Appendix 2.A Some Conceptions of Criteria ofldentity
Appendix 2.B A Negative Free Logic
Appendix 2.C Abstraction on a Partial Equivalence
3. Dynamic Abstraction
3.1 Introduction
3.2 Neo-Fregean Abstraction
3 3 How to Expand the Domain
3.4 Static and Dynamic Abstraction Compared
3.5 Iterated Abstraction
3 6 Absolute Generality Retrieved
3 7 Extensional vs. Intensional Domains
Appendix 3.A Further Questions
3.A.l The higher-order needs of semantics
3 A.2 Abstraction on intensional entities
3.A.3 The need for a bimodal logic
3.A 4 The correct propositional logic
Appendix 3.B Proof of the Mirroring Theorem
Part II. Comparisons
4. Abstraction and the Question of Symmetry
4.1 Introduction
4.2 Identity of Content
4 3 Rayo on "Just is" -Statements
4.4 Abstraction and Worldly Asymmetry
5. Unbearable Lightness ofBeing
5.1 Ultra-Thin Conceptions of Objecthood
5.2 Logically Acceptable Translations
5.3 Semantically Idle Singular Terms
5.4 Inexplicable Reference
Appendix 5.A Proofs and Another Proposition
6. Predicative vs. Impredicative Abstraction
6.1 The Quest for Innocent Counterparts
6.2 Two Forms of!mpredicativity
6 3 Predicative Abstraction
6.3.1 Two-sorted languages
6.3.2 Defining the translation
6.3.3 The input theory
6.3.4 The output theory
6.4 Impredicative Abstraction
Appendix 6.A Proofs
7. The Context Principle
7 .1 Introduction
7 2 How Are the Numbers "Given to Us"?
7 3 The Context Principle in the Grundlagen
7.4 The "Reproduction" of Meaning
7.5 The Context Principle in the Grundgesetze
7.6 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
Part III. Details
8.5
12. Dynamic Set Theory
12. l Introduction
12.2 Choosing a Modal Logic
12.3 Plural Logic with Modality
12.4 The Nature of Sets
12.4. l The extensionality of sets
12.4 .2 The priority of elements to their set
12 .4.3 The extensional definiteness of subsethood
12.5 Recovering the Axioms ofZF
12.5 . l From conditions to sets
12 .5.2 Basic modal set theory
12 .5.3 Full modal set theory
Appendix 12 .A Proofs of Formal Results
Appendix 12.B A Harmless Restriction
Preface
This book is about a promising but elusive idea. Are there objects that are "thin" in the sense that their existence does not make a substantial demand on the world? Frege famously thought so. He claimed that the equinumerosity of the knives and the forks on a properly set table suffices for there to be objects such as the number of knives and the number of forks, and for these objects to be identical. Versions of the idea of thin objects have been defended by contemporary philosophers as well . For example, Bob Hale and Crispin Wright assert that what it takes for "the number of Fs = the number of Gs" to be true is exactly what it takes for the Fs to be equinumerous with the Gs, no more, no less.[ ] There is no gap for metaphysics to plug. 1
The truth of the equinumerosity claim is said to be "conceptually sufficient" for the truth of the number identity (ibid.) . Or, as Agustin Rayo colorfully puts it, once God had seen to it that the Fs are equinumerous with the Gs, "there was nothing extra she had to do" to ensure the existence of the number of F and the number of G, and their identity (Rayo, 2013 , p. 4; emphasis in original).
The idea of thin objects holds great philosophical promise. If the existence of certain objects does not make a substantial demand on the world, then knowledge of such objects will be comparatively easy to attain. On the Fregean view, for example, it suffices for knowledge of the existence and identity of two numbers that an unproblematic fact about knives and forks be known. Indeed, the idea of thin objects may well be the only way to reconcile the need for an ontology of mathematical objects with the need for a plausible epistemology Another attraction of the idea of thin objects concerns ontology. If little or nothing is required for the existence of objects of some sort, then no wonder there is an abundance of such objects. The less that is required for the existence of certain objects, the more such objects there will be. Thus, if mathematical objects are thin , this will explain the striking fact that mathematics operates with an ontology that is far more abundant than that of any other science.
The idea of thin objects is elusive, however. The characterization just offered is imprecise and partly metaphorical. What does it really mean to say that the existence of certain objects "makes no substantial demand on the world"? Indeed, if the truth of "the number of Fs = the number of Gs" requires no more than that of "the Fs are
1 (Hale and Wright, 2009b, pp. 187 and 193). Both of the passages quoted in this paragraph have been adapted slightly to fit our present example
equinumerous with the Gs", perhaps the former sentence is just a fafon de parler for the latter. To be convincing, the idea of thin objects has to be properly explained.
This book attempts to develop the needed explanations by drawing on some Fregean ideas. I should say straight away, though, that my ambitions are not primarily exegetical. I use some Fregean ideas that I find interesting in an attempt to answer some important philosophical questions. By and large, I do not claim that the arguments and views developed in this book coincide with Frege's. Some of the views I defend are patently un - Fregean.
My strategy for making sense of thin objects has a simple structure. I begin with the Fregean idea that an object, in the most general sense of the word, is a possible referent of a singular term. The question of what objects there are is thus transformed into the question of what forms of singular reference are possible. This means that any account that makes singular reference easy to achieve makes it correspondingly easy for objects to exist. A second Fregean idea is now invoked to argue that singular reference can indeed be easy to achieve. According to this second idea, there is a close link between reference and criteria of identity. Roughly speaking, it suffices for a singular term to refer that the term has been associated with a specification of the would-be referent , which figures in an appropriate criterion of identity. For instance, it suffices for a direction term to refer that it has been associated with a line and is subject to a criterion of identity that takes two lines to specify the same direction just in case they are parallel. 2 In this way, the second Fregean idea makes easy reference available. And by means of the first Fregean idea, easy reference ensures easy being. My strategy for making sense of thin objects can thus be depicted by the upper two arrows (representing explanatory moves) in the following triangle of interrelated concepts:
reference
objecthood
(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)
§a = §{3 = a "' f3
2 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.
where a and f3 are variables of some type, § is an operator that applies to such variables to form singular terms, and "' stands for an equivalence relation on the kinds of items over which the variables range. An example made famous by Frege is the aforementioned principle that the directions of two lines are identical just in case the lines are parallel. My preferred way of understanding an abstraction principle is simply as a special type of criterion of identity.
How does my proposed route to thin objects compare with others explored in the literature? My debt to Frege is obvious. I have also profited enormously from the writings of Michael Dummett and the neo-Fregeans Bob Hale and Crispin Wright. As soon as one zooms in on the conceptual terrain, however, it becomes clear that the route to be traveled in this book diverges in important respects from the paths already explored. Unlike the neo-Fregeans, I have no need for the so-called "syntactic priority thesis': which ascribes to syntactic categories a certain priority over ontological ones. And I am critical of the idea of "content recarving': which is central to Frege's project in the Grundlagen (but not, I argue, in the Grundgesetze) and to the projects of the neo-Fregeans as well as Rayo.
My view is in some respects closer to Dummett's than to that of the neo-Fregeans. I share Dummett's preference for a particularly unproblematic form of abstraction, which I call predicative. On this form of abstraction, any question about the "new" abstracta can be reduced to a question about the "old" entities on which we abstract. A paradigm example is the case of directions, where we abstract on lines to obtain their directions. This abstraction is predicative because any question about the resulting directions can be answered on the basis solely of the lines in terms of which the directions are specified. I argue that predicative abstraction principles can be laid down with no presuppositions whatsoever. But my argument does not extend to impredicative principles. This makes predicative abstraction principles uniquely well suited to serve in an account of thin objects My approach extends even to the predicative version of Frege's infamous Basic Law V. This "law" serves as the main engine of an abstractionist account of sets that I develop and show to justify the strong but widely accepted set theory ZF.
The restriction to predicative abstraction results in an entirely natural class of abstraction principles, which has no unacceptable members (or so-called "bad companions"). My account therefore avoids the "bad company problem': Instead, I face a complementary challenge. Although predicative abstraction principles are uniquely unproblematic and free of presuppositions, they are mathematically weak. My response to this challenge consists of a novel account of "dynamic abstraction': which is one of the distinctive features of the approach developed in this book. Since abstraction often results in a larger domain, we can use this extended domain to provide criteria of identity for yet further objects, which can thus be obtained by further steps of abstraction. (This observation is represented by the lower arrow in the above diagram.) The successive "formation" of sets described by the influential iterative conception of sets is just one instance of the more general phenomenon of
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). is an equivalence relation on some of these entities (say, parallelism). Then it is legitimate to adopt an extended language in which we speak precisely as if we have successfully abstracted on (say, by speaking also about the directions of the lines with which we began) I argue we have reason to ascribe to this extended language a genuine form of reference to abstract objects. Since these objects need not be in the domain of the original language, we can introduce yet another language extension, where we talk about yet more objects. In fact, there is no end to this process of forming ever more expressive languages.
Some words about methodology are in order. I make fairly extensive use oflogical and mathematical tools. Formal definitions are provided, and theorems proved I am under no illusions about what this methodology achieves. As Kripke observes, "There is no mathematical substitute for philosophy" (Kripke, 1976, p. 416). Definitions and theorems do not by themselves solve any philosophical problems, at least not of the sort that will occupy us here. The value of the formal methods to be employed lies in the precision and rigor that they make possible, not in replacing more traditional philosophical theorizing. But experience shows that precision matters in the discussions that will concern us. It is therefore scientifically inexcusable not to aspire to a high level of precision. In fact , much of the material to be discussed lends itself to a mathematically precise investigation. While the use of formal methods does not by itself solve any philosophical problems, it imposes an intellectual discipline that makes it more likely that our philosophical arguments will bear fruit. 3
A quick overview of the book may be helpful. Part I is intended as a self-contained introduction to the main ideas developed in the book as a whole . Chapter 1 sets the stage by introducing the idea of thin objects, explaining its attractions as well as some difficulties. This discussion culminates in a detailed "job description" for the idea of thin objects. This job description is formulated in terms of a notion of one claim sufficing for another-although the ontological commitments of the latter exceed those of the former. By formulating some constraints on the notion of sufficiency, I provide a precise characterization of what it would take to substantiate the idea of thin objects. Chapter 2 introduces my own candidate for the job. I explain the Fregean conception of objecthood and the idea that an appropriate use of criteria of identity can suffice to constitute relations of reference. Chapter 3 introduces the idea of dynamic abstraction. The form of abstraction explained in Chapter 2 can be iterated,
3 Compare (Williamson, 200 7)
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.
Many of the ideas developed in this book have had a long period of gestation. The central idea of thin objects figured prominently already in my PhD dissertation (Linnebo, 2002b) and an article (later abandoned) from the same period (Linnebo, 2002a). At first, this idea was developed in a structuralist manner. Later, an abstractionist development of the idea was explored in (Linnebo, 2005) and continued in (Linnebo, 2008) and (Linnebo, 2009b). These three articles contain the germs oflarge parts of this book, but are now entirely superseded by it. The idea of invoking thin objects to develop a plausible epistemology of mathematics has its roots in the final section of (Linnebo, 2006a). The second distinctive feature of this book-namely that of dynamic abstraction-has its origins in (Linnebo, 2006b) and (Linnebo, 2009a) (which was completed in 2007).
Some of the chapters draw on previously published material. In Part I, the opening four sections of Chapter 1 are based on (Linnebo, 2012a), which is now superseded by this chapter. Section 2.3 derives from Section 4 of (Linnebo, 2005), which (as mentioned) is superseded by this book. The remaining material is mostly new. In Part II, Sections 4.2 and 4.3 are based on (Linnebo, 2014), and Section 6.2 on (Linnebo, 2016a). These two articles expand on the themes of Chapters 4 and 6, respectively. Chapter 7 closely follows (Linnebo, forthcoming). In Part III, Chapters 8, 10, and 12 are based on (Linnebo, 2012b), (Linnebo, 2009c), and (Linnebo, 2013), respectively, but with occasional improvements. Chapter 9 and Section 11.5 make some limited use of (Linnebo, 2005) and (Linnebo, 2008), respectively, both of which are (as mentioned) superseded by this book.
There are many people to be thanked. Special thanks to Bob Hale and Agustin Rayo for our countless discussions and their sterling contribution as referees for Oxford University Press, as well as to Peter Momtchiloff for his patience and sound advice. I have benefited enormously from written comments and discussions of ideas
developed in this manuscript; thanks to Solveig Aasen, Bahram Assadian, Neil Barton, Rob Bassett, Christian Beyer, Susanne Bobzien, Francesca Boccuni, Einar Duenger B0hn, Roy Cook, Philip Ebert, Matti Eklund, Anthony Everett, Jens Erik Fenstad, Salvatore Florio, Dagfinn F0llesdal, Peter Fritz, Olav Gjelsvik, Volker Halbach, Mirja Hartimo, Richard Heck, Simon Hewitt, Leon Horsten, Keith Hossack, Torfinn Huvenes, Nick Jones, Frode Kjosavik, J6nne Kriener, James Ladyman, Hannes Leitgeb, Jon Litland, Michele Lubrano, Jonny Mcintosh, David Nicolas, Charles Parsons, Alex Paseau, Jonathan Payne, Richard Pettigrew, Michael Rescorla, Sam Roberts, Marcus Rossberg, Ian Rumfitt, Andrea Sereni, Stewart Shapiro, James Studd, Tolgahan Toy, Rafal Urbaniak, Gabriel Uzquiano, Albert Visser, Sean Walsh, Timothy Williamson, Crispin Wright, as well as the participants at a large number of conferences and workshops where this material was presented. Thanks to Hans Robin Solberg for preparing the index This project was initiated with the help of an AHRC-funded research leave (grant AH/E003753/l) and finally brought to its completion during two terms as a Visiting Fellow at All Souls College, Oxford. I gratefully acknowledge their support.
PART I Essentials
1
In Search of Thin Objects
1.1 Introduction
Kant famously argued that all existence claims are synthetic 1 An existence claim can never be established by conceptual analysis alone but always requires an appeal to intuition or perception, thus mal<ing 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 tal<ing 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 ofFregean 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 speal<ing) 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 speal<ing) 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 See (Kant, 1997 , B622-3).
2 See also (Boolos, 1997, pp. 199 and 214).
3 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
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 Agustin 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 See for instance (Parsons , 1990), (Resnik, 1997), and (Shapiro, 1997) .
5 See for instance (Wright, 1983) and the e ssays collected in (Hale and Wright , 200!a)
6 Philosophers attracted to this view include (Lewis , 1991, Section 3.6) and (Sider, 2007)
7 See for instance (Eklund, 2006a).
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 IR with two infinite numbers -oo and +oo, for example, the extended real number line i = IR U {-oo, +oo} exists. And since it is coherent to supplement IR with the imaginary unit i = .J=T 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 See (Eklund, 2006b) for a discussion of some extremely abundant ontologies that may arise in this way.
9 See (Field, 1989, pp 5 and 79-80). 10 See for instance (Rosen and Dorr , 2002)
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 <p is a coherent formula in a second-order language, then there is a structure that satisfies <p. (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)
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 <p <=> 1/f means that <p and 1/f "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 que stions 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
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 oflines and the identity of their directions:
The judgement "line a is parallel to line b'; or, using symbols, a II b, can be taken as an identity. If we do this, we obtain the concept of direction, and say: "the direction ofline a is identical with the direction of line b". Thus we replace the symbol II 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)
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 oflines and then "recarving" this content so as to yield an identity between directions.
Let <p <=} 1/1 formalize the claim that <p and 1/1 are different "carvings" of the same content-in a sense yet to be explicated. Then (Dir) can be strengthened to: (Dir{?)
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)
§a = §{3 B Ci {3
where a and f3 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= #GB 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
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 §a and §{3 is that the equivalence relation obtain between the entities a and f3. 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 Th eorem, 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 -Frege a ns are pursuing the idea of thin obj ects , as understood here, is provided in Section 6.4 .
17 See (Linnebo , 2017c) for an overview of defenses o f the premises, which , if successful, would support a much stronger conclusion.
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 ofthem? 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 dis t inguished from that of (Balaguer, 1998) As I understand it, Balaguer's "full-blooded platonism" is p r imarily a very generous ontology My present point, however, is that metaontological minimalism promis es to reduce the explanatory burden by equating the existence of mathematical objects with some fact to which epistemic access is less problematic.
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 1/1 is <p <p is (conceptually) sufficient for 1/1 all that God had to do to ensure that 1/1 was to bring it about that <p or the symmetrical analogue
<p is a recarving of the content of 1/1 <p and 1/1 make the same demand on the world for it to be the case that <p just is for it to be the case that 1/1
Some notation will be useful when grappling with this task. Let <p ==? 1/1 formalize the relationship that the first three statements are after, and <p <=:> 1/1, 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.
21 I shall use the word 's tatement' for a formula relative to some contextually salient assignment to its free variables, if any.
It will be convenient to allow the sufficiency operator to take a set of formulas on its left-hand side. Thus, we take r => 1/t to mean that the statements in the set r are jointly sufficient for 1/t. If our language allows infinite conjunctions, then r => 1/t can be regarded as just shorthand for (/\ae A <fJa) => 1/t' where r = {<fJa I a E 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, <p => 1/t must entail <p--+ 1/t; 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
ri => <p; for each i E I
{<p;} ie / => 1/t
U;er r; => 1/t
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 (')we have:
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 <p <=> 1/t as <p => 1/t /\ 1/t => <p. If instead we choose <=> as our primitive, we can define <p => 1/t as <p <=> <p /\ 1/t. (This definition can be motivated in terms of two natural assumptions: first, that <p /\ 1/t suffices for <p; and second, that <p suffices for <p /\ 1/t just in case <p suffices for 1/t .) 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
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 <p 1/t as A(<p 1ft), 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 distin ction 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 dicta 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 dicta. 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, 3x(T =>Ex), where Ex is an existence predicate and Tis 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 dicta, 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 (B o gh os sian , l 996)'s influential distinction , this is a m et aph ys ical rather than an epi stemological notion of analyticity.
23 For the record, I believe Quine succe ssfully undermines the ambitious notion of analyticity championed by the logical positivists , but that a more modest notion may escape his attack. See (Putnam , l 97S a ) for a similar view. In hi s later years, Quine too moved in thi s d irection : see (Quine, 1991) .
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
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 <p --+ 1/1 where the ontological commitments of 1/1 exceed those of </J.
Any such conditional would give rise to an analytic existence statement. To see this, let 3x8 make explicit an ontological commitment that is had by 1/1 but not by <p. We thus have A(<p--+ 3x8). By moving the quantifier 3x out through the parenthesis (which is permissible since <p can be assumed not to contain x free), it follows that A3x(<p--+ 8). 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 <p =} 1/1 where the ontological commitments of 1/1 exceed those of <p
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=?).
2 4 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 3xll 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 co nstraint would , of course, b e needed for the notion of mutual sufficiency. The same obv io usl y goes for the con straints formulated below. Henceforth , I shall not remind readers of this .
