Thin Objects: An Abstractionist Account 0199641315, 9780199641314

Table of contents :
Dedication
Contents
Preface
Part I. Essentials
1. In Search of Thin Objects
2. Thin Objects via Criteria of Identity
3. Dynamic Abstraction
Part II. Comparisons
4. Abstraction and the Question of Symmetry
5. Unbearable Lightness of Being
6. Predicative vs. Impredicative Abstraction
7. The Context Principle
Part III. Details
8. Reference by Abstraction
9. The Julius Caesar Problem
10. The Natural Numbers
11. The Question of Platonism
12. Dynamic Set Theory
Bibliography
Index

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Thin Objects

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Thin Objects An Abstractionist Account

Øystein Linnebo

1

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Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © Øystein Linnebo 2018 The moral rights of the author have been asserted First Edition published in 2018 Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2018931819 ISBN 978–0–19–964131–4 Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work.

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For my daughters Alma and Frida

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Contents Preface

xi

Part I. Essentials 1. In Search of Thin Objects 1.1 1.2 1.3 1.4 1.5 1.6 1.7

Introduction 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 of Identity 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 5 7 9 11 13 17 21 21 23 26 30 33 37 38 38 39 40 41 42 45 46 48 49 51 51 53 55 60 61 64 66

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viii contents Appendix 3.A Further Questions 3.A.1 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 of Being 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 Impredicativity 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

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contents ix

Part III. Details 8. Reference by Abstraction 8.1 8.2 8.3 8.4

Introduction The Linguistic Data Two Competing Interpretations 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

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

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 of language 10.3.3 The objection from lack of directness 10.4 Alleged Advantages of the Cardinal Conception 10.5 Developing the Ordinal Conception 10.6 Justifying the Axioms of Arithmetic

11. The Question of Platonism 11.1 11.2 11.3 11.4 11.5 11.6

Platonism in Mathematics Thin Objects and Indefinite Extensibility Shallow Nature The Significance of Shallow Nature How Beliefs are Responsive to Their Truth The Epistemology of Mathematics

135 135 137 140 143 143 144 146 148 151 153 155 156 157 159 159 160 162 163 166 167 169 171 173 174 176 176 176 178 179 180 181 182 183 185 189 189 191 192 195 197 201

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x contents 12. Dynamic Set Theory 12.1 12.2 12.3 12.4

Introduction Choosing a Modal Logic Plural Logic with Modality The Nature of Sets 12.4.1 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 of ZF 12.5.1 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

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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 Agustín 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.

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xii preface equinumerous with the Gs”, perhaps the former sentence is just a façon 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

§α = §β = α ∼ β

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.

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preface xiii where α and β 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

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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 of logical 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).

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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.

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xvi preface 11

9

12 3

8

5

6

10

2

4

7

1 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 of large 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 Agustín 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

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preface xvii developed in this manuscript; thanks to Solveig Aasen, Bahram Assadian, Neil Barton, Rob Bassett, Christian Beyer, Susanne Bobzien, Francesca Boccuni, Einar Duenger Bøhn, Roy Cook, Philip Ebert, Matti Eklund, Anthony Everett, Jens Erik Fenstad, Salvatore Florio, Dagfinn Føllesdal, Peter Fritz, Olav Gjelsvik, Volker Halbach, Mirja Hartimo, Richard Heck, Simon Hewitt, Leon Horsten, Keith Hossack, Torfinn Huvenes, Nick Jones, Frode Kjosavik, Jönne 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/E003753/1) and finally brought to its completion during two terms as a Visiting Fellow at All Souls College, Oxford. I gratefully acknowledge their support.

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PA R T I

Essentials

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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 making the claim synthetic. This view is boldly rejected in Frege’s Foundations of Arithmetic (Frege, 1953), where Frege defends an account of arithmetic that combines a form of ontological realism with logicism. His realism consists in taking arithmetic to be about real objects existing independently of all human or other cognizers. And his logicism consists in taking the truths of pure arithmetic to rest on just logic and definitions and thus be analytic. Most philosophers now probably agree with Kant in this debate and deny that the existence of mathematical objects can be established on the basis of logic and conceptual analysis alone. This is why George Boolos, only slightly tongue-in-cheek, can offer a one-line refutation of Fregean logicism: “Arithmetic implies that there are two distinct numbers” (Boolos, 1997, p. 302), whereas logic and conceptual analysis—Boolos takes us all to know—cannot underwrite any existence claims (other than perhaps of one object, so as to streamline logical theory).2 However, the disagreement between Kant and Frege lives on in a different form. Even if we concede that there are no analytic existence claims, we may ask whether there are objects whose existence does not (loosely speaking) make a substantial demand on the world. That is, are there objects that are “thin” in the sense that their existence does not (again loosely speaking) amount to very much? Presumably, an analytic truth does not make a substantial demand on the world.3 But perhaps being analytic is not the only way to avoid imposing a substantial demand. Instead of asking Frege’s question of whether there are existence claims that are analytic, we can ask the broader question of whether there are existence claims that are “non-demanding”—in some sense yet to be clarified. A number of philosophers have been attracted to this idea. Two classic examples are found in the philosophy of mathematics. First, there is the view that the existence 1

2 See (Kant, 1997, B622–3). See also (Boolos, 1997, pp. 199 and 214). Analyticity must here be understood in a metaphysical rather than epistemological sense (Boghossian, 1996). I cannot discuss here whether Frege’s rationalism led him to depart from a traditional conception of (metaphysical) analyticity. See (MacFarlane, 2002) for some relevant discussion. 3

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 in search of thin objects of the objects described by a theory of pure mathematics amounts to nothing more than the consistency or coherence of this theory. This view has been held by many leading mathematicians and continues to exert a strong influence on contemporary philosophers of mathematics.4 Then, there is the view associated with Frege that the equinumerosity of two concepts suffices for the existence of a number representing the cardinality of both concepts. For instance, the fact that the knives and the forks on a table can be one-to-one correlated is said to suffice for the existence of a number that represents the cardinality of both the knives and the forks.5 Agustín Rayo nicely captures the idea when he writes that a “subtle Platonist” such as Frege believes that for the number of the Fs to be eight just is for there to be eight planets. So when God created eight planets she thereby made it the case that the number of the planets was eight. (Rayo, 2016, p. 203; emphasis in original)

I am not claiming that there is a single, sharply articulated view underlying all these views, only that they are all attempts to develop the as-yet fuzzy idea that there are objects whose existence does not make a substantial demand on the world. We have talked about objects being thin in an absolute sense, namely that their existence does not make a substantial demand on the world. An object can also be thin relative to some other objects if, given the existence of these other objects, the existence of the object in question makes no substantial further demand. Someone attracted to the view that pure sets are thin in the absolute sense is likely also to be attracted to the view that an impure set is thin relative to the urelements (i.e. non-sets) that figure in its transitive closure. The existence of a set of all the books in my office, for example, requires little or nothing beyond the existence of the books. Moreover, a mereological sum may be thin relative to its parts. For example, the existence of a mereological sum of all my books requires little or nothing beyond the existence of these books.6 I shall refer to any view according to which there are objects that are thin in either the absolute or the relative sense as a form of metaontological minimalism, or just minimalism for short. The label requires some explanation. While ontology is the study of what there is, metaontology is the study of the key concepts of ontology, such as existence and objecthood.7 A view is therefore a form of metaontological minimalism insofar as it holds that existence and objecthood have a minimal character. Minimalists need not hold that all objects are thin. Their claim is that our concept of an object permits thin objects. Additional “thickness” can of course derive from the kind of object in question. Elementary particles, for example, are thick in the sense that their existence makes a substantial demand on the world. But their thickness derives from what it is to be an elementary particle, not from what it is to be an object.

4 5 6 7

See for instance (Parsons, 1990), (Resnik, 1997), and (Shapiro, 1997). See for instance (Wright, 1983) and the essays collected in (Hale and Wright, 2001a). Philosophers attracted to this view include (Lewis, 1991, Section 3.6) and (Sider, 2007). See for instance (Eklund, 2006a).

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coherentist minimalism  Metaontological minimalism has consequences concerning ontology proper. The thinner the concept of an object, the more objects there tend to be. Metaontological minimalism thus tends to support a generous ontology.8 By contrast, a generous ontology does not by itself support metaontological minimalism. The universe might just happen to contain an abundance of objects whose existence makes substantial demands on the world. Just as metaontological minimalists are heirs to the Fregean view that there are analytic existence claims, there are also heirs to the contrasting Kantian view. Hartry Field has attacked the idea that mathematical objects are thin, sometimes mentioning the Kantian origin of his criticism.9 And various metaphysicians reject the idea that mereological sums are thin relative to their parts.10 Just as with the original Kantian rejection of analytic existence claims, this contemporary rejection of thin objects strikes many philosophers as plausible. Metaontological minimalism can come across as a piece of philosophical magic that aspires to conjure up something out of nothing—or, in the relative case, to conjure up more out of less. The chapter is organized as follows. In the next two sections, I outline two influential approaches to the idea of thin objects that are found in the philosophy of mathematics and that were mentioned above. Then, I examine the appeal of the idea. Based on this examination, I formulate some logical and philosophical constraints that any viable form of metaontological minimalism must satisfy. We thus obtain a “job description”, and the task of the book is to find a suitable candidate for the job. The chapter ends with an attempt to dramatically reduce the field of acceptable candidates by rejecting the customary symmetric conception of abstraction in favor of an asymmetric conception. The left-hand side of an abstraction principle makes demands on the world that go beyond those of the right-hand side. Thin objects are nevertheless secured because the former demands do not substantially exceed the latter. For the truths on the left are grounded in the truths on the right.

1.2 Coherentist Minimalism One classic example of metaontological minimalism is the view that the coherence of a mathematical theory suffices for the existence of the objects that the theory purports to describe. Since it is coherent to supplement the ordinary real number line R with two infinite numbers −∞ and +∞, for example, the extended real ¯ = R ∪ {−∞, +∞} exists. And since it is coherent to supplement number line R √ R with the imaginary unit i = −1 and all the other complex numbers, the complex field C exists. All that the existence of these new mathematical objects involves, according to the view in question, is the coherence of the theories that describe the relevant structures. Let us refer to this as a coherentist approach to thin objects. 8 9

See (Eklund, 2006b) for a discussion of some extremely abundant ontologies that may arise in this way. 10 See (Field, 1989, pp. 5 and 79–80). See for instance (Rosen and Dorr, 2002).

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 in search of thin objects This approach enjoys widespread support within mathematics itself and is defended by several prominent mathematicians. In his correspondence with Frege, for example, David Hilbert wrote: As long as I have been thinking, writing and lecturing on these things, I have been saying the exact reverse: if the arbitrarily given axioms do not contradict each other with all their consequences, then they are true and the things defined by them exist. This is for me the criterion of truth and existence.11

As is well known, the word ‘criterion’ is ambiguous between a metaphysical meaning (a defining characteristic) and an epistemological one (a mark by which something can be recognized). Since the context favors the metaphysical reading, the passage is naturally read as an endorsement of metaontological minimalism, not just of an extravagant ontology. A similar view is endorsed by Georg Cantor: Mathematics is in its development entirely free and only bound in the self-evident respect that its concepts must both be consistent with each other and also stand in exact relationships, ordered by definitions, to those concepts which have previously been introduced and are already at hand and established.12

It may be objected that, while this passage defends an extremely generous ontology, it is not a defense of metaontological minimalism. In response, we observe that the passage is concerned with what Cantor calls “immanent reality”, which is a matter of occupying “an entirely determinate place in our understanding”. Cantor contrasts this with “transient reality”, which requires that a mathematical object be “an expression or copy of the events and relationships in the external world which confronts the intellect” (p. 895). He feels compelled to provide an argument that the former kind of existence ensures the latter. The most plausible interpretation, I think, is that Cantor seeks a form of metaontological minimalism with respect to immanent existence but merely a generous ontology concerning transient existence. The coherentist approach to thin objects has enjoyed widespread support among philosophers as well. A structuralist version of the approach has in recent decades been defended by central philosophers of mathematics such as Charles Parsons, Michael Resnik, and Stewart Shapiro.13 For instance, Shapiro includes the following “coherence principle” in his theory of mathematical structures: Coherence: If ϕ is a coherent formula in a second-order language, then there is a structure that satisfies ϕ. (Shapiro, 1997, p. 95)

11 Letter to Frege of December 29, 1899, in (Frege, 1980). See (Ewald, 1996, p. 1105) for another example from Hilbert. 12 See (Cantor, 1883), translated in (Ewald, 1996, p. 896). 13 See the works cited in footnote 4. Also relevant is the “equivalence thesis” of (Putnam, 1967).

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abstractionist minimalism  It is instructive to compare this principle with Tarski’s semantic account of logical consistency and consequence. On Tarski’s analysis, a theory T is said to be semantically consistent (or coherent) just in case there is a mathematical model of T. The coherence principle can be regarded as a reversal of this analysis: we now attempt to account for what models or structures there are in terms of what theories are coherent.14 Shapiro not only endorses the coherence principle but makes some striking claims about its philosophical status. He compares the ontologically committed claim that there is a certain mathematical structure with the (apparently) ontologically innocent claim that it is possible for there to be instances of this structure. These claims are “equivalent” (p. 96), he contends, and “[i]n a sense […] say the same thing, using different primitives” (p. 97). Shapiro’s view is thus a version of coherentist minimalism, centered on the claim that there is a model of T ⇔ T is coherent

where ϕ ⇔ ψ means that ϕ and ψ “say the same thing”. The coherentist approach can be extended to objects that are thin only in a relative sense. Coherence does not suffice for the existence of “thick” objects such as electrons. But given the existence of certain thick constituents, coherence may suffice for the existence of further objects that are thin relative to these constituents. Given the existence of two electrons, for example, their set and mereological sum may exist simply because the existence of such objects is coherent. Is coherentist minimalism tenable? I remain neutral on the question. My present aim is to develop and defend an alternative form of minimalism based on Fregean abstraction. My pursuit of this aim is unaffected by the success or failure of the coherentist alternative.

1.3 Abstractionist Minimalism Another classic example of metaontological minimalism derives from Frege and has been developed by the neo-Fregeans Hale and Wright. Frege first argues (along lines that will be outlined in Section 1.4) that there are abstract mathematical objects. He then pauses to consider a challenge: How, then, are the numbers to be given to us, if we cannot have any ideas or intuitions of them? (Frege, 1953, §62)

That is, how can we have epistemic or semantic “access” to numbers, given that their abstractness precludes any kind of perception of them or experimental detection? 14 This is not to say that we possess a notion of coherence that is independent of mathematics. Our view on questions of coherence will be informed by and be sensitive to set theory. Here we use some mathematics to explicate a philosophical notion, which in turn is used to provide a philosophical interpretation of mathematics. See (Shapiro, 1997, pp. 135–6) for discussion.

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 in search of thin objects Frege’s response urges us to transform the question of how linguistic (or mental) representations succeed in referring to natural numbers into the different question of how complete sentences (or their mental counterparts) succeed in having their appropriate arithmetical meanings: Since it is only in the context of a sentence that words have any meaning, our problem becomes this: To define the sense of a sentence in which a number word occurs. (Frege, 1953, §62)

This response raises some hard exegetical questions, which are discussed in Chapter 7. But the argumentative strategy of the Grundlagen is made tolerably clear a few pages later, where Frege makes a surprising claim about the relation between the parallelism of lines and the identity of their directions: The judgement “line a is parallel to line b”, or, using symbols, a  b, can be taken as an identity. If we do this, we obtain the concept of direction, and say: “the direction of line a is identical with the direction of line b”. Thus we replace the symbol  by the more generic symbol =, through removing what is specific in the content of the former and dividing it between a and b. We carve up the content in a way different from the original way, and this yields us a new concept. (Frege, 1953, §64)

Consider the criterion of identity for directions: (Dir)

d(l1 ) = d(l2 ) ↔ l1  l2

Frege claims that the content of the right-hand side of this biconditional can be “recarved” to yield the content of the left-hand side. The idea is that we get epistemic and semantic access to directions by first establishing a truth about parallelism of lines and then “recarving” this content so as to yield an identity between directions. Let ϕ ⇔ ψ formalize the claim that ϕ and ψ are different “carvings” of the same content—in a sense yet to be explicated. Then (Dir) can be strengthened to: (Dir⇔ )

d(l1 ) = d(l2 ) ⇔ l1  l2

Inspired by this example, Frege and the neo-Fregeans seek to provide a logical and philosophical foundation for classical mathematics on the basis of abstraction principles, which generalize (Dir). These are principles of the form (AP)

§α = §β ↔ α ∼ β

where α and β range over items of some sort, where ∼ is an equivalence relation on such items, and where § is an operator that maps such items to objects. One famous example is Hume’s Principle, which says that the number of Fs (symbolized as #F) is identical to the number of Gs just in case the Fs and the Gs can be one-to-one correlated (symbolized as F ≈ G): (HP)

#F = #G ↔ F ≈ G

As Frege discovered, this principle has an amazing mathematical property. When added to second-order logic along with some natural definitions, we are able to

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the appeal of thin objects  derive all of ordinary (second-order Dedekind-Peano) arithmetic.15 As we shall see, abstraction principles are available not just for directions and numbers but for many other kinds of abstract object as well, such as geometrical shapes and linguistic types. What does all this mean? According to Frege and the neo-Fregeans, the left-hand side of each successful abstraction principle (AP) provides a “recarving” of the content of the corresponding right-hand side.16 All that is required for the existence of the objects §α and §β is that the equivalence relation ∼ obtain between the entities α and β. All that is required for the existence of directions, for example, is the parallelism of appropriate lines. The abstractionist approach to minimalism can be extended to objects that are thin only in the relative sense. It is possible to formulate abstraction principles for sets and mereological sums, for example, which ensures that the existence of sets and sums of thick objects does not make any substantial demand beyond the existence of their thick constituents.

1.4 The Appeal of Thin Objects Why are so many philosophers and mathematicians attracted to the idea of thin objects? The most important reason is that metaontological minimalism promises a way to accept face value readings of discourses whose ontologies would otherwise be problematic. Arithmetic provides an example. The language of arithmetic contains proper names which (it seems) purport to refer to certain abstract objects, namely natural numbers, as well as quantifier phrases which (it seems) purport to range over all such numbers. Moreover, a great variety of theorems expressed in this language appear to be true. These theorems are asserted in full earnest by competent laypeople as well as professional mathematicians. Since the arithmetical competence of these people is beyond question, there is reason to believe that most of their arithmetical assertions are true. But if these theorems are true, then their singular terms and quantifiers must succeed in referring to and ranging over natural numbers. And for this kind of success to be possible, there must exist abstract mathematical objects. The argument is certainly valid. Is it sound? The premises can of course be challenged—like everything else in philosophy. But they have great initial plausibility. It would be appealing to take the apparent truth of the premises at face value, if possible. This would save us the difficult task of showing how both laypeople and experts are wrong about something they take to be obviously true. So the argument provides at least some reason to believe that there exist mathematical objects such as numbers.17 15 This result, which is known as Frege’s Theorem, is hinted at in (Parsons, 1965) and explicitly stated in (Wright, 1983). See (Boolos, 1990) for a nice proof. 16 Further evidence that the neo-Fregeans are pursuing the idea of thin objects, as understood here, is provided in Section 6.4. 17 See (Linnebo, 2017c) for an overview of defenses of the premises, which, if successful, would support a much stronger conclusion.

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 in search of thin objects On the other hand, the ontology of abstract objects is often found to be philosophically problematic. The epistemological challenge brought to general philosophical attention by (Benacerraf, 1973) is well known. Since perception and all forms of instrumental detection are based on causal processes, these methods cannot give us access to abstract objects such as the natural numbers. How then can we acquire knowledge of them?18 Another worry is the perceived extravagance of the huge ontologies postulated by contemporary mathematics. How can we postulate vast infinities of new objects with such a light heart? No physicist would so unscrupulously postulate a huge infinity of new physical objects. Why, then, should mathematicians get away with it? Of course, philosophers are divided over how serious these worries are. But any successful account of mathematical objects needs to have some response to the worries, even if only to explain why they are misguided. The idea of thin objects suggests a promising strategy for responding to the worries. The vast ontology of mathematics may well be problematic when understood in a thick sense. If mathematical objects are understood on the model of, say, elementary particles, there would indeed be good reason to worry about epistemic access and ontological extravagance. But this understanding of mathematical objects is not obligatory. If there are such things as thin objects, then the existence of mathematical objects need not make much of a demand on the world. It may, for instance, suffice that the theory purporting to describe the relevant mathematical objects is coherent. This would greatly simplify the problem of epistemic access. Although our knowledge of the coherence of mathematical theories is still inadequately understood, it is at least not a complete mystery in the way that knowledge of thick mathematical objects would be. More generally, the less of a demand the existence of mathematical objects makes on the world, the easier it will be to know that the demand is satisfied.19 Thin objects would help with the worry about ontological extravagance as well. If mathematical objects are thin, the bar to existence is set very low. So it is only to be expected that a generous ontology should result. This is just an instance of the general phenomenon noted above, namely that metaontological minimalism tends to support generous ontologies. Does this defense of a generous mathematical ontology conflict with Occam’s razor? The answer depends on how the razor is understood. If all the razor says is that objects must never be postulated “beyond necessity” but must earn

18 An improved version of the challenge is developed in (Field, 1989). In (Linnebo, 2006a) I develop a further improvement which I argue survives all extant attempts to answer or reject the challenge. Some ideas about how to answer this improved challenge are found already in that paper but are set out in greater detail below, especially in Section 11.5. 19 This response to Benacerraf ’s challenge must be distinguished from that of (Balaguer, 1998). As I understand it, Balaguer’s “full-blooded platonism” is primarily a very generous ontology. My present point, however, is that metaontological minimalism promises to reduce the explanatory burden by equating the existence of mathematical objects with some fact to which epistemic access is less problematic.

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sufficiency and mutual sufficiency  their keep by enabling better explanations than would otherwise be possible, then the razor may well be compatible with the generous ontology in question. If some objects are metaphysically “cheap”, even a modest contribution to our explanatory power may suffice to justify their postulation.20 It is also important to keep in mind that the explanations in question need not be empirical but can be intra-mathematical. There are other examples too of how thin objects can be philosophically useful. Consider the philosophical debate about the existence of mereological sums. We often speak as if there are various kinds of mereological sums, such as decks of cards, bunches of grapes, crowds of people. And many of these claims appear to be true. But some philosophers find mereological sums problematic, often because of philosophical worries akin to the ones just discussed. If mereological sums are thin relative to their parts—if, that is, little or nothing is required for their existence other than the existence of their parts—then we would be in a good position to assuage these worries. In sum, if some form of metaontological minimalism can be articulated, its explanatory potential will be great.

1.5 Sufficiency and Mutual Sufficiency The principal task confronting any defender of metaontological minimalism is to explain some locutions that we have so far left loose and intuitive, such as all that is required for ψ is ϕ ϕ is (conceptually) sufficient for ψ all that God had to do to ensure that ψ was to bring it about that ϕ

or the symmetrical analogue ϕ is a recarving of the content of ψ ϕ and ψ make the same demand on the world for it to be the case that ϕ just is for it to be the case that ψ

Some notation will be useful when grappling with this task. Let ϕ ⇒ ψ formalize the relationship that the first three statements are after, and ϕ ⇔ ψ, the symmetrical analogue involved in the last three statements.21 Let us refer to these as sufficiency and mutual sufficiency statements, respectively. Our task is to provide a proper explanation of the operator that figures in at least one of these types of statement and to show how the resulting statements can be used to provide the attractive philosophical explanations that we discussed in the previous section.

20

See (Schaffer, 2015) for a related observation. I shall use the word ‘statement’ for a formula relative to some contextually salient assignment to its free variables, if any. 21

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 in search of thin objects It will be convenient to allow the sufficiency operator to take a set of formulas on its left-hand side. Thus, we take  ⇒ ψ to mean that the statements in the set  are jointly sufficient for ψ. If our language allows infinite conjunctions, then  ⇒ ψ can  be regarded as just shorthand for ( α∈A ϕα ) ⇒ ψ, where  = {ϕα | α ∈ A}. Since we mostly work with finitary languages, however, the proposed modification does make a difference. What logical principles should be adopted to govern the new operator(s)? A full answer must obviously await a proper explanation of the operator(s). But a little bit can be said already at this stage. For the idea of thin objects to have any promise, the operator(s) must be at least as strong as the corresponding material conditional or biconditional. That is, ϕ ⇒ ψ must entail ϕ → ψ; and likewise for mutual sufficiency and the biconditional. What else? Consider first the sufficiency operator. It is reasonable to adopt a principle of transitivity. If a set of truths suffice for another set of truths, and if this second set of truths suffice for a third truth, then the first set too suffices for the third truth. We formalize this as the following cut rule: Cut for each i ∈ I i ⇒ ϕi {ϕi }i∈I ⇒ ψ ∪i∈I i ⇒ ψ

We shall not at the outset assume any other logical principles governing the sufficiency operator. Next, consider the mutual sufficiency operator. Since this is meant to be an equivalence relation, we assume logical principles to that effect. A trickier question is for which properties this equivalence is a congruence; that is, for which operators O we have: ϕ ⇔ ψ → (Oϕ ↔ Oψ) We make no assumptions at this stage but shall return to the question shortly. Does it matter which of the two operators we choose as our primitive? Surely, one might think, it must be possible to define either operator in terms of the other. Suppose we choose ⇒ as our primitive. We can then define mutual sufficiency as two-way ordinary sufficiency; that is, define ϕ ⇔ ψ as ϕ ⇒ ψ ∧ψ ⇒ ϕ. If instead we choose ⇔ as our primitive, we can define ϕ ⇒ ψ as ϕ ⇔ ϕ∧ψ. (This definition can be motivated in terms of two natural assumptions: first, that ϕ ∧ ψ suffices for ϕ; and second, that ϕ suffices for ϕ ∧ ψ just in case ϕ suffices for ψ.) However, these arguments make strong assumptions about the logic of the two operators. These assumptions have not been granted. So at least until more has been said, we are not entitled to regard either operator as definable in terms of the other. Thus, the choice of one (or even both) of the operators as primitive may well matter. Indeed, beginning in Section 1.7, I argue that the sufficiency operator is better suited to deliver what we want than the mutual sufficiency operator.

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philosophical constraints



1.6 Philosophical Constraints I now wish to formulate a “job description” for the desired notion of sufficiency or mutual sufficiency. I shall concentrate on the case of sufficiency, for the reason just mentioned, although my discussion is easily adapted to the case of mutual sufficiency. The “job description” should be written so as to ensure that any notion that fits the bill delivers the philosophical benefits discussed in Section 1.4. As discussed, Frege proposed to define ϕ ⇔ ψ as A(ϕ ↔ ψ), where A is an analyticity operator, and where analyticity is understood as truth in virtue of meaning.22 How does this proposal fare? Quinean objections to analyticity pose an obvious threat. Perhaps there is no sharp or theoretically interesting distinction between the analytic and the synthetic. Any attempt to draw such a distinction would then be arbitrary. Can the Quinean objections be countered? The question has been debated at great length. I do not here wish to join this debate.23 Thankfully, there is no need to gauge the general health of the notion of analyticity in order to assess the viability of the Fregean proposal. The proposal faces two serious problems. One problem concerns de re sufficiency statements. Let me explain. Should the formulas that flank the sufficiency operator be permitted to contain free variables? The resulting sufficiency statements can be said to be de re, in obvious analogy with the terminology used in quantified modal logic. This contrasts with de dicto sufficiency statements, where only sentences (that is, formulas with no free variables) are allowed to flank the operators. There is, in fact, strong pressure to accept sufficiency statements that are de re, not merely de dicto. Consider the claim that there are thin objects. To express this, we need to state that there are objects whose existence is undemanding, that is, ∃x( ⇒ Ex), where Ex is an existence predicate and  is some tautology. The same goes for the claim, discussed above, that there are objects (such as impure sets) that are thin only in a relative sense, not absolutely. With this background in place, the problem is easy to state. As (Quine, 1953b) explained, even if we waive all concerns about the distinction between analytic and synthetic sentences, this will only licence analyticity statements that are de dicto, not de re. After all, it is only sentences that are analytic, not open formulas relative to variable assignments. Analyticity is meant to be an entirely linguistic phenomenon, whereas variable assignments typically involve non-linguistic objects. A second problem concerns ontology. To fix the terminology—here and in what follows—let us adopt the usual Quinean notion of ontological commitment, according to which a sentence in the language of first-order logic is ontologically committed to just such objects as must be assumed to be in the domain in order to make the 22 In terms of (Boghossian, 1996)’s influential distinction, this is a metaphysical rather than an epistemological notion of analyticity. 23 For the record, I believe Quine successfully undermines the ambitious notion of analyticity championed by the logical positivists, but that a more modest notion may escape his attack. See (Putnam, 1975a) for a similar view. In his later years, Quine too moved in this direction: see (Quine, 1991).

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 in search of thin objects sentence true. A proponent of thin objects is interested in sufficiency statements where the statement on the right-hand side of the sufficiency operator has ontological commitments that exceed those of the statement on the left-hand side. A possible example is the Fregean claim (Dir⇒ )

l1  l2 ⇒ d(l1 ) = d(l2 )

where the right-hand side is ontologically committed to abstract directions, while the left-hand side avoids such commitments. Our question is whether there are analytically true conditionals ϕ → ψ where the ontological commitments of ψ exceed those of φ. Any such conditional would give rise to an analytic existence statement. To see this, let ∃xθ make explicit an ontological commitment that is had by ψ but not by ϕ. We thus have A(ϕ → ∃xθ). By moving the quantifier ∃x out through the parenthesis (which is permissible since ϕ can be assumed not to contain x free), it follows that A∃x(ϕ → θ). As we saw in the opening of this chapter, however, Boolos—and probably most other contemporary philosophers as well—deny that there are any analytically true existence statements.24 It is not hard to see why. An analytic truth is supposed to be true in virtue of meanings, which are supposed to be cognitively accessible to us. But the objects in question are supposed to be independent of us and our linguistic and cognitive activities; for instance, Frege compares the existence of the natural numbers with that of the North Sea (Frege, 1953, §26). How can meanings that are so closely tied to our minds ensure the existence of objects external to our minds and so robustly independent in their existence? Of course, it was precisely this sort of concern about analytic existence statements that prompted our investigation of thin objects in the first place. So let us set aside analyticity and exploit the extra freedom gained by broadening our perspective to the idea of thin objects, whose existence need not be analytic but nevertheless—in some sense yet to be pinned down—makes no substantial demand on the world. Our discussion motivates a constraint that any viable notion of sufficiency must satisfy:25 Ontological expansiveness constraint There are true sufficiency statements ϕ ⇒ ψ where the ontological commitments of ψ exceed those of ϕ.

Indeed, on an abstractionist approach to thin objects, we probably want true sufficiency statements corresponding to the right-to-left direction of all permissible abstraction principles. This generalizes the Fregean sufficiency claim (Dir⇒ ).

24 We are here setting aside the possible commitment to a non-empty domain, which is often tolerated in order to streamline our logic. This is permissible because ∃xθ can be chosen so as to involve a specific commitment, say to directions, which goes beyond the commitment to a non-empty domain. 25 An analogous constraint would, of course, be needed for the notion of mutual sufficiency. The same obviously goes for the constraints formulated below. Henceforth, I shall not remind readers of this.

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philosophical constraints



Might there be an easy way to obtain the desired sufficiency statements? Perhaps the right-hand side is merely a fancy—and syntactically misleading—rewording of the left-hand side. This proposal would certainly ensure that the right-hand side demands no more of the world than the left-hand side. Whatever its other merits, the proposal is clearly inadequate for our purposes. If the right-hand side is merely a façon de parler for the left-hand side, this would at most justify speaking as if there are abstract objects. But our aim is to make sense of how there in fact are such objects and of how we come to know about them. For a sufficiency claim to have this philosophical payoff, both sentences must be taken at face value. Every singular term must function as such semantically; that is, it must be in the business of referring to an object. And there must be no singular reference other than what is effected by the singular terms that occur in the relevant sentence. These considerations motivate the following constraint: Face value constraint The formulas involved in a sufficiency statement ϕ ⇒ ψ can be taken at face value in our semantic analysis.

Our most significant discovery so far is that it would be too demanding to define the sufficiency statement ϕ ⇒ ψ as the analytic conditional A(ϕ → ψ). We need a less demanding definition. An option that naturally comes to mind is to define ϕ ⇒ ψ as the strict conditional 2(ϕ → ψ) (where ‘2’ stands for metaphysical necessity). On this analysis, any necessarily existing object counts as requiring nothing for its existence. The analysis thus collapses the idea of thin objects into the more familiar idea of necessary existents. This loss of novelty is not itself problematic. The real problem concerns the promised benefits of thin objects, as discussed in Section 1.4. Assume, for instance, that there is a necessarily existing god. Let ψ say so, and let ϕ be any tautology. Then we would have ϕ ⇒ ψ, although knowledge of ϕ would provide no assurance whatsoever that ψ too is knowable. Moreover, ϕ would do nothing to explain ψ or make ψ less mysterious. It follows that any viable analysis of ϕ ⇒ ψ must be more demanding than the strict conditional. Let us therefore try to formulate some further constraints on the sufficiency operator to ensure that the promised benefits materialize. One of the promised benefits is a response to the epistemological challenge made famous by (Benacerraf, 1973). How do we gain knowledge of abstract objects such as directions and numbers, given that no causal interaction with them is possible? The promised answer is nicely illustrated in terms of the Fregean example of directions: (Dir)

d(l1 ) = d(l2 ) ↔ l1  l2

Let ϕ and ψ be the right- and the left-hand sides of this biconditional, respectively. Suppose both parties agree that there is no epistemological mystery concerning the right-hand side ϕ. But since the left-hand side ψ refers to abstract directions, ψ gives rise to an epistemological challenge. According to Frege and his followers, the solution

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 in search of thin objects is to observe that ϕ suffices for ψ. Therefore, it is possible for us to advance from knowledge of ϕ to knowledge of ψ. For this response to succeed, the operator ⇒ must have some capacity for transmitting knowledge from left to right. Ideally, we would like the following. Assume ϕ ⇒ ψ and that ϕ is known. Then ψ too must be known, or at least knowable. We can ensure this transmission, it turns out, by laying down the following constraint: Epistemic constraint If ϕ ⇒ ψ, then it is possible to know ϕ → ψ; and if additionally ϕ is known, then this possible knowledge is compatible with continued knowledge of ϕ.

To confirm that the constraint would have the desired effect, let the operator K represent the epistemic status of being known. It is reasonable to take K to be at least potentially closed under modus ponens, in the sense that K(ϕ → ψ) and Kϕ entail 3Kψ. Now, suppose that ϕ ⇒ ψ and Kϕ. Then the constraint ensures 3(Kϕ ∧ K(ϕ → ψ)). Hence, the mentioned closure property ensures 33Kψ.26 Using Axiom 4 of modal logic, this entails 3Kψ, as desired. How should we understand the modal operator that figures in the constraint? It is important that the possibility in question not be wildly idealized. We should not ascribe to the epistemic agent an ability to carry out supertasks, for example. The possibility needs to be of the same sort as the possibilities that are realized in cases of actual mathematical knowledge. As formulated above, the constraint would give us what we ideally want.27 But needless to say, we can’t always get what we want! Depending on how the sufficiency operator is interpreted, the above formulation may be asking too much. So it is reassuring to observe that our explanatory project allows the constraint to be weakened. It is enough if the operator ⇒ is capable of transmitting knowledge from left to right to some sufficient extent. To weaken the constraint in this way, we would obviously have to clarify what counts as a “sufficient extent”. I shall not attempt to do so here. Indeed, the basic sufficiency statements that I defend satisfy the constraint in its present strong formulation.28 A second promised benefit of thin objects is a response to the worry about the seeming ontological extravagance of modern mathematics and certain other bodies of knowledge, such as classical mereology. How can these sciences get away with postulating such an abundance of objects when ontological economy is otherwise 26 Here we use the fact that in any normal modal logic, the environment ‘3 . . .’ is closed under logical entailment. 27 Or at least something close. We might also want the epistemic constraint to require (possible) preservation of a priori knowledge or justification. But there is no need to commit to that here. 28 For example, the parallelism (or orthogonality) of two lines suffices for the identity (or orthogonality) of their directions; see Chapters 8 and 9 for details. Using sufficiency statements such as these, I also define some less basic forms of sufficiency. First, there are de re sufficiency statements, where the epistemic constraint is satisfied only modulo choice of co-referring names. Second, the immediate sufficiency discussed so far is supplemented with a notion of mediate sufficiency, defined as a form of transitive closure of immediate sufficiency. I do not clam that this mediate notion of sufficiency satisfies the constraint in full generality.

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two metaphysical “pictures”  regarded as a virtue? Again, the minimalist has an answer, namely that the generous ontologies in question either make no substantial demand on the world (in the case of pure abstract objects such as numbers and sets), or their demands on the world do not substantially exceed demands that have already been met (in the case of impure sets or mereological sums). This answer motivates another constraint on ⇒. Assume that ϕ ⇒ ψ. Then any metaphysical explanation of ϕ must also explain ψ, or at least give rise to such an explanation. This ensures that ψ is no more puzzling or mysterious than ϕ. Let us therefore lay down the following explanatory constraint on ⇒: Explanatory constraint If ϕ ⇒ ψ, then ϕ → ψ admits of an acceptable metaphysical explanation.

To see that this constraint has the desired effect, let E ϕ mean that ϕ admits of an acceptable metaphysical explanation. It is reasonable to take E to be closed under modus ponens. This ensures that ϕ ⇒ ψ and E ϕ jointly entail E ψ. Taking stock, we have not yet found an acceptable analysis of the notion of sufficiency. But we now have a far better understanding of where to look. We know that ⇒ must be less demanding than analytic implication but more demanding than strict implication. Moreover, we motivated some philosophical constraints that any viable notion of sufficiency must satisfy. Is there a candidate that satisfies the resulting job description?

1.7 Two Metaphysical “Pictures” Attempts to satisfy our job description can be divided into two broad families, which correspond to two different metaphysical “pictures”. I shall now describe the two pictures and explain why I find one of them more promising than the other. The first picture takes a highly symmetric view of abstraction. Consider the Fregean example: (Dir⇔ )

d(l1 ) = d(l2 ) ⇔ l1  l2

Although one sentence is concerned with directions and the other with lines, the two sentences are nevertheless said to be made true by precisely the same facts or states of the world. As Frege put it, the two sentences are merely different ways to “carve up” a single shared content. Since we are using the operator ⇔ to represent sameness of demands on the world, a mutual sufficiency statement ϕ ⇔ ψ ensures that ϕ and ψ are on a par in every respect that has to do solely with how the world is; that is ϕ ⇔ ψ → (Oϕ ↔ Oψ) for every operator O that is concerned solely with how the world is.29 In particular, the two sides of every legitimate abstraction principle are on a par in every such respect.

29 Of course, a proponent of the symmetric picture can also introduce a non-symmetric operator, where ‘ϕ ⇒ ψ’ is interpreted as ‘the demands that ϕ imposes on the world include those that ψ imposes’.

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 in search of thin objects The competing asymmetric picture denies that the two sides of a legitimate abstraction principle are on a par in every worldly respect. Instead, abstraction is regarded as an inherently asymmetric matter, where abstraction on “old” entities gives rise to “new” objects. While the left-hand side of, say, (Dir) demands of the world that it contain directions, the right-hand side does not. Abstraction therefore involves a worldly asymmetry. Clearly, this means that the mutual sufficiency statement (Dir⇔ ) 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⇒ )

l1  l2 ⇒ d(l1 ) = d(l2 )

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(l2 )’ exceed those of ‘l1  l2 ’. 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, l1  l2 metaphysically explains d(l1 ) = d(l2 ); 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 ϕ metaphysically explains ψ and quite another to spell out the promised explanation.31 My aim—in the cases that concern us—is the latter thing: I want to understand how l1  l2 , for example, metaphysically explains d(l1 ) = d(l2 ). This cannot be done by means of a completely general study of metaphysical grounding but will require a detailed examination of how abstraction works. Second, it is far from clear that the general notion of grounding, as understood in the mentioned literature, satisfies any worthwhile version of the epistemic constraint that was formulated in Section 1.6.32 Which of the two metaphysical pictures is correct? I find the asymmetric picture far more natural and promising. One reason for this view has already been adumbrated 30 Grounding is currently attracting much attention in metaphysics; see e.g. (Fine, 2012a), (Rosen, 2010), and (Schaffer, 2009). In fact, (Rosen, 2011)’s “qualified realism” about mathematical objects has much in common with my own conception of such objects as “thin”. Very roughly, where I develop a particular conception of sufficiency, Rosen proposes that we use the general notion of grounding (but does not develop this proposal in detail). See also (Rosen, 2016). 31 See (Wilson, 2014) for a related complaint and (Schaffer, 2016) for useful discussion. 32 We might add, as a third reason, that grounding, unlike my notion of sufficiency, is factive: if ϕ grounds ψ, then ϕ and ψ. But it is unclear how deep this difference is. A nonfactive notion of grounding is explored in (Litland, 2017); and if desired, my notion of sufficiency could easily be tweaked to ensure factivity.

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two metaphysical “pictures”  and will be developed further in Chapter 4. There appears to be a worldly asymmetry between the two sides of legitimate abstraction principles, for example as concerns their ontological commitments. Another reason has to do with de re statements of sufficiency or mutual sufficiency. As already observed, such statements are needed; for example, in order to express that there are objects whose existence is undemanding, we need the de re formula ∃x( ⇒ Ex), where Ex is an existence predicate and  is some tautology, or an analoguous formula involving the mutual sufficiency operator. Which such de re statements should we accept? And how can we account for the ones that we do accept? I contend that the asymmetric picture is better equipped to answer these questions. To defend this contention, consider any legitimate abstraction principle. Consider first the symmetric theorist’s associated mutual sufficiency statement: (AP⇔ )

§α = §β ⇔ α ∼ β

Are we allowed to “export” the function terms ‘§α’ and ‘§β’ so as to derive the following statement?   (AP⇔ ∃x∃y x = §α ∧ y = §β ∧ (x = y ⇔ α ∼ β) e ) 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⇒ )

α ∼ β ⇒ §α = §β

Suppose we apply exportation to derive:   (AP⇒ ∃x∃y x = §α ∧ y = §β ∧ (α ∼ β ⇒ x = y) e ) 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

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 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 Chapter 4.

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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

identity criteria

objecthood

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 of logical 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 defined 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. 1).

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 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.

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a fregean concept of object  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 specifications— can 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 of logically 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 of singular 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, 1981a, ch. 14).

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 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 language— contingent 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 (Keränen, 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

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a fregean concept of object  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 physical— could 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.

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 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.

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reference to physical bodies  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 of 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 (B1) 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.

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 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. (B5)

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)

15

b(u) = b(v) ↔ u ∼ v

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.

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reference to physical bodies  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 field16 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 b is 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.

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 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 Fregean 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 Fregean 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 → (ϕ(x) ↔ ϕ(y))

As usual, Leibniz’s Law must be restricted to contexts ϕ(. . .) that are transparent, in the sense that the context is only concerned with the referents of the terms that fill

18

Indeed, this inaccessibility plays a key role in an argument I develop in Section 8.4.3. See e.g. (Dummett, 1981a, pp. 73–6, 179–80, 545–6) and (Wiggins, 2001). However, some psychologists make the same departure; see (Xu, 1997). 19

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reconceptualization  the argument place, not with these terms themselves or any non-referential aspects of their meaning. Suppose we have assigned truth-conditions to all identity statements and predications that involve some new singular terms we are trying to introduce, and that the assignments are compatible with the logic of identity—or in other words, that we have satisfied the minimal requirement stated above. In short, suppose an attempted application of the conceptual apparatus for identification and predication is logically coherent. Does this attempted application of our conceptual apparatus succeed in latching on to objects? Two different answers flow from two rival conceptions of ontology, which I shall now canvass with a deliberately broad brush. I later clarify and defend some aspects of one of the conceptions. The rigid conception holds that reality is “carved up” into objects in a unique way that is independent of the concepts that we bring to bear. This conception introduces an element of risk into our proposed application of our conceptual apparatus for identification and predication. Although the attempted application is logically in good order, reality may fail to cooperate. Reality may simply not contain the sorts of objects we are trying to “carve out”. The flexible conception, on the other hand, insists that reality is articulated into objects only through the concepts that we bring to bear. And we often have some choice in this matter.20 Of course, our choice is limited. It is not an option for a starving person to produce more bread by “carving up” reality differently. The choice we have is of a different character. As Frege observed, reality provides different answers to ontological questions about what there is depending on which concepts we bring to bear.21 If you bring to bear the concept of a loaf of bread, some part of reality (say, your kitchen) may answer that there are two such objects. If instead you bring to bear the concept of a molecule of organic material, you will receive a very different answer. This much should be uncontroversial. The flexible conception of ontology gets its bite by adding the controversial claim that there is no unique, privileged set of concepts in terms of which to “carve up” reality, namely the concepts that match some rigid concept-independent articulation of reality into objects. This means there is no risk that reality fails to contain objects answering to some coherent application of our apparatus for identification and predication. This coherent application “carves out” the appropriate objects. The disagreement between the two conceptions hinges on the relative explanatory priority of the articulation of reality into objects and permissible applications of our conceptual apparatus for identification and predication. The rigid conception

20 This distinction has much in common with Hilary Putnam’s distinction between “metaphysical” and “internal” realism. See (Putnam, 1987) for a brief introduction. There are important differences, however, especially as concerns the emphasis that I place on the connection between this ontological debate and the analysis of the concept of an object. 21 See (Frege, 1953, §46).

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 thin objects via criteria of identity contends that first reality provides a range of objects, to which we then apply our conceptual apparatus. On this view, there is only one correct way to apply language to reality, namely to ensure that each singular term is assigned one of the objects that reality provides, and to regard an identity statement as true just in case its two singular terms have been assigned one and the same object. The view operates with a Lego block conception of reality on which the world consists of well-defined objects—“Lego blocks”—that are available independently of any application of our conceptual apparatus for identification and predication.22 As a result, the application of this apparatus to reality is an inherently risky undertaking even when the logical coherence of the application is beyond doubt. We need to ensure that each singular term is assigned a unique “Lego block” or sum thereof, and each predicate, a property (or relation) that may hold of (or obtain between) such blocks or sums. The flexible conception of ontology rejects the idea of a domain of objects—or “Lego blocks”—that are available independently of any application of our conceptual apparatus for identification and predication. It makes no sense to talk about objects until our conceptual apparatus for identification and predication is already in play. So we can make no sense of the rigid conception’s self-individuating “Lego blocks”, which an attempted application of our apparatus for identification and predication must latch on to in order to succeed. On the flexible conception, we need an entirely different explanation of what counts as a permissible application of our apparatus for identification and predication. This explanation must not presuppose the range of objects to which this apparatus is applied. What might the required explanation look like? A radical option is to take the minimal requirement just mentioned to suffice for successful reference. That is, provided that truth-conditions are assigned in a way that accords with the laws of logic and that appropriate sentences come out true, then reference has been achieved. In particular, there is no need to appeal to criteria of identity. Versions of this radical option have been defended by the neo-Fregeans Hale and Wright, as well as by Agustín Rayo. These versions are criticized in Chapter 5 for going too far. This book defends a less radical version of the flexible conception, which does attach special significance to criteria of identity. Where we have a criterion of identity, we can make sense of approaching an object in multiple different ways, of tracking it, and reidentifying it. These capacities mark a watershed in our individuation of objects. Physical bodies provide an example. I proposed an account of how reference to bodies is constituted on the basis of a criterion of identity. As remarked, this account does not presuppose physical bodies, only the parcels of matter of which the bodies are composed and the partial equivalence relation discussed above. Our discourse about books provides another good illustration. Suppose we start with a practice of identifying and distinguishing physical copies of books—or

22

Thanks to Agustín Rayo for suggesting this image.

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reference by abstraction  book tokens, as we may call them—and predicating properties of these. We can meaningfully say, for example, that your token of Anna Karenina weighs more than mine. But we also talk about books in a more abstract way. Suppose you and I have read separate tokens of Anna Karenina. We still regard it as true to say that you and I have read one and the same book. That is, we identify and ascribe properties not only to book tokens but also to book types, such as the book type that you and I have both read. Our practice of identifying book types and predicating properties of them is highly systematic, fully in compliance with the logic of identity, and explicable in a way that makes recourse only to book tokens and their properties. This illustrates the versatility and flexibility of our conceptual apparatus for identification and predication. To avert misunderstanding, let me stress, yet again, that I am not claiming that criteria of identity are necessary for reference. Since my explanations move counterclockwise around the Fregean triangle, I need an appropriate use of criteria of identity to be sufficient for reference. I do not need the corresponding necessity claim.23 As far as this book is concerned, there might be other ways to supplement the minimalist answer so as to explain how objects come to figure as referents of singular terms or values of first-order variables.24 The purpose of this section has (as announced) merely been to locate the account that I develop in a broader philosophical context. Much work obviously remains. In particular, I need to explain the central notion of a permissible application of our apparatus for identification and predication. My next step is to generalize and elaborate on the form of reconceptualization involved in the example of book types.

2.5 Reference by Abstraction Can my account of reference to physical bodies be extended to other kinds of objects, perhaps even to abstract objects? There is reason to be hopeful. My account is based on a form of reconceptualization brought about by an appropriate use of a criterion of identity. And since the notion of a criterion of identity is extremely general, it is applicable to abstract objects as well as to concrete ones. So if an appropriate use of criteria of identity can suffice for reference, this holds out the promise of an extremely general sufficient condition for reference. This approach would liberate the notion of reference from the requirement of any sort of causal connection between the term and

23 Other thinkers inspired by Frege, however, have sometimes been attracted to some version of the necessity claim as well. Perhaps all reference is ultimately based on certain fundamental forms of reference in which criteria of identity play an essential role. For discussion, see (Dummett, 1981a, pp. 231–9), and (Evans, 1982, pp. 109–12). 24 If such alternatives exist, should the objects in question be understood in accordance with the rigid or the flexible conception of ontology? That is, would these objects too be “carved out” by our concepts, but in a way that differs from the one developed in this book? For present purposes, I need not answer these questions—although I confess to a general sympathy with the flexible conception.

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 thin objects via criteria of identity the referent—which would be a good thing. For as we saw in Section 1.4, there appears to be such a thing as reference to abstract objects. This phenomenon of apparent reference to abstract objects calls for a serious investigation, which our approach makes possible, unlike approaches that require some form of causal interaction. One of Frege’s favorite examples of a criterion of identity concerns directions. Suppose we begin with a domain of lines. Parallelism is an equivalence relation on this domain. So we can adopt the following criterion of identity: (Dir)

d(l1 ) = d(l2 ) ↔ l1  l2

where ‘d(l)’ stands for the direction of the line l. That is, two lines specify the same direction just in case they are parallel. On the plausible assumption that lines and facts about parallelism are epistemically accessible to us, this criterion enables us to talk about directions being identical or distinct. What about other properties and relations? Orthogonality provides an example. Two directions are regarded as orthogonal just in case they are specified by two orthogonal lines. Using ⊥ and ⊥∗ as orthogonality predicates for lines and directions, respectively, we formalize this as follows: d(l1 ) ⊥∗ d(l2 ) ↔ l1 ⊥ l2 The orthogonality of two directions is, as it were, “inherited” from the orthogonality of any two lines in terms of which these directions are specified. Of course, this “inheritance” of properties from lines to their directions presupposes that it does not matter which line we choose to specify some given direction. Thankfully, the presupposition is met. Assume that the direction specified by l1 is also specified by l1 , because l1  l1 . If one of these two specifications is orthogonal to another line l2 , so is the other. For the purposes of assessing orthogonality, any line is just as good as any of its parallels. Moreover, the example generalizes. For any formula ϕ on lines that doesn’t distinguish between parallel lines, we can introduce an associated predicate ϕ ∗ on directions by letting ϕ ∗ hold of the directions of some given lines just in case ϕ holds of the lines themselves. What does this account establish? The account indisputably shows how we can use the resources provided by a domain of lines and the relations of parallelism and orthogonality of lines to define a way of speaking that is just like the realist’s. It is true to say that there are directions, related to one another in various ways. My goal is more ambitious, however. I contend that, when properly developed, the account establishes not merely an acceptable way of speaking but explains how reference to abstract directions is constituted through a form of reconceptualization. Before defending this stronger contention, I wish to comment on the scope of the approach. There is a vast family of examples that are structurally similar to the one concerning lines and directions. Any form of type–token distinction provides an example. A syntactic type, for instance, can be specified by means of a syntactic token; and two such tokens specify the same type just in case they count as equivalent by the

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reference by abstraction  relevant standards.25 Geometrical shapes provide another example. A shape can be specified by means of an object that has the shape in question; and two objects specify the same shape just in case they are congruent. All of these examples share a common structure. To identify this structure, we must distinguish between two forms of criteria of identity. A one-level criterion of identity says of two objects of some sort F that they are identical just in case they stand in some relation R:26   (CI-1) Fx ∧ Fy → x = y ↔ R(x, y) Criteria of this form operate at just one level, in the sense that the condition for two objects to be identical is given by a relation on these objects themselves. A good example is the set-theoretic law of extensionality, which says that two sets are identical just in case they have precisely the same elements:   (Ext) Set(x) ∧ Set(y) → x = y ↔ ∀u(u ∈ x ↔ u ∈ y) Clearly, one-level criteria of identity are not connected in any direct way with abstraction principles. A two-level criterion of identity relates the identity of objects of one sort to some condition involving items of another sort. The former sort of objects are given as the value of some function applied to items of the latter sort. So the criterion takes the form (CI-2)

f (α) = f (β) ↔ α ∼ β

where the variables α and β may be of either first or higher order and ∼ is an equivalence relation on the entities over which the variables range.27 This philosophical use of two-level criteria suggests some terminology. Let us call the entities over which the variables α and β range specifications. These entities serve to specify the objects for which the criterion of identity holds.28 Next, let us refer to the equivalence relation ∼ as a unity relation. From a logical point of view, a two-level criterion of identity is just the same as an abstraction principle. Some famous examples derive from Frege. We have already discussed the direction principle (Dir). Another important example is Hume’s Principle 25 Ordinary English may not have a good name for this equivalence relation. But the relation can be grasped without the need for any antecedent knowledge of types. After all, this is how we come to master discourse about types in the first place. See Chapter 8 for more discussion. 26 This conception of criteria of identity goes back at least to Locke, who ponders what various identities “consist in”; see his Essay, Book II, Chapter XXVII. More recently this conception has been explicitly endorsed in (Lowe, 1989) and (Lowe, 1997). 27 An approach based on two-level criteria of identity is found in (Williamson, 1990, ch. 9). 28 I have previously used the word ‘presentation’ but have found that this word has phenomenological and psychological connotations that are avoided by the word ‘specification’, which accordingly seems more apt.

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 thin objects via criteria of identity (HP)

#F = #G ↔ F ≈ G

where F and G are second-order variables and F ≈ G abbreviates the second-order formalization of the claim that the Fs and the Gs are equinumerous, so as to provide a criterion of identity for cardinal numbers. As in the case of directions, we can adopt “inheritance” principles that describe how certain properties of specifications are “inherited” by the objects that are specified. Consider a formula ϕ that does not distinguish between equivalent specifications:   α1 ∼ β1 ∧ . . . ∧ αn ∼ βn → ϕ(α1 , . . . , αn ) ↔ ϕ(β1 , . . . , βn ) For any such formula, we introduce a corresponding predicate ϕ ∗ , defined on all of the objects that can be specified in this way and governed by the following “inheritance” principle: (Inher)

ϕ ∗ (f (α1 ), . . . , f (αn )) ↔ ϕ(α1 , . . . , αn )

In this way, we obtain many useful predicates; for example, an orthogonality predicate on directions and a successor predicate S that holds of two cardinal numbers #F and #G just in case all of the Fs are equinumerous with all but one of the Gs. The unity relation ∼ that figures in a two-level criterion is ordinarily required to be an equivalence relation. But as our discussion of physical bodies revealed, we have reason to relax this requirement slightly. We wish to allow our specification variables to include in their range entities that fail to specify an object of the sort with which we are concerned. Suppose, for example, that we wish to abstract on lines and other directed items so as to obtain directions, while simultaneously including in our universe of discourse objects that cannot be said to be parallel with themselves, such as chairs, electrons, and numbers. We can achieve this by allowing ∼ to be a partial equivalence relation, that is, a relation that is symmetric and transitive but not necessarily reflexive. For instance, in the mentioned example we let ∼ be defined by parallelism on any directed items and be false whenever one or both of the arguments isn’t a directed item. When ∼ is a partial equivalence relation, the function f that figures in (CI-2) must be allowed to be partial as well, defined on all and only items α that are in the field of ∼.29 As in the case of physical bodies, this generalized two-level criterion can still be formalized as (CI-2), provided that this criterion operates in the context of a negative free logic.30 All in all, the proposed technical adjustments are minor and do not materially affect the ensuing philosophical discussion. The distinction between one- and two-level criteria of identity is important, given our project. In a two-level criterion, the objects whose identity conditions we are analyzing do not figure on the right-hand side of the biconditional. This promises a way

29 Since ∼ is a partial equivalence, this is equivalent to saying that f is defined on all and only items α such that α ∼ α. For suppose α ∼ β for some β. Then symmetry and transitivity ensure α ∼ α. 30 See Appendix 2.C for details.

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some objections and challenges



to characterize the identity conditions for potentially problematic objects that does not presuppose such objects. A two-level criterion thus promises a way to single out objects of one sort in terms of items of another—and presumably less problematic— sort. Consider the case of physical bodies. Our toy model of reference to physical bodies explains how such reference is constituted, all in terms that make no explicit appeal to bodies. This is what gives the model its reductionist character. Abstract objects provide further examples. How are directions and other abstract objects “given to us”? Frege’s answer is that we relate the identity of two directions to the parallelism of the two lines in terms of which these directions are specified. Since reference to lines is less puzzling than reference to directions, this is explanatory progress. By contrast, a one-level criterion would be of little or no use when trying to explain our most fundamental way of referring to objects of a certain kind. To apply a one-level criterion, one must already be capable of referring to objects of the sort in question. The stage is now set for an initial formulation of the link between reference and criteria of identity that I defend: Reference by abstraction Consider a two-level criterion of identity (CI-2) which purports to provide identity conditions for some kind F of object in terms that presuppose only some antecedently accepted ontology but not Fs. Assume that an agent has an appropriate grasp of the unity relation ∼ and uses the criterion and property inheritance principles of the form (Inher) to govern her discourse about Fs. Assume that the agent stands in an appropriate relation to specifications α and β, which are in the field of ∼, and uses these specifications as if to make claims about f (α) and f (β). Then:

(i) the agent is in fact referring to objects f (α) and f (β), (ii) α ∼ β suffices for f (α) = f (β), (iii) for any instance of (Inher), the right-hand side suffices for the left-hand side. This initial formulation obviously needs further explication and defense. For one thing, we need to be told what it is for a criterion of identity not to “presuppose” the objects for which it is a criterion.31 For another, we need to be told what it is for an agent to be appropriately related to a unity relation and some specifications.32

2.6 Some Objections and Challenges Let us now consider some objections to the thesis of reference by abstraction, as well as challenges that require further clarification. A more systematic and detailed defense of the thesis is provided in Part III of the book.

31 32

My answer is outlined in Section 2.6.1 and developed in Section 3.3 and Chapter 6. See Section 2.6.3 and, for more details, Chapter 8.

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 thin objects via criteria of identity

2.6.1 The bad company problem An obvious objection is that the good instances of two-level criteria of identity are surrounded by “bad companions” that are inconsistent or otherwise problematic. A notorious example is Frege’s Basic Law V, which states that two concepts share the same extension just in case they are coextensive. We can formalize this as (BLV)

ˆ ˆ x.Fx = x.Gx ↔ ∀x(Fx ↔ Gx)

where ‘F’ and ‘G’ are second-order variables that range over Fregean concepts and ˆ where ‘x.Fx’ stands for the extension of F. As Russell discovered, this “law” gives rise to the paradox now bearing his name. To see this, it is useful to think of Frege’s extensions as classes. The “law” allows us to consider the Russell class r whose members are each and every object that is not a member of itself. We now ask whether r is a member of itself. It is easy to derive the contradiction that r is a member of itself just in case it is not. My response is to dismiss the proposed counterexample on the grounds that one of the assumptions of the thesis of reference by abstraction is violated. Basic Law V attempts to provide a criterion of identity for extensions. But it does so in a way that quantifies over—and thus presupposes—the very extensions that the criterion is meant to govern. The “law” therefore violates the assumption that the identity conditions for the objects to be introduced by abstraction presuppose only an antecedently accepted ontology.33

2.6.2 Semantics and metasemantics What, exactly, is the kind of “account of reference” that I have in mind and in which criteria of identity have an important role to play? To answer the question, we need to distinguish three types of question that can be asked about relations. One type concerns which objects stand in the given relation. Which bicycle is mine? To whom am I married? To whom does the name ‘ØL’ refer? A second type of question concerns the causal and historical process by which the objects came to stand in some relation. When and why did I buy my bicycle? How did I meet my wife, and why did we marry? Why did my parents choose my name, and why have other speakers followed suit? A third type of question arises for all relations that aren’t metaphysically primitive (unlike, perhaps, the relation between an elementary particle and its charge or mass). In virtue of what do I own that bicycle? What makes it the case that I am married to this woman? And what is involved in the relation of reference obtaining between this name and that person? Questions of this third type will be called constitutive and must be carefully distinguished from the extensional and etiological questions of the first and second types, respectively.

33 A more complete explanation of the bad company problem and my response to it is provided in Chapters 3 and 6.

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some objections and challenges



In fact, there is already an established name for the subclass of constitutive questions that concern the relation of reference and other relations studied by semantics. Such questions are said to belong to metasemantics, as opposed to semantics proper.34 Semantics is concerned with how the semantic value of a complex expression depends upon the semantic values of its various simple constituents. We ascribe to each complex expression a certain semantic structure and explain how its semantic value is determined by this structure as a function of the semantic values of its simple constituents, typically in accordance with the principle of compositionality. Metasemantics, on the other hand, is concerned with the constitutive question of what is involved in an expression’s having the semantic properties that it happens to have, such as its semantic structure and its semantic value. The “account of reference” that I develop is concerned with the metasemantic (and thus also constitutive) question of what makes it the case that a singular term refers to a particular object. In particular, Frege’s famous question of how numbers are “given to us” belongs to metasemantics, not to semantics proper.35 The corresponding semantic question would be: To what objects do numerals of various sorts refer? For an ontological realist such as Frege, such questions have easy answers. For instance, both the ordinary decimal numeral ‘7’ and the Roman numeral ‘VII’ refer to the natural number 7. The metasemantic question with which Frege is grappling is much harder: How is it so much as possible to refer to numbers and other abstract objects, given that we cannot have any “ideas or intuitions” of them and (we might add) given that no causal relation to them is possible? The Fregean proposal I am developing holds that criteria of identity have an important role to play in the answers to this and related questions. There is every reason to expect our metasemantic questions to have answers—even if only complex and messy ones. For it seems deeply implausible that relations of reference should be fundamental and irreducible. As (Fodor, 1987, p. 97) observes, when physicists one day draw up a catalogue of “ultimate and irreducible properties of things”, then “the likes of spin, charm, and charge will perhaps appear upon their list [but] aboutness surely won’t”. And the same goes, of course, for reference (or, for that matter, for ownership and marriage).

2.6.3 A vicious regress? Some philosophers believe that the kind of account I am developing falls prey to a vicious regress.36 The account holds that reference is constituted in part by the subject’s being suitably related to a specification of the referent. Suppose this “suitable relation” to a specification has to be one of reference. To refer to this specification, we would then need a further specification and unity relation, which would set off a

34 See (Stalnaker, 1997). This distinction corresponds to Stalnaker’s distinction between “descriptive” and “foundational” semantics. See e.g. (Stalnaker, 2001). See also Section 7.6.3 for further discussion. 35 See (Frege, 1953, §62). 36 See e.g. (Dummett, 1991a, pp. 162–3) and (Lowe, 1998, ch. 7).

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 thin objects via criteria of identity vicious regress. To refer to an object, we would always need a prior ability to refer to some more basic object in terms of which the aforementioned object is specified. A natural response to the alleged regress is to deny that the subject’s relation to a specification needs to be one of reference. Without this supposition, the regress does not get going. How plausible is the supposition? I grant that there are cases where the subject’s relation to a specification is indeed one of reference. Reference to natural numbers provides an example. On the account adumbrated in the Preface and developed in Chapter 10, a natural number is specified by means of a numeral type, which may in turn be specified by a numeral token. The crucial question, however, is whether the subject’s relation to a specification is always one of reference. The regress arises only if the answer is “yes”. But the correct answer is “no”. There is no reason to believe that a subject’s relation to a specification is always one of reference. Our discussion of reference to physical bodies provides one example. Here the subject’s relation to the parcel of matter that serves as a specification is a purely causal one. This stops the regress. Another example is provided by the account of arithmetic based on Hume’s Principle, where a Fregean concept acts as a specification of a number. This too stops the regress, because the appropriate relation to a concept is one of grasping or possessing, not one of referring. While this response is fine as far as it goes, it may be objected that it does not go far enough.37 Granted, a subject’s relation to a specification need not be one of reference. But what about us theorists who are trying to explain how this specification contributes to the constitution of reference? Whatever one says about the subject, it seems that we theorists are referring to the specification in question. When explaining the reference to physical bodies, for example, our relation to a parcel of matter that serves as a specification is not merely a causal one, as we are singling the parcel out and making various claims about it. And since we theorists are referring to parcels of matter, we would like to know how this reference is constituted. This reintroduces the threat of a regress. The objection shows that our initial response cannot stand on its own but needs to be supplemented. Thankfully, a supplementary response is ready to hand. Recall that we are only proposing a sufficient condition for reference, not a necessary one. For the regress to get going, we need two-level criteria of identity to be necessary for reference, or at least for certain fundamental forms of it. But I am not committed to this necessity claim. In particular, I do not claim that we theorists’ reference to parcels of matter comes about through the use of some two-level criterion.

2.6.4 A clash with Kripke on reference? Suppose reference to an object is achieved by means of a specification α. Is my account of reference to this object compatible with the influential view, due to Kripke and

37

Thanks to Tobias Wilsch for pressing this objection.

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some objections and challenges



others, that proper names and demonstratives refer in a direct way? The role of the specification α calls this into question. Isn’t the referent given only indirectly as the f of α? In fact, there is no conflict so long as the thesis of direct reference is properly understood. The thesis belongs to semantics and holds that the semantic value of a singular term such as a proper name or a demonstrative is just its referent. This contrasts with descriptivism about reference, which is a competing semantic thesis according to which the semantic import of such expressions is much richer and, in particular, involves descriptive content. The semantic thesis of direct reference is compatible with a variety of metasemantic views on the constitution of reference, including ones according to which this constitution involves some non-trivial structure. Indeed, when Kripke conjectures that an initial “baptism” and subsequent causal chains serve to constitute certain forms of reference, he proposes a metasemantic account with non-trivial structure. Clearly, Kripke took his conjecture to be compatible with his emphasis on reference as a semantically direct relation.38 We need to distinguish the question of what objects stand in some relation from the entirely different question of what facts make it the case that the objects are so related. The thesis of direct reference concerns the former type of question. A proper name bears the reference relation directly to its bearer rather than to other items in terms of which the bearer is then picked out. By contrast, my sufficient condition for reference concerns the latter type of question. It is not a brute fact that a name refers to its bearer but a fact whose proper explanation sometimes involves a further object, namely a specification of the referent.

2.6.5 Internalism about reference How will a nominalist respond to my thesis of reference by abstraction? Consider the account of reference to directions. We may safely assume that the nominalist has no trouble with lines or parallelism. The lines can be taken to be concrete, and all that is required concerning parallelism is that the relation be defined. We are not committed to any particular metaphysics of predication based on universals or reified relations, which the nominalist might find objectionable. The bone of contention is the claim that a line and parallelism suffice to specify a direction. The nominalist objects to this claim on the grounds that there simply are no directions. The advocate of the Fregean approach will respond that we have specified a direction, namely the direction of some line (reference to which is not in question). There is no alternative, more direct or secure way of specifying a direction than what

38 In fact, my view is not even committed to metasemantic descriptivism, which holds that the referent is fixed by means of a definite description. The specification and unity relation I invoke to explain how the referent is picked out need not be part of a definite description. As we saw in Section 2.6.3, the relation to the specification need not be one of reference; and as emphasized in Section 2.3, the grasp of the unity relation need not be an explicit one, which is accessible to consciousness.

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 thin objects via criteria of identity we have already offered. A direction simply is the sort of thing that is most directly specified by means of a line and subject to the unity relation of parallelism. Clearly, we are confronted with a fundamental disagreement about what it takes to specify a direction. To break the impasse, it is useful to consider a structurally analogous debate that arises in the case of physical bodies. Here too I claim to have provided an account of reference. To specify a physical body, it suffices to have causally interacted with one of its parts and to be operating appropriately with the relevant unity relation. Assume someone challenges me to demonstrate that there really exists a physical body associated with some parcel of matter that both parties admit exists and is in the field of the unity relation. The challenger demands that the alleged referent be shown to her in a more direct or secure way that she too would find acceptable. Clearly, there is nothing I can do that would satisfy the challenger. A physical body just is the sort of thing that is most directly specified by means of a parcel of matter and is subject to the appropriate unity relation. To demand that a body be specified or shown in some altogether different way is to demand the impossible. The comparison with the case of physical bodies brings out some important lessons. First, the nominalist’s challenge is just an instance of a far more general skeptical challenge concerning what it takes to specify an object. It isn’t the abstractness of the desired object that is fueling the challenge but some very general preconceptions about what it takes to specify an object. The Fregean response is to reject these preconceptions as unreasonable. Second, my Fregean view involves a form of internalism about reference. There are certain basic forms of reference instances of which cannot be explained without relying on the very form of reference in question. To have any chance of explaining reference to a physical body, we need to be willing to engage in this very form of reference in the metalanguage. And there is nothing special about physical bodies in this regard. For there to be any hope of explaining reference to a given direction, say, we must be willing to engage in reference to directions in the metalanguage; otherwise, we would be demanding the impossible.39 Once this internalism is taken on board, the Fregean’s claim that the line suffices to specify a direction can be properly defended, or so I claim. My argument is developed in detail in Chapter 8.

2.7 A Candidate for the Job When the idea of thin objects was first introduced in Chapter 1, I characterized an object as thin insofar as a comparatively weak claim—not ontologically committed to

39 Chapter 5 distinguishes my internalism about reference from some more radical “ultra-thin” conceptions of reference, and Chapter 8 develops the internalism in more detail. A view related to mine is found in (Dummett, 1995), where we read: “Frege not only believes in semantics, but constructed a general framework for it. He held, however, that a semantics for a language must be internal to that language: it must describe it as from within, not from some external viewpoint” (p. 17; emphasis in original).

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a candidate for the job  the object in question—nevertheless suffices for the existence of the object. I explained how the idea of thin objects holds great promise for the philosophy of mathematics and metaphysics more generally. The crucial question is how to understand the notion of sufficiency. To make progress, I formulated a “job description” that any notion of sufficiency must satisfy if the resulting account of thin objects is to deliver on its promises. Let us now examine whether we have succeeded in identifying a suitable candidate for the job. The examination can only be preliminary, however, as I have only outlined my account, leaving for later chapters the task of filling in various details. Consider the case of directions. Suppose we start with a domain containing lines (and perhaps other objects) and on which relations of parallelism and orthogonality are defined (but are deemed false whenever one or both of the relata is not a line or some other directed item). I explained how our conceptual apparatus for identification and predication can be used in a novel way such that directions, not lines, are the objects that are identified or distinguished and that serve as subjects of other predications. We add to our language vocabulary for talking about directions and lay down clear and strict rules for how the extended language is to be used. These rules are carefully chosen so as to ensure that the laws of logic remain valid in the resulting extension of the language. My central claim is encapsulated in the stated thesis of reference by abstraction. In essence: when the extended language is used in the described way, we succeed in referring to directions and expressing meaningful statements about them. Let us say that ϕ suffices for ψ when ϕ is a ground for asserting ψ in a permissible language extension of the sort described. In the case discussed above, the statements l1  l2 and l1 ⊥ l2 suffice for d(l1 ) = d(l2 ) and d(l1 ) ⊥∗ d(l2 ), respectively.40 For abstraction more generally, we obtain the universal closures of the following sufficiency statements: α1 ∼ α2 ⇒ f (α1 ) = f (α2 ) ϕ(α1 , . . . , αn ) ⇒ ϕ ∗ (f (α1 ), . . . , f (αn ))

¬α1 ∼ α2 ⇒ ¬f (α1 ) = f (α2 ) ¬ϕ(α1 , . . . , αn ) ⇒ ¬ϕ ∗ (f (α1 ), . . . , f (αn ))

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 ϕ mediately suffices for ψ just in case there is a series of permissible reconceptualizations that takes us from ϕ to ψ 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

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 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 ϕ ⇒ ψ where the ontological commitments of ψ exceed those of ϕ. 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 façon de parler 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 l1  l2 is reconceptualized as d(l1 ) = 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.

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thick versus thin 

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.

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 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).

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appendices  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. Consider 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, 1981a) and (Dummett, 1991a). This view is defended in (Horsten, 2010). (Minimalism about criteria of identity must not be confused with what I have called metaontological minimalism.) 47

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 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 ↔ ∀u(x ∈ u ↔ y ∈ 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 ϕ be a necessary truth and consider:   (2.1) Fx ∧ Fy → x = y ↔ (RF (x, y) ∨ ¬ϕ) It is easy to verify that (2.1) too is a criterion of identity in the minimalist sense—but with the additional feature that this “criterion” entails ϕ!48 To rule out perverse criteria of identity of this sort, defenders of the metaphysical conception need to go beyond minimalism. Given a true identity t1 = t2 , they need to distinguish the statements ϕ that “merely” ensure the truth of the strict biconditional 2(t1 = t2 ↔ ϕ) from those that actually explain or “ground” the identity t1 = t2 . Some philosophers happily take on this explanatory burden (Lowe, 2003). It is not an easy burden to discharge, however. I believe my metasemantic conception explains how criteria of identity are distinguished from other metaphysical truths. Criteria of identity figure at the heart of an account of how certain fundamental forms of reference are constituted. As a result, the criteria of identity come to govern the referents in questions. Recall the Fregean triangle that figures so prominently in this chapter. As this triangle displays, criteria of identity are connected both to reference (as emphasized by the metasemantic conception) and to objecthood (as emphasized by the metaphysical conception).

2.B A Negative Free Logic Throughout this book, we often use a negative free logic, which is so called because any atomic predication involving a non-referring singular term is deemed false. Thus, ‘t = t’ is true just in case there is a true atomic predication involving t. This ensures that we can use ‘t = t’ as an existence predicate. The possibility of non-referring terms requires some changes to the axiomatization of firstorder logic. To begin with, it is only permissible to instantiate a universally quantified formula with respect to a term t on the assumption that t refers. We therefore adopt the following restricted axiom scheme of Universal Instantiation:

48 Clearly, (2.1) has the right logical form. And if (CI-1) and ϕ are necessary truths, then provably so is (2.1). It remains to show that (2.1) entails ϕ. Let x and y be distinct Fs, which by assumption is possible. Then the biconditional on the right-hand side of (2.1) is true. Since the left-hand side of this biconditional is then false, so must be its right-hand side, which means that ϕ must be true. Thus ϕ follows from (2.1). This is a trick familiar from the neo-Fregean literature; see e.g. (Heck, Jr., 1992). See also (Leitgeb, 2013) for a similar argument.

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appendices  ∀v ϕ(v) ∧ A(t) → ϕ(t) where A(t) is any atomic predication in which t occurs. Since we also accept universally quantified versions of all the ordinary laws of identity, including ∀x(x = x), we can prove A(t) → t = t, with A(t) as above. The rule of Universal Generalization is restricted such that we are permitted to universally generalize on a term t only on the assumption that t refers:

Assume  ϕ ∧ t = t → ψ(t), where t is a singular term that does not occur free in ϕ. Then we may infer  ϕ → ∀v ψ(v).

2.C Abstraction on a Partial Equivalence The literature on abstraction is almost exclusively concerned with ordinary equivalence relations.49 However, in the main text we observed that there is good reason to permit abstraction on partial equivalence relations, that is, on relations that are symmetric and transitive but not necessarily reflexive. I now outline three different approaches to abstraction on a partial equivalence relation and comment on how they are related. I use abstraction on objects as my example. The extension to abstraction on various sorts of concepts is straightforward. One option, to which the main text alluded, is to employ a negative free logic, which permits non-referring singular terms. (See Appendix 2.B.) In the context of this negative free logic, abstraction principles can be formulated in the ordinary way, for instance: (AP)

§x = §y ↔ x ∼ y

Had the logic been classical, the reflexivity of identity would have entailed §x = §x and thus also the reflexivity of ∼. But this inference is blocked in our negative free logic. Instead, we can prove the desired result that an object specifies an abstract just in case this object is in the field of ∼; that is: (2.2)

∀x(x ∼ x ↔ ∃y(y = §x))

There are alternative ways to handle abstraction on a partial equivalence which allow us to retain classical logic. Let me describe two options. But in each case, the price of classicality is a reformulation of the abstraction principle. The first alternative is to adopt a conditional version of the abstraction principle: (APc )

x ∼ x ∧ y ∼ y → (§x = §y ↔ x ∼ y)

Of course, to maintain classicality, the operator § must be total. So what is the “abstract” of an object that is not in the field of ∼? In fact, we may help ourselves to such “abstracts” as an innocent convenience. In cases where x ∼ x is false, §x can be anything whatsoever; we just need to ensure that the choice never matters. This is precisely what we achieve by giving the abstraction principle its conditional formulation. In cases where we abstract on objects outside

49

Two noteworthy exceptions are (MacFarlane, 2009) and the response (Hale and Wright, 2009a). See also (Payne, 2013a), which, in addition to providing a useful overview, defends the importance of abstraction on partial equivalence relations and shows how this can be achieved by means of negative free logic.

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 thin objects via criteria of identity of the field of ∼, the antecedent of the abstraction principle is false, thus rendering the relevant instance of the abstraction principle true regardless of how the unintended cases are handled. The second alternative is to “factorize” the abstraction principle (AP) into separate criteria of existence and of identity, namely: (AP∃ ) (AP= )

u ∼ u → ∃xAbs(u, x) Abs(u, x) ∧ Abs(v, y) → (x = y ↔ u ∼ v)

As we shall see in Chapter 9, this approach is particularly useful when we study systems of abstraction principles, where we allow the relation ∼ on which we abstract to vary. The different approaches have their respective advantages and disadvantages. Different choices will therefore be made in different contexts. Thankfully, these choices are ultimately of little philosophical importance, as the approaches are nothing but notational variants of one and the same idea. To substantiate this claim, we can show that the approaches are interpretable in each other and indeed that the translations employed are natural and arguably respect intended meanings. I now describe the translations but leave the task of verifying that they provide interpretations as an exercise for the reader. We begin with a translation from the second approach into the first. The translation maps an atomic predication A to t1 ∼ t1 ∧ . . . ∧ tn ∼ tn → A where t1 , . . . , tn are all the singular terms such that §ti occurs in A. Otherwise, the translation is trivial, in the sense that it commutes with all the connectives and quantifiers. The reverse translation maps an atomic predication A to A ∧ t1 ∼ t1 ∧ . . . ∧ tn ∼ tn where t1 , . . . , tn are as above. Otherwise the translation is trivial. Next, I describe a translation from the third approach to the second. The single non-trivial clause is: Abs(u, x) → x = §u ∧ u ∼ u In particular, other atomic predications are mapped to themselves. The reverse translation behaves non-trivially only on identities involving abstraction terms, where we need to allow the “abstracts” of objects outside the field of ∼ to behave any way they wish. We ensure this as follows. For every abstraction term §u involved in the identity, we make the identity claim conditional on u ∼ u by translating as follows: x = §u

→

u ∼ u → Abs(u, x)

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3 Dynamic Abstraction 3.1 Introduction Any abstractionist approach to thin objects confronts the threat of paradox. The most famous manifestation of the threat is Frege’s Basic Law V, which states that the extension of a concept F is identical to that of a concept G just in case the concepts are coextensive. We can formalize this “law” as (BLV)

ˆ ˆ x.Fx = x.Gx ↔ ∀x(Fx ↔ Gx)

where ‘F’ and ‘G’ are second-order variables that range over Fregean concepts and ˆ where ‘x.Fx’ stands for the extension of F. As Frege learnt from (Russell, 1902), the “law” falls prey to paradox. Once discovered, Russell’s paradox is straightforward. Let us say that x is a member of y just in case x falls under some concept whose extension is y. That is, we define ˆ x ∈ y as ∃F(Fx ∧ y = u.Fu). Consider the concept R defined by (3.1)

∀x(Rx ↔ x ∈ x).

ˆ This definition ensures that the extension of the concept, namely r = u.Ru, is a kind of Russell class whose members are all and only the objects that are not members of themselves. So we ask whether r is a member of itself. Using Basic Law V, it is easy to derive a contradiction. What to do? After Russell’s shocking letter, Frege hurriedly proposed a fix.1 But his proposal has since proven to be flawed, and Frege (who may or may not have been aware of the flaw) appears to have given up on abstraction around 1906.2 This seems an overreaction, however, given the importance of abstraction to our mathematical thought. Let us be more courageous and try to do better! I begin, in the next section, by describing the approach to abstraction that has dominated in recent decades. This is the neo-Fregean approach developed by Bob Hale and Crispin Wright in a series of important works, but which has also benefited from contributions by a large group of other philosophers and logicians (thus making

1

See the appendix to (Frege, 2013). The inconsistency re-emerges provided the domain contains two or more objects. See (Burgess, 2005, pp. 32–34) for a useful discussion. 2

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 dynamic abstraction this a paradigm of collaborative research in philosophy).3 A distinctive feature of this approach is that it is static, in the sense that it operates with a single fixed domain in which one or more abstraction principles are taken to be true.4 As in Frege’s own work, this fixed domain is typically taken to comprise all of reality.5 When abstraction takes place in some fixed domain, the reasoning that leads to Russell’s paradox is incontrovertible. It follows that Basic Law V is beyond rescue. But instead of following Frege into despair, the neo-Fregeans insist that many other abstraction principles are unaffected by the malaise and may therefore continue to serve as the technical engine of a philosophical account of mathematics. This differentiation between acceptable and unacceptable abstraction principles gives rise to a major challenge for the static approach. On what grounds is the differentiation to be made? This is known as the bad company problem.6 The problem is hard, as we shall see, not least because many “bad companions” are unacceptable for more subtle reasons than Basic Law V. Ideally, we would like a technical and philosophical understanding that explains what goes wrong in the bad cases. We would like to exclude the “bad companions” for a principled reason, not merely as an ad hoc expediency. This chapter develops an alternative dynamic approach to abstraction.7 On this approach, abstraction is not tied to a single fixed domain but may take us from one domain to a larger one. When directions are obtained by abstracting on lines under the equivalence of parallelism, for example, there is no reason to assume that the directions are present already in the domain with which we started. The dynamic approach to abstraction gives a radically different answer to the bad company problem: it accepts abstraction on just about any equivalence relation. (The small qualification will be explained below.) Paradox is avoided, not by being selective about which abstraction principles we accept but by allowing the abstracta to lie outside of the domain on which we abstract—much like directions may be taken to lie outside of the domain of lines. Basic Law V provides a good illustration. (If it can be redeemed, surely any abstraction can!) Suppose the extensions that result from abstracting on the concepts on some fixed domain are allowed to lie outside of this domain. Then 3 See (Wright, 1983), (Hale, 1987), and (Hale and Wright, 2001a). Other important contributions include (Boolos, 1998), (Burgess, 2005), (Fine, 2002), (Heck, Jr., 2011), and (Heck, Jr., 2012). 4 An important exception to the predominantly static approach pursued so far is (Fine, 2002), where various more dynamic ideas are explored. See also Fine’s “procedural postulationism” for some related ideas, e.g. (Fine, 2005b). Even in the work of Hale and Wright, however, we find traces of more dynamic ideas; for instance in their discussion of the significance of indefinite extensibility for the possibility of abstraction, as well as in the construction canvassed in (Wright, 1998a). 5 However, there is no expectation that this all-encompassing domain should form a set—which would be a potentially problematic universal set. Indeed, there is no immediate need to reify the domain at all, that is, to regard it as an object. After all, talk about truth in an all-encompassing domain can be replaced by talk about truth simpliciter. 6 See (Linnebo, 2009d) and (Studd, 2016) for introductions to the problem and further references. 7 For earlier work on this approach, see (Linnebo, 2009a), (Payne, 2013b), and (Studd, 2016), as well as the works cited in note 4.

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neo-fregean abstraction  the paradoxical reasoning is blocked, as it becomes impermissible to instantiate the quantifier ‘∀x’ in (3.1) with respect to the “Russell class” r. So provided that the domain is allowed to expand, there is no paradox (cf. Section 3.3). In fact, these domain expansions can be iterated. The resulting dynamic approach to abstraction has important affinities with the influential iterative conception of sets.8 I show how modal operators can be used to represent the domain expansions that are brought about by abstraction and thus also to develop a theory of iterated abstraction (cf. Section 3.5). The modal operators even enable us to retrieve a form of absolute generality, where we generalize across all of the possible domain expansions (cf. Section 3.6). Is the resulting view stable? I argue that it is. When a domain can be extensionally specified (i.e. specified as a plurality of objects), then it can be extended, but not so when the domain can only be intensionally specified (i.e. when it does not admit of specification as a plurality). My form of absolute generality is a case of the latter (cf. Section 3.7).

3.2 Neo-Fregean Abstraction To explain the neo-Fregean approach, it is useful to begin by briefly sketching Frege’s account of arithmetic. The account proceeds in two steps. First, Frege gives an account of the applications and identity conditions of numbers. He argues that counting involves the ascription of numbers to concepts. For instance, when we say that there are eight planets, we ascribe the number eight to the concept “ . . . is a planet”. Frege’s claim is that the number-of operator ‘#’ applies to any concept expression F to form the expression ‘#F’, which we may read as “the number of Fs”. Frege argues that the number of Fs is identical to the number of Gs just in case the Fs and the Gs can be put in a one-to-one correspondence. This is known as Hume’s Principle and is formalized as #F = #G ↔ F ≈ G

(HP)

where F ≈ G is a formalization in pure second-order logic of the claim that the Fs and the Gs are equinumerous. Second, Frege provides an explicit definition of terms of the form ‘#F’. In order to do so, he uses a theory consisting of second-order logic and Basic Law V. He defines #F as the extension of the concept “ . . . is an extension of some concept equinumerous with F”.9 It is straightforward to verify that this definition satisfies (HP). How should we respond to the unfortunate fact that Frege’s approach is inconsistent? A simple but radical answer is proposed in (Wright, 1983). Why not simply abandon the second step of Frege’s approach—which introduces the inconsistent

8 9

See (Boolos, 1971) and (Parsons, 1977) for two classic expositions. ˆ ˆ More precisely, ‘#F’ is defined as ‘x.∃G(x = y.Gy ∧ F ≈ G)’.

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 dynamic abstraction theory of extensions—and make do with the first step? This proposal has given birth to the neo-Fregean approach to the philosophy of mathematics. Wright’s proposal relies on two relatively recent technical discoveries. The first discovery is that (HP), unlike Basic Law V, is consistent. To be precise, let Frege Arithmetic be the second-order theory with (HP) as its sole non-logical axiom. Frege Arithmetic can then be shown to be consistent if and only if second-order DedekindPeano Arithmetic is.10 The second discovery is that Frege Arithmetic and some very natural definitions suffice to derive all the axioms of second-order Dedekind-Peano Arithmetic. This result is known as Frege’s theorem. At least from a technical point of view, the neo-Fregean approach is therefore a success: it is both consistent and strong enough to prove all of ordinary arithmetic. As the case of Basic Law V shows, however, the neo-Fregeans face the bad company problem. They need to demarcate the acceptable abstraction principles from the unacceptable ones. This demarcation has to balance two conflicting pressures. On the one hand, the rot represented by Basic Law V and a wide variety of other unacceptable abstraction principles has to be excised in a way that is both definitive and wellmotivated. On the other hand, we want to leave in place not only Hume’s Principle but also other abstraction principles that can serve as a foundation for analysis and set theory, at least in part. This will require a careful balancing act. The resulting challenge is hard, for at least two reasons. First, abstraction principles that are generally regarded as acceptable have close relatives that are plainly unacceptable. My favorite example is a close, but decidedly evil, cousin of Hume’s Principle.11 Observe that equinumerosity of concepts is just a matter of the concepts’ being isomorphic. So consider the abstraction principle that does to dyadic relations what Hume’s Principle does to concepts. This principle says that the isomorphism types of two dyadic relations are identical just in case the relations are isomorphic: (H2 P)

†R = †S ↔ R ∼ =S

Although this principle is closely related to Hume’s Principle, it is inconsistent, as it allows us to reproduce the Burali-Forti paradox.12 So good principles, such as Hume, have very close relatives that are bad. Second, there are abstraction principles that are bad in more subtle ways than Basic Law V, which at least has the good grace to wear its badness on its sleeves. These principles are therefore more sinister than Law V. My favorite example is Wright’s

10 Proof sketch: Let the domain D consist of the natural numbers. If a concept F applies to n objects, let ‘#F’ refer to n + 1. If F applies to infinitely many objects, let ‘#F’ refer to 0. 11 In fact, Hume has other problematic relatives as well. A wonderful example, due to (Cook, 2009), are “Hume’s big brothers”, which do to monadic concepts of orders two and higher what Hume does to monadic concepts of order one. The satisfiability of these big brothers turns out to be tangled up with the truth of the generalized continuum hypothesis. 12 See (Hodes, 1984a, p. 138) and (Hazen, 1985, pp. 253–4).

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how to expand the domain  “nuisance principle”.13 Let us say that F and G differ finitely just in case there are finitely many things that are F-and-not-G or G-and-not-F. This relation can be expressed in pure second-order logic and is easily seen to be an equivalence. So consider the associated abstraction principle: (NP)

nuisance(F) = nuisance(G) ↔ F and G differ finitely

This principle turns out to be satisfiable on all and only finite domains.14 While this is a curious property, it is not inconsistent: there are, of course, finite domains. But there is a problem. We know that Hume’s Principle is satisfiable in all and only infinite domains.15 Thus, while the nuisance principle and Hume’s Principle are individually satisfiable, they are not jointly satisfiable. So they cannot both be ruled acceptable. In fact, this is just the tip of the iceberg. Technical investigations have revealed a plethora of more complex interactions as well, showing that abstraction principles that appear acceptable on their own are not jointly acceptable. How should the neo-Fregeans respond to the bad company problem? It is hard to deny that a proper response is required. As emphasized by Frege himself, it is part of the very nature of both mathematics and philosophy to seek general explanations wherever such are possible.16 Yet even after decades of work by a number of capable researchers, there is no consensus in sight; on the contrary, new problems keep emerging.17 Most worrisome of all, in my opinion, is the extent to which recent work on the bad company problem has become a largely technical undertaking, which has lost touch with the underlying philosophical question of how abstraction might work. Ideally, we would like our philosophical account of abstraction to motivate, or at least inform, our answer to the bad company problem. In short, it is time to try a new tack.

3.3 How to Expand the Domain The static approach to abstraction is not obligatory. As mentioned, there is no need to assume that the directions obtained by abstraction on lines under the equivalence of parallelism belong to the domain on which we abstract. Let us examine the alternative dynamic approach on which abstraction may result in “new” objects that lie beyond the “old” domain with which we began. 13 See (Wright, 1997, section VI). This style of bad companion was first observed by (Boolos, 1990), who gives a slightly more complicated example known as the parity principle. 14 (NP) is obviously satisfiable in all finite domains, as there will then only be one equivalence class. For the other direction, consider any infinite domain D. Then there can be no one-to-one mapping of the equivalence classes in question to elements of the domain because there are, I claim, as many equivalence classes as there are subsets. To prove this claim, assume, for contradiction, that there were fewer equivalence classes than subsets. Since each equivalence class has |D