Synthese (2009) 170:321329DOI 10.1007/s11229-007-9267-5

Introduction

ystein Linnebo

Accepted: 31 August 2007 / Published online: 22 November 2007 Springer Science+Business Media B.V. 2007

Abstract Neo-Fregean logicism seeks to base mathematics on abstraction principles.But the acceptable abstraction principles are surrounded by unacceptable (indeed of-ten paradoxical) ones. This is the bad company problem. In this introduction I firstprovide a brief historical overview of the problem. Then I outline the main responsesthat are currently being debated. In the course of doing so I provide summaries of thecontributions to this special issue.

Keywords Abstraction Frege Logicism Neo-logicism Paradox

The bad company problem is one of the most serious problems facing one of themost exciting philosophical approaches to mathematics. The approach in questionis neo-Fregean logicism, broadly construed as the project of basing mathematics onabstraction principles, that is, on principles of the form

() () = () where the variables and range over entities of some sort, and where is anequivalence relation on this sort of entity. The problem is that not every abstractionprinciple is acceptable: some are downright inconsistent, while others are unaccept-able for more subtle reasons. Abstraction principles with desirable philosophical andtechnical properties are thus surrounded by bad companions. Some philosophersclaim that this gives rise to a devastating objection to the neo-Fregean programme,while others respond that the challenge it poses is perfectly surmountable.1

1 Since the philosophical and technical interest of the problem posed by the bad companions is inde-pendent of whether or not they give rise to an objection, I prefer the more neutral label bad companyproblem rather than the more common the bad company objection.

. Linnebo (B)Department of Philosophy, University of Bristol, 9 Woodland Rd, Bristol BS8 1TB UKe-mail: oystein.linnebo@bris.ac.uk

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In this Introduction I first provide a brief historical background (whichthe cognoscenti may want to skip). Then I outline the main responses to the badcompany problem which are currently being debated, as well as some broader ques-tions raised by the problem. In the course of doing so I provide summaries of thecontributions to this special issue.

1 Background to the problem

Frege was a logicist about arithmetic and mathematical analysis. That is, he took purelogic to provide a source of knowledge of these branches of mathematics, thus makingthem a priori. His defense of this view proceeds in two steps.

The first step consists in an account of ascriptions of numbers and of their identitycondition. Frege argues that numbers are ascribed to concepts. For instance, when wesay that there are eight planets, we ascribe the number eight to the concept of being aplanet. Let # be the operator the number of. Freges claim is then that # applies toany concept term F to form the expression #F, meaning the number of Fs. NextFrege argues that the number of Fs is identical to the number of Gs if and only ifthe Fs and the Gs can be one-to-one correlated. This principle is known as HumesPrinciple and can be formalized as

(HP) #F = #G F G,where F G is a formalization in second-order logic of the claim that there is arelation R that one-to-one correlates the Fs and the Gs.

The second step of Freges defense of logicism provides an explicit definition ofnumber terms of the form #F. Frege does this in a theory consisting of second-orderlogic and his Basic Law V, which states that the extension of a concept F is identicalto that of a concept G if and only if the Fs and the Gs are co-extensional:

(V) x Fx = xGx x(Fx Gx)In this theory, Frege defines #F as the extension of the concept x is an extension ofsome concept equinumerous with F . That is, he defines

#F = xG(x = yGy F G).

This definition is easily seen to validate (HP). More interestingly, Frege (1964) provesin meticulous detail how this definition and his theory of extensions entail all ofordinary arithmetic.

However, just as the second volume of his magnum opus (Frege 1964) was goingto press in 1902, Frege received a letter from Russell (1902), who reported that he hadencountered a difficulty with Freges theory of extensions. The difficulty Russellhad encountered is the paradox now bearing his name. Freges theory of extensionsis in effect a naive theory of sets. We may thus consider the set of all sets that arenot members of themselves. In Freges theory we can then prove that this set both isand is not an element of itself. Russells paradox eventually led Frege to give up onlogicism. Until the 1980s both logicians and philosophers regarded Fregean logicism

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as a dead end, and people attracted to the idea of logicism pursued other versions ofit, such as Russells or that of the logical positivists.

All this changed in 1983 with the publication of Wright (1983). Crispin Wrighthere suggests that the problem posed by Russells paradox can be avoided by mak-ing do with the first step of Freges approach, abandoning altogether the second stepand its inconsistent theory of extensions. The possibility of this approach is securedby two more recent technical discoveries. The first discovery is that (HP), unlike(V), is consistent. More precisely, let Frege Arithmetic be the second-order theorywith (HP) as its sole non-logical axiom. We can then show that Frege Arithmetic isconsistent if and only if second-order Peano Arithmetic is.2 The second discoveryis that Frege Arithmetic and some very natural definitions suffice to derive all theaxioms of second-order Peano Arithmetic. This important result is known as FregesTheorem.3 For more than a century now, informal arithmetic has almost without excep-tion been given some PeanoDedekind style axiomatization, where the natural num-bers are regarded as finite ordinals, defined by their position in an omega-sequence.Freges Theorem shows that an alternative and conceptually completely different axi-omatization of arithmetic is possible, based on the idea that the natural numbers arefinite cardinals, defined by the cardinalities of the concepts whose numbers they are.

The neo-Fregean programme initiated by Wright (1983) has thus been technicallyquite successful. But much work remains. We need to get clearer on the philosophi-cal significance of the technical results just described. And the programme needs tobe extended beyond arithmetic. The bad company problem is highly relevant to boththese concerns.4

2 The significance of the bad companions

The oldest bad companion is of course Freges inconsistent Basic Law V. However,there are other inconsistent abstraction principle that are even more sinister. Here is anexample.5 Note that equinumerosity of concepts is just a matter of the concepts beingisomorphic. So consider the abstraction principle which does to dyadic relations whatHumes Principle does to concepts, that is, the abstraction principle which says thatthe isomorphism types of two dyadic relations are identical if and only if the relationsare isomorphic:

(H2P) R = S R SAlthough this principle seems just as innocent Humes Principle, it is in fact inconsis-tent, as it allows us to reproduce the BuraliForti paradox.

2 This result was first proved in Geach (1975, pp. 446447). For a nice exposition and discussion, seeBoolos (1987).3 This theorem is first hinted at in Parsons (1965), is explicitly stated and discussed in Wright (1983),and receives a nice proof in Boolos (1990). For a powerful argument that Frege himself was aware of thetheorem, see Heck (1993).4 The problem was noticed already in two reviews of Wright (1983): see Burgess (1984, p. 639) and Hazen(1985, pp. 253254).5 See Hodes (1984, p. 138) and Hazen (1985, pp. 253254).

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This means that attractive abstraction principles such as Humes Principle aresurrounded by bad companions. What is the philosophical significance of this fact?A modest lesson to draw is that we need a better understanding of the conditionsunder which Fregean abstraction is acceptable. Ideally we would like a mathematicallyinformative and philosophically well motivated account of acceptable abstraction. Thereason why this lesson is modest is that it is part of the very nature of both mathematicsand philosophy to seek general explanations wherever such as possible.

Other people have drawn more controversial lessons from the bad companions. Inparticular, Michael Dummett appears to claim that the bad companions are immedi-ately fatal to the neo-Fregean programme. For instance, he claims that

if the context principle, as expounded by Wright, is enough to validate the con-textual method of introducing the cardinality operator [i.e., the introduction of# by means of Humes Principle], it must be enough to validate a similar meansof introducing the [extension] abstraction operator [by means of Basic Law V].6

If this conditional is true, it follows immediately by modus tollens that the neo-Fregeanprogramme is doomed. However, I believe that on a careful and charitable readingDummett is better understood as first drawing the modest lesson mentioned aboveand then arguing that the resulting explanatory demand cannot be met.7 His argumentfor this latter claim has to do with the neo-Fregean programmes ineliminable use ofimpredicative reasoning. The argument is too complex to be analyzed here. But theresulting diagnosis of the bad company problem will be discussed in Sect. 3.4.

3 In search of an account of acceptable abstraction

In this section I describe the main attempts that have been made to develop an accountof acceptable abstraction. I begin with two attempts that have failed for technical rea-sons. Then I turn to the main proposals that are currently being debated. There aremany unresolved questions here, and my aim will merely be to identify these, not toanswer them.

3.1 Two dead ends

A obvious first thought is that an abstraction principle is acceptable if and only if itis consistent. But this suggestion fails, as even consistent abstraction principles canbe unacceptable. For there are consistent abstraction principles that conflict with oneanother.8 For instance, we know that (HP) is satisfiable in all and only infinite domains(assuming the Axiom of Choice). But there are other abstraction principles that aresatisfiable in all and only finite domains. Consider for instance the equivalence relationthat holds between two concepts F and G just in case they are coextensive apart from

6 Dummett (1991, p. 188); but see also the later chapters of Dummett (1991), as well as Dummett (1998).7 See for instance the rest of the paragraph from which the above quote is taken, as well as Dummett (1998,p. 375).8 This was first pointed out by Boolos (1990).

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at most finitely many exceptions. The associated abstraction principle can be shownto be satisfiable in all and only finite domains. So although (HP) and this abstractionprinciple are individually consistent, they contradict one another.9

A more refined suggestion is that an abstraction principle is acceptable if and onlyif it is conservative. An abstraction principle is said to be conservative if (roughly)it doesnt entail any new results about the old ontology consisting of all objectsother than the abstracts being introduced.10 This suggestion suffices to exclude thebad companion mentioned above. For an abstraction principle that is satisfiable onlyin finite domains clearly entails something new about the old ontology, namely that itis finite. Unfortunately, even the desirable property of conservativeness fails to ensurethat an abstraction principle is acceptable. For Alan Weir has described a class ofrestrictions of Basic Law V which, despite being conservative, require the universe tosatisfy incompatible cardinality requirements.11

3.2 Stability

A modified suggestion is that an abstraction principle is acceptable if and only if itis stable, where an abstraction principle () is said to be stable if there is a cardinalnumber such that () is satisfiable in models of cardinality for any > . In fact,stability is equivalent to another proposed requirement on abstraction principles. Saythat an abstraction principle is irenic if it is conservative and compatible with all otherconservative abstraction principles. Alan Weir has proved that an abstraction princi-ple is stable if and only if it is irenic.12 Clearly, the problem with the incompatibleabstraction principles of the previous paragraph is precisely that they fail to be stable.The notion of stability thus provides a promising account of acceptability. In fact, thisis probably the most influential approach these days. But this approach neverthelessfaces some serious technical and philosophical challenges.

One challenge derives from Kit Fines study of systems of abstraction principles inFine (2002). Fine describes systems of abstraction principles each of which is stablebut which are jointly inconsistent (assuming some plausible conditions on how differ-ent abstraction principles interact). Here is an example. For each concept X , considerthe equivalence relation RX which F bears to G just in case either both F and G coin-cide with X or neither does. The associated abstraction principle is clearly stable, as itis satisfiable in any domain with two or more objects. But on the plausible assumption

9 See Boolos (1990). Another reason why consistency alone is insufficient is provided by Heck (1992),who describes how consistent abstraction principles can have consequences that cannot be accepted asmathematically legitimate.10 This response was first suggested in Wright (1997), Sect. IX. Further analysis in Shapiro and Weir (1999)and Weir (2003) has resulted in a better understanding of how to formalize the notion of conservativeness.One good formalization derives from Field (1989). Let T old be the result of restricting the quantifiers of allaxioms of the theory T to objects that arent in the range of the abstraction operator in question. Then anabstraction principle () is conservative over T just in case for any formula in the old language we have:if T old, () | old, then T | . For an alternative formalization, see Weir (2003, p. 23).11 See Weir (2003, pp. 2728).12 See Weir (2003, p. 32). The notion of stability is mentioned already in Heck (1992, p. 494, fn. 5).The requirement of irenicity was first suggested by Wright (1999, p. 328).

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that two abstracts can be identical only if their corresponding equivalence classes ofconcepts are identical, any theory that allows abstraction on each RX is inconsistent.For in any such theory we can interpret the inconsistent Basic Law V by interpretingthe extension of a concept F as the abstract of F with respect to the relation RF .13

Another challenge is developed in Gabriel Uzquianos contribution to this specialissue. Uzquiano first describes an extension of the bad company problem to a structur-alist approach to set theory. The problem is that, when one quantifies over absolutelyeverything, there are different consistent and plausible set theories that require theuniverse to be of different cardinalities. The requirement of stability appears to offeran attractive solution both to the original bad company problem and to this extensionof it. However, Uzquiano proves that no set theory with an axiom of infinity and aweak form of Replacement can be stable. The requirement of stability thus appears tobe incompatible with some of the core principles of set theory. Since these principlesare in good mathematical and philosophical standing, he argues that the best responseis to reject the requirement of stability.14

Finally, a more inchoate worry has to do with the mathematical intractability of thenotion of stability. One example of this intractability is that it can be hard to settle ina non-question begging way whether an abstraction principle is stable, as its stabilitywill hinge on controversial questions about the extent of the set theoretic hierarchy. Adisagreement about the acceptability of an abstraction principle will thus often trans-late into an analogous disagreement in the meta-theory about the extent of the settheoretic hierarchy.15

Another striking example of how very simple questions about stability go far beyondwhat is currently mathematically tractable emerges from Roy Cooks contribution.Cook argues that we can count not just objects but also concepts of different levels.In order to represent this we let the cardinality operator # attach to concept terms oflevels higher than 1, and we let the equivalence relation hold between concepts of agiven level just in case they can be one-to-one correlated. Let HPi be the analogue ofHumes Principle where # attaches to concept terms of level i . Cook then shows thatthe satisfiability of these principles depends on open and extremely difficult questionsin set theory. It is consistent with ZFC that the Generalized Continuum Hypothesisshould hold, in which case each HPi is satisfiable in models of every infinite cardinal-ity . But it is also consistent with ZFC that the Generalized Continuum Hypothesisshould fail so badly that HPi has no models at all for i > 2 and that HP2 fails to bestable.

3.3 Kit Fines proposal

Recall from the previous section Kit Fines example of a system of abstraction prin-ciples each of which seems innocent but which are jointly inconsistent (given aplausible assumption). In response to this problem Fine (2002) proposes that a system

13 This challenge is developed in much greater detail in Linnebo and Uzquiano (Unpublished Manuscript).14 A third challenge is found in Matti Eklunds contribution. But since the target of this challenge is widerthan just the criterion of stability, it will be described below.15 For discussion, see Weir (2003, Sect. 7).

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of abstraction principles is acceptable just in case each of the principles is non-inflationary (in the sense of being satisfiable in the domain of absolutely everything)and each satisfies a logicality constraint (defined la Tarski in terms of permutationinvariance).

This proposal enjoys a number of attractive technical and philosophical features.Can it also serve as a solution to the bad company problem? Unfortunately, the answerappears to be negative. First, this proposal is limited to monadic second-order logic(that is, to second-order logic with quantification over concepts but not over polyadicrelations). Another shortcoming is that its logical strength is limited.16

3.4 Predicativity and well-founded individuation

According to Dummett, the bad companions are due to some illicit impredicativity.This suggests that Fregean abstraction is acceptable precisely insofar as it is predica-tive. One predicativity requirement that can be imposed is on the background second-order logic. More precisely, we can restrict the second-order comprehension scheme

(Comp) Rx1 . . . xn[Rx1, . . . , xn (x1, . . . , xn)]to predicative instances, that is, to instances where does not contain any boundsecond-order variables. Even if we accept all abstraction principles, including BasicLaw V, this restriction is known to ensure consistency.17 However, the logical strengthof the resulting theories is now known to be severely limited.18

Another predicativity requirement that can be imposed concerns the abstractionprinciples themselves. Humes Principle and most other interesting abstraction prin-ciples are impredicative in the sense that their right-hand sides quantify over the veryobjects that the left-hand sides attempt to introduce. It is easy to see that by banning thiskind of impredicativity, consistency is ensured. But this ban would also eliminate themost distinctive feature of the neo-Fregean programme; in particular, it would under-mine the cherished proofs that there are infinitely many mathematical objects. In sum,the two predicativity requirements that have been discussed are too restrictive to pro-vide an attractive analysis of acceptable abstraction. To remove all impredicativitywould be to remove the very heart of neo-Fregeanism.

ystein Linnebos contribution develops a response to the bad company problemwhich is related to the predicative ones but much more liberal. Regarding an abstractionprinciple as a device for individuating one kind of entity in terms of another, Linneboexplores the idea that the process of individuation must be well-founded. This ideais developed in a modal language, where means that we can go on to individuateentities so as to make it the case that . Using this modal framework he argues that allabstraction principles are acceptable and that contradiction should instead be avoided

16 The second-order version of Fines theory is interpretable in third-order arithmetic (and more generally,the nth order version, in n + 1st order arithmetic). See Burgess (2005, Sect. 3.4).17 See Heck (1996).18 See Burgess (2005, Sects. 2.5 and 2.6); Linnebo (2004).

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by restricting what formulas are allowed to define concepts. He then provides someexamples of strong theories to which this approach gives rise.

4 Broader questions

Assume the debate surveyed in the previous section was one day to converge on someaccount of acceptable abstraction. What would the philosophical significance of thisconvergence be? And what philosophical work would the resulting account do for us?

Matti Eklunds contribution investigates what sort of philosophical underpinningcan be provided for whatever technical notion of acceptability on which the discussionmay converge. Eklund considers several possible sources of such an underpinningbut argues that none works. Among the possible sources examined and dismissedare: a general account of a priori justification, the idea that abstraction principles arereconceptualizations, the idea that truth is prior to reference, and relativistic viewsaccording to which different things exist depending on what language we speak.

Philip Ebert and Stewart Shapiros contribution discusses what epistemic status acriterion of acceptable abstraction or implicit definition would have. At one extreme,internalists claim that in order to gain knowledge by abstraction or implicit definition,one will have to know or justifiably believe that the criterion is satisfied. Drawingin part on Gdels second incompleteness theorem, Ebert and Shapiro dismiss this ashopelessly strict. At the other extreme, externalists claim it suffices that the criterion isin fact satisfied. This is dismissed as too lax and as undermined by a problem of easymathematical knowledge: why should we bother to prove mathematical theorems ifsuch knowledge was available much more easily by direct stipulation? Somewherebetween the two extremes lies Wrights current proposal, which is based on the ideaof default entitlement.19 Ebert and Shapiro analyze this idea but are still not satisfied.They end by outlining an alternative holistic answer to their question.

John MacFarlanes contribution discusses Hale and Wrights claim that by stip-ulating Humes Principle, we can simultaneously fix the meaning of the cardinalityoperator # and become entitled to belief in the stipulated principle. MacFarlane exam-ines the semantic and epistemological principles that Hale and Wright appeal to inorder to support this claim. He argues that insofar as these principles succeed, theydont leave any privileged role for abstraction principles. In particular, we would beequally entitled to stipulate the DedekindPeano axioms directly.

Bob Hale and Crispin Wrights contribution responds to MacFarlane, focussing onhis question what distinguishes acceptable abstraction principles from other forms ofimplicit definition. Their answer articulates an epistemological requirement of non-arrogance, where a stipulation is said to be arrogant if it is hostage to the obtainingof conditions of which its reasonable to demand an independent assurance (Sect. 2).Hale and Wright argue that the direct stipulation of the DedekindPeano axiomswould be arrogant, whereas the stipulation of Humes Principle is not. Central to thisargument is the thought that Humes Principle fixes the truth-conditions of canoni-cal statements of numerical identity and in this way provides criteria of identity for

19 See Wright (2004).

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numbers. An appendix responds to Ebert and Shapiros problem of easy mathematicalknowledge.

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IntroductionAbstract1 Background to the problem2 The significance of the bad companions3 In search of an account of acceptable abstraction3.1 Two dead ends3.2 Stability3.3 Kit Fine's proposal3.4 Predicativity and well-founded individuation

4 Broader questionsReferences

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