tensor, but see Section . for a more detailed discussion) essentially. That is, it is metaphysically impossible for a given point to have its metric properties different than those it actually possesses. This implies that when we interpret the posttransformation model d as describing a new possible world, image points dp will generally not be able to represent de re their counterparts p from the pre(barring cases in which d is an isometry).³ Maudlin takes transformation model this as showing that the entire model d is in fact metaphysically impossible. This does not follow from the essentialist thesis alone; we need to introduce an additional premise that points p and dp are to be identified by default.⁴ Once this assumption is represents a in place, we can see that indeed the post-transformation model d contradictory, i.e. impossible world (some of its points should but cannot be identical with their counterparts in ). Several comments can be made in regard to Maudlin’s proposal of how to dismantle the hole challenge.⁵ His variant of essentialism appears very close in spirit to Kripke’s venerated analysis of modal de re contexts (Kripke ). In Kripke’s view, essential properties are necessary but not sufficient for a possible object to be identical with a given actual entity. We construct possible scenarios (possible worlds) by stipulating first which actual objects are present there, and then making sure that these stipulations are consistent with the demand that the appropriate objects possess all the required essential properties. In Maudlin’s interpretation of the hole transformations such stipulations amount to the aforementioned assumption that points p and dp are to be identified. But this assumption seems unmotivated when we adopt a sharp distinction between mathematical models and physical realities represented

³ Model is interpreted here as a mathematical structure hM, g, Ti, where M is a differentiable manifold of points, g is a metric field on M, and T a matter field on M. The mapping d:M!M is a diffeomorphism, meaning that it is smooth enough with respect to the underlying topological features of M. ⁴ The identification of the pre- and post-transformation points may be to a certain extent justified by the active interpretation of diffeomorphisms mentioned earlier. In this interpretation (as opposed to a passive interpretation under which the diffeomorphism changes nothing except the coordinates) we consider the mathematical mapping d: M!M to represent a physical operation consisting of “dragging” all fields (metrical and physical) against the background of the immutable points in the following sense. Suppose ) receive the values Φ(p) and that the points p and dp (both treated as elements of the original manifold as representing the situation in Φ’(dp) of a given field. Now, we interpret the transformed model d which point p obtains a new value Φ’ that was initially associated with dp. Thus the point in the postwhich occupies place Φ’ is considered to be the same as the point p in , and transformation model d hence it is natural to accept the identity p = dp. Obviously, this ‘transworld’ identity connects points from two different models, and not points in one and the same manifold, which are distinct by assumption (as d is not the identity). ⁵ For some critical remarks regarding Maudlin’s solution see e.g. (Bartels ).

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

by these models, as urged by Maudlin himself. A diffeomorphism between two models is in fact a mapping connecting mathematical objects, not their physical counterparts.⁶ If a given point p of the manifold M represents a physical spatiotemporal point s, it is by no means guaranteed that the transformed point dp should represent the very same physical object s. Rather, we should look ‘globally’ at the entire transformed model d and figure out its appropriate physical interpretation. This interpretation ought to be guided by a match between the mathematical and physical qualitative structures, and not by arbitrary point-by-point non-qualitative identifications that smack of metaphysically suspicious haecceities. Maudlin’s version of essentialism has been criticized for its unintuitive analysis of some modal de re statements. Due to the fact that the field equations of General Relativity connect the metric properties of points and regions with their physical contents (e.g. matter distribution), it seems that sentences of the kind “This point (region) could have different matter contents” come out false under Maudlin’s analysis. His reply to this challenge is based on the distinction between metaphysical and nomic necessity, but it is still quite unsettling to admit that it is nomically impossible (even if metaphysically possible) for a point inside the Sun not to be occupied by its actual matter contents. After all, laws should not dictate the physical properties of all spatiotemporal points and regions in the universe. Another problem for Maudlin’s solution is posed by the possibility of ‘alien’ points (or ‘moles’, as they are sometimes called). Maudlin’s variant of essentialism does not exclude a possible scenario in which the place of a given point p (or a region) in the actual spatiotemporal structure of the universe has been taken by a distinct point p’ (‘mole’) that has entirely new metric properties. But this possibility paves the way to a new variant of the hole argument. All we have to do is to imagine a model such that it consists of the very same points as in reality outside the hole, but inside the hole all the transformed points are replaced by moles. It may be argued that this scenario involves some form of ontological indeterminism, as the complete physical state of the region outside the hole does not uniquely determine the nature of the points inside the hole.⁷

.. Glick’s Minimal Structural Essentialism It seems to me that Maudlin’s variant of spacetime essentialism suffers from the above-mentioned ailments mostly because it is not strong (or serious) enough. It still ⁶ Dasgupta maintains that it is possible to apply a given diffeomorphism d directly to the physical manifold, thus bypassing the need for mathematical models (Dasgupta , p. ff). I am skeptical about this claim, though. The reason for my reservation is that the procedure of “dragging” metric and physical fields over the physical manifold in order to get from the actual world W to the possible world dW is not metaphysically neutral, as it presupposes some solutions to the controversial issue of identity across possible worlds. When I point at a given region and say “Imagine that this very region is occupied by that black hole over there”, I have to assume, following Kripke, that possible worlds can be characterized not only qualitatively but individualistically as well. ⁷ It is unclear though whether this indeterminism is of the same caliber as the indeterminism of the original hole argument. In the original variant of the argument the fact of the matter that was not settled by the complete characteristic of the region outside the hole was of the kind “This point has such and such metric properties.” In the new scenario the undetermined states of affairs reduce to facts about whether a given point is alien or not. Some people can claim that this hardly involves a genuine physical difference, even from the substantivalist’s perspective.

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

 

contains some traces of anti-essentialist, haecceitistic sentiments (in particular, in the form of the inter-model identifications p = dp). Thus I believe that we should adopt a more radical version of ontological essentialism in order to tackle all the challenges of the hole-type arguments in GR. However, recently a proposal has been made that goes in an entirely different direction. In his  paper David Glick defends a variant of essentialism called Minimal Structural Essentialism (MSE) which appears to be even weaker than Maudlin’s. Glick claims that his minimalistic conception fares better with respect to the above-mentioned challenges, and we should see how this claim holds up. The main feature of MSE, apart from its strong emphasis on the structural character of essential properties (to which we will return in Secion .), is that it admits that objects can fail to occupy their essential positions in a given structure, if this structure is absent (in the sense of not being instantiated) from a given world. Glick accepts the thesis he calls Weak Sufficiency, which stipulates that if the essential property P of an actual object a is instantiated in a given world, object a is present in this world and possesses P there. But he rejects full-blown Sufficiency, which requires that whatever possesses property P must be identical with object a. The difference between Sufficiency and Weak Sufficiency comes to the surface when we consider cases in which an essential property P is multiply instantiated in a particular possible world. (This happens when an appropriate essential structure admits non-trivial symmetries, in which case there are two or more places in this structure that are indistinguishable from each other.) In this case Sufficiency implies that each object that instantiates property P should be identified as representing a, whereas Weak Sufficiency requires only that one of the objects exemplifying P be identified with a.⁸ Glick’s insistence on the claim that an object a can exist in a possible world w even without possessing its essential property P, if only there is no corresponding structure in w (i.e. property P is not exemplified there), leads to the rejection of Essentialism: the thesis that whatever is identified with a must have P. This surprising move may prompt an immediate objection that MSE is no essentialism at all, as it violates the assumption that seems to be the core of the ontological doctrine of essentialism. But this is purely a matter of terminology with no great import. We may agree to use the term “essentialism” to refer to any doctrine that insists that objects possess certain properties necessarily under some well-defined and reasonable conditions. Glick accepts the conditional claim we may call Weak Essentialism, which stipulates that if place P is instantiated in a given world, then whatever object represents a in this world must possess P. We have to check now what MSE does to the challenge posed by the hole argument. Clearly, MSE avoids the threat of indeterminism, but it achieves this result not by denying the metaphysical possibility of the transformed model d , as in Maudlin’s variant of metric essentialism, but by identifying it with .⁹ As the essential metric structure of is present in d , the original model ⁸ See note  below for my comments regarding the formalization of conditions such as Sufficiency. ⁹ This is the central tenet of all related solutions to the hole argument that are known under the name of sophisticated substantivalism. See Maidens (), Brighouse (), Hoefer (), Belot and Earman (), Pooley (), and Rickles () for various arguments for and against sophisticated substantivalism.

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

Weak Sufficiency implies that original points p will occupy the same positions in the metric structure of the new model d . Thus we no longer identify points p and dp; are isomorphic, it instead we follow the metric structure, and because d and follows that they must represent the same reality.¹⁰ Glick’s minimal essentialism admittedly gets around the problem of contingent statements regarding the physical contents of spatiotemporal points and regions. As we recall, Weak Sufficiency requires that a given actual point occupy its essential place in the metric structure only if this place is instantiated in a new scenario. If the current metric structure of spacetime is not instantiated in an alternative possible world (and, instead, a different metric structure is present, corresponding to different distributions of matter and energy), the actual points are free to take up any place in the new structure, which implies that these points may admit any metric properties without losing their identities. Thanks to that, the actual points can accommodate any material contents without violating the field equations of GR. Thus it is nomically possible that the material fields at this or that region could be different from the actual ones. However, in some special cases the problem persists. Suppose that in the actual world a given spatiotemporal region R has a ‘twin’, i.e. a distinct region R0 which possesses the exact same metric properties (more precisely the world is such that the swapping of R and R0 is a symmetry of its metric structure). Now, in the closest possible world in which R appears to have a different matter distribution, region R0 still possesses the same metric properties as in actuality, and therefore by Weak Sufficiency it represents de re the old region R. Consequently, the counterfactual “This regionwould contain different matter contents” comes out false in this scenario, because now the region we originally referred to as R is represented by R0 , and R0 has its material contents unchanged. Another potential advantage of MSE is its apparent ability to deal with the cases of troublesome ‘moles’. As Glick points out, a mole cannot take the place of a given point in the actual metric structure due to Weak Sufficiency. Once the actual structure is exemplified, Weak Sufficiency implies that the places in this structure must be occupied by the same points. But again there are some peculiar cases in which alien points can be admitted. If in a possible scenario there are two identical copies of a particular place in the actual metric structure, then Weak Sufficiency can only ensure that one of them will be occupied by the same points as in actuality. Thus the other copy is free to be inhabited by new points not present in the actual world. To this it may be replied that a structure which contains two copies of one place from the original structure is not isomorphic with the original structure, and therefore

¹⁰ If the metric structure of the universe admits non-trivial symmetries, we are facing a case of metaphysical underdetermination here, since it is impossible on the basis of Weak Sufficiency to determine which of the points occupying the same places in the metric structure of the transformed model is identical with a given point. This observation may give rise to the following objection against MSE and related corresponds not to one, but many physical proposals: in the symmetric cases the transformed model d possibilities, each associated with a different distribution of actual points over the indistinguishable places in the metric structure. These possibilities will be qualitatively indistinguishable from each other and yet metaphysically distinct, and this fact seems to undermine the identification of pre- and posttransformation models that is the key element of the considered solution to the hole argument. Radical Structural Essentialism, as presented in Section .., is not vulnerable to this problem.

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

 

does not satisfy the precondition of Weak Sufficiency (the original structure is not instantiated). However, I believe that isomorphy is too strong a requirement for the identification of places occupied by particular elements of a given structure. In Section . I will argue that the essential place of an object in a given actual structure is preserved if this structure is embeddable in a suitable sense in a new structure, even if this new structure is not fully isomorphic with the actual one. Thus it is possible to have a structure with more indistinguishable places than the original one had, and some of these places may be occupied by alien points not identical with any actual points. MSE has its own share of troublesome consequences. One of them is the fact that it seems to undermine the principle of General Permutability which insists that two possible worlds that differ only with respect to the permutation of objects should be identified.¹¹ But if we consider a symmetrical essential structure with distinct objects a and b occupying qualitatively identical places, there seem to be distinct possible worlds which differ only in that a and b exchange their places.¹² It has to be noted that a similar problem affects Maudlin’s variant of spacetime essentialism. In my opinion this problem stems from the fact that both approaches allow for nonqualitative, purely haecceitistic identifications of objects across possible worlds. In Maudlin’s approach this sort of identification is first of all present in the presupposed identity between pre- and post-transformation points: p = dp. Moreover, we are allowed (indeed, required) to make such non-qualitative identifications each time we have distinct actual objects that share their essential properties. This remains true under Glick’s proposal, and because he rejects Essentialism, he is forced to admit in addition that there may be possible situations in which a given object a exists even without having its essential properties. This means that in this approach there may be instances of trans-world identity which are not grounded in any qualitative facts.¹³

.. Radical Structural Essentialism An alternative to both Maudlin’s and Glick’s variants of essentialism is to go all the way and adopt radical structural essentialism (RSE). This means that both Essentialism and Sufficiency are admitted as the fundamental assumptions. With Sufficiency we are facing the challenge of symmetric structures in which more than one object ¹¹ According to the original formulation of the principle of General Permutability given by John Stachel, every permutation of a number of entities in a given structure should represent the same possible state of the world (Stachel , p. ). ¹² Glick proposes to deal with this problem by applying the concept of weak discernibility to distinct and yet qualitatively indistinguishable places in a symmetric structure. He claims that weak discernibility of distinct places implies that objects occupying these places will thereby have different essential properties and therefore will not be allowed to be swapped. However, I have my doubts regarding this strategy. Occupying this place rather than that place does not seem to involve a genuine qualitative difference unless these places differ qualitatively. And while mere weak discernibility can ensure that there are two distinct places, it does not make them qualitatively different in the required sense (see Ladyman and Bigaj  for a further discussion of the role of weak discernibility). ¹³ On a related note, we may observe that Glick’s solution commits us to the ontological stance of haecceitism by admitting the existence of two qualitatively distinct possible worlds that differ only in their de re representations. See Fara (), Skow (, ) for the formulations of different variants of haecceitism.

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occupies qualitatively indistinguishable places. It may seem that in this case we are forced to accept the self-contradictory consequence that distinct objects occupying these indistinguishable places are identical with the same actual object, and thus with each other.¹⁴ A well-known strategy to avoid this debacle is to adopt Lewis’s counterpart theory (Lewis , b). In this approach whatever occupies the same essential place as a given object is its counterpart (and therefore represents the given object de re). As the counterpart relation is not identity, there is no logical contradiction in the assumption that more than one entity can be counterparts of one and the same actual object (and, similarly, that two actual objects can have the same counterparts). From a structuralist perspective the case of multiple counterparts is not at all bizarre or troublesome. Structuralists do not believe in ‘bare’ entities existing independently of a particular structure. Why, then, should they insist on identifying them in counterfactual scenarios in addition to an identification of the entire structure they participate in? For the structural essentialist it just doesn’t make much sense to ask which of the qualitatively indistinguishable objects in a possible world really is the actual object that we are interested in. Once the entire actual structure has been identified in a possible world (whether in the form of a perfectly isomorphic copy, or in a weaker sense, defined in Section .) there are no further questions regarding which element in the actual structure corresponds to which element in a possible structure representing the actual one. It is rather straightforward to observe that RSE easily deals with the challenge of to an the hole argument. The hole transformation d takes us from the model isomorphic model d , and therefore these two models represent the same reality consisting of a metric structure of spatiotemporal points and a physical structure encompassed in the matter-energy tensor superimposed on the former one. Furthermore, due to the assumption of Sufficiency there is no worry that a given place in the metric structure could be occupied by an alien point. Finally, RSE rejects the existence of distinct possible worlds that arise by permuting bare elements of appropriate structures. The main challenge that remains for RSE is how to account for our intuitions regarding modal de re statements about alternative physical contents of actual spatiotemporal points and regions. As we have seen, both Maudlin’s and Glick’s approaches struggle with this problem. Maudlin’s response was based on the distinction between metaphysical and nomic possibilities which led to further unintuitive consequences. MSE, on the other hand, promised to solve this conundrum decisively by rejecting Essentialism and hence admitting the possibility that a given point can have metrical properties different from the actually possessed. Still, due to the possibility of the existence of symmetric structures, some cases of alternative matter distribution remain problematic, as explained in Section ... I believe that the problem can and should be approached from an altogether different angle. We should start with the question about the intended meaning of

¹⁴ This is the essence of the “abysmal embarrassment” charge leveled at the structural interpretation of spacetime by Christian Wüthrich (Wüthrich ).

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the intuitively true statement “This region in the center of the Sun could have a different matter distribution.” It seems evident to me that outside the philosophy room this statement means something like this: it is possible that the universe could look very much like ours, except that in the center of a medium-sized star with a planetary system just like our Solar System there might be a different matter distribution. We don’t care much about the true identity of the region in question; all we want is a qualitative characteristic as described above. And it is obvious that such a universe is perfectly possible. Clearly, due to the difference in mass distribution there will be a corresponding difference in the metric properties of the affected region (its curvature will be slightly different). But we are not considering the question of whether it is possible that the metric structure of the universe remains unchanged while the distribution of matter is different—a physicist would respond that this is a plain impossibility, notwithstanding fancy distinctions between metaphysical and nomic possibilities. And she should remain unfazed by the fact that technically the region with a new matter distribution and the associated novel metric properties will not be “the same” as the original one. What counts in this particular case is a broad, qualitative similarity of the entire scenario to the actual world, not a philosophically sharp notion of transworld identity.¹⁵ The philosophical concept of identity across possible worlds does play a crucial role in the hole argument, but observe that for this argument to have any bite at all, it must be presented as a challenge to the metaphysical position of substantivalism, and not to any intuitive view held by a typical working physicist. And if we enter the debate regarding the true nature of fundamental reality, including space-time, we have to be as rigorous as possible in our distinctions. But, again, our intuitive counterfactual judgments like the one considered above do not carry that much metaphysical weight, and therefore can be approached in a more relaxed fashion. If we agree on the proposed solution to the problem of contingent statements regarding alternative matter distributions, then we have to admit that on the remaining counts Radical Structural Essentialism does no worse than its main competitors (and it may be even claimed that on some counts it does better). Thus it may be considered as a serious alternative whose additional benefit is that it eliminates the metaphysically suspicious notions of haecceities and haecceitistic differences. In what follows I will try to further develop and sharpen this approach, starting with the question of the ontological nature of the metric tensor field.

. Metric Essentialism and Structuralism All known variants of metric essentialism seem to presuppose that the essential features of spatiotemporal points are exhausted in the metric field defined on these points. Mathematically, a metric field is an assignment of a particular object called ¹⁵ Lewis would probably say that the broad qualitative similarity is all we need to speak about counterparts and representation de re. I disagree—we need a much more precise and sharp concept of representation de re in order to tackle the hole argument. But the point I am making here is that in order to deal with the intuitive claims involving the concept of ‘sameness’ we don’t have to rely on the strict counterpart relation. Qualitative, rough similarity is sufficient to make sense of these claims.

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metric tensor to every point in spacetime. But this broad characteristic of the essential metric structure can give rise to some misunderstandings as to its exact nature. At first glance it may look as if the metric tensor represented an intrinsic property of individual points rather than a relation connecting them. If this were true, then the adjective “structural” in the acronyms MSE and RSE would be rather empty. Even though a structure consisting of mere monadic properties remains technically a structure, what really excites metaphysicians interested in the structuralist stance is its insistence on the fundamental ontological characters of polyadic relations connecting objects. And we have a strong intuition that the metric structure should be relational; after all, spatiotemporal distance between points is a model example of a relation irreducible to the intrinsic properties of the relata. Thus we should look closer at the proper interpretation of the metric structure encompassed in the metric tensor.¹⁶ The metric tensor can be formally introduced in many ways. One popular method is to define it as the inner product of vectors in a so-called tangent space, associated with every point in the selected manifold (see Friedman ). The tangent space Tp at a given point p is a vector space such that for every curve σ passing through p the vector Tσ(p) tangent to σ at p belongs to Tp. A metric tensor is a real-valued, bilinear, symmetric, and nonsingular function g(X, Y), where X, Y2Tp. If g(X, Y) is in addition positive-definite, it is called a Riemannian metric tensor; otherwise it is referred to as pseudo-Riemannian. The product g(X, X) is naturally interpreted as the squared length of vector X. Given a particular coordinatization xi of a neighborhood of point p, the curve σ is represented as a set of continuous functions xi(u)
1 of one real dxn parameter u, and the vector tangent to σ at p is given as Tσ ðpÞ ¼ dx du ; :::; du . In i dxj these coordinates the inner product g(Tσ(p), Tσ(p)) can be rewritten as gij dx du du , where gij is a matrix representation of the metric tensor in the selected coordinate system. The metric tensor also figures in the formula defining an infinitesimal displacement (‘line element’) in terms of generalized coordinates: ds2 ¼ gij dxi dxj . Thus it is often loosely said that the metric tensor at a given point ‘contains’ the information about distances in the immediate neighborhood of this point. However, this does not immediately guarantee the relational character of the metric tensor. Suppose, for instance, that we consider a flat, Minkowskian spacetime. In such a case the (pseudo-Riemannian) metric tensor at every spatiotemporal point looks exactly the same: ds2 ¼ ðdx1 Þ2  ðdx2 Þ2  ðdx3 Þ2  ðdx3 Þ2 . Thus all points are indistinguishable from each other with respect to their metric properties, and yet the metric relations holding between them will vary (different points will be generally separated by non-equal intervals, some of them space-like, some time-like, and some null, or light-like). To put this differently, the metric relations between two points cannot possibly be reduced to the values of the metric tensor at these points, since in the case of flat geometry these values are identical for all points while the relations obviously vary. This irreducibility of metric relations to the metric tensor may come as a surprise, since it is customary to define the length of

¹⁶ This problem is also considered in Lam and Esfeld (), where it receives a similar though not identical treatment to the one I am presenting here.

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a given curve σ between two selected points in terms of a metric tensor gij by integrating the length of the tangent vector over the whole curve as follows: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi  R R jσj ¼ jTσ ðuÞjdu ¼ gσðuÞ Tσ ðuÞ; Tσ ðuÞ du. Subsequently, in the case of Riemannian metric tensors we can define the distance between two points as the length of the shortest curve (geodesic) connecting these points. If the metric tensor determines the distances between points, it may be hard to understand how points can vary with respect to their metric relations in a flat geometry where points possess the same metric tensor. However, the jump from the metric tensor to the distance relation was too quick. The missing element is the notion of a (smooth) curve, which presupposes the topological structure encompassed in the notion of a neighborhood. We should not forget that a differentiable manifold even without metric is not a structureless collection of qualitatively indistinguishable bare points, but comes already equipped with topological features. A topological space is a set of points where we have identified for each point the sets of points called its neighborhoods.¹⁷ The concept of a neighborhood is clearly a relational one, not reducible to the intrinsic properties of points. This should not come as a surprise, given that with the help of the concept of a neighborhood we can define relations such as the two place-relation “x is in a neighborhood of y”or the three-place relation “x and y belong to the same neighborhood of z”.¹⁸ Once we have the topological (and relational) concept of a neighborhood it is possible to introduce further mathematical structures, including the concept of a homeomorphism (a continuous bijection) between a topological space and the Euclidean space Rn, which leads to the notion of a (differentiable) n-dimensional manifold. This enables us to deploy the apparatus of differential calculus which subsequently paves the way to the concepts such as continuous curves, tangent spaces, and the lengths of curves. The conclusion from this analysis is that the essential structure that identifies points of a manifold must contain the fundamental topological structure defined on these points.¹⁹ And because topology is clearly relational, points in a manifold are characterized structurally in a non-trivial way. The identity of points is fixed not by their intrinsic properties but by their mutual relations, topological and metrical. If we consider two alternative mathematical models of spacetime (such as diffeomorphicallyrelated models of GR), the question of whether these models refer to the same or distinct fundamental objects (i.e. points and regions) must be decided by comparing

¹⁷ The axiomatization of topological spaces in terms of neighborhoods is due to Felix Hausdorff. An alternative, equivalent axiomatization uses the concept of open sets. ¹⁸ Another important relation that can be mentioned here is Hausdorff separation which holds between points x and y iff they possess non-overlapping neighborhoods. A topological space satisfies the Hausdorff condition if all distinct points are separable in this sense. ¹⁹ It may be pointed out that the assignment of a metric tensor to individual points already presupposes the topological structure at the most fundamental level, since it involves the concept of a tangent space and therefore (implicitly) that of a continuous curve. However, I am not so sure about this. We could imagine an entirely formal mapping between structureless and yet numerically distinct points on the one hand and some arbitrary mathematical objects (such as tensors in some abstract space), without claiming that this assignment represents any underlying relational features of these points.

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metrical plus topological structures described in these models.²⁰ However, it remains to be decided how precisely these structures ought to be related in order to identify their underlying elements.

. Structural Criterion of Identity The standard answer to the above question is given in terms of isomorphy. That is, if the pre- and post-transformation essential structures are isomorphic, we can say that they represent the same reality. If we want, we can follow an isomorphism between the structures and identify their individual elements according to the isomorphic mapping, point by point, but in the case of symmetric structures these identifications will not be unique. In order not to worry too much about the underdetermination of transworld identities, we may adopt a full-blown structural perspective and drop altogether the talk about individual points in favor of the identification of entire spatiotemporal structures. However, there remains a problem related to the fact that isomorphy may be too strong a requirement for an identification of an actual structure and its elements in a counterfactual, possible scenario. Two structures that are isomorphic must have the same number of elements, and it seems unintuitive that for an element to occur in a counterfactual scenario it must be accompanied by the exact same number of other elements as in reality. To use a simple example: suppose that in the actual world we have three objects each of which possesses the same essential property. In addition to that let us assume that no further essential relations connect these three objects. In such a case it would be unreasonable to demand that in order for one of these three objects to be present in a counterfactual scenario, the whole three-element structure should reappear there as well. Instead of the condition of isomorphy, I propose the following condition of embeddability to serve as the criterion of representation de re (encompassed in the Lewisian counterpart relation). A relational structure hD, R₁, . . . , Rni is said to be embeddable in another structure hD0 , R₁0 , . . . , Rn0 i of the same type iff there is a map f: D!D0 such that Ri(x₁, . . . , xk) iff Ri0 (f(x₁), . . . , f(xk)) for each Ri. Dropping the condition that f be a surjective bijection distinguishes embeddability from isomorphy.²¹ Now it may happen that the new domain D0 contains fewer or more elements than the original one D, while the intuition that the actual structure is in some sense present in the new situation is still preserved. We can interpret the image f(x) 2D0 as a counterpart of the actual object x, but we have to keep in mind that x may have more than one counterpart due to the existence of more than one ²⁰ Another element that may be necessary to include in the essential structure determining the identity of points is an affine connection (covariant derivative). This is an operator that measures the change of the tangent vector along a given curve, and therefore enables us to compare tangent spaces at different points of the manifold. ²¹ Another option to modify the condition of isomorphy is to replace it with the requirement of homomorphy which is even weaker than embeddability (instead of the equivalence Ri(x₁, . . . , xk) iff Ri0 (f(x₁), . . . , f(xk)) only the implication from left to right is assumed). In my (Bigaj ) I give some reasons why homomorphy may not be the most appropriate way of expressing our intuitions regarding representation de re.

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embedding, and also two actual objects may share a counterpart due to the fact that f is not a bijection. To illustrate the above-introduced criterion of representation de re suppose that we have an actual structure consisting of two objects x and y and one binary, symmetric and reflexive relation R holding between x and y. A one-element structure in which an object z stands in relation R to itself represents one possible scenario where z is a counterpart for both x and y. Thus the modal sentence “Object x might not have stood in relation R to any other objects” comes out true. Another possible scenario involving x and y can be depicted with the help of a three-element structure where all objects are connected by relation R. This structure makes true the following, plausible counterfactual pronouncement “Object x might have stood in relation R to more than one other object.” Let us apply the concept of embeddability to the case of metric essential structures involving geometric points. It is not difficult to observe that in that case two distinct points can never be mapped onto one point. When the metric is Riemannian the reason for that is straightforward, as the distance between distinct points is always non-zero, so this relation could never be preserved by a mapping that ‘glues’ together these points. In the case of a pseudo-Riemannian metric the situation is slightly more complicated. For instance, in a Minkowskian spacetime two distinct points x and y can be separated by a zero interval. However, all we need to secure the bijectivity is to find a third point z for which the interval from x is different from the interval from y. In such a situation the points x, y and z have to be mapped onto distinct points to preserve these metric relations. Thus the mapping between two metric structures which ensures that one structure is embeddable in the other must be bijective. However, the mapping may still not be surjective. If that’s the case, the new model represents a situation in which in addition to old points and regions there are some new points and regions that stand in various new metric relations to the old ones. This is precisely the reason why ‘alien’ points (‘moles’) may be admissible, as stated in Section ... In light of the above-introduced structural criterion of representation de re we can now make precise the assumptions of Essentialism and Sufficiency that form the core of Radical Structural Essentialism. Essentialism can be spelled out in the form of the condition that if R is an essential structure containing some actual objects x₁, . . . xn, then if these objects are to be represented de re in a possible world by objects x₁0 , . . . , xn0 , this world must exemplify a structure R0 containing x₁0 , . . . , xn0 , and such that R is embeddable in R0 , where the embedding f is such that f(xi) = xi0 . Sufficiency proceeds in the opposite direction: if R is embeddable in a possible structure R0 , then for any embedding f the objects f(xi) are to be the counterparts of xi. We keep in mind that the relation of being a counterpart (representing de re) is not identity, and therefore it is admissible for an actual object to be connected, via distinct embeddings, with more than one possible object (and for two actual objects to be represented by the same possible object).²² ²² It is standard in the literature to present the ‘counterfactual’ identity conditions, such as Essentialism and Sufficiency, purely in modal terms, with no direct reference to the presupposed semantics of possible worlds and the relation of representation de re. For instance, Glick formulates the condition of Sufficiency

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. Conclusion At the end of this survey let us return to the hole argument again. The essentialiststructuralist solution to this problem is based on the distinction between two types of structures that are involved in the argument: the essential metric structure (including appropriate topological features on which the metric is built) and the contingent physical structure (including the distribution of matter and energy). In the standard approach these structures are superimposed on the background of bare, nonqualitative and indistinguishable points constituting the basis of the spatiotemporal manifold. In this picture a diffeomorphic hole transformation of the manifold amounts to a shift of both metric and physical objects against the stationary backgrounds of points, so that the same points receive different values of metric and physical fields inside the hole but the same values outside of it. However, from the perspective of the metaphysical essentialist, the transworld sameness of bare points that possess no identifying qualitative features just does not make sense.²³ It is much better to approach the problem in structural terms from the outset, in order to avoid confusions that beset the discussions on the meaning of diffeomorphisms in GR. A diffeomorphism d should not be seen as acting on bare points of the manifold M, but on the entire metric structure containing points. It is straightforward to define the concept of a structure-level transformation induced by a particular object-level mapping (for details see Bigaj ). A structure-level transformation connects entire structures, not points. The image of a given structure is a new structure that stands in a particular relation to the old one. If the object-level mapping f is a bijection, the resulting structure will be isomorphic with

in first-order language with modalities as follows: □8x (P(x) !x = a), and the condition of Essentialism in the form of the opposite implication: □8x (x = a ! P(x)). However, framing these conditions in such a way has some serious disadvantages. First of all, it is a well-known fact that quantified modal logic presents us with several interpretational challenges (see Garson , sec. ). One of them is the following problem: as 9x x = a is a logical theorem, we can derive from it (via the Gödel rule) that □9x x = a, which implies that object a exists necessarily (in all possible worlds). Another problem arises when we take into account the already discussed fact that the essential property P may be multiply exemplified in the actual world. But in that case we have a contradiction, since from Sufficiency it follows that 8x (P(x) !x = a), and this logically implies that there is exactly one actual object satisfying P. If we want to express the intuition that whatever possible object possesses P must represent de re the actual object a, we have to rely on a direct formulation in terms of possible worlds. ²³ Why is the intuition that it is perfectly possible to move the entire universe five feet in one direction so persistent and hard to shake? One way to explain this is via making a distinction between two related but crucially distinct interpretations. The spatial shift can be interpreted as involving a possible universe which differs from the actual one only in that throughout its entire history all actual objects occupy points and regions five feet away from their actual locations. Or, it can involve a possible world in which the material contents of the universe are suddenly shifted by five feet in some direction. The first possibility requires us that we identify actual points and regions in the counterfactual scenario by stipulation, and therefore is not available to the radical essentialist. The second option, on the other hand, is perfectly admissible, as long as we accept the existence of an absolute space. In the second scenario there is a clear qualitative difference between the actual and the possible “shift” worlds (even though this difference may not be empirical). For instance, the qualitative sentence “There is a spatial point that is occupied by some object before but not after a certain moment” is true in the possible world but may very well be false in the actual world. I am grateful to Carolyn Brighouse for stimulating discussions on this topic which helped me improve my arguments.

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the original one; if we drop the requirement of bijectivity, the old structure will be merely embeddable within the new one. Either way, the result of the transformation is a new essential structure. This structure is superimposed on the non-essential (merely contingent) structure which in the currently considered case consists of the stress-energy tensor assigned to individual points. It is crucial to observe that if we wanted to apply a given object-level diffeomorphism d to the entire essential metric-topological structure, and not to bare points, the result would be a new metric-topological structure which is ‘shifted’ against the contingent physical structure. This new model will generally not be a correct model of the field equations of GR, and thus will not represent any possible world. On the other hand, the way both metric and physical fields are dragged in the hole argument, following a given object-level diffeomorphism, clearly shows that from a structural point of view this operation amounts to no change at all (is simply the trivial identity operation), since the essential and contingent structures are not ‘moved’ with respect to one another. The fact that the equations of GR satisfy the condition of general covariance (diffeomorphism-invariance) strongly suggests that the role played by bare spatiotemporal points is secondary (they are mere instruments), and that from the ontological point of view only metric-topological structures enjoy the status of fundamental objects. Thus the lesson from the hole argument is that spacetime is ultimately a structure, and that it is meaningless to talk about individual points in counterfactual scenarios while ignoring the essential metric and topological relations they participate in. The fundamentality of spacetime can be secured not by the dubious assumption that spatiotemporal points and regions are irreducible entities equipped with their unique primitive identities (haecceities), but rather by the fact that the metric and topological relations which constitute the backbone of spacetime are not metaphysically reducible to any physical properties of matter.

Acknowledgments I would like to thank the Editors of this volume for inviting me to participate in this project, and the anonymous referees for their useful comments and suggestions. The work on this article was supported by grant No. //B/HS/ from the National Science Centre, Poland.

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13 When the Actual World Is Not Even Possible Christian Wüthrich

. Introduction Various approaches to formulating a quantum theory of gravity either presuppose or entail that fundamentally, there is neither space nor time. Instead, space and time emerge from the more fundamental, non-spatiotemporal structure of quantum gravity very much in the way that tables and chairs emerge from the more fundamental, non-chairtabular structure of quantum particle physics. This may have cataclysmic consequences for metaphysics: some philosophical analyses of causation, laws of nature, persistence, personal and material identity, and even modality crucially seem to rely on the fundamental existence of space and time. For instance, David Lewis (b) characterizes possible worlds as unified by the spatiotemporal relations among their parts but as spatiotemporally isolated from other possible worlds. If borne out, the disappearance of space and time would motivate a new—fatal—objection to Lewis’s account of modality: his pluriverse, for all its ontological abundance, does not contain our world. Apart from questioning the truth of theories denying the fundamental existence of spacetime, the Lewisian may respond in several ways to this shock. The more promising strategies consist either in questioning the empirical coherence of any theory denying the fundamental existence of space and time, or in arguing that Lewisian modality only requires the existence, but not the fundamentality, of space and time, or even in largely conceding the point and relaxing the conditions on the worldmate relation perhaps to just natural external relations. This chapter contends that the first strategy fails, even though it unveils an important foundational task for the defender of a theory-sans-fundamental spacetime. In order to avoid the charge of empirical incoherence, and thus to salvage the possibility of its own epistemic justification, such a theory must be shown to admit emergent spacetime. Thus, in circumventing the first Lewisian response the assumption of the second response—that spacetime exists at some ontological level—is granted. However, this by no means entails that the second response succeeds. In fact, it is argued that the merely emergent existence of space and time comes at an unpalatably high cost to Lewis. Finally, the third strategy, though generally viable, Christian Wüthrich, When the Actual World Is Not Even Possible In: The Foundation of Reality: Fundamentality, Space, and Time. Edited by: David Glick, George Darby, and Anna Marmodoro, Oxford University Press (2020). © Christian Wüthrich. DOI: 10.1093/oso/9780198831501.003.0014

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will be shown to be so flexible as to drain almost any content from Lewis’s condition of co-inhabiting the same possible world, and thus of being ‘worldmates’, and to potentially sit uneasily with the metaphysics of the fundamental structure. Although this chapter focuses on the Lewisian conception of modality, its point is much more general: theories in quantum gravity promise to have far reaching implications for metaphysics. These implications are difficult to circumvent on even just a mild form of naturalism. The minimal naturalism assumed throughout this chapter asserts that no metaphysical thesis may be in manifest contradiction with facts established by our best science in the sense that the physically possible worlds are a subset of the metaphysically possible ones. On this assumption, then, no physically possible world can be metaphysically impossible; for otherwise, metaphysics would (a priori) deem impossible what physics affirms is possible. This chapter considers metaphysical implications of this minimal form of naturalism if current research in quantum gravity is indicative of our future best science. Section . explicates the need for a quantum theory of gravity and argues that such a theory will be our most fundamental theory of gravity, at least to date. In Section ., I will show how spacetime may not be part of the fundamental furniture of the world by introducing the conceptually simple and clean case of causal set theory. Section . acts as a reminder of the fact that it is spatiotemporal relations that unify and isolate worlds in Lewis’s pluriverse, and Section . parses out the trouble it encounters if those spatiotemporal relations are absent. Sections ., ., and . articulate and discuss the three most promising Lewisian strategies in responding to the challenge presented, respectively. I will argue that they all fail. I offer some brief conclusions in Section ..

. Quantum Gravity and Fundamentality Today, there are two incumbent theories in physics with a serious claim to be not just true, but fundamental theories: the standard model of particle physics and general relativity. The former describes the structure of what are (so far) the smallest scales at which physics makes reliable predictions and concerns the constitution of matter; the latter encodes the large-scale structure of our universe and its history. Both theories make eminently accurate predictions and both stand unrefuted, at least if evidence is restricted to direct tests of these theories. Yet not everything is well in fundamental physics. For starters, the standard model and general relativity stand in significant conceptual tension. The standard model radically reconceptualizes matter and energy from the way they figured in pre-quantum theories, but general relativity relies on these obsolete notions. General relativity proposes an equally radically novel understanding of space and time and their joint interaction with their energy and matter content, departing from the prerelativistic physics of space and time presupposed by the standard model.¹ This conceptual disunity is philosophically unattractive, but the clincher for their joint untenability is that there exist phenomena for which we have compelling reason to ¹ Or at least from the not fully relativistic assumptions made in the standard model.

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believe that their successful description must involve both quantum and relativistic, i.e., gravitational, effects. In other words, a theory is needed which commands the resources to combine the quantum with relativity. This is the theory the discipline of quantum gravity seeks to formulate, and I shall call a quantum theory of gravity any such theory with the explanatory ambition to account for these phenomena. One might complain that the term ‘phenomena’ is ill-chosen given that they concern, e.g., the physics of the very early universe and the hitherto unobserved evaporation of black holes. Of course, it remains true that no observation, which is unambiguously quantum-gravitational in the sense necessary to justify the need for quantum gravity, has knowingly been made to date. But while the case for the evaporation of black holes may be more tenuous, the reasons to believe that our universe started out in a very dense state and that this state can only be correctly captured by a theory attending to both quantum and gravitational effects are firmly anchored in our currently best physical theories, the standard model and general relativity. The argument which translates these reasons into a need for a quantum theory of gravity requires little more in terms of assumptions about the actual world we inhabit beyond these reigning theories. So for present purposes I shall assume that a quantum theory of gravity is needed. A quantum theory of gravity will be a fundamental theory, at least more fundamental than any other theory in physics currently or previously held to be true. Presently, I use fundamentality to denote a relation between theories, partially ordering the true theories of physics, perhaps including merely approximately true theories, or perhaps even including theories which have, or had, some currency in science. The fact that a theory is more fundamental than another in no way entails that the first theory is fundamental simpliciter. The relation I am interested in here is thus more appropriately termed ‘relative fundamentality’, although I will often suppress the qualification in what follows. Let us state what the relevant sort of fundamentality is. To establish whether a given pair of theories exemplifies the relation of relative fundamentality may be a highly non-trivial matter, particularly once one abandons the exclusive focus on physics in favour of a consideration of special sciences. For present purposes, I shall put aside this question and the more general debate on reductionism it invites; both deserve greater care than I can devote to them here. However, we can supply some abstract characterization of relative fundamentality. For the sake of the present argument, I rely on the assumption that the set of all hitherto relevant theories is partially ordered by relative fundamentality. Furthermore, I assume that relative fundamentality is irreflexive and hence that the partial ordering is strict. Since it is a partial ordering, it is of course transitive. Its irreflexivity and transitivity imply that it is asymmetric. Let us call the set of relevant theories T. If T is finite, it contains a ‘minimal’ element, i.e., a theory that is not less fundamental than any other theory in the set of theories considered.² To demand that T contain at least one minimal element does ² If T contains infinitely many theories, then an additional assumption that there is such a minimal element must be made. In that case, though, for T to contain at least one minimal element it suffices that every totally ordered subset of T has a lower bound in T. This follows, mutatis mutandis, from Zorn’s

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not rule out that there exist several distinct theories with a justified claim to being the most fundamental. My insistence that there exists at least one minimal element of T is usually given expression in the stipulation that the partial ordering of T be ‘well-founded’. A binary relation which induces a partial order on its domain is well-founded just in case every non-empty subset of the domain has a minimal element with respect to the relation.³,⁴ Well-foundedness results in the present case from the demand that “all priority claims terminate” (Schaffer a, p. ). In other words, if from anywhere in the relevant set one starts asking the question whether there exists a theory which is relatively more fundamental than the one at hand, and then whether there exists a theory which is relatively more fundamental than that theory, etc., we must reach a negative answer within a finite number of steps.⁵ My characterization of fundamentality differs from others prevalent in the literature.⁶ At least in the philosophical literature, fundamentality is usually understood in ontological or ideological terms rather than as a relation between theories. More specifically, fundamentality typically gets explicated by relations of ontological dependence obtaining among objects or structures or by mereological relations of parthood or by relations of supervenience holding among properties or some combination of these, e.g.in that properties of objects which ontologically depend upon, or are mereological complexes of, more basal or simple objects supervene on the properties exemplified by the basal or simple objects. The relevant relations are then taken to induce a partial ordering on their domain. To take theories to be the primary target of considerations regarding fundamentality instead of the objects they describe and their properties reflects my conviction that in fundamental physics theories indeed (epistemically) precede their ontological commitments and that it is thus more fruitful to address the question at the level of theories. Of course, the fundamentality relations as they obtain among theories will entail relations of ontological dependence among the objects or structures they postulate and relations of supervenience among the properties they ascribe to these objects or structures. In fact, one would hope that judgments about the fundamentality of theories are precisely mirrored by judgments concerning other ways in which fundamentality is considered. In what follows, we will be concerned with the lemma (Zorn ). For a subset A of a partially ordered set ðB; Þ, an element x of B is a lower bound of A just in case for all a 2 A, x  a. To have a lower bound means that for any subset of theories which are totally ordered with respect to fundamentality, there exists a theory t in T—and not necessarily in the subset—which is the most fundamental. Since the relevant ordering relation is irreflexive, t is actually more fundamental, rather than just no less fundamental, than those in the subset considered. ³ More precisely, a partial order is well-founded if and only if the corresponding strict order is induced by a well-founded relation. This precisification, however, adds nothing to the case at stake since ‘relative fundamentality’, as stated above, is an irreflexive relation. ⁴ Note also that in a subset consisting of just two theories, neither of which is more fundamental than the other, both of its elements, rather than none, are minimal. ⁵ The well-foundedness of a partial order may fail because there turns out to be an actually infinite tower of ever more fundamental theories. Ruling out this logical possibility comes at no great cost, since as finite beings, we could never entertain an actual infinity of theories. ⁶ And is arguably not shared by David Lewis, as Ghislain Guigon conveyed to me. This does not affect my argument against his account of modality below.

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fundamentality of an entity—spacetime. An entity will be considered fundamental just in case one of the most fundamental theories entails that it exists (fundamentally, in case the theory has a stratified ontology).⁷ Under the current description, the attempted quantum theory of gravity will certainly be more fundamental than general relativity if its ambition will be actualized. After all, this is its stated goal: to offer a theory of gravity which cannot only deal with the relativistic aspects of gravity, and hence of spacetime, but which also incorporates pertinent quantum effects. In other words, it endeavours to deliver a fundamental theory of gravity, and as such will be more fundamental than our currently most fundamental theory of gravity. Depending on the particular features of a candidate theory of quantum gravity, it may or may not be more fundamental than the standard model of particle physics. As stated above, the standard model offers our currently most fundamental theory of the three non-gravitational forces. Thus, if the quantum theory of gravity to be does not only provide a fundamental theory of gravity, but instead a unified theory of all forces, then it will also be more fundamental than the standard model. Such is the ambition harboured by string theory, for example. If, however, it only amounts to a fundamental theory of gravity, then it will not be more fundamental than the standard model. This will be the case, e.g., for most approaches trying to quantize general relativity such as loop quantum gravity and for most approaches starting out from more revisionary assumptions such as causal set theory. It should be noted, however, that it is also not the case that the standard model is more fundamental than these approaches. In either case, the quantum theory of gravity will be a minimal element of T and thus among the most fundamental theories, at least to date. Let us proceed on this premise.

. The Disappearance of Space and Time According to most approaches to quantum gravity, spacetime is not part of the fundamental furniture of our world, but merely ‘emergent’.⁸ What I mean by this is that whatever fundamental structure a theory of quantum gravity postulates, it is importantly dissimilar from the structure ‘spacetime’ refers to in general relativity or other, non-fundamental theories of gravity or of spacetime. For instance, in a vast class of approaches to quantum gravity, the fundamental structure is discrete.⁹ In socalled canonical approaches to quantum gravity, there is a strong suggestion that at least time is not fundamental.¹⁰ Dualities in string theory may be interpreted to mean that space(time) is not fundamental in string theory either.¹¹ The so-called Weyl symmetry of the ‘internal metric’ already present in plain vanilla string theories is often interpreted, at least among physicists, as rendering unnecessary the background spacetime in which the string was at first assumed to propagate.¹² Taking ⁷ Clearly, this is debatable, e.g. with grounding realists. Since this is not the focus of my chapter, I will refrain from doing so. ⁸ Huggett and Wüthrich (, forthcoming). ⁹ Smolin (, p. ). ¹⁰ Huggett et al. (, §). ¹¹ See also Castellani and de Haro (Chapter , this volume). ¹² Witten (), but cf. also Huggett et al. (, §).

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the theory ontologically seriously, as string theorists tend to, thus means accepting the vanishing of the spacetime at the fundamental level. In non-commutative geometry, an approach related to string theory, the fundamental constituents replacing the familiar spatiotemporal quantities in different dimensions do not commute, i.e.where measuring quantities along different dimensions depends on the order in which these measurements are made.¹³ Despite its spatiotemporal vestige, the non-commutativity of the approach renders the fundamental structure profoundly different from the ordinary conception of spacetime. And the examples could be multiplied. But rather than fully establishing the assertion that approaches to quantum gravity generically deny the fundamental existence of space and time, I shall content myself with offering a representative yet tractable example: causal set theory. Causal set theory is a still inchoate, but conceptually clean approach to quantum gravity, which may serve as a useful illustration for how radically a discrete fundamental structure can differ from our usual conception of spacetime. Causal set theory is based on the assumption that the fundamental structure is a discrete set of featureless basal events partially ordered by causality. It is motivated by theorems in general relativity by Stephen Hawking et al. () and David Malament () which establish that given the causal structure and some volume information, the metric of the spacetime manifold is determined, as is its dimension, topology, and differential structure. In other words, the causal structure determines the geometry, albeit not the ‘size’ of the spacetime. Based on these theorems, causal set theory asserts that the fundamental structure is—or, more cautiously, is best represented by—a ‘causal set’ and thus that causality is prior to space and time. Furthermore, the presupposition of the discreteness of the fundamental structure is justified through its technical and conceptual utility. Slightly more formally, causal set theory postulates that the fundamental structure is best represented by a causal set , i.e. an ordered pair ⟨C; ≼ ⟩ consisting of a set C of elementary ‘events’ and a relation, denoted by the infix ≼, defined on C satisfying two conditions: first, ≼ induces a partial order on C (i.e., ≼ is reflexive, antisymmetric, and transitive); second, is discrete in that the number of elements of C which are causally ‘between’ any two points in C is finite.¹⁴ That the discreteness is stipulated is not in itself a problem, as long as it is ultimately vindicated by the scientific success of the theory. Thus, it is a feature of the theory. Causal set theory is plagued by two major challenges. First, like other discrete relational approaches to quantum gravity, it suffers from what is known as the ‘entropy crisis’, viz. that the vast majority of basic structures satisfying the above postulate cannot be approximated by, or physically related to, a relativistic spacetime. In other words, for most causal sets in causal set theory, no spacetime even remotely resembling ours emerges from it. Second, causal set theory as it has been articulated so far is a classical theory—it does not take quantum interference effects into account. This is hardly satisfactory when the goal was to produce a quantum theory of gravity. ¹³ See Szabo () for a review of non-commutative geometry, and Huggett () for a philosophical assessment. ¹⁴ More technically still, the second axiom demands that 8a; b 2 C; cardfx 2 Cja ≼ x ≼ bg