THE ARCHÉ PAPERS ON THE MATHEMATICS

OF ABSTRACTION

THE WESTERN ONTARIO SERIESIN PHILOSOPHY OF SCIENCE

A SERIES OF BOOKS

IN PHILOSOPHY OF SCIENCE, METHODOLOGY, EPISTEMOLOGY,

LOGIC, HISTORY OF SCIENCE, AND RELATED FIELDS

Managing EditorWILLIAM DEMOPOULOS

Department of Philosophy, University of Western Ontario, CanadaDepartment of Logic and Philosophy of Science,

University of Californina/Irvine

Managing Editor 1980–1997

ROBERT E. BUTTS

Late, Department of Philosophy, University of Western Ontario, Canada

Editorial Board

JOHN L. BELL, University of Western Ontario

JEFFREY BUB, University of Maryland

PETER CLARK, St Andrews University

DAVID DEVIDI, University of Waterloo

ROBERT DiSALLE, University of Western Ontario

MICHAEL FRIEDMAN, Indiana University

MICHAEL HALLETT, McGill University

WILLIAM HARPER, University of Western Ontario

CLIFFORD A. HOOKER, University of Newcastle

AUSONIO MARRAS, University of Western Ontario

JÜRGEN MITTELSTRASS, Universität Konstanz

JOHN M. NICHOLAS, University of Western Ontario

ITAMAR PITOWSKY, Hebrew University

VOLUME 71

THE ARCHÉ PAPERS ONTHE MATHEMATICS OF

ABSTRACTION

Edited by

ROY T. COOK

123

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For Alice, who has kindly tolerated the company ofmany abstractionists, and one in particular

Contents

Foreword ix

Notes on the Contributors xi

Acknowledgements xiii

Introduction xv

Part I The Philosophy and Mathematics of Hume’s Principle

Is Hume’s Principle Analytic? 3G. Boolos

Is Hume’s Principle Analytic? 17C. Wright

Frege, Neo-Logicism and Applied Mathematics 45P. Clark

Finitude and Hume’s Principle 61R. G. Heck, Jr.

On Finite Hume 85F. MacBride

Could Nothing Matter? 95F. MacBride

On the Philosophical Interest of Frege Arithmetic 105W. Demopoulos

Part II The Logic of Abstraction

“Neo-logicist” Logic is not Epistemically Innocent 119S. Shapiro & A. Weir

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viii Contents

Aristotelian Logic, Axioms, and Abstraction 147R. T. Cook

Frege’s Unofficial Arithmetic 155A. Rayo

Part III Abstraction and the Continuum

Reals by Abstraction 175R. Hale

The State of the Economy: Neo-logicism and Inflation 197R. T. Cook

Frege Meets Dedekind: A Neo-logicist Treatment of Real Analysis 219S. Shapiro

Neo-Fregean Foundations for Real Analysis: Some Reflections onFrege’s Constraint 253

C. Wright

Part IV Basic Law V and Set Theory

NewV, ZF and Abstraction 275S. Shapiro & A. Weir

Well- and Non-well-founded Extensions 303I. Jané & G. Uzquiano

Abstraction & Set Theory 331Bob Hale

Prolegomenon to Any Future Neo-logicist Set Theory: Abstraction andIndefinite Extensibility 353

S. Shapiro

Neo-Fregeanism: An Embarassment of Riches 383A. Weir

Iteration One More Time 421R. T. Cook

Foreword

In September 2000 the Arché Centre launched a five-year research project enti-tled the Logical and Metaphysical Foundations of Classical Mathematics. Itsgoal was to study the prospects, philosophical and technical, for abstractionistfoundations for the classical mathematical theories of the natural, real andcomplex numbers and standard set theory. Funding was provided by the thenArts and Humanities Research Board (now the Arts and Humanities ResearchCouncil) for the appointment of full-time postdoctoral research fellows andPhD students to collaborate with more senior colleagues in the project, and atthe same time the British Academy awarded the Centre additional resourcesto establish an International Network of scholars to be associated with thework. This was the beginning of the serial ‘Abstraction workshops’ of whichthe Centre had staged no less than eleven by December 2006. We grate-fully acknowledge the generous support of the Academy and Council, sinequa non.

The project seminars and Network meetings generated—and continue togenerate—a large number of leading-edge research papers on all aspects ofthe project agenda. The present volume is the first of what we hope willbe a number of anthologies of these researches. With two exceptions,—thecontribution by the late George Boolos and the co-authored paper by GabrielUzquiano and Ignacio Jané,—the papers that Roy Cook has collected in thepresent volume are all authored by sometime members of the project team orof the British Academy Network. Their broad focus, as he explains, is on someof the more technical issues thrown up by the Abstractionist project, and it isanticipated that subsequent volumes may have a more purely metaphysical orepistemological emphasis.

I would like to thank Roy Cook for all his hard work putting the vol-ume together, and Bill Demopoulos for sponsoring its publication in theWestern Ontario Series in Philosophy of Science. Special thanks go to themembers of the core team and the Network not just for their direct con-tributions to the researches of the project but for their continuing affirma-tion, by their active participation, of the wider interest and importance of

ix

x Foreword

the neo-Fregean enterprise in the landscape of contemporary philosophy ofmathematics.

CJGWSt. Andrews 6/07

The Logical and Metaphysical Foundations of Classical MathematicsSometime project team members:Crispin Wright, Peter Clark, Roy Cook, Philip Ebert, Bob Hale, FraserMacBride, Paul McCallion, Darren McDonald, Nikolaj Jang Pedersen,Agustin Rayo, Marcus Rossberg, Andrea Sereni, Stewart Shapiro, ChiaraTabet, Robert WilliamsAuditor: Kit Fine

British Academy International Network members:Alexander Bird, Robert Black, Robin Cameron, William Demopoulos,Richard Heck, Keith Hossack, Daniel Isaacson, John Mayberry, Michael Pot-ter, Adam Rieger, Ian Rumfitt, Peter Simons, William Stirton, Peter Sullivan,Alan Weir

Notes on the Contributors

George Boolos was Professor of Philosophy at Massachusetts Institute ofTechnology, and the co-author of Computability and Logic (with Richard Jef-frey, Cambridge 2007) and the author of The Logic of Provability (Cambridge1995).

Peter J. Clark is Professor of the Philosophy of Science and Head of theSchool of Philosophical and Anthropological Studies in the University of StAndrews. He works primarily in the philosophy of physical sciences and math-ematics and was editor of the British Journal for the Philosophy of Science1999–2005.

Roy Cook is Visiting Assistant Professor of Philosophy at Villanova Univer-sity, and an associate fellow of Arché. He has published papers in the phi-losophy of language, logic, and mathematics, focusing primarily on semantic,soritical, and set-theoretic paradoxes, and Fregean and neo-Fregean philoso-phies of mathematics.

William Demopoulos is a member of the Department of Philosophy of theUniversity of Western Ontario and the Department of Logic and Philosophyof Science of the University of California, Irvine. He has published articles indiverse fields in the philosophy of the exact sciences, and on the developmentof analytic philosophy in the twentieth century.

Bob Hale is Professor of Philosophy at the University of Sheffield, and anAssociate Director of Arché. He works mainly on topics in the epistemol-ogy and metaphysics of mathematics and modality. His publications includeAbstract Objects (Blackwell 1987), and, together with Crispin Wright, TheBlackwell Companion to the Philosophy of Language (Blackwell 1997) andThe Reason’s Proper Study: Essays towards a Neo-Fregean Philosophy ofMathematics (Oxford 2001).

Richard Heck is Professor of Philosophy at Brown University and an asso-ciate fellow of Arché. He has published extensively on historical, conceptual,and technical issues emerging from Frege’s philosophy of mathematics. Phi-losophy of language and philosophy of logic are his other main areas of inter-est. He is now working on a book on philosophy of language and another onthe development of Frege’s mature philosophy (co-authored with Robert May).

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xii Notes on the Contributors

Ignacio Jané is Professor of Philosophy in the Department of Logic and theHistory and Philosophy of Science of the University of Barcelona. His maininterests are in the foundations of mathematics, philosophy of mathematics,and philosophy of logic. His recent papers include “Reflections on Skolem’sRelativity of Set-Theoretical Concepts” (Philosophia Mathematica, 2001),“Higher-Order Logic Reconsidered” (The Oxford Handbook of Philosophy ofMathematics and Logic, 2005), and “What is Tarski’s Common Concept ofConsequence” (The Bulletin of Symbolic Logic, 2006).

Fraser MacBride is a Reader in the School of Philosophy at Birkbeck Col-lege, London. He previously taught in the Department of Logic & Metaphysicsat the University of St. Andrews and was a research fellow at UniversityCollege, London. He has written several articles on the philosophy of mathe-matics, metaphysics, and the history of analytic philosophy, and is the editor ofIdentity & Modality (Oxford, 2006) and The Foundations of Mathematics andLogic (special issue of The Philosophical Quarterly, vol. 54, no. 214, 2004).

Agustin Rayo is Associate Professor of Philosophy at MIT and an associatefellow of Arché. He works mainly in the philosophy of language and thephilosophy of logic.

Stewart Shapiro is the O’Donnell Professor of Philosophy at The OhioState University and a Professorial Fellow in the Research Centre Arché atthe University of St. Andrews. His publications include Foundations withoutfoundationalism: a case for second-order logic (Oxford, 1991), Philosophy ofmathematics: structure and ontology (Oxford, 1997), and Vagueness in context(Oxford, 2006).

Gabriel Uzquiano is a Tutorial Fellow in Philosophy at Pembroke Collegeand a CUF lecturer in Philosophy at the University of Oxford. He has pub-lished articles in metaphysics, philosophical logic, and the philosophy ofmathematics.

Alan Weir is Professor of Philosophy, University of Glasgow. He has pub-lished articles on logic and philosophy of mathematics in a number of journalsincluding Mind, Philosophia Mathematica, Notre Dame Journal of FormalLogic, and Grazer Philosophische Studien and contributed chapters to a num-ber of volumes devoted to these areas.

Crispin Wright is Bishop Wardlaw Professor at the University of St Andrews,Global Distinguished Professor at New York University, and Director of theResearch Centre, Arché. His writings in the philosophy of mathematics andlogic include Wittgenstein on the Foundations of Mathematics (Harvard 1980);Frege’s Conception of Numbers as Objects (Aberdeen 1983); and, with BobHale, The Reason’s Proper Study (Oxford 2001). His most recent books,Rails to Infinity (Harvard 2001) and Saving the Differences (Harvard 2003),respectively collect his writings on central themes of Wittgenstein’s Philo-sophical Investigations and those further developing themes of his Truth andObjectivity.

Acknowledgements

The Editor wishes to thank the following:Oxford University Press, Kluwer Academic Publishers, Analysis, The

British Journal for the Philosophy of Science, The Journal of PhilosophicalLogic, The Journal of Symbolic Logic, The Notre Dame Journal of FormalLogic, Philosophia Mathematica, and Philosophical Books for permission toreprint the papers that follow. Detailed individual citations are included withthe papers.

Crispin Wright, Director of Arché: Philosophical Research Centre forLogic, Language, Metaphysics, and Epistemology, for providing the foreword.

William Demopoulos for proposing, and securing, the publication of thiswork in the Western Ontario Series in the Philosophy of Science.

Charles Erkelens and Lucy Fleet at Springer for their guidance and encour-agement.

The administrative staff at the Arché Centre in St Andrews (Gill Gardner,Sylvia Rescigno, and Sharon Coull) and at Villanova University (Elvia Beachand Terry DiMartino) for constant assistance in the practical aspects of prepar-ing this volume.

Marguerite Nesling for converting a number of the papers from hardcopy toelectronic format.

The Arts and Humanities Research Board (now the Arts and HumanitiesResearch Council) for support in the form of a postdoctoral research fellow-ship, which was held by the editor during the initial stages of this volume.

xiii

Introduction

As noted in the preface, the papers included in this volume concentrate (muchof the time, at least) on philosophical questions that are intimately tied up withthe interesting, and sometimes puzzling, mathematical properties of abstrac-tion principles. As a result, the introduction you are about to read will followsuit – concentrating on philosophical issues that have their roots in the mathe-matical characteristics of abstraction principles as well as philosophical prob-lems whose solution would seem to require a somewhat technical approach.

This focus should not be read as any sort of value judgment regarding theworth of technical versus non-technical work on abstraction principles, orwithin the philosophy of mathematics more generally. Instead, this focus onphilosophical problems that are linked to mathematical aspects of abstractionreflects the fact that there has, in the last decade or two, been an immenseamount of valuable work on Fregean-inspired abstraction principles and theirphilosophical importance. To attempt to cover all of this work, or even all suchwork that has some connection to the Arché Centre, would require severalvolumes the size of the present one. Hence the narrower focus.

The volume is divided into four sections. The first contains papers of ageneral sort (which can also serve as a helpful introduction to the subject forthose less familiar with the literature), although the majority of these never-theless address distinctly technical issues, at least indirectly. The remainingthree sections are devoted to three topics which have come under increasingstudy and scrutiny after the apparent success of the account of arithmetic basedon abstraction. The second section (“The Logic Of Abstraction”) containsthree papers that examine the role of logic (in particular, higher-order logic)within the abstractionist framework. The third section (“Abstractionism andthe Continuum”) contains papers that attempt to extend the abstractionistaccount to the theory of the real numbers, as well as papers critically evaluatingsuch attempts. The fourth and final section (“Basic Law V and Set Theory”)is devoted to attempts to reconstruct set theory (or something like it) withinthe abstractionist framework – usually by adopting some consistent variant ofFrege’s original Basic Law V.

Even with our focus narrowed to the more technical aspects of abstractionprinciples, however, the range of topics and problems addressed in the papers

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xvi Introduction

to follow is vast. Therefore, in the interest of providing a reasonably conciseand easily digestible introduction to the subject, there will be no attempt todiscuss every issue that arises in the following chapters. Instead, the remainderof this introduction will proceed as follows: First, a brief sketch of the originof interest in abstraction principles, i.e. Frege’s logicism and its failure, will beprovided. Second, we will briefly examine the philosophical framework under-lying the resurrection of interest in abstraction principles, a view often calledNeo-Fregeanism, Neo-Logicism, or Abstractionism. Next we will look at briefsketches of the philosophical and technical work underlying the abstractionistreconstructions of arithmetic, analysis, and set theory. Then we shall surveythree general types of problem that such reconstructions face, and concludewith a brief discussion of indefinite extensibility, a notion that has become ofcentral importance in much of the work attempting to solve problems of thesort covered in the previous sections.

Before moving on, a comment needs to be made regarding terminology.As already noted, the philosophical view (or views) under discussion in theremainder of this volume have been called, at various times and places,Neo-Fregeanism, Neo-Logicism, and Abstractionism. In the remainder ofthis essay the term “Abstractionism” will be used. The reasons for this aresimple: “Neo-Logicism” is misleading, since it would seem to imply thatthe view is a new version of logicism, while, as we shall see, it is nosuch thing. “Neo-Fregeanism”, while perhaps not misleading in this way,is, in the editor’s opinion, better reserved for the more general view in thephilosophy of language, clearly Fregean in nature, that (usually) underliesthe philosophy of mathematics discussed in the chapters to follow (althougheven here there is further confusion, since this term is also used to refer toa collection of views associated with the work of certain Oxford philoso-phers such as Gareth Evans and John McDowell). One could presumablyhold such Fregean views regarding language without believing in the fun-damental importance of abstraction principles (and vice versa – see AgustinRayo’s “Frege’s Unofficial Arithmetic” [2002], reprinted as chapter 10 below).When reading the essays collected in this volume, however, one shouldkeep in mind that these terms are (unfortunately) used for the most partinterchangeably.

1. Abstraction and logicism

An abstraction principle is any formula of the form:

(∀α)(∀β)(@(α) = @(β) ↔ E(α, β)

where “@” denotes a unary function mapping entities of the type rangedover by α (usually concepts, objects, or sequences of such) to objects, and“E( , )” is an equivalence relation on those same entities. The generalidea behind abstraction principles is that they allow us to introduce new

Introduction xvii

terms (and thus presumably to gain privileged epistemological access to thecorresponding objects) by defining the identity conditions for the referentsof the novel terms using linguistic resources that are already understood (i.e.those resources occurring in the equivalence relation “E( , )” – in most cases“E( , )” is either a purely logical formula, or one composed of logic pluspreviously introduced abstraction operators). Thus, an abstraction principle ismeant to act as an implicit definition of sorts, providing (so the story goes) anaccount of the meaning of novel terms of the form “@(α)”.

Perhaps the first notable occurrence of an abstraction principle occurs inFrege’s attempt at a logicist reconstruction of arithmetic (and, in fact, all ofmathematics). Frege notes, in the Grundlagen [1974] that the standard (higher-order) Peano axioms for arithmetic follow from the abstraction principle nowknown as Hume’s Principle (the explicit derivation of the Peano axiomsfrom Hume’s Principle was “extrapolated” from Frege’s comments in CrispinWright [1983], George Boolos [1990a], and Richard Heck [1993], and Boolos& Heck [1998], among others).

Hume’s Principle (which Hume himself did not state, and whose namederives from a rather charitable reading of a comment in Hume’s Treatise)is the claim that, given two arbitrary concepts P and Q, the number of P’s isidentical to the number of Q’s if and only if the P’s and the Q’s can be put in aone-to-one correspondence. More formally, we have:

HP : (∀P)(∀Q)(NUM(P) = NUM(Q) ↔ (P ≈ Q))

where P ≈ Q abbreviates the second-order formula stating that P and Q areequinumerous.

We can formulate rather natural definitions of arithmetical notions such‘natural number’, ‘successor’, and ‘addition’ in terms of the numerical oper-ator “NUM”. The fact that, given these definitions, the second-order Peanoaxioms for arithmetic follow from Hume’s Principle is quite notable as amathematical theorem independent of any philosophical motivation, and theresult has come to be called Frege’s Theorem (for a detailed examination ofthis result, and various streamlined versions of it, see Richard Heck’s “Finitudeand Hume’s Principle” [1997a], reprinted as chapter 4 below).

Frege, of course, wanted to reduce all of arithmetic to logic, thus defending(at least some of) mathematics from the Kantian charge of being synthetica priori (logic, presumably, being analytic if anything is!). Thus, he rejectedHume’s Principle as the ultimate foundation for arithmetic, since it containsineliminable occurrences of the cardinal number operator. (More famously, healso rejected Hume’s Principle, in its primitive form, since it was susceptibleto the Caesar Problem. See section 7 of this introduction for further discussionof this issue.)

As a result, Frege formulated a second abstraction principle, one thatmapped each concept onto a unique object – its extension. Unlike Hume’s

xviii Introduction

Principle, Frege’s Basic Law V:

BLV : (∀P)(∀Q)(EXT(P) = EXT(Q) ↔ (∀x)(Px) ↔ Qx))

contains only logical vocabulary (as long as talk of extensions is logical). BasicLaw V was, in essence, an early attempt at formulating (in an a priori, logicalmanner) the foundations of what we would now call set theory.

Using Basic Law V, Frege was able to reconstruct Peano Arithmetic on whatappeared to be a purely logical basis. The first step was to define numbers to becertain sorts of extensions – the number of a concept P is the extension of theconcept “(extension of a) concept equinumerous with P”, or, more formally:

NUM(P) =df EXT((∃Y)(x = EXT(Y) ∧ Y ≈ P))

Given this definition, Frege was able to prove Hume’s Principle (now a the-orem of Frege’s logic, and not a primitive non-logical definition of cardinalnumber) and thus prove the second-order Peano axioms for arithmetic. Soif Basic Law V was, as Frege hoped, a logical truth, then logicism (at leastregarding arithmetic) would be demonstrated.

Since not all of us are convinced logicists, something must have gonewrong – something discovered by Bertrand Russell. In a letter dated June16, 1902, Russell wrote to Frege, humbly pointing out that the crucial axiom(Basic Law V) that provided the power needed to reconstruct arithmetic withinlogic also seemed to allow for the derivation of a contradiction. AlthoughRussell’s actual presentation of the paradox that now bears his name is a bitmuddled in the original missive, the derivation of a contradiction from BasicLaw V is well known, and need not be rehashed here.

Frege attempted to fix the problem, but failed to find a convincing replace-ment for Basic Law V. Russell, meanwhile, along with Alfred North White-head, attempted his own reconstruction of mathematics from basic, a prioriprinciples in the monumental Principia Mathematica [1910–13]. Although thePrincipia was (likely) consistent, in the long run it turned out to be no moreconvincing than Frege’s Grundgesetze.

2. Abstractionism

After Frege’s failed attempt at utilizing abstraction principles in a logicistframework, this sort of principle lay unstudied for three-quarters of a century.Interest in abstraction of this sort was rekindled, however, by the publica-tion of Crispin Wright’s Frege’s Conception of Numbers as Objects [1983].Wright noted (perhaps among others, see Parsons [1965] and Hodes [1984]),in essence, that Frege’s project consisted of four basic steps:

(1) Recognize Basic Law V as an axiom of logic.

(2) Formulate suitable definitions of numerical notions in terms of the exten-sions provided by Basic Law V

Introduction xix

(3) Derive Hume’s Principle

(4) Derive arithmetic from Hume’s Principle (i.e. Frege’s Theorem)

Wright revived interest in Frege’s project, founding a new project that hascome to be called Neo-Logicism, Neo-Fregeanism, and Abstractionism, vari-ously, replacing the above blueprint with the following alternate plan:

(1) Lay down Hume’s Principle as an implicit definition of cardinal number

(2) Derive arithmetic from Hume’s Principle (i.e. Frege’s Theorem)

Of course, as was already noted, such a view (misleading nomenclature suchas “Neo-Logicism” notwithstanding) does not deserve the title ‘logicism’, atleast not in the traditional sense of the word as used by Frege, his fans, and hiscritics. Hume’s Principle, with its primitive and ineliminable occurrences ofarithmetical terms (i.e. “NUM”), just does not have the character of a logicallaw or theorem (a point made strenuously and convincingly by George Boolosin “Is Hume’s Principle Analytic?”, the essay that opens this volume). Thus,abstractionists have had to look elsewhere for their defense of Hume’s Princi-ple as something suitably basic as to provide the foundations of arithmetic.

The answer, according to abstractionists, is to note that what is importantabout logicism is not so much the reduction of mathematics to logic, butrather the fact that this reduction (had it been successful) would have gonea long ways towards providing an account of certain aspects of mathematicalknowledge that, ideally, we would like to be able to explain. In particular,the true advantages of logicism were that it purported to explain the a prioricharacter of mathematical knowledge (assuming that the a priori character ofpurely logical knowledge is unproblematic) and it purported to explain theanalyticity of mathematical truths (at least, this is important for those non-Quineans that retain a fondness for the analytic/synthetic distinction in the firstplace). The solution, then, is to retain these goals, while widening the scope ofour means for achieving these goals to something more than pure logic.

Along these lines, Wright and those that follow him deny that Hume’sPrinciple is a logical truth. Instead, Hume’s Principle, so it is argued, is (oris something like) an implicit definition of the “NUM” operator – one thatexplains the meaning of statements of identity of cardinal numbers. SinceHume’s Principle is a definition, we can come to know its consequencesa priori in the same manner (or, at least, in a suitably similar manner) inwhich we obtain a priori knowledge of the consequences of more pedestriandefinitions. Frege’s Theorem insures that all of second-order Peano arithmeticfollows from Hume’s Principle plus standard second-order logic, so (sincepresumably second-order logic is a priori knowable and second-order conse-quence preserves a priori knowledge) it follows that we can, using the abstrac-tionist recipe, obtain a priori knowledge of all of second-order arithmetic. (Thequestion of analyticity is strictly speaking separate from that of aprioricity, and

xx Introduction

has been less of a focus for the abstractionists than it was for Frege himself,although Bob Hale has rekindled interest in this issue.)

There are, unsurprisingly, deep questions regarding how Hume’s Principleand Frege’s Theorem accomplish this epistemological feat. In particular, thereare deep worries regarding the connection between our reconstruction ofarithmetic within the abstractionist framework and actual arithmetic practice:How do we know that the knowledge gained from Frege’s Theorem is, infact, knowledge about the ordinary natural numbers (and not some isomor-phic surrogate)? And how do we determine whether our (supposed) a prioriknowledge of the former allows for an explanation of the a priori status ofeveryday mathematics? William Demopoulos’ “On the Philosophical Interestof Frege Arithmetic” [2003] (reprinted below as chapter 7) develops a sus-tained examination and critique of this aspect of the project (although thereader is encouraged to consult Fraser MacBride’s two contributions to thisvolume as well).

Of course, even if the view in question is not, really, a version of logicism,the above sketch makes it clear that logic plays a crucial role in the abstraction-ist account of mathematical truth and mathematical knowledge. Defending theclaim that second-order logic preserves the relevant epistemological propertiesis one outstanding lacuna in the abstractionist literature, although it is not onethey are unaware of. The most sustained discussion of the issues is to be foundin Stewart Shapiro and Alan Weir’s “Neo-Logicist Logic Is Not EpistemicallyInnocent” [2000], reprinted below as chapter 8.

Setting the role of logic aside, however, there is much of interest to be saidregarding: (a) the notion of implicit definition required for such an abstrac-tionist project, (b) the more general abstractionist accounts of meaning andreference which might allow for such implicit definitions to succeed, and(c) the metaphysical account of abstract objects that would allow for ourepistemological access to them to proceed via such stipulations. Although allof these and more are touched on (and often discussed in some depth) in theessays that follow, they are not the primary focus of this volume or the papersincluded in it. Instead, we are here interested in those philosophical problemsthat stem from mathematical issues arising within the abstractionist project.Thus, we shall move on to examine those aspects of abstractionism that areof a more technical nature (the reader interested in more straightforwardlyphilosophical aspects of the abstractionist project, such as the topics mentionedat the top of this paragraph, can do no better than to consult Fraser MacBride’s“Speaking with Shadows: A Study of Neo-logicism” [2003], although the firsttwo essays in this volume, both titled “Is Hume’s Principle Analytic?”, byGeorge Boolos [1997] and Crispin Wright [1999], also provide much usefulphilosophical background material).

Arithmetic, it would seem, is, in one sense, the big success story of abstrac-tionism, since the technical results, at least, seem to be for the most partsettled – all that remains is sorting out the philosophical problems and issues

Introduction xxi

that result from this abstractionist reconstruction. As we shift our attentionto abstractionist accounts of other mathematical theories, however, we shallsee that things are not always so successful even within the purely technicalaspects of the project. Thus, our next task is to quickly survey the extension ofthis project to set theory and real analysis.

3. Abstractionist real numbers

One of the two obvious test cases for extending any philosophy of mathe-matics past an initial account of the natural numbers is to attempt to reconstructthe continuum (the other test case is to provide an adequate account of sets orsomething like them, the subject of the next section). Abstractionism is noexception here, and it did not take long for both believers and critics to wonderwhat shape an abstractionist account of real analysis might take. Althoughvarious accounts differ in the details (and this difference tends to depend onvarying attitudes towards Frege’s Constraint, see below), Bob Hale’s initialreconstruction (as found in his “Reals By Abstraction” [2000], reprinted aschapter 11 below) and those that follow are similar to the following, at leastfrom a mathematical perspective.

The first step in an abstractionist account of the real numbers is to notethat we are already provided with the natural numbers via Hume’s Principle.We can obtain the integers from these by adding an additional abstractionprinciple to our theory – something like the following Difference AbstractionPrinciple (the universal quantifiers here are restricted to natural numbers andthe arithmetical operators on the right-hand side of the biconditional are thestandard operations on the natural numbers):

DAP : (∀x)(∀y)(∀z)(∀w)(DIFF(x, y) = DIFF(z, w) ↔ x + w = y + z)

This principle provides us with an object corresponding to the differencebetween two natural numbers – in other words, DAP provides us with (a prioriaccess to) the integers.

With the integers in hand, we can obtain the rational numbers from theseby adding another abstraction principle to our theory – the following QuotientAbstraction Principle will do the trick (here, the initial universal quantifiersare restricted to integers, i.e. objects that are in the range of the DIFF operator,and the arithmetic operators on the right-hand side of the biconditional are thestandard operations on the integers – these are definable in terms of DIFF andsecond-order logic):

QAP : (∀x)(∀y)(∀z)(∀w)(QUO(x, y) = QUO(z, w) ↔((y = 0 ∧ w = 0) ∨ (y �= 0 ∧ w �= 0 ∧ x × w = y × z)))

Note that the Quotient Abstraction Principle provides us, not only with therational numbers, but also with an extra, ‘bad’ object: QUO(a, 0) for anyinteger a. This object results from the fact that we assume that our abstraction

xxii Introduction

operators are total functions, and thus certain unintended instances (such asdivision by zero in the present instance) nevertheless result in abstracts. Thepresence and role of ‘bad’ objects will be discussed in section 6 of thisintroduction.

Now that we have the integers, we can obtain the reals by applying anabstraction principle that simulates Dedekind-style cuts on the rationals, suchas the following Cut Abstraction Principle (here the universal quantifiers arerestricted to non-empty bounded concepts holding only of non-‘bad’ rationals,i.e. objects in the range of the QUO operator other than the ‘bad’ object):

CAP : (∀P)(∀Q)(REAL(P) = REAL(Q) ↔(∀x)((∀y)(P(y) → y < x) ↔ (∀y)(Q(y) → y < x)))

It is possible (although non-trivial) to prove that the objects provided by CAPare a complete ordered field, i.e. that they are isomorphic to the standardclassical continuum (see Shapiro’s “Frege Meets Dedekind: A NeologicistTreatment of Real Analysis” [2000], reprinted as chapter 14 below, for details).Thus, the abstractionist position can account for not only the natural numbers,but the classical theory of the real numbers as well (from a technical perspec-tive, at least).

As a technical note, there seems to be no reason why, at this last step, wecould not have applied, instead of CAP which encodes Dedekind’s notion ofcut within the abstractionist framework, an abstraction principle that, whenapplied to sequences of rationals, provides the real numbers along the lines ofCauchy’s methodology. There is no formal reason why we could not formulatesuch a principle (e.g. let the abstraction principle in question map functionsfrom the naturals to the rationals onto objects). The possibility of such alter-nate constructions raises a host of philosophical issues, however. Not leastamong them are the following questions: If it turns out that both CAP and anappropriate Cauchy-sequence principle are legitimate abstraction principles,then how are we to determine whether they provide us with access to the sameobjects? If not, then which one delivers the genuine real numbers (as opposedto merely an isomorphic copy)? Such questions are intimately tied up bothwith Frege’s Constraint and with the Caesar Problem, both of which will bediscussed below.

4. Abstractionist sets

The second natural extension of a foundational account of mathematics is toproduce some account of set theory (or, at the very least, to provide some othertheory that can do the work for which we normally invoke the theory of sets).The most notable attempt to provide such an account within the abstractionistframework is due to George Boolos, one of the most outspoken critics of theabstractionist view itself.

Introduction xxiii

In “Iteration Again” [1989], Boolos compared and contrasted the iterativeand limitation-of-size conceptions of sets. The former proposes to solve theproblem posed by Russell’s paradox by claiming that sets must be formed in aninfinitary step-by-step process, while the latter avoids paradoxes by claimingthat only collections that are (in some sense) not too ‘big’ determine sets.One version of the limitation-of-size conception (the one Boolos used) can beformulated by defining ‘X is too big’ as ‘there is a bijection between X and theentire domain’.

Boolos formulated an abstractionist version of the limitation-of-size con-ception of set along these lines. Letting “Big(P)” abbreviate the second-orderformula asserting that there is an onto function from P to the entire domain,Boolos’ abstraction principle for extensions, called New V, is:

New V : (∀P)(∀Q)(EXT(P) = EXT(Q) ↔((∀x)(Px ↔ Qx) ∨ (Big(P) ∧ Big(Q))))

New V provides a distinct object (an extension, or, more loosely, set) foreach collection of objects provided that collection is smaller than the entiredomain – concepts that hold of as many objects as there are in the domain,however, all receive the same abstract, the ‘Bad’ object (again, see below fordiscussion of ‘bad’ objects).

Given New V, we can define a set to be the extension of a small concept:

Set(x) =df (∃P)(x = EXT(P) ∧ ¬Big(P))

One object is the member of another object if and only if the second object isthe extension of a concept which holds of the first object, or, in symbols:

x ∈ y =df (∃P)(y = EXT(P) ∧ P(x))

(Note that ‘Bad’ objects can have, and be, members.) Given these definitions,New V entails many of the standard set theoretic axioms – extensionality,empty set, pairing, separation, replacement, and choice all follow (the unionaxiom does not follow on the above definitions, since the union of the singletonof the ‘bad’ object is not a set. Slight reformulations of this axiom do follow,however – for details see the chapters in section IV of this volume). In addition,the axiom of foundation holds if restricted to the pure sets (i.e. those sets thatcan be ‘built up’ from the empty set – see Gabriel Uzquiano and Ignacio Jané’s“Well- and Non-Well-Founded Extensions” [2004], reprinted as chapter 16 ofthis volume, for an in-depth examination of non-well-founded sets within theabstractionist framework). Thus, the only axioms that fail to follow, in somesense or another, are the powerset axiom and the axiom of infinity.

It is easy to see why the axiom of infinity fails – if we take as our domainthe hereditarily finite sets built from a single urelement (to serve as the ‘bad’object), then the resulting model satisfies New V (since all ‘small’, i.e. finite,concepts receive extensions – the corresponding sets – while all ‘big’, i.e.infinite, concepts can be mapped onto our single urelement). In other words,

xxiv Introduction

although NewV entails that there must be infinitely many objects, it does notentail that there need be any non-‘Bad’ concept that holds of infinitely manyobjects.

The proof that powerset fails is non-trivial, however – readers are encour-aged to consult chapter 15 of this volume, “New V, ZF, and Abstraction”[1999] by Stewart Shapiro and Alan Weir, for the technical details.

Given that New V does not allow us to reconstruct all of standard ZermeloFraenkel set theory, work has been done exploring other abstractionist routesto set theory. Among these are Roy T. Cook’s “Iteration One More Time”[2004], which formulates an abstractionist version of the iterative conceptionof set based on an abstraction principle called Newer V. Newer V entails theextensionality, empty set, pairing, separation, powerset, and choice axioms,but fails to imply both the axiom of infinity and the replacement axiom. Otherapproaches include Bob Hale’s “Abstraction and Set Theory” [2000], whichformulates an alternative version of the limitation of size conception, andStewart Shapiro’s “Prolegomenon to Any Future Neo-Logicist Set Theory:Abstraction and Indefinite Extensibility” [2003], which examines the generalconditions under which a restricted version of Basic Law V (such as NewV) will entail various set-theoretic principles (these three papers are reprintedbelow as chapters 20, 17, and 18 respectively).

Although one could debate how much we should worry about abstractionprinciples failing to imply powerset or replacement, there is no ignoring thefact that the failure of natural abstractionist accounts of set theory to providea proof of the axiom of infinity is just that, a failure. Presumably, a suc-cessful defense of abstractionism will require a development of a set theory(or surrogate for it) that is stronger than any of the existing proposals, sinceany set theory which fails to guarantee the existence of any infinite sets isunlikely to be adequate to our needs. To be fair, there are abstraction principlesthat imply all the axioms of second-order Zermelo–Fraenkel set theory (AlanWeir considers such principles in his “Neo-Fregeanism: An Embarrassment ofRiches” [2004], reprinted here as chapter 19). Unlike New V or even NewerV, however, these principles do not seem to codify plausible ‘definitions’ ofthe notion of set or collection – the sort of conception that could underliesuccessful a priori introduction of the notion of set or extension into ourdiscourse. Instead, these principles seem tailor made to provide all of theset theoretic axioms, and it is unlikely that anyone could have conceived ofthem without extensive prior knowledge of advanced set theoretic methods(e.g. formulation of typical ‘distractions’ requires an understanding of notionssuch as strong inaccessible cardinal).

Thus, unlike the case of arithmetic and real analysis, there seems to be muchmore work needed of a purely technical nature before the abstractionist canmake any claim to have explicated the aprioricity and analyticity of set theory.The main technical problem is to find an appropriate abstraction principlefor extensions that is satisfied only on uncountable domains of the right sort

Introduction xxv

(presumably, something like an inaccessible rank). As of the time of writingthis introduction, there does not seem to be any plausible abstraction principlethat will do the job, although there is interesting work leading in this direction(e.g. see Shapiro’s “Prolegomenon to Any Future Neo-Logicist Set Theory. . . ”and the later sections of Cook’s “Iteration One More Time”).

5. The first problem: too many abstraction principles

The first general problem plaguing the abstractionist project is that thereseem to be too many abstraction principles. What is required, and what we,at present, fail to have, is some general criteria for distinguishing betweenacceptable and unacceptable abstraction principles. Clearly, Basic Law V,being inconsistent, is on the unacceptable side of the field, while Hume’sPrinciple, the pride and joy of abstractionism, is (it is hoped) on the acceptableside (if not, then presumably some suitably modification of it is, such as FiniteHume, discussed in the next section). The problem, however, is that mereconsistency is not enough for acceptability, and as a result, we need somefurther guide to distinguishing the good from the bad.

The initial formulation of this problem is (as is almost always the case inthese debates) due to George Boolos (in Boolos [1990a]), who pointed out thatthere are abstraction principles that are consistent, but which are neverthelessincompatible with Hume’s Principle (or, in fact, with any abstraction principleguaranteeing the existence of infinitely many objects). Assuming that Hume’sPrinciple is acceptable if anything is, it follows that inconsistency, while suf-ficient for rejecting an abstraction principle as unacceptable, is not necessary.

Crispin Wright [1997] provided perhaps the most well-known exampleof such an abstraction principle: his aptly-named Nuisance Principle (hereFSD(P,Q) abbreviates the second-order formula asserting that the symmetricdifference of P and Q, that is, the collection of objects that are either P-and-not-Q or are Q-and-not-P, is finite):

NP : (∀P)(∀Q)[NUI(P) = NUI(Q) ↔ FSD(P, Q)]The Nuisance Principle can be satisfied on domains of any finite cardinality(in which case all objects receive the same nuisance), but can be satisfied onno infinite domain. Thus, the Nuisance Principle, although consistent, is asunacceptable an abstraction principle as is Basic Law V. The reason for theunacceptability is different, however. At first glance, the Nuisance Principleappears to derive its unacceptability, not solely in terms of its own formalproperties, but rather in terms of its interaction with other principles (such asHume’s Principle).

The existence both of inconsistent abstraction principles, and of pairs ofindividually consistent but incompatible abstraction principles, has given riseto a collection of problems that have been labeled The Bad Company Objec-tion. Chief amongst the concerns falling under this heading are:

xxvi Introduction

(1) The Existential Challenge: Given the existence of problematic principles ofthe same general form as Hume’s Principle (such as Basic Law V and theNuisance Principle), what reason do we have for thinking that there are anygood abstraction principles (including Hume) which have the privilegedstatus the abstractionist claims for them?

(2) The Epistemological Challenge: Even if one is convinced that there aregood abstraction principles that can play a foundational role such as theone envisioned for Hume’s Principle, in general how do we tell the goodprinciples from the bad?

The epistemological challenge, although clearly important, will be sidesteppedhere, since we are interested in those problems that are intimately connectedto the mathematics of abstractionism. The existential challenge, however, is,or at least can be easily approached as, a logical/mathematical issue – i.e. whatproof- or model-theoretic features will guarantee that an abstraction principleis acceptable?

One common response to this version of the Bad Company Objection (onefirst put forward by Wright’s “Is Hume’s Principle Analytic?” [1999] andfinessed by Shapiro and Weir in “New V, ZF, and Abstraction” [1999], bothreprinted below) is to require that an abstraction principle be conservative in acertain sense. The intuitive philosophical idea is this: An acceptable abstrac-tion principle is meant to be a definition of the abstracts that it introduces, butit is also meant to be no more than this. As a result, the principle in questionshould have no substantial consequences for those objects in the domainthat are not abstracts. Put simply, Hume’s Principle might entail all sorts ofinteresting claims about numbers, and even interesting claims regarding thenumbers corresponding to certain collections of cats, but Hume’s Principleshould not imply any substantial non-numerical claim about cats (a numericalclaim would be one containing at least one occurrence of the NUM operator).

Hume’s Principle can be proven to be conservative, as we would expect. Onthe other hand, the Nuisance Principle turns out to be non-conservative, as wewould hope (since it entails, for example, that there must be only finitely manycats). So at first glance the conservativeness constraint would seem to be doingthe job that it was designed to do.

There are problems, of course. For one, New V, the most promising abstrac-tionist reconstruction of set theory so far (even if far from fully satisfactory),is non-conservative. The technical details can be found in Shapiro and Weir’s[1999] paper, but the informal idea is easy to grasp. Within the languageof New V we can define the ordinals in the usual way – ordinals are justtransitive pure sets, well-ordered by membership. By the familiar reasoningof the Burali-Forti paradox, we can conclude as usual that there is no set of allordinals. Within the context of New V, however, this means that the collectionof ordinals is ‘Big’ – i.e. there is an onto function from the ordinals to theentire universe. But, since the ordinals are well-ordered by membership, this

Introduction xxvii

imposes a well-ordering on the entire universe. So New V is not conservative,since it implies that the universe can be well-ordered (and, within second-orderlogic, we can express this claim using no set-theoretic terminology). It is worthnoting that Cook’s [2004] iterative variant of abstractionist set theory fares nobetter on this score.

The problems with the conservativeness requirement do not stop with thefact that it would seem to rule out principles (such as New V) that we mightotherwise have wished to be acceptable, In addition, it turns out, as AlanWeir shows in his “Neo-Fregeanism: An Embarrassment of Riches” [2004],that there are consistent yet incompatible abstraction principles that pass theconservativeness constraint. Weir calls such principles distractions, and heshows, further, that if we try to strengthen the conservativeness constraint invarious natural ways in order to avoid such pairs of distractions, analogousproblems arise in the meta-theory.

There are a number of other criteria that have been proposed for narrowingdown the list of potentially good abstraction principles. One suggestion is thatthe equivalence relation on the right-hand side of the biconditional accuratelyreflect the mathematical content mastered when we actually first learn themathematical theory in question. In other words, the criterion for identity ofthe mathematical objects in question, provided by the abstraction principle,should clearly reflect the criterion by which we actually learned to identifyand distinguish the objects in question. (The various abstractionist set theoriesare of particular relevance here, since there does not seem to be one singlenotion of set underlying our mathematical practice, but a number of competingnotions, which are reflected in the competing reconstructions such as New Vand Newer V.) Critics of this approach, however, have suggested that suchrequirements confuse something like the order of discovery with the orderof explanation (e.g. see MacBride’s “On Finite Hume” [2002] and “CouldNothing Matter” [2003], reprinted as chapters 5 and 6 below). Accordingto this line of thought, abstraction principles are intended to provide a storyabout how we might come to know mathematical truths a priori, but there isno reason to think that the actual route that we took in first coming to knowthese truths is necessarily anything like the privileged route provided by theabstraction (since the initial knowledge could even have been a posteriori!).

6. The second problem: Too many objects

One of the main (supposed) advantages of abstractionism is that abstractionprinciples imply the existence of more objects than we would expect fromlogic and definitions alone. Some (including, of course, Boolos) have objectedto this, on the grounds that logic (or analytic statements, or a priori knowledgemore generally) should not imply the existence of all (or most) of the objectsstudied by working mathematicians:

xxviii Introduction

. . . It was a central tenet of logical positivism that the truths of mathematicswere analytic. Positivism was dead by 1960 and the more traditional view, thatanalytic truths cannot entail the existence either of particular objects or of toomany objects, has held sway ever since. (Boolos [1997], pp. 249–250)

Nevertheless, abstractionism is hopeless without the assumption that at leastsome existential claims are analytic, or a priori knowable, or somethingsimilar – the position in question is (on one reading) nothing more than adetailed philosophical account of how such is possible. So for our purposeshere we will ignore such general worries regarding ontological excess.

This ontological success seems to come at a price, however, since the veryabstraction principles (or, sometimes, natural generalizations of them) thatprovide us with the ontology of standard mathematics have a tendency to implythe existence of more objects than are strictly needed for the reconstruction ofthe mathematical theory in question. Unfortunately, these additional objectsare often unwanted or inconvenient.

The first such unwanted object is ‘anti-zero’. Hume’s Principle implies that,in addition to the countable infinity of finite numbers, at least one other numberexists, namely the number of the universal concept denoted by “x = x” – this‘number’ is anti-zero. The standard account of cardinal numbers as developedin ZFC implies that there is no largest cardinal number, however. Thus, aswas first pointed out by George Boolos, the theory of cardinals derived fromHume’s Principle seems to contradict the spirit, if not the letter, of the standardtheory of cardinality as derived in Zermelo-Fraenkel set theory, where therecan be no cardinal number of all objects.

It is important to note that there is no formal contradiction here. Onecan easily construct a model which satisfies both Hume’s Principle and the(second-order) axioms of Zermelo–Fraenkel set theory – just take any set-theoretic model of second-order ZFC, and interpret the numerical operator inHume’s Principle as mapping each concept onto the appropriate ZFC cardinal,if the concept’s extension is set-sized, and mapping all other concepts ontosome other object. Boolos’ point, rather, must be that there is no model ofHume’s Principle plus second-order ZFC where the ZFC cardinal numbersare exactly the cardinal numbers as defined by Hume’s Principle (under thesame ordering).

There are a number of obvious moves one could make here, although eachhas its problems. Among them are: (a) We might deny that ZFC providesan account of all the cardinal numbers, arguing instead that through thismeans we only get a model of the cardinal numbers corresponding to set-sized concepts (while Hume’s Principle provides us with a theory of all thecardinal numbers). While attractive, this option seems to challenge the ideathat set theory (however it is formalized) can play the foundational role tradi-tionally ascribed to it (a role that abstractionists presumably would prefer it toretain, hence the interest in set-theoretic abstraction principles such as New V).

Introduction xxix

(b) We might adopt a (positive) free logic, so that some instances of the numer-ical operator fail to designate objects (such as the instance that purports to referto anti-zero). This strategy, however, seems open to two problems. First, giventhe abstractionist’s rather lenient criteria for when a term refers (that it occurin a true statement of the appropriate sort), this response seems somewhat adhoc. Second, if abstractionists make it part of their official view that somenumerical terms can fail to refer, then this allows the critic of abstraction toask why it is not possible that all numerical terms fail to refer (or to argue thatsince some numerical terms fail to refer, then it seems unlikely that we canknow a priori that other numerical terms do refer). For a detailed discussionof free logic within the abstractionist context, see Shapiro and Weir’s “Neo-Logicist Logic Is Not Epistemically Innocent” [2000], reprinted below.

A final strategy, however, is to replace Hume’s Principle with some suitablymodified version, such as Finite Hume (here “Inf(P)” abbreviates the second-order claim that there are infinitely many P’s):

FHP : HP : (∀P)(∀Q)(NUM(P) = NUM(Q) ↔((Px ≈ Q) ∨ (Inf(P) ∧ Inf(Q))))

Finite Hume’s Principle provides a cardinal number for each finite concept, butmaps any concept with an infinite number of instances onto the same, ‘Bad’object (assuming that the logic is not free). Frege’s Theorem still holds forFinite Hume’s Principle (since the finite cardinals, i.e. the natural numbers,behave just as they do in the case of Hume’s Principle). There is no largestcardinal number, however, since the ‘Bad’ object that is the value “NUM”assigns to any infinite concept cannot be interpreted coherently as a numberat all (to see this, it is enough to note that Finite Hume maps concepts ofdiffering cardinalities onto the ‘Bad’ object in any uncountable model). SoFinite Hume does not entail the existence of any strange cardinal numbers,such as anti-zero. It does, however, provide us with a generic ‘Bad’ object,just as the Quotient Principle QAP and New V were seen to do earlier. Whilesuch an additional object does not, like anti-zero, seem to violate any intuitionsregarding the order-type of the cardinal numbers, they do bring with themdifferent, yet equally serious problems of their own.

In examining such ‘Bad’ objects, however, let us return our attention to the‘Bad’ object provided by New V and similar extensions-forming principles,as it is this object that has attracted the most attention in the literature. Now,all existent abstraction principles that purport to provide us with somethinglike extensions or sets also provide at least one unwanted, ‘Bad’ object – infact, if the extensions-forming operator EXT is a total function then they must,since the claim that each concept receives a unique extension is contradictory.Typically (as is the case with New V) all of the concepts which are too‘badly behaved’ to determine sets get mapped on to a single object, the ‘Bad’extension, and in the case of New V, the ‘Bad’ object is the extension of anyconcept that is equinumerous to the entire domain.

xxx Introduction

Now, unlike anti-zero, where the jury is still out regarding whether or notit is a genuine number, the ‘Bad’ extension is clearly not a genuine extensionor set at all. It is merely an artifact of the particular abstractionist means forobtaining the things that we do want, i.e. the other extensions. Since treatingthe ‘Bad’ extension as some novel, until now unrecognized, yet real set seemsimplausible, the other option would be to treat it exactly as just described – asan artifact of the fact that we are treating our abstraction operators as definingtotal functions.

The first problem is, of course, the seeming unavoidability of such ‘Bad’objects in the first place. Why should our account of set theory (or rationalnumbers, or perhaps natural or cardinal numbers) seem to require the existenceof an additional, and unwanted, object in the first place? Shouldn’t it bepossible to provide a foundational account of any mathematical theory thatentails the existence of all, and crucially, only, the objects required by thattheory?

This sort of question, while important and intuitively quite troubling, isalso rather loosely formulated. There are other problems associated with theexistence of ‘Bad’ objects (in particular, the ‘Bad’ extension) that are a goodbit more precise, however. If the ‘Bad’ extension is merely an artifact of ourtheory, and not a ‘real’ extension in some sense, then presumably any proofof a crucial set-theoretic result based on New V should not depend on theexistence of the ‘Bad’ extension. In other words, any set theoretic axiom thatturns out to be true (given New V) should be true solely in virtue of the set-theoretic ‘behavior’ of the genuine extensions (and thus should not depend forits truth on the existence or ‘behavior’ of the ‘Bad’ extension).

This requirement seems reasonable. Unfortunately, at least for the presentattempts at reconstructing set theory within an abstractionist framework, itseems like a requirement that cannot be met. The problem is that, if we arenot allowed to make use of the existence of the ‘Bad’ extension, we lose theproof that there are infinitely many sets.

Boolos provides the following proof that New V entails the existence of atleast two objects (this is the initial part of his proof that New V entails theexistence of infinitely many objects):

Let Ø be the concept [x : x �= x] . . . since there is at least one object (e.g. EXT

(x = x) or EXT (x �= x)), Ø is small, Ø �= V, and EXT(x �= x) �= EXT(x = x).([1989], p. 90, notation modified to fit that used here)

Notice that the proof makes explicit reference to the ‘Bad’ extension (i.e. EXT

(x = x)). This is not accidental – the result depends on the existence of the‘Bad’ extension in order to guarantee that finite concepts are not ‘Big’. Theredoes not exist any proof from New V to the pairing axiom that does not, insome way, make use of the ‘Bad’ extension.

To see why, consider New V interpreted in a free logic that allows EXT to failto take on a value when applied to ‘Big’ concepts. In such a logic, there will be

Introduction xxxi

a one-element model of New V – just let the domain contain (for example) theempty set. Then there are two concepts – the empty one, and the one holdingof the empty set. Since the latter is ‘Big’ (it is, in fact, the entire universe), itneed not receive an extension, so we can just map the empty concept onto theempty set, and we have our model.

Thus, there seems to be a real problem in the way that abstraction principlesfor extensions such as New V behave. On the one hand, they seem to imply theexistence of unwanted objects – in particular, the ‘Bad’ object. On the otherhand, this object seems necessary in order to ‘bootstrap’ our way up to a proofthat there are infinitely many sets. No satisfactory solution to this dilemma hasbeen presented as of yet.

Although the reader might be forgiven for thinking that this is, already, morethan enough problems, a look at abstractionist reconstructions of real analysisis in order. As noted above, the abstraction principle generating quotients ofintegers also generated a ‘Bad’ object, but this seems less worrisome thananti-zero or the ‘Bad’ extension, since the traditional theory of the rationalswas already plagued with a similar problem (i.e. the ill-definedness of 1/0.

More troubling, however, is the use of cut abstraction in the final step of theconstruction. Now, applying this particular abstraction principle to the domaindoes not, at first glance, seem to present any problems – we obtain, in fact,exactly the real numbers (or something isomorphic to them) and nothing else.

The problem possibly arises, however, when we ask the following question:If we can use abstraction to take cuts on the rationals, as we did to obtain thereals, then is it permissible to take, as objects, the cuts on any linear order, byapplying an appropriate abstraction principle?

If the answer is “No”, the we seem faced with another particularly dif-ficult instance of the Bad Company Objection – how are we to determinewhen we can, and when we cannot, apply an abstraction principle to a linearorder to take cuts as objects? If the answer is “Yes”, however, then we arebesieged by another worry: Generalizing such cut abstraction to any linearordering whatsoever generates a large ontology (in the worse case, properclass sized). This is, at best, extremely surprising in a view that emphasizes itsepistemic conservativeness. Additionally, some of the more powerful versionsof generalized cut abstraction are incompatible with other, somewhat attractiveabstraction principles, such as New V and variants of it (for a discussion ofgeneralized versions of cut abstraction see Cook’s “The State of the Economy:Neo-Logicism and Inflation” [2002], reprinted as chapter 12 below, and criti-cism of it in Bob Hale’s “Reals by Abstraction”, [2000] chapter 11).

Thus, in a number of ways the ontological power of abstractionism seemsto backfire – the very strength of the view, the fact that it purports to provideus with an account of how we can have a priori knowledge of the existenceand properties of those abstract objects studied by mathematicians, also is oneof its weaknesses, since it also seems to provide us with a priori knowledgeof the existence of until now unrecognized and, once recognized, unwanted

xxxii Introduction

objects such as anti-zero and the ‘Bad’ extension. Like the Bad Companyobjection before it, a satisfactory solution to this problem (or, better, thisfamily of problems) would seem to be a matter of determining where to drawcertain lines: How do abstractionists determine which principles (and whichformulations of certain principles) will provide them with access to the objectsrequired for mathematics without also entailing the existence of additionalobjects that are both unnecessary and, at times, inconvenient?

7. The third problem: What objects?

The final major problem of interest here is the notorious Caesar Problem.Frege first points out the problem in the Grundlagen, where he considers anabstraction principle introducing directions (here the initial quantifiers rangeover lines, and “//” is the relation of parallelism):

(∀a)(∀b)(DIR(a) = DIR(b) ↔ a//b)

After pointing out that this definition provides us with the means for identi-fying directions, and distinguishing distinct directions from one another, hepoints out that:

. . . this means does not provide for all cases. It will not, for instance, decide forus whether England is the same as the direction of the earth’s axis – if I maybe forgiven an example which looks nonsensical. Naturally no one is going toconfuse England with the direction of the Earth’s axis; but that is no thanks toour definition of direction. (Frege, [1974], §66, pp. 77–78)

Looking at this from a technical perspective, we can see the problem asfollows: Abstraction principles, such as Hume’s Principle and New V, whoseright-hand side can be expressed in purely logical vocabulary, place no con-straints on which object in a particular domain plays the role, say, of seven, orthe empty set (things are slightly more complicated in the case of abstractionprinciples, such as those used to construct the reals, where the equivelencerelation on the right contains other abstraction operators). All that determineswhether a particular set can serve as the domain of a model of either of theseprinciples is the cardinality of the set – if the set is the right size, then anyobject in the set can be any number or set (the only requirement is that eachobject can play the role of at most one number, or one set).

Much has been written on the Caesar Problem, but approaches to it gener-ally take one of three routes: First, we can deny it is a problem, adopting a sortof structuralist approach to abstractionism where it does not matter whetherCaesar turns out to be the number two, as long as we are guaranteed that someobject plays this role. (Although the Caesar problem is not his main target,Crispin Wright’s “Neo-Fregean Foundations for Real Analysis: Some Reflec-tions on Frege’s Constraint” [2000], chapter 13 below, draws connectionsbetween ante rem structuralism and abstractionism, and is particularly relevant

Introduction xxxiii

here.) Second, we can attempt to reformulate our abstraction principles in morecomplicated ways (e.g. by inserting modal operators in appropriate places orthe like) so that the reference of numerical terms is more determinate. Third,we might argue that although abstraction principles alone do not determinewhich object, in particular, is picked out by a certain numeral, abstractionprinciples plus other background constraints do determine numerical referenceuniquely.

Which of these approaches is most promising has yet to be determined. Infact, as the literature grows, new variations on the Caesar Problem seem tosprout up at least as fast as attempts to solve them. Most important amongthese are:

The Counter-Caesar Problem: How do we guarantee that particularFregean numerals denote the same object as their natural languagecounterparts (e.g. does NUM(x �= x) denote the same thing as theEnglish locution “zero”)?

The Julio Cesar Problem: How do we guarantee that the cardinal numbersprovided by Hume’s Principle denote the same kind of objects as aredenoted by mathematical terms occurring in natural language (e.g. doesNUM(x �= x) denote the same kind of thing as the English locution“zero”)?

The C-R Problem: How do we determine whether abstracts provided bydistinct abstraction principles are identical or distinct (e.g. is thecomplex number 0, provided by the appropriate abstraction principle,identical to the real number 0, provided by a different abstractionprinciple)?

Although the Caesar Problem (and its cousins) results from certain formalcharacteristics of abstraction principles, responses to it tend to be less techni-cal. Nevertheless, a number of the chapters included below contain extendeddiscussions of it. (The reader is also encouraged to consult MacBride [2005]and Cook and Ebert [2005] for further discussion of variants of the CaesarProblem.)

8. Indefinite extensibility

As the literature on these problems and other issues has grown, the notionof indefinite extensibility has become more and more central to purportedsolutions. One promising line of attack on both the ‘too-many-abstraction’principles class of problems and the ‘too-many-objects’ class of problems hasbeen to suggest that we restrict our attention to those abstraction principlesthat provide abstracts only for concepts which are not indefinitely extensible.

Of course, this does little to help us until we know what indefinite exten-sibility is. Bertrand Russell seems to be the first person to discuss this notionwhen considering the cause of the various set-theoretic paradoxes:

xxxiv Introduction

The contradictions result from the fact that . . . there are what we may call self-reproductive processes and classes. That is, there are some properties such that,given any class of terms all having such a property, we can always define a newterms also having the property in question. Hence we can never collect all of theterms having the said property into a whole; because, whenever we hope we havethem all, the collection which we have immediately proceeds to generate a newterm also having the said property. ([1906], p. 144)

The term “indefinite extensibility” is due to Michael Dummett, however, whoextended Russell’s idea as follows:

An indefinitely extensible concept is one such that, if we can form a definiteconception of a totality all of whose members fall under the concept, we can, byreference to that totality, characterize a larger totality all of whose members fallunder it. ([1993], p. 441)

It has become standard to use the term ‘definite’ for those concepts that are notindefinitely extensible.

The ordinal numbers provide perhaps the clearest example of an indefinitelyextensible collection. Consider any definite collection of ordinals (i.e. a set ofordinals). Given such a collection, we can immediately form a conception of anordinal not in that collection (i.e. the ‘next’ ordinal, (i.e. either the successorof the greatest ordinal in the collection in question, or the supremum of thecollection in question). As a result, there seems to be a sense in which wecan never collect together all of the ordinals into a definite totality, since wecould repeat this reasoning on such a collection to obtain an ordinal that is notin such a collection of all ordinals – contradiction (this is essentially just thereasoning behind the Burali-Forti paradox).

An indefinitely extensible concept is thus one which allows for a certain sortof iteration – any time we have collected together some definite sub-collectionof things falling under that concept, we can find a new object that is not in thatcollection. In fact, the ordinals are not only a clear example of the notion inquestion, but their structure seems to be fundamental to indefinite extensibilityitself, since this iterability suggests that any indefinitely extensible collectionwill contain a structure isomorphic to the ordinals (it is worth noting, how-ever, that Dummett would reject this Russellian characterization of indefiniteextensibility).

Thus, one way of characterizing indefinitely extensible concepts is “thoseconcepts that are like the ordinals in relevant ways”. As Peter Clark points outin his contribution to this volume (“Frege, Neo-logicism and Applied Mathe-matics” [2004], chapter 3 below), another way of picking out the indefinitelyextensible concepts is to just note that they are the ones whose extensions donot form sets. But neither of these suggestions, intuitively helpful as they are,do the abstractionist any real good. The abstractionist, remember, wishes to usethe notion of indefinite extensibility in order to formulate a restricted version ofBasic Law V (and other abstraction principles) which will provide an adequate

Introduction xxxv

set theory. As a result, no characterization of indefinite extensibility (such asthose above) which uses set-theoretic notions can be of use, since using settheoretic notions to formulate one’s implicit definition of set would introducea rather vicious circle into the picture. Thus, the abstractionist needs someneutral formulation of the notion in question.

As of yet, no completely adequate account of indefinite extensibility hasbeen found, at least none that is of the sort that could be mobilized by theabstractionist wishing to use it in formulating various abstraction principles.This is not to say, of course, that no work of interest has been carried out– on the contrary, at least half of the papers in the present volume make atleast passing reference to the importance of this problem, and almost all ofthe papers in the last section, on set theory, contain detailed discussion ofthe issue. Of particular interest is Stewart Shapiro’s “Prolegomenon to AnyFuture Neo-Logicist Set Theory. . . ” (chapter 18 below), which contains botha detailed examination of indefinite extensibility as discussed by philosopherssuch as Dummett and Russell, as well as a sustained technical examination ofwhat formal characteristics a successful abstractionist account of the notionrequires.

9. One last thing

Although the bulk of the literature on abstraction and its mathematics, andthe majority of the papers to follow, focus either on the actual formalization ofarithmetic, analysis, and set theory, or on the three major sorts of problemjust outlined, there are of course many other crucial questions regardingabstraction to be answered and many other avenues to be explored. Whilespace considerations preclude detailed discussion of them here, at least oneof them deserves brief mention before moving on to the papers themselves.The question in question is this: In what ways can the abstractionist’s formalresults be adopted or adapted by their philosophical opponents?

For example, in “Frege’s Unofficial Arithmetic” [2002] (chapter 10 below)Agustin Rayo utilizes Frege’s Theorem (and corollaries of it) to provide adistinctly non-Fregean (in fact, somewhat Quinean) account of arithmetic(and, in particular, applied arithmetic). While Rayo suggests that the accounthe provides is at least inspired by Frege’s own views (views Frege heldafter he abandoned logicism), the project he sketches is worked out againsta philosophical background quite different from the one assumed by mostabstractionists.

The point, to put it bluntly, is this: even if abstraction principles are notdefinitions in the sense the abstractionist suggests, they might neverthelessplay some crucial role in our epistemological account of mathematics. It is thushoped that this collection will serve as a repository of work on the technicalaspects of abstraction principles which can be utilized by both the abstrac-tionist himself and also by adherents of different, competing philosophical

xxxvi Introduction

accounts of mathematics (even if the majority of the actual papers are workingwithin the standard Fregean abstractionist picture).

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