Symmetry, Indiscernibility, and the Generalist Picture Thomas Møller-Nielsen Balliol College, University of Oxford Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Michaelmas Term, 2015 i Abstract This thesis consists of four independent but thematically interrelated papers, plus an addendum to one of these papers. Chapter 1 defends the view that Leibniz subscribed to generalism, or the view that the world is fundamentally purely qualitative in character. In particular, I respond to Cover & O’Leary-Hawthorne’s (1999) claim that Leibniz’s use of symmetry considerations in the correspondence with Clarke reveals that he was not a generalist. In addition, I claim that what reveals Leibniz’s generalism is not — as many scholars seem to believe — his mere commitment to his theory of truth, theory of complete individual substances and Principle of the Identity of Indiscernibles (PII). Rather, I argue, it is the fact that Leibniz claimed to derive the PII from his theories of truth and complete individual substances which ultimately demonstrates his commitment to generalism. Chapter 2 examines the putative relationship between two notions: symme- try and qualitativity. I argue that, on the standard metaphysical construal of the notion of qualitativity, a widely-held view about the relationship between these two notions is mistaken. However, I also argue that on a nonstandard construal of the notion of qualitativity due to Ismael & van Fraassen (2003), the alleged relationship between the two notions holds much more promise. I finish by expanding upon Ismael & van Fraassen’s own account of the notion of qualitativity relevant to the notion of symmetry, and in doing so I attempt to elucidate the methodology of symmetry reasoning in physics more generally. This chapter is followed by a brief addendum. Here, I discuss two recent independent proposals — due to Adam Caulton (forthcoming) and Shamik Dasgupta (forthcoming, a) — for understanding the methodology of symmetry reasoning that are prima facie very similar to the view that I defend at the end of Chapter 2. I argue that both proposals differ from my own, and each other, in subtle but nevertheless highly non-trivial respects. Chapter 3 argues that the focus of much contemporary discussion concerning the proper role and philosophical significance of a modern-day version of the PII is misplaced. More specifically, I argue that metaphysicians’ criticism of the notion of weak discernibility on the basis of its alleged inability to ground objects’ numerical diversity is orthogonal to Saunders’ (2003a) main concern in his original paper, which is to use the notion of weak discernibility as part of a broader program for interpreting physical theories. I subsequently assess this interpretative program, and suggest some reasons for thinking it implausible. Chapter 4 discusses a variety of recent objections to generalism. The objec- tions are related by the fact that all of them appeal, at some stage in the relevant line of argument, to the possibility of worlds containing qualitatively indis- ii cernible individuals. I argue that none of these objections ultimately succeed. Thus, I claim that a significant class of difficulties for the view is obviated. Lightly-edited versions of some of these chapters have already been accepted for publication. At the time of writing, Chapter 1 is forthcoming in Studia Leibnitiana; Chapter 2 (without the addendum) has appeared in Tomasz Bigaj and Christian Wuthrich’s¨ edited collection Metaphysics in Contemporary Physics (Amsterdam/New York, NY: Rodopi, 2015); and Chapter 3 is forthcoming in Ergo. Chapter 4 and the addendum to Chapter 2 are currently unpublished. This thesis is approximately 60,000 words long. iii Acknowledgements First and foremost, I wish to thank my current supervisors, Oliver Pooley and Simon Saunders. I am very grateful to both of them for reading my work so carefully, for being so generous with their time, and for their support. It has been a great privilege to know, and learn from, both of them. I am also grateful to several other Philosophy Faculty members at Oxford. Gonzalo Rodriguez-Pereyra provided the original inspiration and encourage- ment to work on Chapter 1, while Harvey Brown, Chris Timpson and David Wallace have each, in their own ways, extended their kindness and support over the years. Of the philosophy graduate students here at Oxford, Neil Dewar, Joel En- twistle, Niels Martens, Joshua Melville, Tushar Menon, Carina Prunkl, James Read, Alex Robertson, Teru Thomas, Aron Vallinder, and Andy Yu all deserve special mention for providing helpful feedback on various aspects of this thesis. There are many other philosophers outside of Oxford who deserve my thanks. In particular, I have greatly benefitted from discussions and comments relating to material in this thesis from Gordon Belot, Adam Caulton, Shamik Dasgupta, Sam Fletcher, James Fraser, Steven French, Nick Huggett, Dennis Lehmkuhl, Keizo Matsubura, Patrick McGovern, Ed Perez, Joe Rachiele, Jim Weatherall, and Alistair Wilson. Comments from anonymous referees for the journals Studia Leibnitiana and Ergo have also greatly helped improve Chapters 1 and 3 respectively. As this thesis’ external examiner, James Ladyman de- serves special mention here: together with David Wallace—this thesis’ internal examiner, and a person to whom I am also especially grateful—he provided wonderfully detailed and insightful feedback on numerous parts of the thesis. Jeff Russell also deserves special mention, but for a different reason: he super- vised me for my first two years as a DPhil student. Undoubtedly, much of what is good about this thesis I owe to him. Finally, very special thanks to my family, and in particular to my parents. Needless to say, this thesis is dedicated to them. Contents Abstract i Acknowledgements iii Introduction 1 1 Was Leibniz a Generalist? 10 1.1 Generalism vs Singularism . 12 1.2 Generalist and Singularist Pressures in Leibniz . 15 1.2.1 Generalist Pressures in Leibniz . 15 1.2.2 Singularist Pressures in Leibniz . 29 1.3 Weak Haecceitism to the Rescue? . 34 1.4 The Received View Defended . 46 1.4.1 The claim that Leibniz took de re modal claims at face value 47 1.4.2 The claim that Leibniz’s use of “switching” considerations in his correspondence with Clarke reveals that he was not a generalist . 52 1.5 Conclusion . 63 2 Symmetry and Qualitativity 64 2.1 More on Symmetry and Qualitativity . 67 2.1.1 Symmetry . 67 2.1.2 Qualitativity . 71 2.2 Symmetries Do Not Only Relate Qualitatively Indiscernible So- lutions . 77 2.3 Symmetry and the Nonqualitative . 87 2.3.1 The First Problem . 87 2.3.2 The Second Problem . 92 2.4 A New Notion of “Qualitative”? . 101 iv Contents v Addendum to Chapter 2 110 Caulton . 111 Dasgupta . 121 3 Weak Discernibility, Again 129 3.1 Methodology, Not Metaphysics . 132 3.2 Three Questions . 142 3.3 The Na¨ıve Question . 143 3.4 The Epistemological Question . 144 3.5 The Metaphysical Question . 148 3.6 Conclusion . 155 4 Some General Worries, Discharged 157 4.1 The Standard Objection . 158 4.2 The Na¨ıve Objection . 161 4.3 The Metaphysical Discontinuity Objection . 164 4.4 The Motivation Objection . 167 4.5 The Greavesian Objection . 171 4.6 Lewisian Determinism? . 176 4.7 The Kmentian Objection . 183 References 198 Introduction This thesis is not a research monograph. Instead, it is comprised of four chapters (plus an addendum to one of these chapters), each of which is intended to be readable independently of the others. The chapters are, however, united by a common theme — more precisely, by three interrelated sub-themes: symmetry, indiscernibility, and the generalist picture. Let me explain. First, symmetry. We are all familiar with the notion of symmetry in its standard, geometrical sense. For instance, rotations by 90◦ are symmetries of a perfect square; rotations by 45◦ are not. This notion of symmetry will prove salient in Chapter 4, where I consider worlds which display a kind of geometrical symmetry up to a given moment in time, but which fail, as a matter of physical law, to exhibit the same symmetry thereafter. For the majority of this thesis, however, I will be interested in a quite different notion of symmetry: dynamical symmetry. Dynamical symmetries are slightly more abstract than geometrical symmetries. In brief, they are transformations which map solutions to solutions of a particular physical theory — they “preserve the laws”, in a (somewhat) sim- ilar manner to which geometrical symmetries preserve the structure of a given geometrical object. A large part of this thesis will focus on such symmetries’ “invariantist” construal, according to which only that structure which is left invariant under them is genuinely real. This invariantist construal of symmetry 1 Introduction 2 plays a key role in Chapters 1 and 2 of this thesis; it plays a smaller but still significant role in Chapter 3 as well (see page 134, fn 4 below). Second, indiscernibility. This notion is a notoriously slippery one in philos- ophy. A number of different kinds of thing can be indiscernible; and things can be indiscernible in a number of different ways. In this thesis, I will talk both of worlds and of various things within them as being indiscernible. My main focus, however, will be on the notion of qualitative indiscernibility: that is, indis- cernibility by qualitative predicates alone. In particular, much of this thesis will focus on versions of a principle most famously advocated by Leibniz, namely the Principle