Thin Objects

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Thin 0 bj ects An Abstractionist Account

0ystein Linnebo

OXFORD UNIVERSITY PRESS

For my daughters Alma and Frida

Contents Preface

xi

Part I. Essentials 1. In Search of Thin Objects 1.1 Introduction

1.2 1.3 1.4 1.5 1.6 1.7

Coherentist Minimalism Abstractionist Minimalism The Appeal of Thin Objects Sufficiency and Mutual Sufficiency Philosophical Constraints Two Metaphysical "Pictures"

2. Thin Objects via Criteria of Identity 2.1 2.2 2.3 2.4 2.5 2.6

My Strategy in a Nutshell A Fregean Concept of Object Reference to Physical Bodies Reconceptualization Reference by Abstraction Some Objections and Challenges 2.6.1 The bad company problem 2.6.2 Semantics and metasemantics 2.6.3 A vicious regress? 2.6.4 A clash with Kripke on reference? 2.6.5 Internalism about reference 2.7 A Candidate for the Job 2.8 Thick versus Thin Appendix 2.A Some Conceptions of Criteria ofldentity Appendix 2.B A Negative Free Logic Appendix 2.C Abstraction on a Partial Equivalence

3. Dynamic Abstraction 3.1 3.2 3.3 3.4 3.5 3.6 3.7

Introduction Neo-Fregean Abstraction How to Expand the Domain Static and Dynamic Abstraction Compared Iterated Abstraction Absolute Generality Retrieved Extensional vs. Intensional Domains

3 3

s 7 9

11 13 17

21 21

23 26 30 33 37 38 38 39 40 41 42 4S

46 48 49 Sl Sl

S3 SS

60 61

64 66

viii

CONTENTS

Appendix 3.A Further Questions 3.A.l The higher-order needs of semantics 3.A.2 Abstraction on intensional entities 3.A.3 The need for a bimodal logic 3.A.4 The correct propositional logic Appendix 3.B Proof of the Mirroring Theorem

70 70 70 71 73 74

Part II. Comparisons 4. Abstraction and the Question of Symmetry 4.1 4.2 4.3 4.4

Introduction Identity of Content Rayo on "Just is" -Statements Abstraction and Worldly Asymmetry

5. Unbearable Lightness ofBeing 5.1 Ultra-Thin Conceptions of Objecthood 5.2 Logically Acceptable Translations 5.3 Semantically Idle Singular Terms 5.4 Inexplicable Reference Appendix 5.A Proofs and Another Proposition

6. Predicative vs. Impredicative Abstraction 6.1 The Quest for Innocent Counterparts 6.2 Two Forms of!mpredicativity 6.3 Predicative Abstraction 6.3.1 Two-sorted languages 6.3.2 Defining the translation 6.3.3 The input theory 6.3.4 The output theory 6.4 Impredicative Abstraction Appendix 6.A Proofs

7. The Context Principle 7.1 7.2 7.3 7.4 7.5 7.6

Introduction How Are the Numbers "Given to Us"? The Context Principle in the Grundlagen The "Reproduction" of Meaning The Context Principle in the Grundgesetze Developing Frege's Explanatory Strategy 7.6.1 An ultra-thin conception of reference 7.6.2 Semantically constrained content recarving 7.6.3 Towards a metasemantic interpretation 7.7 Conclusion Appendix 7.A Hale and Fine on Reference by Recarving

77

77 79 81 83

87 87 89 90 92 94 95 95

96 98 98 100 100 102 103

106 107 107 108 110 114 117

123 123 124 127 129 129

CONTENTS

ix

Part III. Details 8. Reference by Abstraction 8.1 Introduction 8.2 The Linguistic Data 8.3 Two Competing Interpretations 8.4 Why the Non-reductionist Interpretation is Preferable 8.4.1 The principle of charity 8.4.2 The principle of compositionality 8.4.3 Cognitive constraints on an interpretation 8.5 Why the Non-reductionist Interpretation is Available 8.6 Thin Objects Appendix 8.A The Assertibility Conditions Appendix 8.B Comparing the Two Interpretations Appendix 8.C Internally Representable Abstraction Appendix 8.D Defining a Sufficiency Operator

135 135 137 140 143 143 144 146 148 151 153 155 156 157

9. The Julius Caesar Problem 9.1 Introduction 9.2 What is the Caesar Problem? 9.3 Many-sorted Languages 9.4 Sortals and Categories 9.5 The Uniqueness Thesis 9.6 Hale and Wright's Grundgedanke 9.7 Abstraction and the Merging of Sorts Appendix 9.A The Assertibility Conditions Appendix 9.B A Non-reductionist Interpretation Appendix 9.C Defining a Sufficiency Operator

159 159 160 162 163 166 167 169 171 173 174

10. The Natural Numbers 10.1 Introduction 10.2 The Individuation of the Natural Numbers 10.3 Against the Cardinal Conception 10.3.1 The objection from special numbers 10.3.2 The objection from the philosophy oflanguage 10.3.3 The objection from Jack of directness 10.4 Alleged Advantages of the Cardinal Conception 10.5 Developing the Ordinal Conception 10.6 Justifying the Axioms of Arithmetic

176 176 176 178 179 180 181 182 183 185

11. The Question of Platonism 11.1 Platonism in Mathematics 11.2 Thin Objects and Indefinite Extensibility 11.3 Shallow Nature 11.4 The Significance of Shallow Nature 11.5 How Beliefs are Responsive to Their Truth 11.6 The Epistemology of Mathematics

189 189 191 192 195 197 201

X

CONTENTS

12. Dynamic Set Theory 12. l 12.2 12.3 12.4

Introduction Choosing a Modal Logic Plural Logic with Modality The Nature of Sets 12.4. l The extensionality of sets 12.4.2 The priority of elements to their set 12.4.3 The extensional definiteness of subsethood 12.5 Recovering the Axioms ofZF 12.5.l From conditions to sets 12.5.2 Basic modal set theory 12.5.3 Full modal set theory Appendix 12.A Proofs of Formal Results Appendix 12.B A Harmless Restriction

Bibliography Index

205 205 206 208 211 211

212 213

214 214 216 217

219 222 223 233

Preface This book is about a promising but elusive idea. Are there objects that are "thin" in the sense that their existence does not make a substantial demand on the world? Frege famously thought so. He claimed that the equinumerosity of the knives and the forks on a properly set table suffices for there to be objects such as the number of knives and the number of forks, and for these objects to be identical. Versions of the idea of thin objects have been defended by contemporary philosophers as well. For example, Bob Hale and Crispin Wright assert that what it takes for "the number of Fs = the number of Gs" to be true is exactly what it takes for the Fs to be equinumerous with the Gs, no more, no less.[ ... ] There is no gap for metaphysics to plug. 1

The truth of the equinumerosity claim is said to be "conceptually sufficient" for the truth of the number identity (ibid.). Or, as Agustin Rayo colorfully puts it, once God had seen to it that the Fs are equinumerous with the Gs, "there was nothing extra she had to do" to ensure the existence of the number of F and the number of G, and their identity (Rayo, 2013, p. 4; emphasis in original). The idea of thin objects holds great philosophical promise. If the existence of certain objects does not make a substantial demand on the world, then knowledge of such objects will be comparatively easy to attain. On the Fregean view, for example, it suffices for knowledge of the existence and identity of two numbers that an unproblematic fact about knives and forks be known. Indeed, the idea of thin objects may well be the only way to reconcile the need for an ontology of mathematical objects with the need for a plausible epistemology. Another attraction of the idea of thin objects concerns ontology. If little or nothing is required for the existence of objects of some sort, then no wonder there is an abundance of such objects. The less that is required for the existence of certain objects, the more such objects there will be. Thus, if mathematical objects are thin, this will explain the striking fact that mathematics operates with an ontology that is far more abundant than that of any other science. The idea of thin objects is elusive, however. The characterization just offered is imprecise and partly metaphorical. What does it really mean to say that the existence of certain objects "makes no substantial demand on the world"? Indeed, if the truth of "the number of Fs = the number of Gs" requires no more than that of "the Fs are

1 (Hale and Wright, 2009b, pp. 187 and 193). Both of the passages quoted in this paragraph have been adapted slightly to fit our present example.

xii

PREFACE

equinumerous with the Gs", perhaps the former sentence is just a fafon de parler for the latter. To be convincing, the idea of thin objects has to be properly explained. This book attempts to develop the needed explanations by drawing on some Fregean ideas. I should say straight away, though, that my ambitions are not primarily exegetical. I use some Fregean ideas that I find interesting in an attempt to answer some important philosophical questions. By and large, I do not claim that the arguments and views developed in this book coincide with Frege's. Some of the views I defend are patently un -Fregean. My strategy for making sense of thin objects has a simple structure. I begin with the Fregean idea that an object, in the most general sense of the word, is a possible referent of a singular term. The question of what objects there are is thus transformed into the question of what forms of singular reference are possible. This means that any account that makes singular reference easy to achieve makes it correspondingly easy for objects to exist. A second Fregean idea is now invoked to argue that singular reference can indeed be easy to achieve. According to this second idea, there is a close link between reference and criteria of identity. Roughly speaking, it suffices for a singular term to refer that the term has been associated with a specification of the would-be referent, which figures in an appropriate criterion of identity. For instance, it suffices for a direction term to refer that it has been associated with a line and is subject to a criterion of identity that takes two lines to specify the same direction just in case they are parallel. 2 In this way, the second Fregean idea makes easy reference available. And by means of the first Fregean idea, easy reference ensures easy being. My strategy for making sense of thin objects can thus be depicted by the upper two arrows (representing explanatory moves) in the following triangle of interrelated concepts: reference

/~

objecthood - - - - - - - - - . identity criteria (The lower arrow will be explained shortly.) My concern with criteria of identity leads to an interest in abstraction principles, which are principles of the form: (AP) 2

§a

= §{3 = a

"'

f3

Admittedly. we would obtain a better fit with our ordinary concept of direction by considering instead directed lines or line segments and the equivalence relation of "co-orientation': defined as parallelism plus sameness of orientation. We shall keep this famous example unchanged, however, as the mentioned wrinkle does not affect anything of philosophical importance.

PREFACE

xiii

where a and f3 are variables of some type, § is an operator that applies to such variables to form singular terms, and "' stands for an equivalence relation on the kinds of items over which the variables range. An example made famous by Frege is the aforementioned principle that the directions of two lines are identical just in case the lines are parallel. My preferred way of understanding an abstraction principle is simply as a special type of criterion of identity. How does my proposed route to thin objects compare with others explored in the literature? My debt to Frege is obvious. I have also profited enormously from the writings of Michael Dummett and the neo-Fregeans Bob Hale and Crispin Wright. As soon as one zooms in on the conceptual terrain, however, it becomes clear that the route to be traveled in this book diverges in important respects from the paths already explored. Unlike the neo-Fregeans, I have no need for the so-called "syntactic priority thesis': which ascribes to syntactic categories a certain priority over ontological ones. And I am critical of the idea of "content recarving': which is central to Frege's project in the Grundlagen (but not, I argue, in the Grundgesetze) and to the projects of the neo-Fregeans as well as Rayo. My view is in some respects closer to Dummett's than to that of the neo-Fregeans. I share Dummett's preference for a particularly unproblematic form of abstraction, which I call predicative. On this form of abstraction, any question about the "new" abstracta can be reduced to a question about the "old" entities on which we abstract. A paradigm example is the case of directions, where we abstract on lines to obtain their directions. This abstraction is predicative because any question about the resulting directions can be answered on the basis solely of the lines in terms of which the directions are specified. I argue that predicative abstraction principles can be laid down with no presuppositions whatsoever. But my argument does not extend to impredicative principles. This makes predicative abstraction principles uniquely well suited to serve in an account of thin objects. My approach extends even to the predicative version of Frege's infamous Basic Law V. This "law" serves as the main engine of an abstractionist account of sets that I develop and show to justify the strong but widely accepted set theory ZF. The restriction to predicative abstraction results in an entirely natural class of abstraction principles, which has no unacceptable members (or so-called "bad companions"). My account therefore avoids the "bad company problem': Instead, I face a complementary challenge. Although predicative abstraction principles are uniquely unproblematic and free of presuppositions, they are mathematically weak. My response to this challenge consists of a novel account of "dynamic abstraction': which is one of the distinctive features of the approach developed in this book. Since abstraction often results in a larger domain, we can use this extended domain to provide criteria of identity for yet further objects, which can thus be obtained by further steps of abstraction. (This observation is represented by the lower arrow in the above diagram.) The successive "formation" of sets described by the influential iterative conception of sets is just one instance of the more general phenomenon of

xiv

PREFACE

dynamic abstraction. Legitimate abstraction steps are iterated indefinitely to build up ever larger domains of abstract objects. Dynamic abstraction can be seen as a development and extension of the famous iterative conception of sets. A second distinctive feature of my approach is the development of the idea of thin objects. Suppose we speak a basic language concerned with a certain range of entities (say, lines). Suppose ~ is an equivalence relation on some of these entities (say, parallelism). Then it is legitimate to adopt an extended language in which we speak precisely as if we have successfully abstracted on ~ (say, by speaking also about the directions of the lines with which we began). I argue we have reason to ascribe to this extended language a genuine form of reference to abstract objects. Since these objects need not be in the domain of the original language, we can introduce yet another language extension, where we talk about yet more objects. In fact, there is no end to this process of forming ever more expressive languages. Some words about methodology are in order. I make fairly extensive use oflogical and mathematical tools. Formal definitions are provided, and theorems proved. I am under no illusions about what this methodology achieves. As Kripke observes, "There is no mathematical substitute for philosophy" (Kripke, 1976, p. 416). Definitions and theorems do not by themselves solve any philosophical problems, at least not of the sort that will occupy us here. The value of the formal methods to be employed lies in the precision and rigor that they make possible, not in replacing more traditional philosophical theorizing. But experience shows that precision matters in the discussions that will concern us. It is therefore scientifically inexcusable not to aspire to a high level of precision. In fact, much of the material to be discussed lends itself to a mathematically precise investigation. While the use of formal methods does not by itself solve any philosophical problems, it imposes an intellectual discipline that makes it more likely that our philosophical arguments will bear fruit. 3 A quick overview of the book may be helpful. Part I is intended as a self-contained introduction to the main ideas developed in the book as a whole. Chapter 1 sets the stage by introducing the idea of thin objects, explaining its attractions as well as some difficulties. This discussion culminates in a detailed "job description" for the idea of thin objects. This job description is formulated in terms of a notion of one claim sufficing for another-although the ontological commitments of the latter exceed those of the former. By formulating some constraints on the notion of sufficiency, I provide a precise characterization of what it would take to substantiate the idea of thin objects. Chapter 2 introduces my own candidate for the job. I explain the Fregean conception of objecthood and the idea that an appropriate use of criteria of identity can suffice to constitute relations of reference. Chapter 3 introduces the idea of dynamic abstraction. The form of abstraction explained in Chapter 2 can be iterated, 3

Compare (Williamson, 2007).

PREFACE

XV

resulting in ever larger domains. I argue that this dynamic approach is superior to the dominant "static" approach, both philosophically and technically. Part II compares my own approach with some other attempts to develop the idea of thin objects. I begin, in Chapter 4, by describing and criticizing some symmetric conceptions of abstraction according to which the two sides of an acceptable abstraction principle provide different "recarvings" of one and the same content. In Chapter 5, I explain and reject some "ultra-thin'' conceptions of reference and objecthood, which go much further than my own thin conception. One target is Hale and Wright's "syntactic priority thesis", which holds that it suffices for an expression to refer that it behaves syntactically and inferentially just like a singular term and figures in a true (atomic) sentence. The ultra-thin conceptions make the notion of reference semantically idle, I argue, and give rise to inexplicable relations of reference. The important distinction between predicative and impredicative abstraction is explained in Chapter 6. I argue that the former type of abstraction is superior to the latter, at least for the purposes of developing the idea of thin objects. Only predicative abstraction allows us to make sense of the attractive idea of there being no "metaphysical gap" between the two sides of an abstraction principle. Finally, in Chapter 7, I discuss a venerable source of motivation for the approach pursued in this book, namely Frege's context principle, which urges us never to ask for the meaning of an expression in isolation but only in the context of a complete sentence. Various interpretations of this influential but somewhat obscure principle are discussed, and its role in Frege's philosophical project is analyzed. Part III spells out the ideas introduced in Part I. I begin, in Chapter 8, by developing in detail an example of how an appropriate use of criteria of identity can ensure easy reference. Chapter 9 addresses the Julius Caesar problem, which concerns crosscategory identities such as "Caesar = 3''. Although logic leaves us free to resolve such identities in any way we wish, I observe that our linguistic practices often embody an implicit choice to regard such identities as false. Chapter 10 examines the important example of the natural numbers. I defend an ordinal conception of the natural numbers, rather than the cardinal conception that is generally favored among thinkers influenced by Frege. The penultimate chapter returns to the question of how thin objects should be understood. While my view is obviously a form of ontological realism about abstract objects, this realism is distinguished from more robust forms of mathematical Platonism. I use this slight retreat from Platonism to explain how thin objects are epistemologically tractable. The final chapter applies the dynamic approach to abstraction to the important example of sets. This results in an account of ordinary ZFC set theory. The major dependencies among the chapters are depicted by the following diagram. The via brevissima provided by Part I is indicated in bold.

Xvi

PREFACE

11

/ i i 9

12

i

8

'~f/10

I~' I

Many of the ideas developed in this book have had a long period of gestation. The central idea of thin objects figured prominently already in my PhD dissertation (Linnebo, 2002b) and an article (later abandoned) from the same period (Linnebo, 2002a). At first, this idea was developed in a structuralist manner. Later, an abstractionist development of the idea was explored in (Linnebo, 2005) and continued in (Linnebo, 2008) and (Linnebo, 2009b). These three articles contain the germs oflarge parts of this book, but are now entirely superseded by it. The idea of invoking thin objects to develop a plausible epistemology of mathematics has its roots in the final section of (Linnebo, 2006a). The second distinctive feature of this book-namely that of dynamic abstraction-has its origins in (Linnebo, 2006b) and (Linnebo, 2009a) (which was completed in 2007). Some of the chapters draw on previously published material. In Part I, the opening four sections of Chapter 1 are based on (Linnebo, 2012a), which is now superseded by this chapter. Section 2.3 derives from Section 4 of (Linnebo, 2005), which (as mentioned) is superseded by this book. The remaining material is mostly new. In Part II, Sections 4.2 and 4.3 are based on (Linnebo, 2014), and Section 6.2 on (Linnebo, 2016a). These two articles expand on the themes of Chapters 4 and 6, respectively. Chapter 7 closely follows (Linnebo, forthcoming). In Part III, Chapters 8, 10, and 12 are based on (Linnebo, 2012b), (Linnebo, 2009c), and (Linnebo, 2013), respectively, but with occasional improvements. Chapter 9 and Section 11.5 make some limited use of (Linnebo, 2005) and (Linnebo, 2008), respectively, both of which are (as mentioned) superseded by this book. There are many people to be thanked. Special thanks to Bob Hale and Agustin Rayo for our countless discussions and their sterling contribution as referees for Oxford University Press, as well as to Peter Momtchiloff for his patience and sound advice. I have benefited enormously from written comments and discussions of ideas

PREFACE

xvii

developed in this manuscript; thanks to Solveig Aasen, Bahram Assadian, Neil Barton, Rob Bassett, Christian Beyer, Susanne Bobzien, Francesca Boccuni, Einar Duenger B0hn, Roy Cook, Philip Ebert, Matti Eklund, Anthony Everett, Jens Erik Fenstad, Salvatore Florio, Dagfinn F0llesdal, Peter Fritz, Olav Gjelsvik, Volker Halbach, Mirja Hartimo, Richard Heck, Simon Hewitt, Leon Horsten, Keith Hossack, Torfinn Huvenes, Nick Jones, Frode Kjosavik, J6nne Kriener, James Ladyman, Hannes Leitgeb, Jon Litland, Michele Lubrano, Jonny Mcintosh, David Nicolas, Charles Parsons, Alex Paseau, Jonathan Payne, Richard Pettigrew, Michael Rescorla, Sam Roberts, Marcus Rossberg, Ian Rumfitt, Andrea Sereni, Stewart Shapiro, James Studd, Tolgahan Toy, Rafal Urbaniak, Gabriel Uzquiano, Albert Visser, Sean Walsh, Timothy Williamson, Crispin Wright, as well as the participants at a large number of conferences and workshops where this material was presented. Thanks to Hans Robin Solberg for preparing the index. This project was initiated with the help of an AHRC-funded research leave (grant AH/E003753/l) and finally brought to its completion during two terms as a Visiting Fellow at All Souls College, Oxford. I gratefully acknowledge their support.

PART I

Essentials

1

In Search of Thin Objects 1.1 Introduction Kant famously argued that all existence claims are synthetic.1 An existence claim can never be established by conceptual analysis alone but always requires an appeal to intuition or perception, thus mal to represent sameness of demands on the world, a mutual sufficiency statement

1/1 ensures that

) is inappropriate. Proponents of the asymmetric picture nevertheless insist that the right-hand side of a legitimate abstraction principle suffices for the left-hand side, for example: (Dir=>) How should this sufficiency statement be understood? It cannot straightforwardly be understood in terms of demands on the world. For according to the asymmetric picture, the demands of'd(l1) = d(/2)' exceed those of'/1 II 12'. Of course, a proponent of thin objects will deny that the demands of the former statement substantially exceed those of the latter-but this is itself in need of explanation. I find it useful to understand the desired notion of sufficiency as a species of metaphysical grounding. 30 That is, /1 1112 metaphysically explains d(/1) = d(/2); the directions are identical in virtue of the lines' being parallel. However, these remarks can at most serve to locate the desired notion of sufficiency in a broader philosophical landscape, not to define it. I would resist any identification of the notion of sufficiency with that of grounding, for two reasons. First, the notion of grounding is highly schematic. It is one thing to say that

a ,....., f3

Are we allowed to "export" the function terms ' §a' and ' §{J' so as to derive the following statement?

3x3y(x = §a /\ y

= §{3 /\ (x = y {:> a ,....., fJ))

I believe we should be wary of accepting the resulting statements-whether obtained by exportation or in some other way. For example, while the parallelism of two lines suffices for the identity of the corresponding directions, the converse sufficiency statement is problematic. Why should the self-identity of a certain direction suffice for the parallelism of two particular lines? While abstraction on any suitably oriented line yields the relevant directions, there is no way to "retrieve" any particular line from this direction. Abstraction is a one-way road. What distinguishes one line from any of

its parallels is irretrievably lost in the abstraction that takes us from a line to its direction. Compare now the asymmetric theorist's sufficiency statement associated with our abstraction principle, namely: (AP=})

a ,....., f3

=>

§a = §{3

Suppose we apply exportation to derive:

3x3y(x = §a /\ y

= §{3 /\ (a

,....., f3

=> x = y))

In this case, the resulting statements are independently plausible. For example, the parallelism of two lines suffices for the self-identity (and thus also, as we shall see, for the existence) of their direction. And in this case, there is no commitment to the problematic converse sufficiency statement. In short, on the asymmetric picture, a class of plausible de re sufficiency statements can be obtained by exportation from sufficiency statements already accepted. By contrast, on the symmetric picture, the de re statements that would result from exportation are problematic. This raises tricky questions about which de re statements

20

IN SEARCH OF THIN OBJECTS

of sufficiency or mutual sufficiency are acceptable on this picture and how this class of acceptable statements should be accounted for. The next order of business is to introduce my own version of the asymmetric picture. I do this in Chapter 2, while Chapter 3 shows how the asymmetric abstraction steps can be iterated. My argument against the symmetric picture is continued in Chapter4.

2 Thin Objects via Criteria of Identity 2.1 My Strategy in a Nutshell My strategy for developing a viable form of metaontological minimalism belongs to the same broad family as the abstractionist approach discussed in Chapter 1. The main source of inspiration is therefore Frege. But my aim is not primarily exegetical. I wish to explore and develop some Fregean ideas that may help us make sense of the idea of thin objects, regardless of whether Frege himself would have agreed with this development. My argument is structured around the Fregean triangle, which consists of three concepts central to philosophical logic and metaphysics-object, reference, and criterion of identity. Frege believed these concepts to be connected in deep and interesting ways. I develop explanations whose directions are represented by the arrows in the following diagram: reference

objecthood - - - - - - - - identity criteria What is an object? There is a certain audacity to the question. Can we possibly say anything general and informative about such a fundamental concept? As Frege observes, the concept is "too simple to admit oflogical analysis" (Frege, 1891, p. 140), and "what is logically simple cannot have a proper definition" (Frege, 1892, p. 182). Frege is no doubt right that a "proper definition" of the concept of an object is out of the question. Nevertheless, I believe an explication of the concept is possible. Even when a concept cannot be de.fined in more basic terms, it can still be glossed or characterized, for instance by relating it to other concepts and by explaining its role in our thought and reasoning. There can be no doubt that an explication of the concept of an object has the potential to be philosophically valuable. In the absence of some general account, our thinking about objects will be in danger of being distorted by intuitions formed in response to paradigm examples of objects, such as ordinary physical bodies. 1 This may dispose us to think of all objects as thick, that is, as making 1

Compare (Hale, 2013, ch. !).

22

THIN OBJECTS VIA CRITERIA OF IDENTITY

a substantial demand on the world. The explication of the concept of an object to be developed below pays particular attention to the role that objects play in our semantic theories. This leads us to the concept of reference. Singular terms and corresponding mental representations refer to objects with which they have no intrinsic connection. For instance, the inscription 'Socrates' refers to an ancient Greek philosopher, although neither the inscription nor the philosopher depends on or in any way resembles the other. Likewise, the current configuration of neurons in my brain allows me to entertain thoughts about the laptop computer directly in front of me. What endows ink marks on paper and configurations of neurons in brains with this power to refer to objects with which they have no intrinsic connection? That is, what does the relation of reference consist in? The problem is particularly serious in the case of abstract objects. There certainly appears to be such a thing as reference to abstract objects. Even a casual examination of natural language reveals lots of apparent examples. But it is not at all clear what such reference might consist in. It certainly cannot be based on any form of causal interaction. The final concept in our triangle is that of a criterion of identity, by which I mean a principle which in a systematic and informative way relates the identity or distinctness of a certain class of objects to certain other facts. For example, two sets are identical just in case they have the same elements. This is often taken to provide metaphysical information about what the identity or distinctness of the relevant objects "consists in''. 2 Criteria of identity are sometimes thought to do epistemological work as well, namely by underlying and guiding people's judgments about the identity and distinctness of objects. Why do I judge that the computer I am now seeing is the same as the one I saw a minute ago? And why do I identify the object that my fingers are currently touching with the object that I am currently seeing? A natural answer is that I am relying-perhaps unconsciously-on a criterion of identity for physical bodies that informs me that two parcels of matter belong to the same body just in case they are spatiotemporally connected in some suitable way. It is far from obvious that the three fundamental concepts that make up the Fregean triangle should be connected in any deep or interesting ways. But according to Frege, they are. He connects the first two concepts by characterizing an object as a possible referent of a singular term. It is important to notice that the singular term in question need not be available in our current language. The claim is rather that an object is the sort of entity that could be referred to by a singular term in some language or other. Next, Frege connects the last two concepts by suggesting that singular reference can be

explained in terms of the concept of a criterion of identity. This book develops and defends versions of these Fregean links and thus shows that the three concepts do indeed belong to a tightly integrated triangle. In this way

2

For discussion, see e.g. (Lowe, 2003) as well as Appendix 2.A below.

A FREGEAN CONCEPT OF OBJECT

23

I hope to shed light on the attractive idea of thin objects. My explanatory strategy is easy to describe now that the stage has been set. By explaining reference in terms of criteria of identity, we make reference comparatively easy to achieve, including reference to abstract objects. For example, it suffices for a direction term to refer that it has been associated with a line and is subject to a criterion of identity that takes two lines to specify one and the same direction just in case the lines are parallel. I show how this kind of sufficient condition for reference-based on a specification of the would-be referent and a criterion of identity operating on such specificationscan be generalized to cover a vast range of cases. Furthermore, because an object is characterized as a possible referent of a singular term, easy reference ensures easy being. The result is that the obtaining of an appropriate criterion of identity becomes sufficient for the existence of an object governed by this criterion. For example, two lines' being parallel is sufficient for the existence of a direction that is shared by the two lines. A truth about parallelism is thus reconceptualized in a way that reveals a new object, namely a direction, which was not involved in the original truth. 3 Finally, the new objects obtained by reconceptualization may enable us to formulate criteria of identity for yet more objects. The formation of sets provides an example. Given some domain, we can reconceptualize so as to "form'' sets of objects from this domain. This makes available more objects than those with which we began, and these objects can be used to specify yet further sets. In this way, our explanations can proceed around the Fregean triangle one more time. This form of iterated abstraction will be the focus of Chapter 3.

2.2 A Fregean Concept of Object Let us take a closer look at the Fregean links on which my strategy is based, namely between objecthood and reference, and between reference and criteria of identity. To understand the first of these links, it is useful to ask what role objects play in our semantic theories. 4 Like Frege, however, I am not concerned with the semantics of natural languages, which tends to be complex and quirky. It is better to focus on the semantics oflogically regimented languages, such as that of first- or higher-order logic, which systematically and perspicuously represents a logical core of natural languages. The concept of an object that we explore is therefore a logico-semantic one. Thus refined, our question has a clear answer. Objects serve as referents ofsingular terms and as values of bound first-order variables. Frege's characterization of an object as a possible referent of a singular term focuses on the former role. Objects are the

3

This reconceptualization must not be conflated with the notion of recarving of content. Where recarving is tied to the symmetric idea that one and the same worldly fact can be "carved up" as two different contents, reconceptualization (as I use the term) is tied to an asymmetric conception of abstraction; in particular, ontological commitments can increase as a result of reconceptualization. 4 Cf. (Dummett, 198la, ch. 14).

24

THIN OBJECTS VIA CRITERIA OF IDENTITY

kind of entity to which singular terms are suited to refer. 5 By contrast, Quine focuses on the latter role of objects as values of bound variables, as encapsulated in his famous slogan that "to be is to be the value of a variable''. 6 It may be objected that both the Fregean and the Quinean explication make the concept of an object excessively semantic. Even if there were no singular terms or variables, there would still have been objects. How, then, can the concept of an object be characterized in terms of the semantics of singular terms and variables? But the objection is based on a fairly elementary misunderstanding. It is true that both explications exploit the contingent fact that we have a language-and thus also singular terms and variables, or at least their natural language equivalents-to explicate one of our fundamental logico-metaphysical concepts. But nothing prevents the concept thus explicated from being applied to other possible circumstances, including ones where there is no language. As always, when evaluating counterfactuals, we keep the interpretation of our own language fixed in order to pronounce on how things would have been in alternative circumstances. Even if these circumstances involve there being no language, it is still permissible for us to rely on our languagecontingent though it may be-to pronounce on how things would have been under these circumstances. In particular, we can ask what entities of the sort to which singular terms are suited to refer can be found in these circumstances. So it is simply incorrect that the proposed semantic characterization of the concept of an object makes all objects language dependent. A more interesting question concerns the relation between the Fregean and Quinean explications of the concept of an object. Since the referent of a singular term can always serve as the value of a bound variable, it follows that an object in Frege's sense is also an object in Quine's sense. Somewhat surprisingly, however, the converse conditional turns out to be controversial. A problem with the converse arises in connection with mathematical structures with non-trivial automorphisms.7 Consider the imaginary units i and -i in complex analysis, which are swapped by the automorphism of complex conjugation. 8 If there really are such things as sui generis complex numbers-rather than just set-theoretic simulacra that exhibit the structure of the complex field-then it is hard to see how a singular term could refer to one of the imaginary units rather than the other. Nothing internal to the structure could

5

See also (Hale, 2013, ch. 1). By 'variable' Quine here means bound first-order variable, as these are the only variables he thinks can legitimately be bound. See (Quine, 1986). But the Quinean gloss on the concept of an object is independent of this: we just need to add that the slogan is concerned with bound first-order variables. 7 An automorphism is a permutation of the objects of the structure which leaves the structure intact. An automorphism is said to be non-trivial if it is distinct from the identity automorphism. Structures with non-trivial automorphisms have received much attention in the debate about non-eliminative or ante rem mathematical structuralism, where the phenomenon is used as the basis for an objection to the mentioned form of structuralism; cf. (Burgess, 1999) and (Keranen, 2001). My present concern is not with this objection but with the relation between the two mentioned explications of objecthood. 8 Complex conjugation maps the complex number a + bi to a - bi. 6

A FREGEAN CONCEPT OF OBJECT

25

ensure that the term refers to one rather than the other. For the existence of nontrivial automorphisms entails that the two units are structurally indiscernible, in the sense that every structural property that holds of one also holds of the other. And it is hard to see how facts external to the structure-whether mathematical or physicalcould be responsible for a reference relation of the sort in question. 9 The upshot is that the imaginary units appear not to qualify as objects in the Fregean sense: each unit appears not to be a possible referent of a singular term. 10 By contrast, it seems possible to quantify over the complex numbers, including the imaginary units, which would thus qualify as objects in the Quinean sense. This suggests that Quine's concept of object-if coherent-is strictly more general than Frege's. Quineans will no doubt regard this as an advantage of their explication. I see two lines of response to this argument. An uncompromising response would be to deny that proper sense can be made of quantification over objects to which singular reference is impossible. The standard truth-condition for a quantified formula is based on the successive assignment of each object in the domain to the variable bound by the quantifier. Can we really make sense of such an assignment where the domain contains objects to which singular reference is impossible? Notice, moreover, that a negative answer to this question would have no direct effect on the practice of mathematics. The practice of complex analysis, for instance, does not require that the complex field be regarded as a sui generis mathematical structure, as was presupposed in the argument under discussion. It suffices, for all practical purposes, to define the complex numbers as pairs of reals with operations of addition and multiplication, as in fact is the official definition in most textbooks. On this approach, we rule out all non-trivial automorphisms. Indeed, this observation generalizes. For any purported structure with non-trivial automorphisms, we can find an isomorphic copy based on pure sets, which permit no such automorphisms. 11 A far more ecumenical response is also possible. Recall that my intention is to use the Fregean concept of an object as part of an explication and defense of metaontological minimalism. All that this argumentative strategy requires is that being a possible referent of a singular term suffices for objecthood. This establishes one of the links associated with the Fregean triangle. There is no need to insist on the necessity of this Fregean criterion. Philosophers who prefer the Quinean conception are invited to develop their own account of lightweight objects. If they succeed, their account would supplement mine, not threaten it. I prefer this second response because of its smaller commitments. Thus, although my arguments in this book rely on the Fregean concept of an object as the kind of entity to which a singular term is suited to refer, I happily leave the door open 9

Indeed, non-eliminative or ante rem structuralists, who are among the foremost defenders of

sui generis complex numbers, deny that the units have any mathematical properties other than their structural ones. 10 See (Brandom, 1996) for relevant discussion. 11 See (Parsons, 2004, Section IV) for a discussion of this response.

26

THIN OBJECTS VIA CRITERIA OF IDENTITY

to developments of the Quinean concept of an object that might supplement my own account.

2.3 Reference to Physical Bodies I turn now to the second link, which connects the concepts of reference and criteria of identity. The idea, we recall, is that criteria of identity provide an easy route to reference of singular terms. Combined with the link already established, this will ensure easy existence of objects. How should the second link be developed? Since the Fregean ideas we are discussing are very abstract and theoretical, I begin with an example. In later sections, I describe the form of reconceptualization involved in the second link and explain how our example can be generalized. The example concerns the simplest and most direct form of reference to ordinary physical bodies, such as sticks and stones, and tables and chairs. (A more detailed characterization of the concept of a physical body will be provided shortly.) Let me state straight away that my aim is a toy model of such reference, not an account that captures every aspect of this immensely complex problem. What we need is a model that explains how a criterion of identity can play a role in the constitution of such reference. Since our aim is merely a toy model of reference to physical bodies, it is useful to carry out our investigation in terms of robots that are embedded in, and interacting with, a physical environment. This allows us simply to stipulate how the robots function, rather than to engage in speculation about the intricacies of actual human psychology. 12 We can simplify our task further by focusing on the senses of sight and touch, and on some very fundamental cognitive processes. Other senses, such as smell, taste, and hearing, appear to play a less fundamental role in reference to physical bodies and are therefore set aside in our toy model. Consciousness too-in the sense of subjective awareness of what it is like to have various sorts of experiences-is put to one side, as it appears inessential to the core notion of reference. So consider a robot equipped with senses of sight and touch. Such a robot interacts with its environment by detecting light reflected by surrounding surfaces and by having a capacity for touching and grasping things in its vicinity. Our question is what it takes for such a robot to refer to physical bodies in its environment. At the very least, the robot must "perceive" the body, in the very undemanding sense that it receives light from part of its surface or touches some part of it. The robot thus receives information from some part of the body. These parts need not have natural boundaries in either space or time; they are simply the sumtotals of the physical stuff with which the robot causally interacts in this rudimentary perception-like way. While these simple causal interactions are certainly necessary for 12 My account is inspired, however, by the account of infants' reference to physical bodies defended by some developmental psychologists; see note 14.

REFERENCE TO PHYSICAL BODIES

27

the robot to refer to bodies, they are not sufficient. The rudimentary perception-like interactions with parcels of matter enable the robot to develop an internal map of the distribution of matter in its environment and cif the purely qualitative properties instantiated by various parcels of matter. But such interactions alone do not enable the robot to engage with any questions involving identity, such as to determine whether two parcels of matter belong to one and the same physical body or to count the number of bodies in its environment. For the robot to be engaged with questions such as these, it needs to possess some more sophisticated capacities, involving the representation of space, time, continuous trajectory, and physical cohesiveness. Only then will the robot be capable of determining whether two parcels of matter belong to one and the same body. The characterization of the robot's additional capacity requires some care. We are surrounded by bodies that are wholly or partially hidden, and that move in and out of view. A stick can be partially buried, and a stone can be mostly covered by other stones piled up around it. There are always many different ways of "getting at" one and the same physical body, both from different spatial points of view and at different moments of time. All these situations have to be handled by the robot's capacity for grouping together parcels of matter as belonging to one and the same body. What ultimately matters is that the parcels of matter with which the robot interacts through its rudimentary forms of perception are spatiotemporally connected in some appropriate way. Assume for instance that the robot establishes visual contact with part of a stick that emerges from the ground and that one of its "arms" simultaneously probes into the ground nearby and encounters something hard. What should we "teach" the robot about the conditions under which the two parcels of matter with which it interacts belong to the same body? Roughly, the kind of connectedness that matters has to do with solidity and motion. The two parcels of matter must be related through a continuous stretch of solid stuff, 13 all of which belongs to the same unit of independent motion-roughly in the sense that, if you wiggle or pull one of the parcels, the other one follows along or at least has some disposition to do so. To produce a more precise answer, let us continue our investigation as an exercise in robotics. I submit that the following fundamental principles are part of an analysis of the concept of a physical body, and therefore have to be implemented in the robot: 14 (Bl)

Bodies are three-dimensional solid objects.

13 I mean "solid" in the ordinary, loose sense in which a stick or a stone is said to be solid. Of course, physics tells us that even sticks and stones aren't solid in the stricter sense of filling up all space at an atomic level. 14 These principles are also constitutive of the concept of what psychologists sometimes call "Spelkeobjects''. This concept corresponds closely to the concept of physical body that I am employing here. See e.g. (Spelke, 1993) and (Xu, 1997). See also (Burge, 2010, pp. 437-71) for a useful discussion and further references.

28

THIN OBJECTS VIA CRITERIA OF IDENTITY

Thus, a cloud of gas doesn't qualify as a body in the present sense. This means that not all spatiotemporal objects are bodies. (B2)

Bodies have natural and relatively well-distinguished spatial boundaries.

For instance, the (undetached) lower half of a rock fails to be a body because it lacks sufficiently natural boundaries, and a mountain fails because its boundaries are insufficiently well distinguished. 15 (B3)

Bodies are units of independent motion.

Thus, although a book is a body, a pile of papers is not. Bodies need to be cohesive enough to be disposed to move as a unit. (B4)

Bodies move along continuous paths.

Consider the object that came into being with the birth of Bill Clinton, coincided with Clinton until the end of his presidency, and thenceforth coincides with George W Bush. By (B4), this object cannot be a body. Bodies move along continuous paths. (BS)

Bodies have natural and relatively well-distinguished temporal boundaries.

So arbitrary temporal parts of bodies are not themselves bodies. I believe this relatively simple model captures the core of the most direct, perceptual way of referring to physical bodies. What matters is that our agents-whether humans or robots-receive sensory information from parcels of matter and have a capacity for grouping together such pieces of information just in case these pieces derive from parcels that are spatiotemporally connected in the way just outlined. Let ,...., be the relation that obtains between two parcels of matter just in case they are spatiotemporally connected parts of a cohesive and reasonably well-delimited whole. This relation is clearly symmetric. For if two parcels of matter u and v specify the same body, then so do v and u. A similar defense can be given of transitivity. We should not require that the relation be reflexive, however, since not every parcel of matter belongs to a coherent and well-delimited whole. For instance, if I point to some partially solidified mud in a field, I will probably fail to specify a unique physical body. Were it not for the possibility of failures of reflexivity, it would have been straightforward to formulate a criterion of identity for bodies, namely (CI-B)

b(u)

= b(v) *+

u

rv

v

15 Precisely how well distinguished must the boundaries of a body be? Presumably, a shedding cat is still a physical body despite all the hairs that are in the process of falling off it. Although I doubt that our question admits a precise answer, I am hopeful that an approximate answer can be given by empirical investigation of ordinary people's concept of a body.

REFERENCE TO PHYSICAL BODIES

29

where b is a "body building" function that maps a parcel of matter to the body to which it belongs. How should we accommodate the possible reflexivity failures, however? The question requires a brief logical digression, which this paragraph will provide. A relation that is symmetric and transitive but not necessarily reflexive is known as a partial equivalence relation. Such a relation does to the objects in its field 16 what an equivalence relation does to the entire domain: it partitions the relevant objects into disjoint and mutually exhaustive equivalence classes. In this way, the partial equivalence relation "" on parcels of matter determines a partial function b that maps a parcel u to the physical body, if any, of which u is a part. This suggests a way to tweak (CI-B). We can accommodate reflexivity failures by letting the "body builder" b be a partial function and by letting (CI-B) operate in the context of a negative free logic. 17 This allows b(u) = b(u) to be false for any argument u for which the function bis undefined-which comes to the same thing as an argument u on which "" fails to be reflexive. With this understanding of (CI-B) in place, let us return to our main discussion. I claim that it suffices for our robot to refer to a body that the robot is appropriately related to some parcel of matter and that it treats two such parcels as specifications of the same body just in case they are related by the appropriate partial equivalence relation. There is no more direct way for it to 'get at" a physical body. The most direct form of reference to bodies is constituted on the basis of a specification and a partial equivalence relation that provides a criterion of identity. Thus, our toy model of reference to physical bodies nicely illustrates my general claim that there is a tight connection between reference and criteria of identity. By operating with an appropriate criterion of identity, the robot comes to refer to physical bodies in its environment. It may be objected that this toy model does not even approximate what is going on in us humans because bodies are directly given to us in perception in a way that bypasses the need for any criterion of identity. As a phenomenological point, this is no doubt correct. In our ordinary perception of bodies, there is no need to actively and consciously apply a criterion of identity. This does not contradict my claims, however. A lot of subpersonal processing takes place between the stimulation of our sense organs and what is phenomenologically given to us in perception. What the toy model suggests is that these subpersonal processes must involve a criterion of identity in order for the resulting representation to refer to a physical body. I do not claim that this process is accessible to our consciousness or ever rises above the subpersonal level. On the contrary, the lack of conscious or explicit access to the process is something that

16 Recall that x is said to be in the field of a relation R just in case x bears R to, or is borne to R by, some object or other. 17 This logic is explained in Appendix 2.C, which also provides an elaboration of this and some other approaches to abstraction on partial equivalence relations.

30

THIN OBJECTS VIA CRITERIA OF IDENTITY

I welcome. 18 This view represents a major departure from most earlier philosophical work on criteria of identity. 19

2.4 Reconceptualization My account of the constitution of reference to physical bodies has a reductionist character that may be surprising. I set out to explain how things have to be for a robot to refer to a physical body. But my explanation mentions only parcels of matter with which the robot causally interacts and a partial equivalence relation that the robot uses to determine when two such parcels should be regarded as parts of one and the same body. There is no mention of the body that is the referent! Although bodies figure in our explanandum-namely, how reference to physical bodies is constituted-bodies are strangely absent from the proposed explanans. It is as if we were offered an account of the constitution of marriage which leaves out one of the spouses! I believe the key to understanding the surprising reductionist character of our example-and of a vast family of generalizations described in the next section-is the idea of reconceptualization. The material made available by the explanans is reconceptualized in a way that brings out the existence of bodies. This reconceptualization is possible because there is not a unique right way to apply the Fregeau concept of an object to reality. We can always reconceptualize by applying this concept in a new and different way. Of course, these claims are highly programmatic and need to be spelled out and defended. The present section makes a start by locating the approach to be developed in the book as a whole within a broader philosophical landscape. Our question is how the Fregeau concept of an object is applied to reality. Suppose we want to introduce new singular terms referring to a range of objects that we have not yet recognized. What does it take to succeed? A minimal requirement is that truthconditions have been assigned to all identity statements and other predications that involve the new singular terms to be introduced. Moreover, this assignment must be done in a way that respects the laws of logic. The most important part of logic in this connection is what we may call the logic of identity, which describes the identity relation and its interaction with predication. In brief: identity is an equivalence relation that interacts with predication as described by Leibniz's Law, namely, if x is identical with y, then x and y have precisely the same properties: (LL)

x =

y

-+ ( ....,rp*(j(a1), ... ,f(an))

Notice that in all these formulas, the statement on the left-hand side of the sufficiency operator is concerned only with the "old" entities with which we began, which are accepted by all parties, while the statement on the right-hand side is concerned with "new" objects that are obtained from the "old" entities by abstraction. 41 40 A general definition is provided in Chapters 8 and 9. Beginning in Chapter 3, we shall iterate the abstraction-based language extensions just described. This makes it necessary to define a more general notion of mediate sufficiency (contrasted with the immediate notion just sketched). We say that rp mediately suffices for i{I just in case there is a series of permissible reconceptualizations that takes us from rp to i{I by a chain of relations of immediate sufficiency. 41 As discussed in Section 1.7, my notion of sufficiency is closely related to a form of metaphysical grounding. Let me now be more specific. Suppose at some stage of extending the interpretation of our language we have established the truth of the left-hand side, but not of the right-hand side, of one of the displayed sufficiency statements. Then a further language extension establishes the truth of the right-hand side as well, in such a way that the left-hand side serves as a (strict, full, and immediate) ground for the

44

THIN OBJECTS VIA CRITERIA OF IDENTITY

I contend that this notion of sufficiency is a good candidate for our job-but hasten to add that a final assessment will have to await the detailed development of my account in later chapters. To defend my contention, let us consider the constraints that make up the job description. First, we required that there be true sufficiency claims cp ==> 1/1 where the ontological commitments of 1/1 exceed those of cp. This constraint is clearly satisfied, as shown by the examples involving directions. 42 Second, we required that the sentences flanking the sufficiency operator can be taken at face value; in particular, that the statement on the right-hand side is not just a faron de par/er for the one on the lefthand side. If my account works at all, this constraint will be satisfied: for the account was designed specifically to show how reference to the new objects that figure on the right-hand side of the sufficiency operator is constituted. We now come to the epistemological and explanatory constraints. Roughly, these constraints require that for any true sufficiency claim, the corresponding material conditional must enjoy-or potentially enjoy-a privileged epistemological and explanatory status. The material conditional must be within our epistemic reach and admit of an adequate metaphysical explanation. Are these constraints satisfied? A proper answer would require a careful analysis of the relevant kinds of epistemic and explanatory status. I have no such analysis to offer and anyway prefer my account to be independent of the finer details of any such analysis. For now, I therefore content myself with observing that we are in a good position to show that the constraints are satisfied. Consider, once again, the case of directions. I explained how the criterion of identity for directions can be used to effect a reconceptualization, where for example the statement 11 II 12 is reconceptualized as d(lt) = d(l2 ). I also argued that, because of its predicative character, this reconceptualization carries no metaphysical or epistemological presuppositions. When we use the extended language with direction vocabulary in the described way, we reconceptualize reality in a novel way, and there is no sense in which reality might fail to cooperate. Moreover, there is nothing about this reconceptualization that is deeply hidden from us. By appropriate reflection, we can therefore come to understand how it works. 43 All that remain are the logical constraints from Section 1.5. These constraints are shown to be satisfied in Chapters 8 and 9 (in a two- and one-sorted setting, respectively).

right-hand side. In this way, the sufficiency statements can be seen as recording grounding potentials: if the left-hand side of such a statement becomes "available" before the right-hand side, then the former provides a (strict, full, and immediate) ground of the latter. 42 Recall that we are relying on an ordinary Quinean notion of ontological commitment. A non-Quinean alternative will be discussed, and rejected, in Section 4.4. 43 I have more to say about the epistemology of abstraction in Chapter 11.

THICK VERSUS THIN

45

2.8 Thick versus Thin In closing, I wish to return to the distinction between thick and thin objects. How, exactly, does the existence of physical bodies make a more substantial demand on the world than the existence of natural numbers? We now have the resources to answer the question. On the account I have developed, the question of what it takes for an object a to exist is a matter of what it takes for there to be a specification of a which is in the field of the relevant unity relation. Some examples will help to convey the idea. Consider a physical body a, say the table at which I am sitting. Then the candidate specifications of a are the spatiotemporal parts of the table, and the unity relation is that of belonging to the same cohesive and naturally bounded whole. Consider any spatiotemporal part u of a. For this specification to succeed, u must belong to a cohesive and naturally bounded whole. In contemporary metaphysical jargon, u must be a part of some stuff arranged "tablewise". The upshot is that the existence of my desk makes a substantial demand on a particular region of spacetime-just as one would expect. Other objects are thinner. For a direction to exist, it suffices that there be an appropriately oriented line. This is far less demanding than what we found in the case of physical bodies. There is no requirement on any particular region of spacetime, since the witnessing line could be located anywhere. In fact, I later argue that the existence of a direction does not require that there must actually exist a line that instantiates it; it suffices that there possibly exists such a line. 44 In general, abstract objects are thinner than concrete objects because they do not make demands on any particular region of spacetime. Surprisingly, it turns out that not all abstract objects are equally thin. To see why, let us follow (Parsons, 1980) and say that an object is pure abstract if it lacks both spatiotemporal location and any kind of intrinsic relation to space or time. The natural numbers and pure sets are examples. Let us then say that an object is quasiconcrete if it lacks spatiotemporal location but nevertheless has canonical realizations in spacetime (and only there). Letters and geometrical figures provide examples, as these have canonical realizations in the form of tokens and concrete figures with the shapes in question. The surprising discovery is that quasi-concrete objects are somewhat thicker than pure abstract objects. The existence of a quasi-concrete object makes a non-trivial demand on spacetime, however weak and indirect: there must be, or at least possibly be, concrete realizations of the object somewhere or other in space and time. There are presumably no quasi-concrete twenty-dimensional geometrical figures because spacetime is not rich enough to allow realizations of such figures. Nor are there quasi-concrete Euclidean triangles; for the spacetime that we inhabit is curved, not Euclidean. (Of course, I do not deny that such figures and triangles exist 44

See Section 11 . 1.

46

THIN OBJECTS VIA CRITERIA OF IDENTITY

as pure abstract objects.) Far more dramatic conclusions would follow if spacetime turned out to be granular, as suggested by some recent physical speculations. Then there would be surprisingly few quasi-concrete geometrical figures. There is an interesting connection between how thin an object is and the modal robustness of its existence. The less that the existence of an object demands of the world, the more modally robust the object is. Since my desk makes a substantial demand on the world, for example, there are nearby circumstances in which it fails to exist: the carpenter who made the desk might instead have used the wood to make some chairs. So my desk is modally fragile. A pure abstract object, by contrast, makes no demand on the world and therefore exists by necessity. The cardinal number 0 would have existed however the circumstances had been, as there would still have been empty collections from which this number could be obtained by abstraction. Quasiconcrete objects are intermediate as concerns the modal robustness of their existence. A linguistic type, for example, would still have existed in any circumstances in which it could be instantiated. Its existence thus depends on the existence of a certain sort of space, but not on anything specific about the existence or organization of matter in this space.

Appendices 2.A Some Conceptions of Criteria of Identity Philosophers have defended some very different conceptions of criteria of identity. I shall now compare and contrast my own favored conception with two important alternatives. Some philosophers regard criteria of identity as metaphysical principles, which provide information about the nature of the objects in question. For instance, the criteria of identity for sets and directions say something important about the nature of sets and directions, respectively; and an account of personal identity or the persistence of ordinary material objects over time would provide valuable information about such objects. Some philosophers even claim that criteria of identity explain "what grounds the identity and distinctness" of objects. 45 An example frequently used to illustrate this claim are sets, whose identity or distinctness is said to be "grounded" in accordance with the principle of extensionality. Another view regards the principal interest of criteria of identity as epistemic. For instance, (Geach, 1962, p. 39) describes a criterion of identity as "that in accordance with which we judge as to the identity" of certain objects. How do we find out whether two encounters with an object are encounters with one and the same object? Assume, for instance, that you and I both extend our right arms to demonstrate a direction. Then the criterion of identity for directions informs us that the directions we have demonstrated are identical just in case our arms are "co-directed" (where two directed items are said to be "co-directed" when they are not just parallel but also oriented in the same way) .

45

See e.g. (Lowe, 2003).

APPENDICES

47

Finally, the conception defended in this chapter focuses on the role that criteria of identity play in a metasemantic account of reference. 46 What is it about the world that makes a certain expression or representation refer to a certain physical body or abstract direction, as opposed to all of the objects to which it might have referred? I have argued that criteria of identity play an essential role in the answer. The three conceptions of criteria of identity differ radically in what they emphasize. My own view is that, while criteria of identity do have metaphysical and epistemological aspects, it is advantageous to regard them first and foremost as metasemantic principles. This view allows us to avoid some difficulties associated with the epistemic and metaphysical conceptions. Let us start with the epistemic conception. Imagine that you on two different occasions are presented with an object. How can you find out whether the objects with which you were presented are identical or distinct? The means by which you can find out depend in part on facts about reference and the nature of the objects in question-that is, on the matters that are emphasized by the metasemantic and metaphysical conceptions. If you are referring to sets rather than to properties, for example, it will be appropriate for considerations of coextensionality to figure in your deliberations. Or, if you are referring to persons and some sort of bodily continuity provides the right criterion of personal identity, then this will have consequences for your ways of answering the question. To a limited extent one may even proceed in the opposite direction and glean information about the nature of the objects in question from the means available for answering questions about identity and distinctness. If we know that coextensionality is such a means, this suggests we are talking about sets rather than properties. All this is unsurprising. The epistemic conception understands "criterion'' as a mark by which we identify some phenomenon, or a symptom. And clearly, the marks or symptoms of any phenomenon will depend in large part on the nature of the phenomenon, and to a limited extent, may even shed light on the phenomenon itself. If a phenomenon has enough of a nature to admit of a systematic investigation, however, it is misguided to focus on the marks rather than on the phenomenon itself. Why focus on the symptoms rather than on the underlying disease? Doing so can only hamper the investigation. This general point applies to criteria of identity as well. Our means for finding out about identity and distinctness are exceedingly complex and messy. Co!lsider our identification of people we have met. The vast majority of our knowledge of such people is based not on tracing bodies through space and time (or, for that matter, an investigation of their memories) but on a wealth of indirect evidence involving looks, utterances, and a vast variety of other cues. Any attempt to identify a set of canonical ways of finding out about the identity and distinctness of a certain class of objects will only serve to shift the focus back to where the two competing conceptions prefer to locate it. The metaphysical conception of criteria of identity faces difficulties of its own. What distinguishes a criterion of identity, on this conception, from other metaphysical truths? A criterion of identity must presumably have a logical form of the sort represented by (CI-1) or (CI-2) above. And the criterion must presumably be metaphysically necessary. Let minimalism be the metaphysical conception of criteria of identity that comprises just these two requirements.47

46

Similar ideas are found in (Dummett, 198la) and (Dummett, 199la). This view is defended in (Horsten, 2010). (Minimalism about criteria of identity must not be confused with what I have called metaontological minimalism.) 47

48

THIN OBJECTS VIA CRITERIA OF IDENTITY

Although the minimalist conception is appealingly simple and well understood, it is insufficient to distinguish criteria of identity from other necessary truths. The account is unable, for example, to explain why we should prefer extensionality as the criterion of identity for sets to the principle that two sets are identical just in case they are elements of the exactly same sets, or in symbols: SET(x) /\ SET(y) ~ (x

= y ++ 'v'u(x E u ++ y E u))

After all, this principle too is a metaphysical truth concerning the identity of sets. In fact, this example is just the tip of the iceberg. It turns out that any necessary truth can be made to follow logically from a criterion of identity in the minimalist sense. Consider a criterion of the form (CI-1), and assume that there are at least two objects of the relevant kind F. (The case of (CI-2) is analogous.) Let . . . ,Xn):::} 1p*(§x1, . .. ,§x n)

= §x2

-.1p(X1> ... ,Xn):::} -.1p* (§X1> . . . , §xn)

Proof sketch. (a) is immediate from the second clause of our definition. For (b) , consider such that first the case where N+ Fa T ( 1/1) :::} 1/1 for some variable assignment a for a (y;) = §a (.xj) for any y; that occurs free in 1/1. Let p be the restriction of a to .Co. Assume that N Fp T(1f!). Since T(1f!) is in .Co, this means that I Fp T(1f! ) . Hence by Proposition 8.3, we have R Fa' 1/1 for every variable assignment a ' for .C1 based on Do such that a ' (x;) = p(x;) and a ' (y;) = p (xj) for each i E w. We now apply Proposition 8.2 to establish N F§a' 1/1. Since §a ' agrees with a on any y; that occurs free in 1/1, our proof of this particular case is done. The

.ct

general claim now follows by observing that N validates Leibniz's Law. Finally, (c) is immediate from our definition. -l It is important to understand the significance of clause (a). This clause ensures that it is safe to quantify into the position occupied by §t in the sufficiency statement so as to form a corresponding de re sufficiency statement. For example, we may infer as follows (by so-called "exportation"): xi ~ xz :::} §x1

= §xz ,

therefore

This makes good on a promise made in Section 1.7.

3y1 (xi ~ x2 :::} Y1

= §x2)

9

The Julius Caesar Problem 9.1 Introduction Is the natural number 3 identical with the Roman emperor Julius Caesar? Some peculiar questions of this sort are raised in Frege's Grundlagen. This is the so-called Julius Caesar problem. 1 There are two kinds of reaction one may have to the opening question. One reaction is that such questions are not only pointless but downright meaningless. However much arithmetic you learn, no answer to the opening question will be forthcoming. Arithmetic tells you that 3 is the successor of 2 and that it is prime, but not whether it is identical with Caesar. Why does our opening question not receive an answer? The reason, it seems, is not that the question is hard but that it simply has not been "provided for''. The other reaction one may have is that the opening question is to be answered negatively. Since the number 3 is an abstract object and Caesar is not, it follows immediately by Leibniz's Law that they are distinct. In light of these two kinds of reaction, what are we to say about "mixed" identi,ty statements such as the one with which we began? An influential response due to Bob Hale and Crispin Wright attempts to do justice to both of the mentioned reactions: Within a category, all distinctions between objects are accountable by reference to the criterion of identity distinctive of it, while across categories, objects are distinguished by just that-the fact that they belong to different categories. (Hale and Wright, 200la, p. 389)

On this view, our ontology divides into categories, each with its own criterion of identity for the objects in the category. 2 For instance, the identity or distinctness of natural numbers is determined by considerations of equinumerosity, whereas in the case of physical bodies, we appeal to facts about spatiotemporal continuity. However, since natural numbers and physical bodies belong to different categories, the statement that 3 is identical with Caesar is not determined as true or false in

1 Actually, Frege never raises this particular question. However, since the first such question he discusses involves Caesar (Frege, 1953, §56), this Roman emperor has become the stock example and has lent his name to the problem. 2 Hints of this view are found in (Dummett, !98la) and (Dummett, 199la). (Perry, 2002) and (Williamson, 1990, ch. 9) contain more explicit anticipations.

160

THE JULIUS CAESAR PROBLEM

any such way. 3 So the first reaction-that there is something amiss with the question of whether 3 is identical with Caesar-is correct when restricted to the canonical grounds for identity and distinctness that are provided by a criterion of identity: no such ground exists for either the identity or the distinctness of 3 and Caesar. But according to Hale and Wright, objects belonging to different categories are ipso facto distinct; in particular, 3 is distinct from Julius Caesar. So the second reaction is correct when all grounds for the identity or distinctness of objects are taken into account. The plan for this chapter is as follows. After an initial clarification of the Caesar problem, I consider and reject the response that all mixed identities can be dismissed as meaningless. This response fails to do justice to the second of the mentioned reactions. Then I turn to the response by Hale and Wright outlined above. I have considerable sympathy for this response and share the broadly Fregean approach to reference and objecthood on which it is based. But I deny that there is any strong logical or conceptual support for the claim that every object belongs to a unique category. 4 I show that it is logically coherent for categories to overlap. It follows that, when we develop our linguistic practices, we have some degree of choice about whether or not to allow categories to overlap. To handle mixed identity statements, we often need conceptual decisions, not just factual discoveries. This analysis has important consequences for how the Caesar problem should be approached. My analysis emphasizes the importance of choice and the insufficiency of always looking to the world for an answer. When our ancestors first confronted Caesar-style questions, they had a choice which way to go; and this choice played a role in shaping the concepts that they thereby forged. Today we find ourselves in a different situation, since many choices are already implicit in the linguistic practices that we have inherited. Of course, insofar as we are willing to revise these practices, we still have the same choice as our ancestors had. But we face an important additional question not encountered by our pioneering ancestors, namely what conceptual decisions are implicit in our inherited linguistic practices. I shall argue that these practices have by and large legislated against the overlap of categories. But exceptions are certainly possible and very likely even actual.

9.2 What is the Caesar Problem? The Caesar problem arises in the Grundlagen when Frege observes that the direction abstraction principle (Dir) does not

3 I shall henceforth assume that persons belong to the category of physical bodies. My purpose in doing so is merely to simplify the discussion, not to make any controversial metaphysical claim. Readers are free to substitute their own favorite view of persons and adjust tbe argument of this chapter accordingly. 4 In fact, Hale and Wright too express some reservation at tbe end of the article where the quoted passage occurs.

WHAT IS THE CAESAR PROBLEM?

161

decide for us whether England is the same as the direction of the Earth's axis-if I may be forgiven an example which looks nonsensical. Naturally no one is going to confuse England with the direction of the Earth's axis; but that is no thanks to our definition of direction. (Frege, 1953, §66) This passage may suggest that the Caesar problem is purely internal to Frege's program or its abstractionist successors. Of course we all know that England is distinct from the mentioned direction. But since this cannot be derived from the relevant abstraction principle, the principle must be deemed inadequate as a "definition of direction': This reaction is unwarranted. The Caesar problem is both deeper than it initially appears and more general. Let us start with its depth. How do we actually come to know that Caesar and the natural number 3 are distinct? A natural answer is that this knowledge is based on an application of Leibniz's Law to the prior knowledge that one of the objects has a property that the other lacks. For instance, Caesar was assassinated in 44 BC, whereas the number 3 was not. But this is too quick! How do we actually know that the number 3 was not assassinated and perhaps lives on in a different guise? It is hard to see how this fact can be known by ordinary empirical means. What kind of knowledge is this, then? How do we know that the number 3 does not have properties that far outstrip our conception of it? 5 The Caesar problem is also more general than it initially appears. The problem arises whenever we have a many-sorted language and wonder whether to "merge its sorts" by formulating some associated one-sorted language. 6 One example is the twosorted language£ 1 from Chapter 8, which has one sort for "basic" objects and another for the new objects introduced by an abstraction principle. Should these two sorts be merged, and if so, how? These are questions of fundamental importance to any abstractionist approach to mathematics. Another example concerns systems of abstraction principles rather than individual such principles, which is all we have considered thus far. Let ABs(x, R, a) mean that x is the R-abstract of a; that is, that x is the result of abstraction on a under the partial equivalence relation R. Suppose abstraction is permitted on some family of partial equivalence relations R: Raa

~

3xABs(x, R, a)

ABs(x, R, a ) /\ ABs(y, R, f3)

~

(x

=y #

Raf3)

As a convenient shorthand, we shall sometimes represent such systems of abstraction principles in the more familiar functional notion:

5

See (MacBride, 2006) for a related observation. See (Enderton, 2001 , pp. 296- 9) for a simple way of doing so, which also corresponds to the approach to be described in Section 9.6. 6

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THE JULIUS CAESAR PROBLEM

It must be borne in mind, however, that each operator §R corresponds to a partial function whose domain is the field of R. In any event, our official notation is the relational one displayed in (ABsR3) and (ABSR =).What should we say about mixed identities of the form §Ru = §sv? We would like either a principled reason to dismiss such questions as meaningless, or alternatively a systematic answer of the form

for some suitable condition .•• , Cn from D1> Pis assertible of ci. ... , Cn iff {3p(ci. . .. , Cn). We make the following further assumptions. • Suppose a predicate P, other than'=: is in Lo and that ap is the satisfaction condition for Pon the interpretation I of L 0 . Then for any {a;, I;) from Dz we have:

(That is, a predicate of the base language, other than '=: is regarded as assertible only of objects from Do, and as assertible of a string of such objects iff the predicate is true of that string on the interpretation I.) • For any elements al> ... , an and b; of Do and any labels lo, ... , In E {O, 1}, we have: I;= 1 /\a;~ b;--+ (f3p( .. .) ++ {3p( .. .)[(b;, 1)/{a;, l)l)

where' .. .' abbreviates the string '(ai. 11), ... , (an, In)'. (That is, if P is regarded as assertible of a string of specifications, the i'th of which is an extended specification, then P is also regarded as assertible of any other string of specifications that specify the same objects, that is, where the first coordinate of any extended specification may be replaced by any ~-equivalent token.) • Suppose f3= is the condition associated with the identity predicate =. Then: f3=((a1,Ji), (a2,Ji)) ++ ((11=12 = 0 /\ a1 = a1) V (I,= 12

= 1 /\ a1

~ a1))

(That is, two specifications are regarded as specifying identical objects just in case both are basic specifications with identical first coordinates, or both are extended specifications with ~-equivalent first coordinates. 25 )

Singular terms. Let a be an assignment of elements of D1 to each variable of L1. Relative to a, every singular constant t of L1 is assigned an element c of Di. subject to the following constraints. • Suppose tis a constant of Lo. Then c = (a, 0) where a is the referent oft under I. • Suppose t is a variable. Then t receives its assignment from a. • Suppose tis a term of the form §sand thats is assigned (a, I) relative to a. Then tis assigned (a, 1) relative to a. 26

Formulas. The assertibility conditions for formulas of L1 are as follows.

25 This way of handling mixed identities-and thus of resolving the Caesar problem-is plausible in the case we are considering, as argued in the main text. However, what follows can easily be adapted to any other formally acceptable way of resolving the Caesar problem. 26 What matters is the case where I = 0. Allowing the operation to be defined where I = 1 is merely a matter of convenience, corresponding to identifying a "double abstraction" with the corresponding single abstraction (e.g. the letter type of this letter type= this letter type).

APPENDICES

173

• Suppose rp is an atomic formula P(t1> . .. , tn) and that each singular term t ; is assigned an element c; of D1 relative to a. Then rp is assertible relative to a iff f3p(ci. . .. , en) . • The clauses for truth-functional connectives are the obvious compositional ones. • Suppose rp is of the form Vv 1/f, where v is a variable. Then rp is assertible relative to a iff for every variable assignment a ' that differs from a at most in its assignment to v, 1/f is assertible relative to a'.

9.B A Non-reductionist Interpretation Fix an interpretation I of Lo. I contend that the assertibility conditions for L 2 that we defined in Appendix 9.A function precisely as if the standard of correctness was truth on some non-reductionist interpretation that extends I. I wish to define the desired non-reductionist interpretation N of L2 and to use this definition to prove the mentioned contention. As in the Chapter 8, we work in a metalanguage that is governed by the same sort of assertibility conditions and in which we can therefore avail ourselves of the very form of abstraction in question. Thus, when Do is the domain associated with I, we have the following abstraction principle in our metalanguage: (APc)

a ~ a /\ b ~ b-+ (§a= §b

~

a ~ b)

where the variables range over Do . We begin by defining a domain D2 which extends Do by adding §a for each a E Do such that a ~ a. Moreover, just as in the object language, we assume that all Caesar-style questions are answered negatively, that is, that a =!= §b for every a, b E Do. The predicates of L2 are interpreted as follows . The identity predicate is interpreted standardly on D2. Any other predicate that is present already in Lo retains its old interpretation. Any predicate P that is present only in L2 is interpreted in accordance with its assertibility condition f3p , in the following sense. Let us say that b E D2 is the value of a labeled specificiation c = (a, l) iff either l = 0 and b = a, or l = 1 and b = §a; we symbolize this as [ell = b. Then, for any string C1> .•• , Cn from D1> we lay down: Pis trueof[c1Il,. .. , [ cnD onN

iff

f3p(c1,. . .,en)

Notice that this is well-defined because of the requirements we have imposed on f3p (cf. Appendix 9.A). When a is a variable assignment based on Di. we let [a Il be the assignment based on D2 that is defined by [a Il(x;) = [a(x;)Il. We can now state the desired result, which is easily proved by induction on the complexity of rp. Proposition 9.1 Let rp be any formula of L2 and a be any assignment based on D1. Then:

N F[uD rp iff rp is assertible relative to a This proposition shows that, relative to suitably coordinated variable assignments, a formula is true on the non-reductionist interpretation just in case the formula is regarded as assertible. The proposition thus shows that an important feature of the two-sorted language L 1, discussed in Chapter 8, extends to our single-sorted language L2 as well; see especially Corollary 8.1 .

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THE JULIUS CAESAR PROBLEM

9. C Defining a Sufficiency Operator We wish to define a sufficiency operator for the one-sorted language L2. As we recall from Appendix 8.D, our definition for the two-sorted language L 1 makes crucial use of a translation from L1 to Lo. No such translation is available for the single-sorted language L2· We shall therefore restrict ourselves to literals of L2, that is, to atomic formulas and negations thereof. We shall define a sufficiency operator:=} such that, for every true literal if! of L2, there is a true formula 1/t must ensure that the corresponding material conditional rp -+ 1/t is knowable. Let us now verify that this constraint is satisfied. As we recall, 'rp ==> 1/t' means that either there is a predicative language extension where