Handbook of Philosophy of Mathematics

Constructivism in Mathematics 333puter programs, perhaps as R. Constable and colleagues have done in the NuPRLProject [Constable et. al., 19861. Bishop encouraged an analogy between suitableformalizations of new constructivism and high-level languages for specification andprogramming. The idea was that, in the future, proofs in the formalism would becompiled more or less directly into implementable code [Bishop, 1985, 14-15]. Bishop did not recommend a development of constructive mathematics in iso-lation from conventional mathematics, but a careful and elaborate recreation ofthe latter. He deemed this project \"the most urgent task of the constructivist.\"[Bishop, 1970, 541 In practice, its completion often requires that extra hypothesesbe added to the statements of conventional theorems so that their proofs are pos-sible using strictly constructive reasoning. For example, the classical least upperbound theorem states that every nonempty set of real numbers that is boundedabove has a least upper bound. With extra hypothesis, Bishop's version of thetheorem is that every inhabited set of real numbers that is order located and hasan upper bound has a least upper bound. Here, a set A of real numbers is orderlocated if, for any real numbers r and s with r < s, either s is an upper boundfor A or there exists a real number x E A such that r < x. In fact, A being orderlocated is also a necessary condition for A having a least upper bound [Bishop andBridges, 1985, 371. The reasoning accepted by Bishop's new constructivists is formalizable in Heyt-ing's first-order predicate logic. Their mathematics can be captured in intuition-istic Zermelo-Frankel set theory IZF or the weaker CZF [Aczel, 19781. HarveyFriedman has argued [Friedman, 19771 that the work of the new constructivistscan also be formalized within the even weaker set theory B, which is provablyconservative over the elementary intuitionistic formal arithmetic HA or HeytingArithmetic [Beeson, 1985, 3211. 7 PREDICATIVISMPredicativism manifests itself as a restriction of mathematics, primarily set theory,class theory or analysis, imposed upon the linguistic means by which supposedhigher-order entities such as logical classes, real numbers and infinitary functionsare defined by intension or abstraction. Roughly put, a specification of a classC via a class abstract {x : @(x))is impredicative when @(x)contains a variableranging over elements of either a collection that has C as a member or another classrequiring C for its proper definition. When the abstract @(x)fails to contain sucha variable, the specification is predicative, and can be allowed by predicativists.Therefore, a class C is deemed to be predicatively well-defined by {x : @(x))whenall variables in @(x)are restricted to ranging over collections of classes or otherentities already known to be well-defined predicatively prior to the moment of C'sdefinition. In standard Zermelo-Fraenkel set theory, impredicativity is ubiquitous:the familiar definition of the set w of natural numbers as the C-least inductive set, w = {x : Vy (y is inductive -+ x E y)),

334 Charles McCartyis impredicative since w is itself an inductive set. Although the committed predicativist does not call for any large restriction inclassical logic beyond predicative limits imposed upon schemes of comprehensionor definitions of classes and sets, predicativism does count as a species of con-structivism as delimited infra. Quantification over numbers or finite strings isnot often thought subject to predicativistic restriction. (However, Nelson [I9861argued that, because the definition of natural number via inductive sets is im-predicative, a consistent predicativist should call for a predicative arithmetic inwhich bounded number quantifications play the starring role.) Normally, existen-tial quantification over classes 3x.@(x)is crucial for predicativists and a proof ofsuch a quantified statement is admissible if the proof gives a predicatively speci-fiable class C such that @ ( C )is also admissibly provable. One can think of aclass within a predicative universe of classes as an abstract collection constructedthrough a well-founded process of definition involving conventional class opera-tions like union, intersection, and relative complement, as well as quantification,subject to predicative restrictions on variables. Henri Poincark (1854-1912) and, after him, Bertrand Russell (1872-1970) ini-tially advanced the idea that use by mathematicians of impredicative definitionsis fallacious and that mathematics should be reconstructed so that all definitionsare strictly predicative in form. Mathematician and physicist Poincark was cousint o Raymond Poincar6, prime minister and president of France. After working asa mining engineer, he completed a doctorate in mathematics in 1879 under thedirection of Charles Hermite, submitting a dissertation on differential equations.From 1886 until his death, Poincark held chairs at the Sorbonne and the ~ c o l ePolytechnique. He introduced into complex analysis the study of automorphicfunctions, discovered and developed basic ideas of algebraic topology, includingthat of homotopy group, and made major contributions t o the fields of analyticfunctions, number theory, and algebraic geometry. In physics, he receives jointcredit, with Einstein and Lorentz, for discovering the special theory of relativity.In addition to his membership in the Acadkmie Francaise and the Acadkmie desSciences, he was a corresponding member of scientific societies in Amsterdam,Berlin, Boston, Copenhagen, Edinburgh, London, Munich, Rome, Stockholm, St.Petersburg, and Washingston. Poincark's views on the foundations of mathematics inspired Brouwer and theDutch intuitionists; he emphasized that only potential, rather than completed oractual infinities exist, and that the principle of induction over the natural numbersis known to us exclusively by the exercise of an intuition irreplaceable by deductionfrom logical axioms. Poincark anticipated Bishop in requiring that, to be adequate,infinitary mathematics must retain a clear numerical meaning; he wrote, Every theorem concerning infinite numbers or particularly what are called infinite sets, or transfinite cardinals, or transfinite ordinals, etc., etc., can only be a concise manner of stating propositions about finite numbers. [Poincark, 1963, 621

Constructivism in Mathematics 335 PoincarQthought to see a fallacy of impredicativity (which he explicitly called 'avicious circle') underlying both foundational paradoxes such as Richard's and whathe believed a questionable extension, by Georg Cantor, of mathematics into thetransfinite. Reinforcing PoincarQ'sobjections to impredicativity was his vision ofthe existence of mathematical entities as time-dependent and, accordingly, variableover time: a mathematical object can exist only if it is properly defined, and comesto exist once it is defined. Therefore, collections of mathematical objects are notimmutable when it comes to members. A mathematical object like a real numbermay be a logical class C that does not exist unless and until it has been specifiedusing an abstract. Hence, the collection of real numbers is constantly growing insize as more real numbers are defined. It was in terms of the growth of classes and their membership relations over timethat Poincar6 first explicated the terms 'predicative' and 'impredicative' as appliedto classes. For him, a class is predicative when it is so defined that its membershipis stable: any new members that get added to the class remain permanently inthe class. The use of an abstract defining C and featuring a variable construedto range over an infinite collection D containing C makes the false assumptionthat C already exists and is well-defined. Furthermore, since membership in Cis determined with reference to all members of D, C's impredicative definition islikely t o be a cause of instability, membership in C being dependent upon elementsof D that are not yet defined. PoincarC objected in his [I9631to Zermelo's proof of the Well-Ordering Theoremon the grounds that the set-theoretic operation of arbitrary union, as exploited byZermelo, is impredicative. By today's standards, he was right on the last point:U xthe union of a set x is given by the abstract { z : 3y ( z E y A y E s ) )in which the unrestricted bound set variable y is intended t o range over all sets,U x .including For example, in standard set theory, the union U n of a nonzero(von Neumann) natural number n is always a member of n itself.Russell formulated the demand that all logical classes be predicatively definedin his Vicious Circle Principle:I recognise, however, that the clue to the paradoxes is to be found inthe vicious-circle suggestion; I recognise further this element of truth inM. PoincarQ'sobjection to totality, that whatever in any way concernsall or any or some (undetermined) of the members of a class mustnot be itself one of the members of a class. In M. Peano's language,the principle I wish to advocate may be stated: \"Whatever involvesan apparent variable must not be among the possible values of thatvariable.\" [Russell, 1906, 1981 By Russell's lights, classes are naturally organized into a noncumulative hierar-chy of orders or types in such a way that \"any expression containing an apparent[bound] variable is of higher type than that variable.\" [Russell, 19081 Here, the

336 Charles McCartymeans to implement a ban upon impredicativity is to conceive of classes falling oftheir own accord into mutually disjoint types or orders or levels, and to adopt asan official means of expression and deduction a many-sorted formal system withvariables for such classes indexed with symbols for those types, orders or levels. Insuch a system, a quantifier with a bound variable carrying type index a can onlybe replaced, in universal instantiation or existential generalization, by a variableor parameter carrying that same index a . Formal systems of this kind include theramified type theory of Principia Mathernatica [Whitehead and Russell, 1910-19131and that of Hao Wang's systems C [Wang, 19641 [Chihara, 19731. One thinks of a standard model of a predicative type theory as a subuniverseof the standard model of simple type theory over the natural numbers, but hav-ing classes further divided or ramified into a sequence of orders or levels indexedby natural numbers or constructive ordinal numbers. Classes of natural numberson level 1 are those that are specifiable using variables ranging only over naturalnumbers. Classes of level 2 are those specifiable using variables ranging exclusivelyover natural numbers or classes of level 1. Classes of level 3 have, in their specifi-cations, variables ranging only over natural numbers or classes of levels 1 and 2,and so on. Every class is required to exist in some level. Such requirement is in-tended to rule impredicativity out. Predicativists often imagine that the levels areformed by some kind of definitional process evolving in discrete stages over time,with new classes and levels appearing on the bases of classes and levels already inexistence. Of this process, Wang wrote, \"[Nlew objects are only to be introducedstage by stage without disturbing the arrangement of things already introducedor depending for determinedness on objects yet to be introduced at a later stage.\"[Wang, 1964, 6401 Consequently, latter-day predicativists often followed Poincarkin thinking that the mathematical universe of classes expands over time as newdefinitions and specifications become available. If real numbers are classes of rational numbers, e.g., Dedekind cuts, then inramified analysis, there is no single class containing all real numbers and there maybe 'new' real numbers appearing at every level from some point onward. Therefore,if one cleaves strictly to the ramified conception, there can be no single variablethat ranges over all real numbers. As Russell came to realize, ramification appearsto block all satisfying formulations of Dedekind's Theorem that every nonemptycollection of cuts that is bounded above has a least upper bound. The least upper+bound of a nonempty class of order a of cuts, defined as it is by a union over all cutsof order a, must be a cut of order at least a 1. To circumvent such drawbacks tohis type theory, Russell reluctantly adjoined to his system the controversial Axiomof Reducibility: in the above terms, the assumption that every class of any levelis coextensive with some class of level 1. Hermann Weyl (1885-1955), a distinguished mathematician and a leading stu-dent of David Hilbert, championed the cause of predicativism in his monographDas Kontinuum [Weyl, 19181. Weyl studied mathematics and physics, first at Mu-nich and later under Hilbert at Gottingen. After obtaining his doctorate, Weyltook up a professorial post at the Swiss Federal Institute of Technology in Ziirich.

Constructivism in Mathematics 337He later replaced Hilbert at Gottingen, before emigrating to America -t o the Ad-vanced Institute at Princeton - in 1933. As mathematician and physicist, Weylmade notable contributions not only to the foundations of mathematics but also tothe theories of integral and differential equations, geometric function theory, dif-ferential topology, analytic number theory, gauge field theory, group theory, andquantum mechanics. In his [1918],Weyl sidestepped the technical issues besettingRussell's formulation of ramified analysis by constructing a predicative versionof analysis using strictly arithmetic comprehension, that is, taking the naturalnumbers as given and permitting classes of numbers at level 1 only. He wrote, A \"hierarchical\" [ramified]version of analysis is artificial and useless. It loses sight of its proper object, i.e., number. . .. Clearly, we must take the other path -that is, we must restrict the existence concept to the basic categories (here, the natural and rational numbers) and must not apply it in connection with the system of properties and relations (or the sets, real numbers, and so on, corresponding to them). [Weyl, 1918, 321By such means, Weyl was able to obtain a sequential version of Dedekind's The-orem. For that, he treated real numbers not as cuts but as Cauchy sequences,and used strictly level 1 definitions to prove that every Cauchy sequence of realnumbers has a real number as its limit. Close metamathematical study of predicativity, using the formal tools forged byGodel, Kleene, Tarski and others, began anew in the 1950s with efforts to extendthe hierarchy of arithmetically definable sets into the transfinite. Prominent hereare the contributions of Wang [1954],Lorenzen [1955], and Kreisel [1960]. Morerecently, Solomon Feferman (b. 1928) has been largely responsible for the detailedproof-theoretic study of the depth and extent of predicative mathematics, writinga number of papers (some published with coworkers) from his classic [I9641 upthrough the retrospective [2005].Independently of Kurt Schiitte [1965],Fefermanidentified the precise upper bound on the predicatively provable ordinal numbers. 8 FINITISMFinitists demand that mathematicians avoid all reference, explicit or implicit, toinfinite totalities, including the totality of natural numbers. Sometimes, as wasthe case with the finitism of David Hilbert [Hilbert, 19261, this avoidance is alliedwith nominalism and a desire, epistemically motivated, to replace abstract notionswith notations that are relatively concrete and physically realized. In addition tothe finitism of Hilbert, one should count Skolem's primitive recursive arithmetic[I9231 and Yessenin-Volpin's ultra-intuitionism [I9701 among influential versionsof finitism in the 20th Century. Only natural numbers or items simply and fully encodable as natural num-bers count as finitistically acceptable. The natural numbers are not deemed to

338 Charles McCartyconstitute a completed infinite totality, but are permitted as concrete, readily vi-sualizable notations. Hilbert [I9261 insisted that finitistic talk of operations onnumbers or symbols (the only sort of talk permitted in his metamathematics forproof theory) must be understood entirely in terms of performable manipulationsupon the intuitable forms of strings of tally marks. Hilbert wrote, The subject matter of mathematics is, in accordance with this theory, the concrete symbols themselves whose structure is immediately clear and recognizable. [Hilbert, 1926, 1421Finitists maintain that the customary use of unbounded existential quantifica-tion over natural numbers in mathematics commits the user to the existence ofcompleted infinite totalities and, hence, existential claims require, for their fulllegitimacy, finitistic reconstrual, perhaps by the imposition of explicit numericalbounds on all arithmetic quantifiers. Such completely bounded quantifications overthe numbers are usually finitistically admissible without further ado. According toHilbert, thoroughly finitistic statements express mathematical propositions thatare contentual. Since the kinds of claims recognized as finitistic are so narrowlydelimited as always to be decidable, classical logic reigns in finitistic mathematics,whether in the style of Hilbert or of Skolem. Hilbert proposed to construe some unbounded numerical quantifications as 'in-complete statements.' The completion of an unbounded existential statement withprimitive recursive matrix includes the provision of a correct bound on the existen-tial quantifier. In the case of universal quantification Vn.@(n)with @(x) primitiverecursive, completion requires a finitistic proof of each instance of the free-variablescheme a(%)H. ilbert treated the claims of analysis and set theory that do not ad-mit finitistic reconstrual as ideal. These ideal statements lack denotative meaningbut can be manipulated by the deductive apparatus of a theory containing them. One can speak of constructions or operations in finitistic mathematics, butthey do not constitute an absolutely infinite collection of functions, and are notconceived as bearing with them infinite domains or ranges of input and outputvalues. An operation on natural numbers is finitistic whenever it can be seen asa transformation that can be carried out on concrete signs so that there is anexplicit, uniform, humanly calculable bound on the number of steps required tocomplete the transformation in any given case. Natural candidates for numericalfunctions that fit this bill are the primitive recursive functions, since these can bedefined as computation routines that never require of the computer an unboundedsearch. Whether the finitistic functions of Hilbert include - either conceptuallyor historically - more than the primitive recursive functions has been a subjectof some dispute among the cognoscenti. W. Tait [I9811 argued that Hilbertianfinitistic mathematics is to be identified with primitive recursive mathematics,but see his [2002] and [2005a] for qualification. Thoralf Skolem (1887-1963), a pioneer in model theory and set theory, andprofessor of mathematics a t Oslo, admitted as finitistic only those assertions whosetruth or falsity is determinable in a finite number of steps via calculations that

Constructivism in Mathematics 339are primitive recursive. Hence, he allowed simple equations between primitiverecursive terms and statements obtainable from such equations via combinationswith sentential connectives and bounded quantifiers. R. L. Goodstein (1912-1985), who studied under L. Wittgenstein and J.E. Lit-tlewood, developed, in his [I9571 and [1961],finitistic mathematics further in thefashion established by Skolem. Goodstein examined various conceptions of realnumber, among them the notion of a primitive recursive Cauchy sequence at-tached to a primitive recursive modulus of convergence. Here, as in Hilbert's andSkolem's versions of finitism, nonclassical axioms are not accepted, so there canbe no proof of any theorem such as ~ e i t i n ' sor Brouwer's Continuity Theoremthat flouts laws of classical analysis. Since finitistic mathematics is also intu-itionistically correct, there can be no finitistic theorem that contradicts Brouwer'sintuitionism. According to Goodstein, a Skolemite finitist cannot prove that anybounded, monotonically increasing sequence of rational numbers has a real numberas its limit. The ultra-intuitionism of A. S. Yessenin-Volpin is substantially different fromthe finitistic outlooks just described. Yessenin-Volpin (b. 1924) is the son of theRussian poet Sergei Esenin, once the husband of Isadora Duncan, and NadezhdaVolpin, a writer and translator. In 1949, he was arrested by the Soviet authoritiesfor his poetry, and was committed to a mental institution. In 1950, he was exiledto Khazakhstan. Yessenin-Volpin emigrated to the United States in 1972. In hismathematical work, he rejected both the standard notion of a natural numbersystem closed under the successor operation and the ideas for a primitive recur-sive mathematics set out by Skolem. Numbers are not thought to be potentiallyrealizable in terms of concrete notations; only those numbers that are feasible, lit-erally displayable, are to be accepted. Following P. Bernays [1935], scholars referto views like Yessenin-Volpin's as 'strict finitism.' C. Kielkopf [I9701 and otherinvestigators have thought to see in the writings of Wittgenstein, principally his[1956], an endorsement of a form of strict finitism. BIBLIOGRAPHY[Aberth, 19801 0. Aberth. Computable Analysis. New York, NY: McGraw-Hill International Book Company. xif187, 1980.[Aczel, 19781 P. Aczel. T h e type-theoretic interpretation of constructive set theory. A. Macintyre et al. (eds.) Logic Colloquium '77. Amsterdam, NL: North-Holland Publishing Company. pp. 55-66, 1978.[Beeson, 19851 M. Beeson. Foundations of Constructive Mathematics. Berlin, DE: Springer- Verlag, 1985.[Benacerraf and Putnam, 19641 P. Benacerraf and H. Putnam, eds. Philosophy of Mathemat- ics. Selected Readings. First Edition. Englewood Cliffs, NJ: Prentice-Hall, Inc, 1964. Second Edition. Cambridge, UK: Cambridge University Press. viii+600, 1983.[Bernays, 19351 P. Bernays. Sur le platonism duns les mathe'matiques. L'enseignement Mathkmatique. Volume 34. pp. 52-69, 1935. Reprinted C. Parsons (tr.) O n platonism i n mathematics. [Benacerraft and Putnam 19641. pp. 274-286, 1964.[Bishop, 19671 E. Bishop. Foundations of Constructive Analysis. New York, NY: McGraw-Hill. xiii+370, 1967.

340 Charles McCarty[Bishop,19701 E. Bishop. Mathematics as a numerical language. A . Kino et al. (eds.) Intu- itionism and Proof Theory. Proceedings o f t h e Summer Conference at Buffalo, N.Y. 1968. Amsterdam, NL: North-Holland Publishing Company. pp. 53-71, 1970.[ ~ i s h o1~98,5) E. Bishop. Schizophrenia i n contemporary mathematics. M. Rosenblatt (ed.) Er- rett Bishop: Reflections on Him and His Research. Contemporary Mathematics. Volume 39. Providence, RI: American Mathematical Society. pp. 1-32, 1985.[Bishopand Bridges, 19851 E. Bishop and D. S . Bridges. Constructive Analysis. Grundlehren der mathematischen Wissenschaften. Volume 279. Berlin, DE: Springer-Verlag. xii+477, 1985. [Bridgesand Richman, 19871 D. S . Bridges and F. Richman. Varieties of constructive math- ematics. London Mathematical Society Lecture Notes Series. Volume 97. Cambridge, U K : Cambridge University Press. x+149, 1987.[Brouwer, 19051 L. E. J . Brouwer. Leven, Kunst en Mystiek. Delft, NL: Waltman. 99 pp., 1905. Reprinted W . P . van Stigt (ed.) Life, art and mysticism. T h e Notre Dame Journal o f Formal Logic. Volume 37. 1996. pp. 381-429, 1996. [Brouwer, 19071 L. E. J . Brouwer. Over de grondslagen der wiskunde. [ O n the Foundations of Mathematics.] Amsterdam, NL: University o f Amsterdam dissertation. 183 pp, 1907. [Brouwer, 19081 L. E. J . Brouwer. De onbetrouwbaarheid der logische principes. [ T h e unrelia- bility of the logical principles.] Tijdschrift voor Wijsbegeerte. Volume 2. pp. 152-158, 1908. [ ~ r o u w e r1,9091 L. E. J . Brouwer. Het wezen der meetkunde. [The Nature of Geometry.] Inau- gural Lecture a s Privaat Docent. Amsterdam, NL. 23 pp, 1909. [Brouwer, 19131 L. E. 3. Brouwer. Intuitionism and fomalism. A. Dresden (tr.) Bulletin o f t h e American Mathematical Society. Volume 20. pp. 81-96, 1913. Reprinted [Bernacerrafand Putnam 19641. pp. 66-77, 1913. [Brouwer, 19181 L. E. J . Brouwer. Begriindung der Mengenlehre unabhangig vom logischen Satz vom ausgeschlossenen Dritten. Erster Teil, Allgemeine Mengenlehre. [Foundationof set the- ory independent of the logical law of the excluded middle. Part One, general set theory.] Verhandelingen der Koninklijke Akademie van wetenschappen t e Amsterdam. First Section. Volume 12. Number 5. pp. 1-43, 1918. [Brouwer, 19211 L. E. J . Brouwer. Besitzt jede reelle Zahl eine Dezimalbruchentwichlung? [Does every real number have a decimal ezpansion?] Mathematische Annalen. Volume 83. pp. 201- 210, 1921. [Brouwer, 19241 L. E. J . Brouwer. Beweis dass jede volle Funktion gleichmassigstetig ist. [Proof that every total function is unifomly continuous.] Koninklijke Nederlandse Akademie van Wetenschappen. Proceedings o f t h e Section o f Sciences. Volume 27. pp. 189-193, 1924. [ ~ r o u w e r1,9481 L. E . J . Brouwer. Essentieel negatieve eigenschappen. [Essentially negative properties.] Indagationes Mathematicae. Volume 10. pp. 322-323, 1948. [Brouwer. 19541 L. E. J. Brouwer. Points and spaces. Canadian Journal o f Mathematics. Volume 6. pp. 1-17, 1954. [Browder, 19761 F. E. Browder, ed. Mathematical Developments arising from Hilbert Problems. Proceedings o f Symposia in Pure Mathematics. Volume XXVIII. Providence, RI: American Mathematical SocieJy. xiif628, 1976. [ ~ e i t i n1,9591 G . S. Ceitin. A l g o r i t m i ~ ~ s k iokperatory v konstruktivnyk polnyk skparabklnyk me'tritkskyk prostranstvak. [Algorithmic operators i n constructive complete separable metric spaces.] Doklady Akadkrnii Nauk SSR. Volume 128. pp. 49-52, 1959. [Chihara, 19731 C . Chihara. Ontology and the Vicious-Circle Principle. Ithaca, N Y : Cornell University Press. xv+257, 1973. [Constable, 19861 R . Constable et al. Implementing Mathematics with the NuPRL Proof Devel- opment System. Englewood Cliffs,NJ: Prentice-Hall. x+299, 1986. [Diaconescu, 19751 R. Diaconescu. Axiom of choice and complementation. Proceedings o f t h e American Mathematical Society. Volume 51. PI?:175-178, 1975. [Du Bois-Reymond, 18861 E. Du Bois-Reymond. Uber die Grenzen des Naturerkennens. [ O n the limits of the knowledge of nature.] Reden von Emil Du-Bois Reymond. Erste Folge. Leipzig: Verlag von Veit und Comp. viii+550, 1886. [Du BoisReymond, 18771 P. Du Bois-Reymond. ~ b e dr ie Paradoxon des Infinitarcalculs. [ O n the paradoxes of the infinitary calculus.] Mathematische Annalen. Volume 10. pp. 149-167, 1877. [DU Bois-Reymond, 19661 P. Du Bois-Reymond. ~ b e rdie Grcmdlagen der Erkenntnis in den exakten Wissenschajten. [Onthe Foundations o f Knowledge in the Exact Sciences.] Sonder- ausgabe. Darmstadt, DE: Wissenschaftliche Buchgesellschaft. vi+130, 1966.

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342 Charles McCarty reis is el, 19601 G . Kreisel. La predicativite' . Bulletin de la Societe Mathematique de fiance 88. pp. 371-391, 1960.[Kreisel, 1962al G. Kreisel. O n weak completeness of intuitionistic predicate logic. T h e Journal o f Symbolic Logic. Volume 27. pp. 139-158, 1962. reis is el, 1962bl G . Kreisel. Foundations of intuitionistic logic. E. Nagel et. al. (eds.) Logic, Methodology and Philosophy o f Science I . Stanford, C A : Stanford University Press. pp. 198- 210, 1962.[Kreisel et al., 19591 G . Kreisel, D. Lacombe and J . Shoenfield. Partial recursive functions and effective operations. A. Heyting (ed.)Constructivity in Mathematics. Proceedings o f the Col- loquium held at Amsterdam 1957. Studies in Logic and the Foundations o f Mathematics. Amsterdam, NL: North-Holland,Publishing Company. pp. 290-297, 1959.[Kronecker, 18871 L. Kronecker. Uber den Zahlbegrifl. Philosophische Aufsatze, Eduard Zeller zu seinem funfzigjahrigen Doctorjubilaum gewidmet. Leipzig, DE: Fues. pp. 261-274, 1887. Reprinted W . Ewald (tr.) O n the concept of number W . Ewald (ed.) From Kant t o Hilbert: A Source Book in t h e Foundations o f Mathematics. Volume 11. 1996. Oxford,U K : Clarendon Press. pp. 947-955, 1996. [ ~ o r e n z e n1,955) P. Lorenzen. Einfuhrung i n die operative Logik und Mathematik. [Introduction to Operative Logic and Mathematics.] Berlin, DE: Springer-Verlag. VII+298, 1955. [Markov, 19581 A. A. Markov. O n constructive functions. American Mathematical Society Translations ( 2 ) . Volume 29. pp. 163-196, 1963. Translated from Trudy Matematicheskogo Instituta imeni V A Steklova. Volume 52. pp. 315-348, 1958. [Markov, 19711 A. A. Markov. O n constructive mathematics. American Mathematical Society Translations. Series 2. Volume 98. pp. 1-9, 1971. Translated from Trudy Matematicheskogo Instituta imeni V A Steklova. Volume 67. pp. 8-14, 1962. [Martin-Lof,19841 P. Martin-Lof. Intuitionistic Type Theory. Naples, IT: Bibliopolis. iv+91, 1984. [ ~ c C a r t1~98,61 C. McCarty. Subcountability under realizability. Notre Dame Journal o f Formal Logic. Volume 27. Number 2. April 1986. pp. 210-220, 1986. [McCarty, 19881 C . McCarty. Constructive validity is nonarithmetic. T h e Journal o f Symbolic Logic. Volume 33. Number 4. December 1988. pp. 1036-1041, 1988. [McCarty, 19961 C . McCarty. Undecidability and intuitionistic incompleteness. T h e Journal o f Philosophical Logic. Volume 25. pp. 559-565, 1996. [Nelson, 19861 E. Nelson. Predicative Arithmetic. Mathematical Notes. Volume 32. Princeton, NJ: Princeton University Press. v i i i f l 9 0 , 1986. [Poincare,19631 H . PoincarB. The Logic of Infinity. J . Bolduc (tr.) Mathematics and Science: Last Essays. New York, N Y : Dover Publications, Inc. pp.45-64, 1963 [Pringsheim,1898-19041 A. Pringsheim. Inationalzahlen und Konvergenz unendlicher Prozesse. [Irrational numbers and the convergence of infinite processes.] W .F. Meyer (ed.)Encyklopadie der mathematischen Wissenschaften. Erster Band in zwei Teilen. Arithmetik und Algebra. [Encyclopedia o f t h e Mathematical Sciences. First Volume in T w o Parts. Arithmetic and Analysis.] Leipzig, DE: Druck und Verlag von B.G. Teubner. pp. 47 - 146, 1898-1904. [Russell,19061 681 B. Russell. Les paradoxes de la logique. Revue de M6taphysique et la Morale. Volume 14. September 1906. pp.627-650, 1906. Reprinted O n 'insolubilia' and their solution by symbolic logic. D. Lackey (ed.) Bertrand Russell. Essays in Analysis. New York, N Y : George Braziller. pp. 190-214, 1973. [Russell, 19081 B. Russell. Mathematical logic as based on a theory of types. American Journal o f Mathematics. Volume 30. pp.222-262, 1908. Reprinted [van Heijenoort, 19671. pp. 150-182, 1967. [Schutte, 19651 K . Schiitte. Predicative well orderings. J . Crossley and M. Dummett (eds.) For- mal Systems and Recursive Functions. Amsterdam, NL: North-Holland Publishing Company. pp. 279-302, 1965 [Shanin, 19581 N. A. Shanin. O n the constructive interpretation of mathematical judgments. American Mathematical Society Translations. Series 2. Volume 23. pp.108-189, 1958. Trans- lated from 0 konstruktiviom ponimanii matematicheskikh suzhdenij. Trudy Ordena Lenina Matematicheskogo Instituta imeni V . A . Steklova. Akademiya Nauk SSSR. Volume 52. pp. 226-311, 1958. [Shanin, 19681 N. A. Shanin. Constructive real numbers and constructive function spaces. E. Mendelson (tr.)Translations o f Mathematical Monographs. Volume 21. Providence, RI: Amer- ican Mathematical Society. i v f 3 2 5 , 1968.

Constructivism in Mathematics 343[Skolem, 19231 T . Skolem. Begriindung der elementaren Arithmetik durch die rekurrierendeDenkweise ohne Anwendung scheinbarer Veranderlichen mit unendlichem Ausdehnungsbere-ich. Skrifter utgit av Videnskapsselskapet I Kristiania, I . Matematisk-naturvidenskabeligKlasse 6. pp. 1-38, 1923. Reprinted S. Bauer-Mengelberg (tr.) The foundations of elemen-tary arithmetic established by means of the recursive mode of thought, without the use ofapparent variables ranging over infinite domains. [van Heijenoort, 19671. pp. 302-333, 1967.[Specker, 19491 E. Specker. Nicht konstruktiv beweisbare Satze der Analysis. [Nonconstructivelyprovable sentences of analysis.] T h e Journal o f Symbolic Logic. Volume 14. pp. 145-158, 1949.[ ~ a i t1,9811 W . Tait. Finitism. T h e Journal o f Symbolic Logic. Volume 78. Number 9. pp. 524-546, 1981. Reprinted [Tait,2005bl. pp. 21-41, 2005.[ ~ a i t2,0021 W . Tait. Remarks on finitism. W . Sieg et. al. (eds.) Reflections on t h e Foundationso f Mathematics. Essays in honor o f Solomon Feferman. Assocation for Symbolic Logic. LectureNotes in Logic. Natick, M A : A.K. Peters, Ltd. pp. 410-419, 2002. Reprinted [Tait,2005bl. pp.43-53, 2005.[ ~ a i t2,005al W . Tait. Appendix to Chapters 1 and 2. [Tait, 2005bl. pp. 54-60, 2005.[Tait,2005bl W .Tait. The Provenance of Pure Reason. Essays in the Philosophy of Mathemat-ics and Its History. Oxford, U K : Oxford University Press. 2005. viiit-332, 2005.[Troelstra, 19811 A. Troelstra. Arend Heyting and his contribution t o intuitionism. Nieuw Archief voor Wiskunde. Volume XXIX. pp. 1-23, 1981.[ ~ r o e l s t r and van Dalen, 19881 A. S. Troelstra and D. van Dalen. Constructivism in Mathe-matics. A n Introduction. Volume I . Studies in Logic and t h e Foundations o f Mathematics.Volume 121. xx+342+XIV. Volume 11. Studies in Logic and t h e Foundations o f Mathematics.Volume 123. xvii+345-879+LII. Amsterdam, NL: North-Holland, 1988.[Turing, 1936-371 A . M. Turing. O n computable numbers with an application to the Entschei-dungsproblem. Proceedings o f t h e London Mathematical Society. Series 2. Volume 42. pp.230-265, 1936-37.[van Atten, 20041 M. van Atten. O n Brouwer. London, U K :Thomson/Wadsworth. 95 pp, 2004.[van Dalen, 19991 D. van Dalen. Mystic, Geometer, and Intuitionist. The Life of L. E. J.Brouwer. Volume I: The Dawning Revolution. Oxford, U K : T h e Clarendon Press. xv+440,1999.[vanDalen, 20051 D. van Dalen. Mystic, Geometer, and Intuitionist. The Life of L. E. J.Brouwer. Volume II: Hope and Disillusion. Oxford,U K : T h e Clarendon Press. x+946, 2005.[van Heijenoort, 19671 J . van Heijenoort. (ed.) From Frege to Godel: A Source Book i n Mathe-matical Logic, 1879-1931. Cambridge, MA: Harvard University Press. xi+665, 1967.[Wang, 19541 H. Wang. The formalization of mathematics. T h e Journal o f Symbolic Logic.Volume 19. pp. 241-266, 1954.[Wang, 19641 H. Wang. A Survey of Mathematical Logic. Amsterdam, NL: North-Holland Pub-lishing Company. x+651, 1964.[Weber, 18931 H. Weber. Leopold Kronecker. Jahresberichte der Deutschen Mathematiker Vere-inigung. Volume 11. pp. 5-31, 1893.[ w e y l , 19181 H. Weyl. Das Kontinuum. Kritische Untersuchungen fiber die Grundlagen derAnalysis. Leipzig, DE: V o n Veit. vi+83, 1918. Reprinted S. Pollard and T . Bole (trs.) TheContinuum. A Critical Examination of the Foundation of Analysis. New York, NY: DoverPublications, Inc. xxvi+l30, 1994.[Whitehead and Russell, 1 9 1 ~ 1 3 1A . N . Whitehead and B. Russell. Principia Mathematica. Volumes I , I1 and 111. Cambridge, U K : Cambridge University Press, 1910-13.[Wittgenstein, 19561 L. Wittgenstein. Remarks on the Foundations of Mathematics. G. vonWright et. al. (eds.) G . Anscombe (tr.) New York, N Y : T h e Macmillan Company. xix+204,1956.-~ ~~[Yessenin-Volpin, 19701 A . S. Yessenin-Volpin. The ultra-intuitionistic criticism and the anti-traditional program for foundations of mathematics. A. Kino et al. (eds.) Intuitionism andProof Theory. Proceedings o f t h e Summer Conference at Buffalo.New York. 1968. Amsterdam,NL: North-Holland Publishing Company. pp. 3-45, 1970.[Zaslavskii, 19551 I . D. Zaslavskii. The refutation of some theorems of classical analysis i n con-structive analysis. [in Russian] Uspehi Mat. Nauk. Volume 10. pp. 209-210, 1955.

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FICTIONALISM Daniel Bonevac Fictionalism, in the philosophy of mathematics, is the view that mathematical dis-course is in some important respect fictional: mathematical objects such as T or 0 havethe same metaphysical status as Sherlock Holmes or Macbeth. Admittedly, this definitionis vague. But it hard to do better without ruling out the positions of some philosopherswho consider themselves fictionalists. Hartry Field [1980, 21, who is largely responsiblefor contemporary interest in fictionalism as a strategy in the philosophy of mathematics,defines it as the view that there is no reason to regard the parts of mathematics that involvereference to or quantification over abstract entities such as numbers, sets, and functions astrue. I do not adopt that definition here, for some fictionalists (e.g., Stephen Yablo [2001;2002; 20051) think of fictional discourse as in some important sense true. Zoltan Szabo[2003] treats fictionalism about F s as the belief that 'There are Fs' is literally false butjctionally true. This rules out the possibility that fictional statements lack truth valuesand introduces a concept of fictional truth that only some fictionalists endorse. A number of writers, influenced by Kendall Walton [1978; 1990; 20051, treat fiction-ality as a matter of attitude: a discourse is fictional if its participants approach it with acognitive attitude of pretense or make-believe. As John Burgess, Gideon Rosen [1997],and Jason Stanley [2001] point out, however, there is little evidence that participants inmathematical discourse approach it with such an attitude. One might expect, moreover,that philosophers who diverge in their accounts of mathematics would also diverge in theirattitudes toward mathematical discourse: fictionalists might approach it with an attitudeof make-believe, while realists approach it with an attitude of discovery. To avoid a choicebetween relativism and trivial falsity, therefore, fictionalists about mathematics had betternot take attitude as definitive of fictionality. Mark Kalderon [2005] defines fictionalism at a higher level of abstraction: \"The dis-tinctive commitment of fictionalism is that acceptance in a given domain of inquiry neednot be truth-normed, and that the acceptance of a sentence from an associated region ofdiscourse need not involve belief in its content\" (2). This seems to make fictionalism aclose relative of noncognitivism. But mathematics is unquestionably a realm of inquirythat admits rational discourse and determination; indeed, it seems a paradigm of rationalinquiry. It does not express emotion; it does not issue commands. Taking this into ac-count, the fictionalist, we might say, sees some kinds of discourse that count as rationalinquiry as aiming at something other than truth and as accepted in a sense weaker thanbelief. In what follows, I will usually speak of success rather than acceptance to make itclear that the relevant notion of acceptance is one of being accepted by the participants in a discourse as a successful contribution to that discourse. Fictional sentences do not aim at truth (whether or not they in some sense achieve it) and succeed in playing their roles Handbook of the Philosophy of Science. Philosophy of Mathematics Volume editor: Andrew D. Irvine. General editors: Dov M. Gabbay, Paul Thagard and John Woods. @ 2009 Elsevier B.V. All rights reserved.

346 Daniel Bonevacin discourse even if they are not believed -or, at any rate, could play their roles even ifthey were not believed. 1 KINDS OF FICTIONALISMTo understand what fictionalists mean to assert about mathematics, consider a simpleinstance of fictional discourse, from Nathaniel Hawthorne's Twice Told Tales: I built a cottage for Susan and myself and made a gateway in the form of a Gothic Arch, by setting up a whale's jawbones.Hawthorne's utterance, conceived as fictional, succeeds, even if there are no cottage, gate-way, whale, and jawbones related in the way described. Conceived as nonfictional, incontrast, the nonexistence of any of those entities would prevent the utterance from suc-ceeding. In ordinary, nonfictional discourse, pragmatic success requires truth, and truth,for existential sentences, at least, requires that objects of certain kinds exist. In fiction,pragmatic success and existence come apart. Existential utterances can succeed even if noobjects stand in the relations they describe. The pragmatic success of a fictional discourse,that is, is independent of the existence of objects to which the discourse is ostensibly com-mitted (or, perhaps, would be ostensibly committed if it were asserted as nonfiction). The fictionalist about mathematics, then, maintains that the pragmatic success of math-ematical discourse is independent of the existence of mathematical objects. Mathematicscan do whatever it does successfully even if there are no such things as numbers, sets,functions, and spaces. Not everyone who holds this view, however, counts as a fictionalist. Fictionalism is avariety of mathematical exceptionalism, the view that the success of mathematical state-ments is exceptional, depending on factors differing from those upon which the success ofordinary assertions depends. There are many other versions of exceptionalism: (a) Reduc-tionists, for example, maintain that mathematical statements are exceptional in that theyare about something other than what they appear to be about; they translate into statementswith different ontological commitments. Those statements they take to be more meta-physically revealing than the originals. (See [Link, 20001.) (b) Supervenience theoristsdissent from the translation thesis, but contend nevertheless that the success of mathemat-ical statements depends on facts about something other than what they appear to be about,namely, nonmathematical entities. (c) Logicism (e.g., that of Russell [I9181 or Hempel[1945]) maintains that the success of mathematical statements depends solely on logic -classically, because mathematical truths translate into truths of logic. (d) Putnam's [I9671deductivism (or if-thenism) maintains that the success of mathematical statements is de-termined not according to the truth conditions of the statements themselves but insteadaccording to those of associated conditionals. (e) Hellman's [I9891 modal structuralismmaintains that mathematics, properly understood, makes no existence claims, but speaksof all possible realizations of structures of various kinds. Resnik [I9971 advances anotherversion of structuralism. (f) Chihara [I9901replaces traditional existential assertions withconstructibility theorems. (g) Schiffer's [2003] account of pleonastic propositions and

entities that result from \"something-from-nothing transformations\" (see also Hofweber[2005a; 2005b; 2006; forthcoming]) might offer a foundation for an account of mathe-matical statements without the need to invoke extra-mental mathematical entities. (Note,however, that fictional entities are just one kind of pleonastic entity.) These generally donot count as fictionalist accounts of mathematics in the contemporary sense - thoughthey are all in a sense anti-realist, though Schiffer's might count as a generalization of fic-tionalism, perhaps, and though, as we shall see, several (including, especially, the reduc-tionist account) might have been considered fictionalist throughout considerable stretchesof philosophical history. What distinguishes fictionalism from these kinds of exceptionalism, all of which arekinds of anti-realism? ( I ) It does not necessarily endorse the thesis that mathematicalstatements, properly understood, are true. Some fictionalists hold that fiction is in a sensetrue and that mathematics is true in the same or at least in an analogous sense. Butfictionalism per se carries no such entailment. And that by virtue of which mathemat-ical statements are true is not the same as that by virtue of which ordinary statementsabout midsize physical objects, for example, are true. It seems fair to summarize this, asKalderon does, by saying that mathematics is not truth-normed; it aims at something otherthan truth. Reductionists, supervenience theorists, logicists, deductivists, etc., in contrast,all take mathematical statements (properly understood or translated) as true and as aim-ing at truth. (2) The success of mathematics is independent of belief. The reductionist,supervenience theorist, deductivist, modal structuralist, constructibility theorist, etc., allbelieve that mathematical statements, as true, are worthy objects of belief, even if theirsurface forms are misleading. The fictionalist, however, sees belief as inessential to thesuccess of mathematical statements. (3) Fictional entities, if it is proper to speak of themat all, are products of free creative activity. A fictionalist about mathematics maintainsthat the same is true of its entities. More neutrally, we might say that according to thefictionalist mathematical statements, like those in fiction, are creative products. Amongthe facts by which their success is determined are facts about human creative activity.(4) Fiction is nevertheless in some sense descriptive. It describes objects and events. Itdiffers from ordinary descriptive discourse only in that the nonexistence of its objects andthe nonoccurrence of its events does not detract from its success. If fictionalists agree about that much, they disagree about much else. I shall classifyfictionalists here according to their views on three issues: truth, interpretation, and elimi-nation.1.1 TruthTo say that mathematics is not truth-normed is not to say that it is not truth-evaluable.Mathematics might aim at something other than truth but nevertheless be evaluable astrue or false. Indeed, that seems to be just the status of fiction. A work of fiction aims atsomething other than truth. But we can still ask whether the sentences it comprises aretrue or false. In nonfiction, success requires truth, which in turn requires the existenceof objects. If success in fiction is to be independent of the existence of objects, however,either success must be independent of truth, or truth must be independent of the existence

Daniel Bonevacof objects. Fictionalist strategies, then, divide naturally into two kinds, depending on their attitudetoward truth. According to one sort, Hawthorne's sentence succeeds despite its literalfalsity. Although fictionalists and their critics sometimes speak this way - see Szabo[2003] -it seems not quite right; literal contrasts most naturally withfigurative, but theliterallfigurative distinction does not line up neatly with the nonfiction/fiction distinction.It would be more accurate to say that fiction is not realistically true. Yablo [2001; 20051advances a version of fictionalism he callsfiguralisn~w, hich maintains that mathematicalstatements are analogous specifically to fictional, figurative language. On his view, thekind of truth appropriate to mathematics does contrast naturally with literal truth. But itseems contentious to build such a view into our terminology from the beginning. Accord-ing to the other, Hawthorne's sentence is in some sense true, despite the nonexistence ofthe objects it describes, precisely because it occurs in a fictional context. Correspondingly, some fictionalists, such as Field [1980], contend that mathematicssucceeds without being true. Others contend that mathematics can be true even if theobjects it seems to describe do not exist. Among the latter are some whose views comeclose to Putnam's; they contend that mathematics is true in a purely deductive, \"if-then\"sense, mathematical truth being simply truth in a story (the standard set-theoretic hierar-chy story, for example), much as fictional truth is truth in a story (the Twice Told Tales,for example). Whether such views remain fictionalist depends on the details. Others holdthat mathematics, and fiction in general, are true in a more full-blooded sense.1.2 InterpretationBurgess and Rosen [I9971 distinguish hermeneutic from revolutionary nominalists. Vari-ous writers draw the same distinction among fictionalists. Hermeneutic fictionalists aboutmathematics maintain that we do interpret mathematics as fictional; revolutionary fic-tionalists, that we should. As Yablo puts it: \"Revolutionary nominalists want us to stoptalking about so-and-so's; hermeneutic nominalists maintain that we never started\" [2001,851. Both versions of fictionalism face serious problems. Most mathematicians and scien-tists, not to mention nonspecialists, do not appear to interpret mathematics fictionally. Thevarious schemes for reinterpreting mathematics that fictionalists or nominalists have pro-posed, moreover, are complex and difficult to use; it is hard to see what sort of mathemat-ical, scientific, or practical advantage they could possess that would justify that assertionthat we ought to reinterpret mathematics in accordance with them. Indeed, they seem tohave severe disadvantages by those measures. The revolutionary fictionalist (e.g., [Leng,20051) would presumably claim philosophical (specifically, epistemological) advantages.But, even if those claims can be sustained, it is hard to see why philosophical advantagesshould outweigh mathematical, scientific, or practical disadvantages. Fortunately for fictionalism, Burgess and Rosen's distinction is not exhaustive. Fic-tionalists can argue, not that we do or should interpret mathematics fictionally, but thatwe can. Imagine a fictionalist about ancient Greek gods and goddesses, for example. Fic-tionalism about Zeus, Hera, etc., seems an entirely reasonable position. Such a person

Fictionalism 349might contend that ancient Greeks adopted an attitude of make-believe toward the gods,but might not; he or she might contend that they should have adopted such an attitude,but, again, might not. (Belief in the gods might have been crucial to social stability, forexample.) The fictionalist in this instance holds that we can best understand ancient Greekreligion by interpreting it in fictional terms. Alternatively, think of Russell's [I9181 view of ontology as the study of what we mustcount as belonging to the basic furniture of the universe. A fictionalist interpretation ofa discourse, such as Russell's no-class theory, on his view shows that we do not havethe count the objects the discourse ostensibly discusses as among the basic furniture ofthe universe. Whatever attitude about the objects to which such a discourse is ostensiblycommitted we happen to have, and whatever attitude about them might be best for uspractically, scientifically, psychologically, or linguistically, the theoretical viability of afictionalist attitude is enough to show that we are capable of avoiding ontological com-mitment to them. From Russell's perspective, fictionalism pays ontological dividendseven if no one does or should (except perhaps in a technical ontological sense) adopt afictionalist attitude toward the pertinent discourse. Similarly, a fictionalist about mathematics can hold that we can best understand theontological commitments forced upon us by mathematics by interpreting it in fictionalterms. The view implies nothing at all about how mathematicians themselves do or oughtto interpret their subject. Mathematics can accomplish its purposes, according to the fic-tionalist, even if there are no mathematical objects. To show this, the fictionalist needs to(a) specify the purposes of mathematics that must be accomplished, and (b) demonstratethe possibility of accomplishing them with a theory that makes no commitment to math-ematical entities. Field, for example, takes the application of mathematics in physicalscience as the purpose of mathematics that the fictionalist must explain, and attempts toshow the physical application of mathematical theories can be understood without appealto distinctively mathematical objects. In addition to hermeneutic and revolutionary fictionalism, then, we should distinguishdeflationary fictionalism, which maintains that there is no need for a substantive meta-physics or epistemology for mathematics. There is no deep mystery, the deflationaryfictionalist insists, about how we know that Dr. Watson was Holmes's associate. Neitheris there a deep mystery about how we know that T z 3. We are not committed to theexistence of Dr. Watson, Holmes, T , or 3. Nor must we postulate any strange facultyof intuiting objects with which we stand in no causal relation. Mathematics serves itsfunction without any such assumption. Mark Balaguer expresses the spirit of deflationary fictionalism concisely: ... w e use mathematical-object talk in empirical science to help us accurately depict the nature of the physical world; but we could do this even if there were no such things as mathematical objects; indeed, the question of whether there exist any mathematical objects is wholly irrelevant to the question of whether we could use mathematical-object talk in this way; therefore, the fact that we do use mathematical-object talk in this way does not provide any reason whatsoever to think that this talk is true, or genuinely referential. [Balaguer, 1998, 1411

350 Daniel BonevacWe do not need to show that people actually adopt an attitude of make-believe in mathe-matics. All we need is to show is that they could use mathematics just as successfully ifthey did. Deflationary fictionalism bears some resemblance to the indifferentism propounded byEklund [in press]. On that view, the ontological commitments of our statements are non-serious features of them, features that are beside the point of the statements. In general,Eklund argues, \"we do not commit ourselves either to its literal truth or to its truth in anyfiction; we are, simply, non-committed.\" Think of a picture of content similar to that ofStalnaker [1978], in which statements in a context function to restrict the class of possibleworlds that constitutes that context. Speakers in nonphilosophical contexts may not beinterested in ruling out possibilities that differ only metaphysically. Their statements havea content, therefore, that is properly understood as metaphysically neutral. The similaritybetween this view and deflationary fictionalism emerges in this passage: \"It can be that Ido not in fact make, say, mathematical statements in a fictional spirit, but when I come torealize this is a possibility I can also realize that doing so would all along have satisfied allof my conversational aims\" [Eklund, in press]. Deflationary fictionalism needs to showonly that a fictional interpretation of a discourse is possible, and establishes thereby thatthe discourse is ontologically neutral.1.3 EliminationWhat does it take to show that mathematics can fulfill its purposes even if there are nomathematical objects? The oldest tradition historically falling under the heading of fic-tionalism maintains that the fictionalist must reduce mathematical to nonmathematicalobjects. William of Ockham, for example, contends that universals are$cta; they do notexist in any real sense. Everything that really exists is particular. He seems willing to dis-card some talk of universals as incorrectly assuming their real existence. Most discourseinvoking universals, however, he seeks to reinterpret. Socrates has wisdom, for example,is true by virtue of the fact that Socrates is wise. It seems to be committed to the existenceof a universal, wisdom, but in fact requires the existence of nothing more than Socrates.Nominalists (sometimes calling themselves fictionalists) and others have employed re-ductive strategies in a wide variety of contexts. consider, for example, David Hume'saccount of necessary connection, Bertrand Russell's \"no-class\" theory, and Rudolf Car-nap's [I9281 phenomenological construction of the world. Opponents of nominalist reinterpretation such as Burgess, Rosen, and Hofweber ar-gue as follows: Socrates has wisdom can be interpreted as Socrates is wise only if theyare equivalent. But, by the nominalist's own lights, the former entails the existence ofuniversals, while the latter does not. So, they cannot be equivalent, and the nominalis-tic interpretation fails. Nominalists, in response, tend to deny that interpretation requiresequivalence tout court; it requires only equivalence relative to the purposes of the kind ofdiscourse in question. Broadly speaking, it seems fair to say, nominalism takes its inspiration from some-thing like empiricism. The purpose of discourse in the broadest sense is to account forour experience. Call sentences A and B experierztially equivalent if and only if they have

Fictionalism 351exactly the same implications for experience. (We might suppose that there is an experi-ential language E adequate to and restricted to describing experience such that A and Bare experientially equivalent just in case, for any sentence C of E, A k C w B C.)The claim is then that Socrates is wise and Socrates has wisdom are experientially equiv-alent; they have exactly the same implications for our experience. They differ in theirostensible commitments -that is to say, their prima facie commitments, what they seemto be committed to independent of any considerations about the possibility of translation,paraphrase, or elimination, etc. So, they are not equivalent all things considered. But theyare empirically equivalent. The upshot, according to the nominalist: we would suffer nodecline in our ability to describe our experience if we were to assert Socrates is wise inplace of Socrates has wisdom. We have no reason, therefore, to take on the additionalontological commitments of the latter. We might think in model-theoretic terms, as follows. M and N are elementarily equiv-alent with respect to E if and only if they agree on every sentence C of E: M k C u N kC. Suppose that M is a platonistic model including abstracta as part of its domain andthat N is a nominalistic model with a domain consisting solely of concreta. If M and Nare elementarily equivalent with respect to E, then they satisfy exactly the same experien-tial sentences. So, we may safely replace M with N, avoiding M's worrisome platonisticcommitments, without adversely affecting our ability to account for our experience. Experiential equivalence in this sense is weaker than reduction. Suppose that for everyplatonistic model M of our best theory of the world there is an experientially equiva-lent nominalistic model N of that theory. It does not follow that the theory reduces toone having only nominalistically acceptable entities as ostensible commitments, unlessthere is a function from platonistic to experientially equivalent nominalistic models meet-ing stringent criteria (see e.g., [Enderton, 19721). So, there is plenty of logical space inwhich fictionalists may adopt a strategy based on experiential equivalence without com-mitting themselves to reductionism. Mounting an argument that every platonistic modelhas an associated experientially equivalent nominalistic model without specifying such afunction, on the other hand, presents a challenge. (See the discussion of Field below.)Fictionalists must steer a path between the Scylla of reductionism and the Charybdis ofdeductivism. 2 MOTIVATIONS FOR FICTIONALISM IN THE PHILOSOPHY OF MATHEMATICSMathematics does not appear to be a species of fiction. Why, then, insist on the possibil-ity of construing it fictionally? Fictionalists fall into two camps. Just as debaters advancecases either on the basis of needs or on the basis of comparative advantage, so fictional-ists argue for fictional interpretations either because alternative interpretations raise philo-sophical puzzles or because fictionalist interpretations simply provide better explanationsfor mathematical success.

Daniel Bonevac2.1 Benacerraf 's DilemmaThe traditional argument for fictionalism is that nonfictional interpretations of math-ematics raise insuperable philosophical difficulties. The locus classicus of the argu-ment, though presented with a different intent, is Paul Benacerraf's \"Mathematical Truth\"[1973]. Benacerraf argues that we can devise a successful semantics or a successful epis-temology for mathematics, but not both. We cannot reconcile the demands of an accountof mathematical truth with the demands of an account of mathematical knowledge. ... accounts of truth that treat mathematical and nonmathematical discourse in relevantly similar ways do so at the cost of leaving it unintelligible how we can have any mathematical knowledge whatsoever; whereas those which attribute to mathematical propositions the kinds of truth conditions we can clearly know to obtain, do so at the expense of failing to connect these con- ditions with any analysis of the sentences which shows how the assigned conditions are conditions of their truth. [Benacerraf, 1973,6621Benacenaf makes two assumptions. First, he assumes that we should maintain a unifiedTarskian semantics for mathematical as well as nonmathematical discourse. Second, heassumes a causal theory of knowledge. The first assumption implies that mathematicalobjects exist; mathematical discourse succeeds only to the extent that it is true, and it istrue only to the extent that the objects over which it quantifies exist. The second impliesthat we can have mathematical knowledge only by causally interacting with mathematicalobjects. But that, evidently, is what we do not and cannot do. (Some abstract objects arenevertheless dependent on concrete objects and events, as Szabo (2003) observes. Storiesmay be abstract but depend on concrete people and events; concrete events may in turndepend on them. So, it is a mistake to think of all abstracta as causally isolated. It seemsdoubtful that enough mathematical objects could be dependent in this sense to groundmathematical knowledge. But see Maddy [1990; 1992; 19971.) The fading popularity of causal theories of knowledge may make Benacerraf's dilemmaseem like something of a period piece, no longer compelling a choice between semanticand epistemological adequacy. There are various ways, however, of weakening these as-sumptions. Here I will present just one (developed at length in [Bonevac, 19821). So longas (a) pragmatic success requires truth, (b) truth is to be explained in terms of referenceand satisfaction, and (c) we must have epistemic access to the objects we take our dis-course to be about (as Benacerraf puts it, we must have \"an account of the link betweenour cognitive faculties and the objects known\" (674)), we face the same problem. Theseassumptions are weaker than Benacerraf's in several respects. They do not assume thata single semantic theory must apply to mathematical and nonmathematical language, aswell as all other forms of discourse. Most crucially, they do not assume a causal theoryof knowledge. Demanding epistemic access requires that there be a relation between usas knowers and the objects of our knowledge that allows for the possibility of an empir-ical cognitive psychology. This might be causal, but it need not be. The central idea ismotivated by a naturalized epistemology: In short, our ability to have knowledge concerning the objects assumed to

Fictionalism exist must itself be capable of being a subject for empirical, and preferably physiological, investigation. [Bonevac, 1982,9]I have summarized this by demanding an empirically scrutable relationship between our-selves and the objects postulated by theories we accept. It must be possible to explainour knowledge of those objects in a naturalized epistemology. One may spell out the re-quired relationship differently: Field, for example, simply says that our knowledge mustbe explicable. It should not be chalked up to coincidence: The key point, I think, is that our belief in a theory should be undermined if the theory requires that it would be a huge coincidence if what we believed about its subject matter were correct. But mathematical theories, taken at face value, postulate mathematical objects that are mind-independent and bear no causal or spatio-temporal relations to us, or any other kinds of relations to us that would explain why our beliefs about them tend to be correct; it seems hard to give any account of our beliefs about these mathematical objects that doesn't make the correctness of the beliefs a huge coincidence. [Field, 1989, 71.Mathematicians are reliable; surely that fact needs to be explained. We might distinguish two kinds of mathematical theories: existential theories, whichpostulate the existence of mathematical entities such as 0, r,or the exponentiation func-tion, and algebraic theories, which d o not, but instead speak only of objects related incertain ways, e.g., as groups, rings, fields, and so on. A structuralist or deductivist anal-ysis of the latter seems natural, though they too make existence claims, which might beseen as derivative from the claims of existential theories or as sui generis and needing sep-arate explanation. Benacerraf's dilemma seems most acute for existential theories, suchas arithmetic, analysis, and set theory, which postulate the existence of numbers, sets,and functions. Classical mathematics, in Quine's words, \"is up to its neck in commit-ments to an ontology of abstract entities\" [Quine, 1951, 131. But how can we know aboutsuch entities? Must our epistemology of mathematics remain nothing but \"a mysteriousmetaphor\" [Resnik, 1975, 30]? Since the argument from Benacerraf's dilemma has fallen under widespread attack, letme try to spell it out somewhat more explicitly, restricting it to existential theories andkeeping its assumptions as weak as possible: 1. Some existential mathematical theories - arithmetic and set theory, for example -are successful. 2. Mathematics is successful only to the extent that it is true. 3. An adequate theory of truth for mathematics must be continuous with the theory of truth for the rest of language. 4. An adequate theory of truth in general must be Tarskian, proceeding in terms of reference and satisfaction.

354 Daniel Bonevac 5. Any Tarskian theory interprets existential sentences as requiring the existence of objects in a domain. 6. Existential mathematical theories contain existential sentences. 7. Therefore, mathematical objects exist. 8. We know about objects only by standing in an explicable epistemic relation to them. 9. We stand in no explicable epistemic relation to mathematical objects. 10. Therefore, we cannot know anything about mathematical objects -even that they exist.We can thus conclude that mathematical objects exist -a conclusion which, having beenrationally justified, appears to be known. But we can also conclude that we cannot haveany such knowledge. Any account of mathematics must confront the problem this poses.The fictionalist focuses on the first three of the above premises. Most deny the second,maintaining that mathematics may be successful without being true. Some deny the first,holding that all mathematical theories can be given an algebraic interpretation. And somedeny the third, seeking a non-Tarskian semantics that can apply to fictional and mathe-matical discourse in a way that frees them from ontological commitment and associatedepistemological difficulties. We interpret sentences in and about works of fiction in waysthat do not seem to commit us to fictional characters in a way that raises serious meta-physical and epistemological difficulties. We have no trouble, for example, explaininghow it is possible to know that Sherlock Holmes is a detective. That suggests to some thatour semantics for fiction is non-Tarskian.2.2 Yablo's Comparative Advantage ArgumentThe traditional argument just outlined faces an obvious problem, even if it is not quite\"dead and gone,\", as Yablo [2001,87] says. It rests on a thesis requiring epistemic accessto objects to which we make ontological commitments. Epistemic access need not bespelled out in terms of empirical scrutability or even explicability. Say simply that theobjects of our commitments must exhibit property P. The problem then arises becausemathematical objects lack P. But an opponent can turn this argument on the fictionalistby using the fact that mathematical objects lack P to refute the premise that the objects ofour commitments must exhibit P. Mathematical objects, that is, can be used as paradigmcases undermining any epistemology a fictionalist or other anti-realist might use (see, e.g.,[Hale, 19941. It is hard to see how to resolve the resulting impasse. Yablo argues for fictionalism on different grounds. The traditional argument focusessolely on the problems facing a platonistic account of mathematics, effectively grantingthat otherwise a platonistic theory would be preferable. But that, Yablo, insists, ignoresthe real advantages of a fictionalist approach while allowing the false advantages of pla-tonism. Yablo begins with the Predicament: \"One, we find ourselves uttering sentences thatseem on the face of it to be committed to so-and-so's - sentences that could not be

Fictionalism 355true unless so-and-so's existed. But, two, we do not believe that so-and-so's exist\" (72).We do not have to insist that so-and-so's are unknowable; it is enough to observe that itis coherent to speak of them without really believing in them. Nonplatonists adopt thisattitude toward mathematical and other abstract objects, but we adopt it in all sorts ofeveryday contexts as well, in speaking of sakes, petards, stomach-butterflies, etc. Quineoutlines three ways out of the predicament: reduction, elimination, and acceptance. Yabloargues that fictionalism constitutes a fourth way. Let's begin with platonism. Waive epistemological objections to abstracta. What doesthe platonist explain by invoking them? Presumably, the objectivity of certain kinds ofdiscourse. Nominalists have long thought that \"explaining\" the truth of Socrates is wiseby pointing out that Socrates has wisdom is no explanation at all. Yablo sharpens theargument by constructing a dilemma. Consider our conception of the numbers, for ex-ample. It is either determinate (in the sense that it settles all arithmetical questions) orindeterminate. If it is determinate, \"Then the numbers are not needed for objectivity.Our conception draws a bright line between true and false, whether anything answers toit or not\" (88). If it is indeterminate, how do we manage to pick out one of the manypossible models of our conception as the intended model -that is, as the numbers? Inshort, the numbers themselves are either unnecessary or insufficient for objectivity. (Fora perceptive treatment of determinacy in mathematics, see [Velleman, 19931.) By itself, this argument seems little better than the Benacerraf-inspired argument wehave been considering. A platonist who believes in determinacy will surely insist thatour conception of the numbers manages to settle all mathematical questions preciselybecause it is the conception of a determinate reality. Like a photograph that settles aset of questions (about who won the race, say) because it is the photograph of an eventor state of affairs (the finish of the race), our conception of mathematical objects maysettle questions because it is a conception of those objects. Such a conception does notdemonstrate that the objects are not needed to account for the objectivity of mathematicsany more than the photograph would show that the event or state of affairs is not neededto account for the objectivity of our judgment about who won the race. In short, we mayneed the objects themselves to account for the determinacy as well as the objectivity ofour conception. What of a platonist who believes in indeterminacy? Yablo's question -how then dowe pick out one model as the intended model? -bears some similarity to Benacerraf'squestion of how we manage to refer to or have knowledge of mathematical entities withouthaving any causal contact with them. Yablo seems to be asking for an explanation of ourability to pick out the objects of mathematics. But this seems to have exactly the status ofField's request for an explanation of the correctness of our mathematical beliefs. The platonist, of course, might also say that we cannot pick out the intended model; theindeterminacy of our mathematical conceptions may entail the indeterminacy of referencein mathematics, which is arguably a conclusion of Benacerraf [1965]. This may do nomore to disrupt mathematical discourse than the indeterminacy of reference in generaldisrupts our discourse about rabbits [Quine, 19601. The upshot of Quine's argumentsseems to be that it is indeterminate whether the natives (or, by [Quine, 19691, we) arespeaking of rabbits, undetached rabbit parts, rabbithood, etc. But they are speaking of

356 Daniel Bonevacsomething with rabbit-like characteristics. Similarly, it may be indeterminate whethermathematicians are speaking of categories, sets, classes, numbers, etc., but clear that theyare speaking of something with abstract characteristics. (It is perhaps enough to observethat, whatever they are speaking about, they are committing themselves to infinitely manyof them.) Let's turn to the positive portion of Yablo's argument. Yablo's fictionalist account,he contends, in contrast to a platonistic account, succeeds in explaining a great deal. Itexplains why numbers are \"thin,\" lacking any hidden nature; for mathematical objects,nominal and real essence coincide. It explains why numbers are indeterminate with re-spect to identity relations involving nonnumbers; it is determinately true that 0 x 2 = 0, anddeterminately false that 0 x 2 = 2, but indeterminate whether 2 = ((0))or 2 = (0,(0)).Itexplains why applied arithmetical statements are translucent; we immediately see throughthem to recognize their implications for concreta. It explains why people are impatientwith any objection to mathematics on ontological grounds, since the ontological statusof its objects makes no difference to what a mathematical theory is communicating. Itexplains why mathematical theories are excellent representational aids, applying readilyto the world. And it explains why mathematical assertions strike us as necessary and apriori; they do not depend on the actual existence or contingent circumstances of theirobjects. 3 A BRIEF HISTORY OF FICTIONALISMThe problem fictionalism addresses is ancient: the problem of knowing the forms. Platohypothesized the existence of forms, universals standing outside the causal order but ex-plaining its structure. If they remain outside the causal order, however, how is it possibleto know anything about them? Aristotle, arguably, had formulated a version of Benacer-raf's dilemma in his criticisms of the theory of forms in Metaphysics I, 9: \"if the Formsare numbers, how can they be causes?'Fictionalism might be read into some forms ofancient skepticism and into an alternative outlined (but not accepted) by Porphyry, thatuniversals are nuda intellects. But the earliest philosopher to endorse a fictionalist strat-egy is probably Roscelin. The father of nominalism, Roscelin sought to solve the problemposed by knowing the forms by denying their existence. He maintained that abstract termsare Jatus vocis, puffs of the voice, empty noises, reflecting Boethius's thought that Nihilenim aliud est prolatio (vocis)quam aeris plectro linguae percussio. It is not clear ex-actly what position Roscelin meant to endorse by this claim; we do not know whether hethought abstract terms were empty of meaning or empty of reference. If the latter, whichis in any case a more plausible position to hold, Roscelin might reasonably be considereda fictionalist about universals. Here are some highlights in the history of fictionalism, de-signed to illustrate some important types of fictionalism. (For an alternative history thatfills in many gaps in the following, see Rosen [2005].)

Fictionalism3.1 William of Ockham: Reductive FictionalismThe earliest fictionalism of which we have any detailed record is probably William ofOckham's. Ockham explicitly speaks of universals asjcta, and makes it clear that, in hisview, they do not exist; everything that exists is particular. Whether we interpret this as aform of nominalism, as most traditional commentators have, or as a form of conceptual-ism, as some more recent scholars have, it is in any case the thesis that sentences seem-ingly referring to universals can succeed in their linguistic function even though nothing inmind-independent reality corresponds to a universal. (See, for example, [Boehner, 1946;Adams, 1977; 1987;Tweedale, 1992; Spade, 1998; 1999a; 1999b].) It is important to note that, from Ockham's point of view, one must do more to breakthe success-existence link than interpret discourse involving abstract terms as fictional.Ockham assumes that sentences involving abstract terms play a role in discourse, andthat, to make the case the case that such terms need not be taken seriously from an on-tological point of view, one must explain how it is possible for them to play such a role.Ockham initiates one of the traditional strategies for doing so, arguing that abstractionsare shorthand for expressions that fill the same semantic and pragmatic role but withoutinvoking the existence of objects lying outside the causal order. Socrates exempliJies wis-dom is an inefficient way of saying that Socrates is wise (Ockham 1991, 105ff); Courageis a virtue is an efficient way of saying that courageous people and courageous actions areceteris paribus virtuous. To say Socrates and Plato are similar with respect to whiteness,Ockham says, is just to say that Socrates is white and Plato is white (572). In general,every true sentence containing abstract terms translates into a true sentence containingonly concrete terms, and vice versa: \"it is impossible for a proposition in which the con-crete name occurs to be true unless the [corresponding] proposition in which its abstract[counterpart] occurs is true\" (432). Ockham thus outlines the reductive strategy: we may break the link between successand existence by arguing that the use of abstract terms, for example, is unnecessary. Any-thing that can be said with abstract terms can be said without them. A language withoutabstract terms could thus in principle fill every role in discourse that a language with ab-stract terms can fill. Of course, it might do so inefficiently; there may be good practicalreasons to use abstract language. Nevertheless, its in-principle eliminability shows thatthe ontological commitments such language seems to force upon us are also in princi-ple eliminable. If abstract language is unnecessary, then so is a commitment to abstractentities. Note that the reductive strategy breaks the chain from success to existence notby denying the truth of the sentences with troublesome ontological commitments but bydenying that they form an essential part of an accurate description of the world. Thoughthe'sentences may make commitments to objects to which we stand in no empiricallyscrutable relation, we even in asserting them make no such commitment, because we cantreat them as optional abbreviations for sentences making no such commitment. Though Ockham considers reductionism a version of fictionalism, and though Ben-tham, Russell, and others have agreed, he is probably wrong to do so. At any rate, areductive fictionalism seems not to take advantage of anything distinctive of fictionalism.(1) The analogy with fiction is not very strong [Burgess, 20041. On a reductive account,

358 Daniel Bonevacsentences ostensibly making commitments to abstract objects are better understood as be-ing about something else. But fictional statements, it seems plausible to claim, are notabout something other than fictional characters. Sherlock Holmes is a detective by virtueof the fact than Sir Arthur Conan Doyle described him that way. But it would be strangeto say that Sherlock Holmes is a detective should be translated, from an ontological pointof view, into Sir Arthur Conan Doyle described Sherlock Holmes as a detective, and notonly because the latter still seems to refer to Holmes. It would be even stranger to saythat the former sentence is really about descriptions. In any case, such translations arenot recursive in the way that reductive translations ought to be. (2) However this may be,reductive theories interpret purported truths about objectionable entities as truths aboutacceptable entities. They translate sentences with ostensible commitments to abstractainto sentences without such commitments. The criterion for the translation's success istruth preservation. Not only does the discourse thus emerge as truth-normed, though inan unexpected way; the discourse still commits one to objects, even if not to the objectsone initially took it as being about. Existential sentences translate into other existentialsentences. Hawthorne's sentence in Twice Told Tales, viewed as fictional, however, seemsto force no commitments at all.3.2 Jeremy Bentham: Instrumentalist FictionalismJeremy Bentham is perhaps the first philosopher to have advocated fictionalism explic-itly. His theory of fictions, which exerted a powerful influence on Bertrand Russell atan early stage of his thought, appears over the course of seven of Bentham's works andstill receives surprisingly little attention. But Bentham thought of it as one of his chiefachievements, something upon which most of his other philosophical conclusions depend. Bentham distinguishes fictitious entities not only from real entities but also from fab-ulous entities, \"supposed material objects, of which the separate existence is capable ofbecoming a subject of belief, and of which, accordingly, the same sort of picture is capableof being drawn in and preserved in the mind, as of any really existing object\" [Bentham,1932, m - x r x v i ] . Fabulous objects, in other words, are nonexistent objects of existentkinds (legendary kings, for example, or countries such as Atlantis or El Dorado) or nonex-istent objects of nonexistent kinds (dragons, elves, or the Loch Ness Monster); they wouldpose no particular metaphysical or epistemological problems if they were to exist. Ordi-nary fiction thus introduces, primarily, fabulous objects by way of referring expressionsthat would, if they were to denote at all, denote substances. Fictitious objects, in contrast, do not \"raise up in the mind any corresponding im-ages\" (xxxvi); their names function grammatically as if they were names of substances,but would not refer to substances even if their referents were to exist. We speak as ifsuch names refer to real objects, \"yet in truth and reality existence is not meant to be as-cribed\" (12). \"To language, then -to language alone - it is, that fictitious entities owetheir existence; their impossible, yet indispensable, existence\" (15). Every language mustspeak of a fictitious object as existing - in that sense, their existence is indispensable-\"but without any such danger as that of producing any such persuasion as that of theirpossessing, each for itself, any separate, or, strictly speaking, any real existence\" (16).

Fictionalism 359 Distinguishing fabulous from fictitious objects might serve as an argument against fic-tionalism, for it suggests that the objects of mathematics, for example, are not very closelyanalogous to ordinary fictional entities. But Bentham nevertheless applies his theory offictions directly to mathematics. Quantity, he contends, is the chief subject matter ofmathematics, and is fictitious. \"The ink which is in the ink-glass, exists there in a certainquantity. Here quantity is a fictitious substance - a fictitious receptacle - and in thisreceptacle the ink, the real substance, is spoken of as if it were lodged\" ( x m i i i ) . Puremathematics, Bentham holds, \"is neither useful nor so much as true\" (WorksIX, 72). Thismakes it sound as if Bentham were breaking the success-existence chain by denying thetruth of mathematics in the way that most contemporary fictionalists do. In fact the pic-ture is more complicated. \"A general proposition which has no individual object to whichit is truly applicable is not a true one\" (Works VIII, 163),Bentham says, denying that cer-tain branches of pure mathematics are true. But he concedes that geometry can be givenan interpretation in which it applies, for example, to all spherical bodies, in which caseit is properly viewed as a true or false theory of those physical entities. If such entitiesare \"capable of coming into existence, it may be considered as having a sort of potentialtruth\" (Works VIII, 162). So, in Bentham's view, there are three kinds of mathematical theories: the theories ofapplied mathematics, which are empirical theories of the world; theories with potentialtruth, which would be true if their objects, which are capable of existing, were to exist;and theories with no kind of truth at all, of objects that are not even capable of existing.Corresponding to these three kinds of theories are three strategies for reconciling thesemantics and epistemology of mathematics: the empirical strategy, the strategy of JohnStuart Mill and W. V. 0. Quine, which holds that mathematics is an empirical theoryand is thus in no sense fictional; the modal strategy, the strategy of Cantor, Poincare,and Chihara, which holds that mathematics is a theory of possible objects and is thusfictional in the sense that fiction, too, describes possible objects; and the instrumentaliststrategy, the strategy of Hartry Field, which holds that mathematics is not true but rather aninstrument or, as Bentham would say, a successful system of contrivances for a practicalpurpose. There is considerable merit, Bentham maintains, in the reductive strategy, which lo-cates the success of mathematics in \"mere abbreviation ... nothing but a particular speciesof short-hand\" (183),but ultimately, he insists, that is not enough; \"Newton, Leibnitz,Euler, La Place, La Grange, etc., etc. -on this magnificent portion of the field of sci-ence, have they been nothing more than so many expert short-hand writers?'(37) Heemphasizes, therefore, the view of mathematics as a system of contrivances. It is notclear what Bentham means by contrivance, but he does give examples: \"the conversionof algebraic method into geometrical ... the method of Juxions ... and the differentialand integral calculus\" (169). The success of such contrivances consists in their satisfyingtwo conditions. First, there must be kinds of empirical circumstances to which the theorywith its attendant fictions would be applicable. Second, there must be some advantageto using such a theory in such circumstances. Mathematics is not true, but it succeedsto the extent that it serves as a successful instrument in reasoning about empirical statesof affairs, specifically, in proceeding from the known to the unknown. Mathematics in-

360 Daniel Bonevactroduces objects -infinite collections, for example, limits, derivatives, integrals, tangentlines, etc. -that cannot be reduced to nonmathematical entities, but which prove usefulin deducing conclusions about nonmathematical entities.3.3 C. S. Peirce: Representational FictionalismThe fictionalism of nominalists from Roscelin to Bentham has been motivated by an epis-temological attitude we might construe as at least a proto-empiricism. Two other forms offictionalism stem from rather different perspectives: Peirce's pragmatism and Vaihinger'sKantianism. C. S. Peirce developed a view of mathematics that might be counted as a variety offictionalism. \"Mathematics,\" Peirce says, \"has always been more or less a trade\" 11898,1371. We can understand the nature of mathematics only by understanding \"what serviceit is\" to other disciplines. Its purpose, he writes, is to draw out the consequences ofhypotheses in the face of complexity: An engineer, or a business company (say, an insurance company), or a buyer (say, of land), or a physicist, finds it suits his purpose to ascertain what the necessary consequences of possible facts would be; but the facts are so com- plicated that he cannot deal with them in his usual way. He calls upon a mathematician and states the question. . . . He [the mathematician] finds, however in almost every case that the statement has one inconvenience, and in many cases that it has a second. The first inconvenience is that, though the statement may not at first sound very complicated, yet, when it is accu- rately analyzed, it is found to imply so intricate a condition of things that it far surpasses the power of the mathematician to say with exactitude what its consequence would be. At the same time, it frequently happens that the facts, as stated, are insufficient to answer the question that is put. 11898, B 137-38, CP 3.3491Think of a child measuring a line segment as, say, 5 cm long. Almost certainly, themeasurement is an approximation; the line is slightly shorter or longer. Similarly witha surveyor's angle. An economist predicting a firm's profits seems to be in a bit betterposition, since money comes in discreet units. But the complexity of predicting profits isimmense, since many factors influence income and expenditure and do so in complicatedways. A physicist calculates the trajectory of a projectile using approximate values forforce, velocity, distance, etc., and ignoring the effects of friction, the moon's gravitationalpull, and so on. In short, the real world is overwhelmingly complicated. This providesthe chief motivation but also the chief difficulty for applying mathematics to real-worldproblems. Essential to the application of mathematics to the world, therefore, is idealization. Likethe economic principles linking variables such as population growth, personal income, theinflation rate, etc. to income and expenditures, the physical laws governing the motion ofprojectiles successfully relate quantities in an idealized, simplified version of the real sit-uation, not the real situation itself. The world is full of friction, vagueness, imprecision -

Fictionalism 361in short, noise. The noise itself can be studied and classified. But it cannot be eliminated.One can deduce consequences only by abstracting from it: Accordingly, the first business of the mathematician, often a most difficult task, is to frame another simpler but quite fictitious problem (supplemented, perhaps, by some supposition), which shall be within his powers, while at the same time it is sufficiently like the problem set before him to answer, well or ill, as a substitute for it. This substituted problem differs also from that which was first set before the mathematician in another respect: namely, that it is highly abstract. [1898, B 138, CP 3491This \"skeletonization\" or \"diagrammatization\" serves \"to strip the significant relations ofall disguise\" (B 138, CP 349) and thus to make it possible to draw consequences. In idealizing a problem in this way, Peirce writes, The mathematician does two very different things: namely, he first frames a pure hypothesis stripped of all features which do not concern the drawing of consequences from it, and this he does without inquiring or caring whether it agrees with the actual facts or not; and, secondly, he proceeds to draw necessary consequences from that hypothesis. (1898, B 138, CP 349-350)A pure hypothesis \"is a proposition imagined to be strictly true of an ideal state of things\"[1898, B 137, CP 3481. The principles used by the engineer, the economist, or even the physicist are not likethat; \"in regard to the real world, we have no right to presume that any given intelligi-ble proposition is true in absolute strictness\" [1898, B 137, CP 3481. The real world iscomplicated; principles tends to hold of it only ceteris paribus, or in the absence of anycomplicating factors. Some laws of nature may be simple enough to be easily intelligiblewhile also holding absolutely, as Galileo thought, but we have no right to expect that tobe the case in general. In some areas, \"the presumption in favor of a simple law seemsvery slender\" [1891, 3181. \"We must not say that phenomena are perfectly regular, butthat their degree of regularity is very high indeed\" (unidentified fragment, 1976, xvi);\"The regularity of the universe cannot reasonably be supposed to be perfect\" (\"Sketch ofa New Philosophy,\" 1976,376). Peirce offers an evolutionary account of laws that explic-itly rejects the assumption that the ultimate laws of nature are simple and hold withoutexception: This supposes them [laws of nature] not to be absolute, not to be obeyed precisely. It makes an element of indeterminacy, spontaneity, or absolute chance in nature. Just as, when we attempt to verify any physical law, we find our observations cannot be precisely satisfied by it, and rightly attribute the discrepancy to errors of observation, so we must suppose far more minute discrepancies to exist owing to the imperfect cogency of the law itself, to a certain swerving of the facts from any definite formula. [1891, 3181In sum, \"There are very few rules in natural science, if there are any at all, that will bearbeing extended to the most extreme cases\" [1976, 1581. Mathematics, however, abstracts

362 Daniel Bonevacaway from complications, using principles that hold absolutely of idealized states of af-fairs. Since idealization or diagrammatization involves two components, the constructionof a pure hypothesis or ideal state of affairs and the derivation of consequences from it,so mathematics can be defined in two ways, as the science \"of drawing necessary conclu-sions\" or \"as the study of hypothetical states of things\" (1902, 141). The latter conceptionis essentially fictionalist. They are equivalent, Peirce sometimes maintains, because thefact \"that mathematics deals exclusively with hypothetical states of things, and asserts nomatter of fact whatever\" alone explains \"the necessity of its conclusions\" 11902, 1401. Peirce's account of the nature of mathematics comprises, then, the following theses: 1. The real world is too complicated to be described correctly by strictly universal principles. 2. At best, laws governing the real world hold ceteris paribus or in the absence of complicating factors. 3. In applying mathematics to the real world, we (a) build idealized models that abstract away from many features of the world but highlight others; (b) deduce consequences of assumptions using the 'pure hypotheses,' i.e., strictly universal principles holding in those models; and (c) use those consequences to derive consequences for the real world. On this Peircean theory, which I will term representationalJictionalism,the applicabil-ity of mathematics to the real world is no mystery; mathematical theories are designed tobe applicable to the world. Mathematics is the science of constructing idealized modelsinto which aspects of the real world can be embedded and using rules to derive conse-quences from the models that, ceteris paribus vel absentibus, apply to the real world. Thatmeans that measurement and, more generally, the representability of features of the worldin mathematics is essential to mathematical activity. There is no need to worry about theinterpretability of mathematics in concrete terms; the issue is the interpretability of ourtheories of the concrete in mathematical terms. In Peirce's terms, \"all [combinations]that occur in the real world also occur in the ideal world.. .. [Tlhe sensible world is but afragment of the ideal world\" (1897, 146). Consequently, \"There is no science whateverto which is not attached an application of mathematics\" [1902, CP 1121. There are, in first-order languages, at least, two equivalent ways of thinking about theinterpretability of theories of the real world in mathematics. We might interpret theo-ries of the concrete in mathematics by translating nonmathematical language into math-ematical language. More naturally, we might represent nonmathematical objects and re-lations mathematically by mapping them into mathematical objects and relations. Weapply mathematics by embedding nonmathematical structures into mathematical ones. Itis natural to identify mathematical theories, therefore, by the structures (e.g., the naturalnumbers, the integers, the reals) they describe, and to think of mathematical theories asdescribing intended models. It is likewise natural to think of mathematics as a science of

Fictionalism 363structure or patterns; to apply mathematics to a real-world problem is to find a mathemat-ical structure that encompasses the relevant structure of the real-world situation. One may think of mathematics, consequently, as a universal container -a body of the-ory into which any actual or even possible concrete structure could be embedded. Theremay be a single mathematical theory -set theory or category theory, perhaps -generalenough to serve by itself as a universal container. But this is not essential to the Peirceantheory. What matters is that mathematics collectively be able to play this role. The moregeneral a mathematical theory is, of course, the more useful it is, and the more basic itcan be taken to be within the overall organization of mathematics. Because mathematics strives to be a universal container in which any concrete structurecan be embedded, mathematical theories tend to have infinite domains. This accounts forthe dictum that mathematics is the science of the infinite. But mathematics does not studythe infinite for its own sake; it studies the infinite because it studies structures into whichother finite and infinite structures can be embedded. The application of mathematics does not require that it be true. In some ways, Peirce'saccount of mathematics and its application presages Field's (1980) account. But there areimportant differences. For Peirce, there is no reason to assume that the real world can bedescribed in nonmathematical terms. We map certain structures into mathematical struc-tures so readily that we may lack any nonmathematical language for describing them.Also, for Peirce, mathematics need not be conservative; it may be possible to use a math-ematical model to derive conclusions concerning a real-world problem that are false inthe real world. This may happen because the mathematical theory contains structure thatgoes beyond the structure of the corresponding situation in the real world. The densityand continuity of the real line, for example, may permit us to obtain conclusions about anactual concrete line segment that are false. It may also happen because of the complicatedcharacter of the real world. The real world is unruly, but the 'ideal world' into which it isembedded is rule-governed. Inevitably, some things that hold of the idealized model willnot hold in the real world. That is why a ceteris paribus proviso is needed.3.4 Hans Vaihinger: Free-range FictionalismFrom a very different epistemological standpoint is the fictionalism of Hans Vaihinger,who elaborated his particular form of Kantianism (called \"Positivistic Idealism\" or \"Ide-alistic Positivism\") in The Philosophy of As-If, the chief thesis of which is that \"'Asif', i.e. appearance, the consciously-false, plays an enormous part in science, in world-philosophies and in life\" (xli). Vaihinger seems to think of fictionalism in attitudinalterms; he holds that \"we operate intentionally with consciously false ideas.\" \"Fictions,\"he maintains, \"are known to be false\" but \"are employed because of their utility\" (xlii).Fictions are artifices, products of human creative activity and as such \"mental structures\"(12). They act as \"accessory structures\" helping human beings make sense of \"a worldof contradictory sensations,\" \"a hostile external world.\" Vaihinger, inspired by Schiller'sphrase, \"In error only is there life,\" finds fictions throughout our mental life, and interpretsKant's Critique of Pure Reason as showing that many of the concepts of metaphysics andethics, not to mention ordinary life, are fictional. The same is true of mathematics.

364 Daniel Bonevac How do we identify something as a fiction, particularly when there is disagreementabout the attitude one ought to take toward it? The paradigms of a demonstration offictional status, for Vaihinger, are Kant's antinomies of pure reason. In short, the mark offiction is contradiction. Reality is entirely consistent; anything that successfully describesit must be consistent. But fiction need not obey any such constraint. Of course, a writermay and typically does seek consistency in a work of fiction, the better to describe aworld that could be real. But consistency is inessential, and, in philosophically interestingcases, highly unusual. We resort to fictions to make sense of an otherwise contradictoryexperience, and characterize them to meet contradictory goals in generally contradictoryways. In the case of mathematics, specifically, The fundamental concepts of mathematics are space, or more precisely empty space, empty time, point, line, surface, or more precisely points without ex- tension, lines without breadth, surfaces without depth, spaces without con- tent. All these concepts are contradictory fictions, mathematics being based upon an entirely imaginary foundation, indeed upon contradictions. (51) Most fictionalists have thought of consistency as the only real constraint on fiction and,so, on mathematical existence. But Vaihinger, impressed perhaps with the inconsistenciesin the theories of infinitesimals, infinite series, functions, and even negative numbers thatprompted the great rigorization projects of Cauchy, Dedekind, Peano, and others in thenineteenth century, and certainly taking as an exemplar the theory of limits, held that The frank acknowledgment of these fundamental contradictions has become absolutely essential for mathematical progress. (51 )The point, from his perspective, is not to recognize contradictions in order to get rid ofthem but rather to understand the fictional nature of the objects supposedly being de-scribed. There is therefore no object in trying to argue away the blatant contradictions inherent in this concept [of pure absolute space]. To be a true fiction, the concept of space should be self-contradictory. Anyone who desires to \"free\" the concept of space from these contradictions, would deprive it of its char- acteristic qualities, that is to say, of the honour of serving as an ideal example of a true and justified fiction. (233)In addition to the items already listed, Vaihinger counts as contradictory theses that acircle is an ellipse with a zero focus; that a circle is an infinite-sided polygon; and thata line consists of points. He would no doubt have taken Russell's paradox, the Burali-Forti paradox, Richard's paradox, and so on as lending strong support to his thesis. (ThePhilosophy ofAs-lf was published in 1911, but he wrote Part I, in which he developed hisfictionalism, in 1877.) Mathematics is an instance of the method of abstract generalization, \"one of the mostbrilliant devices of thought\" ( 5 3 , but one which easily generates contradictions and in

Fictionalism 365general produces fictions. \"The objects of mathematics are artificial preparations, arti-ficial structures, fictional abstractions, abstract fictions\" (233); \"they are contradictoryconstructs, a nothing that is nevertheless conceived as a something, a something that isalready passing over into a nothing. And yet just these contradictory constructs, thesefictional entities, are the indispensable bases of mathematical thought\" (234). It is tempting to see Vaihinger's idealistic fictionalism as quaint, a product of nineteenth-century German idealism that has little relevance to contemporary discussions of fiction-alism. As Vaihinger's analysis of the Leibniz-Clarke correspondence shows, however,that is not so. The fact that we might imagine everything in space displaced some dis-tance to the right, or everything in time displaced some period into the future or past,but find the results of those thought experiments indistinguishable from the actual stateof affairs shows, according to Vaihinger, that absolute space and time are fictions. Thecontradiction here, like the contradiction he finds in the idea of an extensionless point, alimit, and infinitely large or infinitely small quantity, etc., is not strictly speaking logicalbut epistemological. There is no contradiction in the idea of everything being displacedtwo feet to the right, or everything have been created two minutes before it actually was;but such hypotheses also seem pointless, for there would be no way to tell whether theywere true or not. Vaihinger does not elaborate the exact nature of the contradiction thissituation entails. On one interpretation, absolute space and time imply the possibility of different statesof affairs that are in principle (and not merely as a result of limitations of our own cog-nitive capacities) not empirically distinguishable. The contradiction is thus not reallyself-contradiction, though Vaihinger sometimes describes it in those terms. It is a con-tradiction with the positivistic part of his positivistic idealism. But that again describesit too narrowly, for the contradiction is with a thesis that can seem appealing for reasonsindependent of positivism, namely, that the objects a theory postulates ought to be in prin-ciple empirically scrutable. The objects of mathematics are not logically incoherent butepistemically incoherent; their inaccessibility makes them philosophically objectionableeven if practically indispensable. On another interpretation, the contradiction stems from our inability to distinguish iso-morphic structures. It is characteristic not only of mathematics but of all discourse, Vai-hinger believes, that isomorphic structures are indiscernible. Any concept, proposition, orentity that depends for its sense on the discernibility of isomorphic structures is fictional;its contradictory character lies in its presumption of the discernibility of indiscernibles.Vaihinger's view, so construed, bears an interesting relation to contemporary structuralistaccounts of mathematics. The structuralist, from his perspective, is essentially correct,but misses part of the story. Mathematics, properly understood, is structuralist in thesense that its objects can be nothing more than roles in a certain kind of structure. Butit purports to be something else; it presents its objects as if they were substances anal-ogous to concrete objects in the causal order. The argument of Benacerraf 1965, thatmathematics can characterize its objects only up to isomorphism even within the realm ofmathematics itself, Vaihinger would no doubt take as demonstrating the fictional natureof mathematics.

Daniel Bonevac 4 SCIENCE WITHOUT NUMBERSHartry Field single-handedly revived the fictionalist tradition in the philosophy of math-ematics in Science Without Numbers (Field 1980). Field begins with the question of theapplicability of mathematics to the physical world, something about which most earlierversions of fictionalism had little to say. (Peirce's representational fictionalism is an ob-vious exception.) Field offers an extended argument that \"it is not necessary to assumethat the mathematics that is applied is true, it is necessary to assume little more than thatmathematics is consistent\" (vii). Since \"no part of mathematics is true ...no entities haveto be postulated to account for mathematical truth, and the problem of accounting for theknowledge of mathematical truths vanishes\" (viii). Field's fictionalism is plainly of the instrumentalist variety; mathematics is an instru-ment for drawing nominalistically acceptable conclusions from nominalistically accept-able premises. He shows that, for mathematics to be able to perform this task, it need notbe true. It must, however, be conservative: Anything nominalistic that is provable fromra nominalistic theory Aw,itwh htheerehMelpisofammaaththeemmaatitcicsalisthalesoorypraonvdabrleawndithAoumtaikt.e Roughly, no com- u M A B r t=mitment to mathematical entities. Usually, a purely nominalistic proof would be far lessefficient than a platonistic proof. Mathematics is thus practically useful, and perhaps evenheuristically indispensable, since we might never think of certain connections if confinedto a purely nominalistic language. But mathematics is theoretically dispensable; anythingwe can do with it can be done without it. Field gives a powerful argument for the conservativeness of mathematics, though thereis a limitation that points in the direction of representational fictionalism. (For an ex-cellent discussion of the technical aspects of Field's work, see [Urquhart, 19901.) LetZFUvcT)be Zermelo-Fraenkel set theory with urelements, with the vocabulary of a first-order, nominalistic physical theory T appearing in instances of the comprehension axiomschema. Let T* be T with all quantifiers restricted to nonmathematical entities. If TcZTZtiavFhFneeUinbcsV,oecin(fomTsnZi+)ossFtdieTseUntl*eceVnydist(,,rwisZn)+oitFte,hNrbiipfL*nrZCe+ZtFoFa-bAn,isl(t*eNhceio*innsn+ZsiZniTFsFctAUoeI-nn*vst)C(,i.srZoHt)e,Fnein(nUntTdcv-)ee(e,(rdt)Nth,h+e*iantTc+oZi*sTnF,isAs.iifIs*cntZoectnenFarsncUpiynsrVotesett(tnaarbtb)t.+eeilmLmiNteeyo*tndetTte-sfltAoeabdbre*lT-iiNns)h,ZuetahFsn{e,rdlneaA,lnsai}do-f.so N* t A*. On the assumption that ZFU is strong enough to model any mathematicaltheory that might usefully be applied to the physical world, this yields the conclusion thatmathematics is conservative. Mathematics is conservative, that is, with respect to theoriesthat are interpretable within it. Field treats that assumption as safe, given set theory's foundational status in mathemat-ics. It is not, however, unassailable; second-order set theory and ZFU + Con(ZFU) aretwo theories that cannot be modeled in ZFU. Second-order set theory, unlike ZF, is cat-egorical; all its models are isomorphic, which means that the continuum hypothesis, forexample, is determinately true or false within it. (We do not know which.) Perhaps noth-ing physical will ever turn on the truth of the continuum hypothesis. We know, however,that whether certain empirical testing strategies are optimal depends on the continuum hy-

Fictionalism 367+pothesis [Juhl, 19951, so empirical consequences should not be ruled out. Similarly, ZFU Con(ZFU) might seem to add nothing of physical relevance to ZFU. But there may wellbe undecidable sentences of ZFU that do not involve coding but have real mathematicaland even physical significance, just as there are undecidable sentences of arithmetic withreal mathematical significance (Paris and Harrington 1977). The difficult part of Field's argument lies in showing that we do not need mathemat-ics to state physical theories in the first place. Given nominalistic premises, we can usemathematics without guilt in deriving nominalistic conclusions. But why should we haveconfidence that we can express everything we want to say in nominalistically accept-able form? Malament [I9821 argues, for example, that Field's methods cannot apply toquantum field theory. It is hard to evaluate that allegation without making a full-blownattempt to rewrite quantum field theory in nominalistically acceptable language (but see[Balaguer, 1996; 19981). There are similar problems involving relativity; the coordi-nate systems of Riemannian and differential geometries cannot be represented by bench-mark points as Euclidean geometry can, and such geometries have not been formalizedas Hilbert, Tarski, and others have formalized Euclidean geometry [Burgess and Rosen,1997, 117-1 181. Even if all of current science could be so rewritten, however, there seemsto be no guarantee that the next scientific theory will submit to the same treatment (see[Friedman, 1981; Burgess, 1983; 1991;Horgan, 1984; Resnik, 1985; Sober, 19931). Con-sequently, as impressive as Field's rewriting of Newtonian gravitation theory is -and itis impressive -it is hard to know how much confidence one should have in the generalstrategy without a recipe for rewriting scientific theories in general. There is an obverse worry as well. Modem theories of definition (as in, for example,[Suppes 19711) generally have criteria of eliminability and noncreativity or, in Field'slanguage, conservativeness. That is, an expression is definable in a theory if (1) all oc-currences of it can be eliminated unambiguously without changing truth values and if(2) one cannot prove anything in the remainder of the language by using the expres-sion that one could not also prove without it. For an n-ary predicate R, these condi-tions hold if and only if the theory contains a universalized biconditional of the formVxl ...x,(Rxl ...x, ++ A(xl ...x,)), where A is an expression with n free variables that doesnot contain R. Now suppose that R is a mathematical predicate. Field's program requiresthat it be eliminable -in the strong sense of being replaceable salva veritate by nominal-istically acceptable expressions -and conservative. It seems to follow that R is definablein nonmathematical terms. Generalizing to all mathematical expressions, the worry is thatmathematics meets Field's conditions if and only if mathematics is reducible to nominal-istically acceptable theories. As Stanley [2001] puts the objection, \"a pretense analysisturns out to be just the method of paraphrase in disguise\" (44). Does Field's instrumentalist fictionalism collapse into a reductive fictionalism? To un-derstand what Field is doing, we must understand the ways in which his approach fallsshort of a reduction of mathematics to nonmathematical theories. The key is the express-ability of physical theories in nominalistically acceptable terms. In three respects, wemight interpret Field's eliminability requirement as weaker than that invoked in theoriesof definition. First, the eliminability need not be uniform. We must be able to write physical theories,

368 Daniel Bonevacfor example, in forms that make no use of mathematical language and make no commit-ment to mathematical objects. But that might fall short of a translation of a standardphysical theory into nominalistic language, for we might substitute different nominalisticlanguage for the same mathematical expression in different parts of the theory. Presum-ably we can specify the contexts in which the mathematical expression is replaced in agiven way. But we may not be able to express that specification within the language ofthe theory itself. Second, even if we were to have a reduction of a physical theory expressed mathemat-ically into a physical theory expressed nominalistically, we would not necessarily have areduction of mathematics to a nominalistic theory. We might, for example, know how toexpress the thought that momentum is the integral of force in nominalistic terms withoutbeing able to translate intgrul in any context whatever. Any definitions of mathematicalterms that emerge from this process, in short, might be contextual definitions, telling ushow to eliminate mathematical expressions in a given context without telling us how todefine them in isolation. For an analogy, consider Russell's theory of descriptions. Russell stresses that he givesus, not a definition of the,but instead a contextual definition of a description in the contextof a sentence. We can represent The F is G as 3xVy((Fy* x = y) & G x ) ,but we have notranslation of the or even the F in isolation. By using the lambda calculus, we can providesuch definitions; we might define the F, for example, as RG3xVy((Fy * x = y) & G x )andthe as RFRG3xVy((Fy * x = y) & G x ) . But such definitions are not expressible in theoriginal first-order language. Moreover, an analogous strategy for mathematics does notseem particularly plausible. It may be true that integration is a relation that holds betweenmomentum and force, for example, but it is hardly the only such relation, or even the onlysuch mathematical relation. Even if it were, characterizing it in those terms would not bevery helpful in eliminating integration from theories about work or electrical force, not tomention volume or aggregate demand. Third, given the potentially contextual nature of nominalistic rewritings of physicaltheories, the best we might hope for within our original language is a translation of a math-ematical expressions into universalized infinite disjunctions. The context dependence ofthe rewriting, in short, presents a problem similar to that of multiple realizability. Mathe-matical concepts are multiply realizable in physical theories, and we might not be able todo better than to devise an infinite disjunction expressing the possible realizations. Thatwould yield a reduction of mathematics not to a nominalistically acceptable theory butinstead to an infinitary extension of that theory. As I have argued elsewhere [1991; 199.51,this would be equivalent to showing that mathematics supervenes on nominalistically ac-ceptable theories. At most, then, Field's fictionalism commits him to the claim that mathematics super-venes on theories of the concrete. That would be enough, however, to cast doubt on hisclaim to be advancing a version of fictionalism, for it would commit him to the claim thatmathematics is true and moreover true by virtue of exactly what makes concrete truthstrue. To evaluate this objection, we need to characterize Field's method more precisely. Field[1980, 89-90], presents his method as including the following steps:

Fictionalism 3691. We begin with a physical theory T expressed using mathematics. We define a nom- inalistic aisxieoqmuivsyalsetenmt toSreaqnuyirminogdethlaotfSwrheidcuhceistohoZmt)o(Rm4o)r:pShi<caZllyh(eRm4b).eddable in @. This2. We expand S to S f > S by adding statements that express a nominalistic physical theory in a language making +noMcokmTmaitnmdeTnts+tMo mkatShfe,mwahtiecrael entities. This ex- panded theory is such that S' M > Zt)(@). If S' and T are both finitely axiomatizable by, say, &S' and & T , then ,M b & S f H & T .3. LTe+t AMbe a AnomainaSli'st+icaMllybstaAtabale physical truth. By the conservativeness of M, S' k A. Any nominalistically statable truth that follows from ordinary physical theory follows from a nominalistically stated theory. So, \"mathematical entities are theoretically dispensable\" [Field, 1980,901.The crucial step here is the second. Not only is it the difficult step from a technical point ofview, the topic of Field's central chapters; it is the point at which a prospective nominalistis likely to become faint of heart. What justifies confidence that such an S' exists? To answer this question, we need to examine Field's method in those central chapters.He begins, addressing step (I), by using Hilbert's axiomatization of geometry to providea theory of space-time interpretable in Zt)(@). Hilbert's approach proceeds by way ofrepresentation theorems, which show that the structure of phenomena under certain oper-ations and relations is the same as the structure of numbers or other mathematical objectsunder corresponding mathematical operations and relations. A representation theorem fortheories T and T' shows, in general, that any model of T can be embedded in a modelof T'. (Generally, we would want to show in addition that the embedding is unique orat any rate invariant under conditions, something Field proceeds to do.) A representationtheorem for T and T' thus establishes that T 5 T'. Throughout his treatment of Newto-nian gravitation theory -that is, throughout his treatment of step (2) -Field employsthe same method. He seeks \"a statement that can be left-hand side of the representationtheorem\" (71), where the right-hand side is given by the mathematically expressed phys-ical theory. In short, he seeks a nominalistic theory whose models are embeddable intosmToo,+WdineMhlssatbtoe,fpShto(hf2we.),esvtwaener,dnjaueresdtdipfaiheysthsietchoaerlycthSlae'iomr5y.tThJau+tstSMafs.+, TinMhasttefpacT(tl?)s,uWwffeihcanetes,ettdohajauttshitsief,oyjruytshtSiefiIcelsaTiFm~ie)tlh(daR'st ~ ) ,claim that \"the nominalistic formulation of the physical theory in conjunction with stan-dard mathematics yields the usual platonistic formulation of the theory\" (go)? We mustchoose the nominalistic statements that expand S to S' in such a way that \"if these fur-ther nominalistic statements were true in the model then the usual platonistic formulation.. .would come out true\" (90). Our nominalistic statements must give \"the full invariantcontent\" of any physical law (60). And why should we believe that this can be done? Thethought, inspired by the theory of measurement (e.g., [Krantz et al., 1971]), is that theremust be intrinsic features of a physical domain by virtue of which it can be representedmathematically. Those intrinsic features can be expressed nominalistically. If the intrinsicfeatures of the objects -and, thus, the statements we build into S' -were insufficient to

370 Daniel Bonevacentail the mathematical description of them resulting from such representation, then theplatonistic physical theory of those objects would unjustifiably attribute to them a struc-ture they do not in fact have. So, any mathematical representation of physical featuresof objects must, if justifiable, be entailed by intrinsic features of those objects. But thatimplies that S' + M k T. We are now in a position to understand in what respect Field's strategy falls short ofestablishing the supervenience of mathematics on a theory of concreta. Nothing in theabove implies that M 5 S f or even that there is some nominalistically statable theoryS\" 2 S' such that M 5 S\". We do, nevertheless, get the result that M k &S' o &T if S'and T are finitely axiomatizable. Given mathematics, that is, we can demonstrate not onlythe reducibility of the nominalistic theory S' to our ordinary physical theory T , which isrequired for the representation theorem underlying the applicability of mathematics to therelevant physical phenomena, but also the equivalence of S' and T modulo our mathe-matical theory. To fall back on the account of reduction in Nagel [1961]:reducibility isequivalent to definability plus derivability. Since M k &S' t.t & T , we have derivability,but not definability. We can define the expressions of our nominalistic language in termsof the mathematical language of our standard physical theory, but not necessarily viceversa. None of this is surprising, given Field's outline of his method. But carrying it outoften gives rise to the temptation to think that we have definability as well. Consider,for example, his treatment of scalar quantities, as represented by a function T : I+ Rrepresenting temperature, gravitational potential, kinetic energy, or some other physicalquantity. Suppose that we have a bijection 4 : Ds H R4 and a representation function9 from a scalar quantity into an interval, each unique up to a class of transformations.Field observes that T = 9 0 4-'. He concludes (emphasis in original): \"This suggests thatlaws about T (e.g. that it obeys such and such a differential equation) could be restatedas laws about the interrelation of 4 and 9; and since q!~ and t,b are generated by the basic[nominalistic]predicates. ..it is natural to suppose that the laws about T could befurtherrestated in terms of these latter predicates alone\" (59-60). In practice, then, Field oftenoperates by translating mathematical into nonmathematical expressions. The result is nota wholesale reduction of mathematics to a theory of concreta. But it is a reduction offragments of mathematics employed in a physical theory to something nominalisticallyacceptable. Stewart Shapiro observes that, since Field provides a model of space-time isomorphicto R ~w,e can duplicate basic arithmetic within Field's theory of intrinsic relations amongpoints in space-time. But that allows us to duplicate the Godel construction and so devisea sentence in that theory that holds if and only if it is not provable in the theory. Within thetheory S of space-time, that is, we can construct a sentence G such that S t- G t,7 P r [ G ] ,where Pr is the space-time correlate of the provability predicate and [GI is the code of G.As usual, we can show that G is equivalent to Con(S),the consistency statement for S ,thus showing that S bL Con(S).So, some mathematical truths are not derivable from S. Godel's incompleteness theorems are widely recognized as having dealt a serious blowto Hilbert's program, which depended on being able to show the consistency of infinitarymathematics by finite means. Shapiro rightly recognizes the analogy between Hilbert's

Fictionalism 37 1program and Field's strategy of using the conservativeness of mathematics tojustify math-ematical reasoning. It is not clear, however, whether the analogy is strong enough to gen-erate a serious problem for Field. (For discussion, see [Shapiro, 1983a; 1983b; 1997;2000; Field, 1989; 19911.) It does follow, it seems, that Field cannot demonstrate theconservativeness of mathematics by strictly nominalistic reasoning. We need to employmathematics to prove its own conservativeness. But Field denies that this is troublesome,for he sees his project as a reductio of the assumption that mathematical reasoning isindispensable. Still, we can show in set theory that G and Con(S) are true, even though neither canbe demonstrated in Field's space-time theory. That seems to show that mathematics al-lows us to prove some truths about space-time that Field cannot capture. We might putthe point simply by saying that if space-time is isomorphic to R4,the theory of space-time is not axiomatizable. It is, like arithmetic, essentially incomplete. How heavily thiscounts against Field's program seems to depend on how adequately he can account for set-theoretic reasoning in metamathematics, something no one has investigated in any detail.For any illuminating discussion of the issues, see [Burgess and Rosen, 1997, 118-1231. Field's program has encountered other, less technical objections. An excellent sourcefor discussion of the issues is Irvine [1990], which contains discussions of Field's workby many of the leading figures in the philosophy of mathematics. Field [I9891 developsField's view further, partly in response to various criticisms. It diverges in complex waysfrom Field 1980 and will not be discussed in any depth here. Bob Hale and Crispin Wright [1988; 1990; 19921 observe that Field's theory, espe-cially as developed in Field 1989, takes consistency as a primitive notion, and raise twoobjections on that basis. First, both standard mathematical theories and their denials areconsistent and, as conservative, cannot be confirmed or disconfirmed directly. So, theyconclude, Field should be agnostic with respect to the existence of mathematical objects.Field [I9931 points out that this ignores his argument that mathematics is dispensable,which constitutes indirect evidence against the existence of mathematical objects. (Ifmathematics were indispensable in physics, that would constitute indirect evidence in fa-vor of their existence; Field accepts and, indeed, starts from the Quine [1951]-Putnam[I9711 indispensability argument. See Colyvan [2001].) Second, on Field's view, theexistence of mathematical objects is conceptually contingent. But what could it be con-tingent on? Hale and Wright argue that Field needs an answer. Without one, the claim ofconceptual contingency is not only empty but incoherent. Field [1989; 19931responds bydenying the principle that, for every contingency, we need an account of what it is contin-gent on. It is conceptually possible that God exists; it is conceptually possible that Goddoes not exist. But we have no account of what the existence of God might depend on, norcan we even imagine such an account. The same is true of many conceptual contingen-cies: the existence of immaterial minds, the n-dimensionality of space-time, the values offundamental physical constants, and the amount of matter in the universe, for example. Yablo [2001] raises three additional problems for Field: +The problem of real content: What are we asserting when we say that 2 2 = 4? According to Field, we are saying something false, since there are no objects standing in those relations. A fictionalist carrying out Field's program may be well

Daniel Bonevac aware of that. We might say that we are quasi-asserting that 2 + 2 = 4 without really asserting it. Is there anything we are really asserting? I take it that Field's answer is no. We do not have to be viewed as asserting mathematical statements at all. What we are doing is closer to supposing them. It is not clear why this generates a problem, or why we must be asserting anything at all. +The problem of correctness: I assert that 2 2 is 4, not 5, even though there may be a consistent and thus conservative theory of the numbers according to which 2 + 2 = 5. (Let 4 and 5 switch places in the natural number sequence, for example. On some theories of mathematics, this suggestion makes no sense. But I take it that Field's is not one of them.) How can I distinguish correct from incorrect assertions about the numbers? If the purpose of a mathematical theory, however, is to characterize up to isomorphism a model into which we can embed aspects of reality, or even a class of such models, we can distinguish correct from incorrect characterization of that model, even if there is some arbitrariness about which model we use. The problem of pragmatism: Fictionalists seem to assert sentences, put forward evidence for them, attempt to prove them, get upset when people deny them, and so on - all of which normally accompany belief. How, then, does the fictional- ist's attitude toward mathematical utterances fall short of belief? This raises the complex issue of trying to delineate an account of acceptance such as that of Bas van Fraassen [1980]; see Horwich [1991]. Placing this in the context of success in mathematical discourse, however, may make the problem easier. We need an ac- count of what constitutes mathematical success. Arguably, the instrumentalist and representational aspects of Field's fictionalism provide a detailed answer.Yablo [2002] raises an additional objection. Mathematics, if true at all, seems to be truenecessarily. Most properties of mathematical objects seem to be necessary. Being odd,for example, seems to be a necessary property of 3. The relations between mathemati-cal objects - that rr > 3, for example, or, in set theory, that Vx 0 E p(x) - appear tohold necessarily. Field thinks he has an explanation: the conservativeness of mathematicsentails the applicability of mathematics in any physical circumstance. Conservativeness,he quips, is \"necessary truth without the truth\" [1989, 2421. Yablo objects to this expla-nation. First, any physical circumstance has a correlate with numbers; any circumstancewith numbers has a correlate without them. If the first leads us to treat mathematics asnecessary, why doesn't the second lead us to treat it as impossible? This asymmetry,however, seems easy to explain. The conservativeness of mathematics implies that, inany physical circumstance, it is safe to assume mathematics and use it in reasoning aboutphysical situations and events. We cannot conceive of a situation in which mathematicalreasoning fails. That any circumstance with numbers has a correlate without them seemsto have no corresponding implication. Second, mathematical truths seem necessary on their own. For Field, however, math-ematical theories are conservative. That rr > 3, however, seems necessary simpliciter,not merely relative to a theory. It is hard to evaluate this objection, since we learn aboutmathematical objects in the context of a theory; rr > 3 is a necessary truth of arithmetic.

Fictionalism 373 Third, inconsistent statements can both be conservative. But they cannot both be nec-essarily true. Both the axiom of choice and its negation are conservative over physics,presumably, but they cannot both be necessary (unless they are taken as holding of dif-ferent parts of the domain of abstracta, as in the Full-Blooded Platonism of Balaguer[1998]). But this seems to relativize mathematics. It conflicts with our sense that n > 3,period, not merely relative to a certain quite specific mathematical theory -that is, notjust relative to set theory, say, but relative to ZFC. Again, however, it is not clear howmuch force this objection has against Field's view. If a class of theories such as variantsof set theory are all conservative over physics, and there is no other basis for choosingamong them, it seems plausible to say of the sentences (e.g., the axiom of choice or thecontinuum hypothesis) on which they disagree not that they are necessarily true but thatthey are neither true nor false. Imagine a work of fiction in which there are variant read-ings in various manuscripts. (This is actually the case with The Canterbury Tales, forexample; there are about eighty different versions.) A supervaluation over the variantsseems the only reasonable po!icy. This may complicate Field's picture slightly, but onlyslightly; it explains our sense of necessity with respect to those sentences on which theappropriate conservative theories agree while also explaining our unwillingness to assertor deny those on which they disagree. 5 BALAGUER'S FICTIONALISMMark Balaguer [I9981develops versions of platonism and fictionalism bearing some affin-ity with Vaihinger's free-range fictionalism. He does so to show that \"platonism andanti-platonism are both perfectly workable philosophies of mathematics\" ( 4 ,emphasis inoriginal). Platonists might feel vindicated, since they see the burden of proof as being onthe anti-realist; anti-realists might conclude that they were right all along to insist that acommitment to abstracta is unnecessary. Balaguer himself, however, concludes that thereis no fact of the matter. To see what Balaguer takes fictionalism to accomplish, it is useful to begin with hispreferred version of platonism, Full-Blooded Platonism (FBP). FBP's central idea is that\"all possible mathematical objects exist\" ( 5 ) . For abstract objects, possibility sufficesfor existence. He observes that this solves many of the philosophical problems facedby platonism. How is it possible, for example, to know anything about mathematicalentities? It's easy; any consistent set of axioms describes a realm of abstract entities. Nocausal contact, empirically scrutable relationship, or explanation of how knowledge is notcoincidental is required. FBP is committed to abstract entities, but the entities themselvesplay no role in our explanation of our knowledge of them. How is it possible to refer to aparticular entity? Again, it's easy (at least up to isomorphism); we simply need to makeclear which kind of abstract entity we have in mind. Benacerraf [I9651 points out thatnumbers are highly indeterminate, and seem to be nothing but places in an w-sequence.Balaguer finds this easy to explain; in characterizing a kind of abstract object, we do notspeak of a unique collection of objects. Since any possible collection falling under thekind exists, the theory speaks of all of them. Fictionalism shares the advantages of FBP. Just as authors may tell stories however they

374 Daniel Bonevaclike, mathematicians may characterize mathematical domains however they like, providedthe characterization they give is consistent. Any fiction gives rise to a collection of fic-tional characters. Fictionalism is thus full-blooded in just the way FBP is, but withoutFBP's commitment to abstracta. Shapiro [2000] complains that consistency is itself a mathematical notion; Balaguer'sproject is in danger of circularity. Balaguer responds that his concept of consistencyis primitive. So long as consistency can be understood pre-mathematically - whichseems plausible, since it seems a logical rather than specifically mathematical notion,although theories of logical relationships of course use mathematics -Balaguer's projectavoids circularity. An alternative response would be to free both fiction and mathematicsfrom the constraint of consistency, as Vaihinger does, perhaps employing a paraconsistentlogic as the underlying logic of mathematics. Given the continuing utility of inconsistentmathematical theories such as naive set theory, this option may have other attractions aswell. See, for example, [Routley and Routley, 1972; Meyer, 1976; Meyer and Mortensen,1984; Mortensen, 1995; Priest, 1994; 1996; 1997; 20001. Balaguer's preferred version of fictionalism is Field's, which may seem disappoint-ing, since it is highly constrained by the need to nominalize scientific theories and thusquite different in spirit from FBP. But he develops the underlying picture in a pragma-tist fashion reminiscent of Peirce, bringing out more fully the representational aspects ofField's fictionalism. His key premise: \"Empirical theories use mathematical-object talkonly in order to construct theoretical apparatuses (or descriptive frameworks) in whichto make assertions about the physical world\" (137). We do not deduce features of thephysical world from features of mathematical objects alone; we understand the structureof physical states of affairs in terms of related mathematical structures. That is to say, wemodel features of the physical world mathematically, mapping intrinsically physical fea-tures into mathematical models and using our knowledge of those models to infer featuresof the physical world. The reductionist thus has the picture exactly backwards. We should think of mathe-matics not as something that reduces to the nonmathematical but as something, by design,to which the nonmathematical reduces. Galileo famously defined mathematics as the lan-guage in which God has written the universe. From the pragmatist point of view, this isnot so remarkable; we create mathematics to be a language in which the relationships wefind in the physical universe can be expressed. Mathematics is a container into which anyphysical structure might be poured, a coordinate system on which any physical structuremight be mapped. This is why mathematical domains are typically infinite. Field as-sumes a physical infinity, an infinite collection of space-time points. But, if mathematicsis designed to be a \"universal solvent,\" a theory to which any theory of physical phenom-ena reduces, and if there is no known limit to the size of physical structures, we needmathematical domains to be infinite if they are to serve their purpose reliably. 6 YABLO'S FIGURALISMYablo [2001; 2002; 20051 develops an alternative to Field's fictionalism which he refers toas figuralism or, sometimes, \"Kantian logicism.\" Yablo draws an analogy with figurative

Fictionalism 375speech. \"The number of F s is large iff there are many Fs,\" for example, he likens to \"yourmarital status changes iff you get married or ...,\" \"your identity is secret iff no one knowswho you are,'' and \"your prospects improve iff it becomes likelier that you will succeed.\"In every case, a figure of speech quantifies over an entity we do not need to take to bea real constituent of the world. We speak of marital status, identity, prospects, stomachbutterflies, pangs of conscience, and the like not by describing a distinct realm of objectsbut instead describing a familiar realm in figurative ways. The unusual entities serve asrepresentational aids. The point of the figurative discourse is not to describe them butto use them to describe other things. They may describe them truly; on this view, thefictional character of the figurative description does not contradict its truth.Mathematical discourse similarly invokes a realm of specifically mathematical entities,typically using them as representational aids to describe nonmathematical entities. We usestatements about numbers, for example, to say things about objects; \"the number of as-teroids is greater than the number of planets\" shuoclhdsasif2an+d2on=ly4,ifetxhperreessarleogmicoarel asteroidsthan planets. Statements of pure arithmetic, truths (inthis case, that (3zxFx& 3zyGy & -3z(Fz&Gz)) -+ &(FxVGx); see, e.g., Hodes (1984)).Statements of pure set theory are logically true over concrete combinatorics, that is, areentailed by basic facts (for example, identity and distinctness facts) about concrete ob-jects. Figuralism thus explains why mathematics is true necessarily and a priori. It alsoexplains why mathematics is absolute rather than relative to a particular theory (until, atany rate, one reaches the frontiers of set theory).Yablo refers to his figuralist view as a kind of fictionalism, specifically, relative re-flexive fictionalism. It is reflexive to recognize that fictional entities can function in twoways, as representational aids (in applied mathematics, for example) or as things repre-sented (e.g., in pure mathematics). They may even, in self-applied contexts, function inboth ways, as when a fictionalist, speaking ontologically, says that the number of evenprimes is zero (on the ground that there are no numbers, and afortiori no even primes). This example brings out the need for a relative reflexive fictionalism. The relativityis not to a particular mathematical theory but to a perspective. Rudolf Carnap [I9511would draw the contrast as one between internal and external questions. Internally, thenumber of even primes is one; externally, by the fictionalist's lights, that number is zero.Yablo draws it in terms of engaged and disengaged speech. However the distinction is tobe drawn, we must distinguish the perspective of those engaged in the relevant languagegame from the perspective of those talking about the game.Stanley [2001], directing himself primarily at Yablo's theory, argues that \"hermeneuticfictionalism is not a viable strategy in ontology\" (36). He advances five objections againstYablo's fictionalism and against hermeneutic fictionalism as such. First, a fictionalistaccount does not respect compositionality: \"there is no systematic relationship betweenmany kinds of sentences and their real-world truth conditions\" (41). Second, we moverapidly from pretense to pretense; is there any systematic way of understanding how we doit? Third, pretense is an attitude that evidently must be inaccessible to the person engagedin it, since people do not generally think of themselves as approaching mathematics inan attitude of make-believe. Fourth, the claim that people have the same psychologicalattitude toward games of make-believe and mathematics is empirically implausible. Fifth,

376 Daniel Bonevacfictionalism splits the question of a speaker's believed ontological commitments fromwhat our best semantic theory postulates in the domains of the models it uses to interpretthe speaker's discourse. But what should interest us is the latter, which is the key toexplaining a speaker's actual commitments. The first two objections pertain to compositionality, and may well afflict certain ver-sions of fictionalism, including Yablo's. Yablo does not spell out any compositional wayof generating the real-world truth conditions of various mathematical sentences. It is hardto say whether he thinks compositionality is not necessary or whether he takes Frege,Russell, and others as having already spelled it out (with implications he might not like;see [Rosen, 1993]),though he seems, in response to Stanley's criticism, to deny the needfor compositionality in any strong sense. But hermeneutic fictionalism is after all a the-ory of what we mean in a certain realm of discourse. It does not seem unreasonable todemand that hermeneutic fictionalism meet the criteria we impose on any other semantictheory of what we mean. As Stanley interprets it, at any rate, hermeneutic fictionalism isby definition -as hermeneutic - a one-stage theory. It is a semantic theory intendedto characterize what sentences within a certain part of language mean and to show, inthe process, by virtue of that semantic theory, that those sentences lack the ontologicalcommitments they seem to have on the basis of an analogy with other kinds of discourse. From Yablo's perspective, this is a misconstrual. Faced with a sentence in a work offiction such as that from the Twice Told Tales, we do not use an alternative semantics; weinterpret the sentence as we always would, but recognize that it is not a literal descriptionof reality. In short, Yablo seems to intend his theory as a two-stage theory. The first stageis that of semantic interpretation, and it proceeds in standard fashion. The second in-volves a recognition that the context is fictional, which leads us to reinterpret the seemingontological commitments of the discourse. It is not clear that Yablo can escape the objection so easily. First, he may have to sac-rifice his claim to the \"hermeneutic\" moniker; a two-stage theory, arguably, is no longera theory of meaning, but a supplement to a theory of meaning. Second, most fiction doesnot seem to fit his analysis, for most fiction is not figurative. Hawthorne does not in anyobvious way invoke a gate made from a whale's jawbone as a representation of somethingelse. Third, and most seriously, a demand for compositionality is not out of place evenin the second stage of a two-stage theory. Reductionists have set out to accomplish theirontological aims by interpreting one theory in another in a fully recursive fashion. It isnot clear that an ad hoc interpretive scheme that cannot be cast in compositional formdeserves to be taken seriously. Does the demand for compositionality tell against other versions of fictionalism in thephilosophy of mathematics'? A theory like Field's or the one I sketch in the last sectionof this paper provides a compositional semantics for mathematical sentences; there is noasystematicity about how such sentences are to be interpreted. It seems plausible, more-over, that mathematics constitutes a language game of its own - consider, for example,learning to count -and fictionalists such as Yablo can explain why it is in fact easy forus to switch into and out of that game. The second two objections address questions of psychological attitude, and, again, mayapply to hermeneutic versions of fictionalism, particularly if one defines fictionalism in

Fictionalism 377terms of an attitude of pretense or make-believe. A deflationary fictionalist, however,advances a view that remains independent of the attitude a speaker takes toward his or herdiscourse. These objections thus miss deflationary fictionalism entirely. Finally, the fifth objection is straightforwardly metaphysical, and raises an interestingissue about the significance of semantic theory for ontology. It tells against one-stagetheories, but not against their two-stage counterparts. Any two-stage theorist must drawa distinction between two senses of ontological commitment. Semantic theory is a goodguide to our ostensible commitments, the commitments we prima facie seem to have.That is Stage I; that is where ontology starts. But it is not where it ends. Semantic theoryis not a reliable guide to our real commitments, the commitments we are bound to em-brace once, at Stage 11, the relations among various theories and discourses is taken intoaccount. Someone who reduces everyday talk of medium-size physical objects to talkof atomic simples, microparticles, sense data, or object-shaped gunk avoids real com-mitment to ordinary objects as a distinct ontological kind. The deflationary fictionalistsimilarly avoids real commitment to objects to which he or she is ostensibly committedby a discourse that can successfully be interpreted as fictional. 7 SEMANTIC STRATEGIESThe fictionalist's goal is to resolve Benacerraf's dilemma by breaking the chain of rea-soning that leads from the success of mathematics to its truth and then to the existenceof mathematical objects. Most fictionalists have chosen to break the chain at the firstlink, denying that mathematics is true. But it is also possible to question the second link,granting that mathematics is true while denying that its truth requires countenancing dis-tinctively mathematical objects. Reductionism is of course one attempt to accomplishthis. Reinterpreting mathematics so that its existential sentences might be true withoutthe existence of abstract objects is another.7.1 Truth in a FictionThe simplest strategy is probably to think of each sentence of a fiction as preceded bya fiction operator F meaning something like \"it is true in the story that.\" It is harder tomake this thought precise, however, than one might initially suppose. There is an obviousworry about compositionality akin to that Dever [2004]raises against modal fictionalism.Since that raises some technical issues, however, and tracks debates concerning modalfictionalism quite closely -see, for example, [Brock, 1993; Nolan, 1997; 2005; Nolanand Hawthorne, 1996; Divers, 1999; Kim, 2002; 2005; Rosen, 1990; 1995; 20031 - letme concentrate on another problem. What is a story? How can we give a semantics for the F operator? In ordinary casesof fiction, this is not difficult to do; a story is a finite sequence of sentences. So, we+can understand Fp as having the truth condition S p (or equivalently, if S is first-order, & S + p). Notice, however, that this immediately collapses fictionalism intodeductivism. It analyzes the (fictional) truth of a statement as being the logical truth ofan associated conditional. (Actually, the story as articulated by the author also certainly

378 Daniel Bonevacneeds supplementation with frame axioms to derive what are intuitively consequencesof the story, but I will ignore this complication.) Mathematical stories (e.g., first-orderPeano arithmetic or ZF) are often infinite stories. In the context of a first-order language,or for that matter in any compact logic, Fp will still be true if and only if an associatedconditional is true, since S k p if and only if there is a finite So c S such that So b p. We might alternatively think of a story as a model or class of models: we could countFp as true if and only if, for every model M E K, M k p. But it would be hard todistinguish this version of fictionalism from platonism. How do we pick out the relevantmodel or model class? How do we know anything about it? Is our knowledge of it acoincidence? Any argument the fictionalist might use against platonism -even Yablo'scomparative advantage case -might be turned against this version of fictionalism. The only viable version of this simple semantic approach to fictionalism that remainstruly fictionalist seems to be to second-order. Assume that we tell our mathematical storiesin a second-order language. Since we can replace axiom schemata with higher-orderaxioms, we can safely assume that our stories are finite. We might then say that Fp holdsif and only if S b p, where S is the appropriate mathematical story and b is a second-order relation. This yields a theory like those of Hellman [1989], Shapiro [1997; 20001,and the later Field [1989]. Set aside the Quinean worry that the use of second-order logicsmuggles mathematics in through the back door. Strikingly, a second-order approach tooturns out to be a version of deductivism. It is not surprising that Hellman and Shapiro donot consider their views fictionalist.7.2 Constructive Free-range FictionalismThe obvious semantic route to fictionalism, then, in fact leads away from it. Is thereanother way of construing the semantic strategy? Reading quantification in mathematics (and perhaps not only in mathematics) as sub-stitutional rather than objectual, for example, might offer a way of accepting existentialsentences in mathematics as true without being forced to recognize the existence of num-bers, sets, functions, and other abstracta. I explored this possibility in a series of papers inthe 1980s [Bonevac, 1983; 1984a; 1984bl. While this seemed to some a \"wild strategy\"(Burgess and Rosen 1997), Kripke [I9761 had already cleared away the most serious ob-jections to interpreting quantifiers substitutionally. Burgess and Rosen nevertheless arguethat any strategy based on a nonstandard reading of the quantifiers is bound to fail, since\"'ontological commitment' is a technical term, introduced by a stipulative definition, ac-cording to which, nearly enough, ontological commitment just is that which ordinary lan-guage quantification, in regular and paradigmatic cases, expresses\" (204). It is far fromclear that this is correct. First, ontological considerations have figured in philosophicaldiscussion at least since the time of Plato; figures throughout philosophical history havedebated questions of ontological commitment without using those words. Second, Quine[1939; 1951; 19601treats his thesis that to be is to be a value of a variable as a substantiveclaim, not as a stipulative definition. Third, as Szabo [2003] emphasizes, it is not clearthat ordinary language existential expressions are univocal, a point already emphasized inParsons [l980].

Fictionalism 379 It may seem that a substitutional strategy nevertheless succeeds too easily and much toobroadly. If substitutional quantification avoids ontological commitment, and, as Benacer-raf's semantic continuity requirement seems to demand, ordinary language quantificationis substitutional, then it would seem that ordinary language quantification avoids ontolog-ical commitment, which is absurd, since commitment is defined in those very terms. But substitutional quantification does not avoid commitment; it transfers the ontolog-ical question to the level of atomic sentences. The strategy means not to avoid meta-physical questions or assume that nothing at all requires the existence of objects but onlyto shift metaphysical questions from quantified and specifically existential sentences toquantifier-free sentences and their truth conditions. On a substitutional approach, theinteresting metaphysical problem arises at the atomic level - why count those atomicsentences as true? -and there no longer seems to be any reason to assume that such aquestion must have a uniform answer that applies no matter what the atomic sentenceshappen to be about. In \"regular and paradigmatic cases\" ordinary language quantifica-tion expresses ontological commitment because, in such cases, the truth values of atomicsentences are determined in standard Tarskian fashion and so depend on the existence ofobjects. On my view, in short, not only is it true that ordinary quantification carries onto-logical commitment in paradigm cases, but there is an explanation of which cases countas paradigmatic and why. Nevertheless, a central idea behind this strategy is independent of substitutional quan-tification. Mathematical domains are generally infinite; one cannot assume that there areenough terms in the language to serve the purposes of a traditional substitutional account.One must therefore think in terms of extensions of the language, counting an existen-tial sentence true if it is possible to add a term to the language serving as a witness tothe sentence. That, however, introduces a modal element to the theory, which can beisolated from a substitutional interpretation of the quantifiers. Here I shall present therevised-semantics strategy in a form that emphasizes its modal character, relying in parton unpublished work that Hans Kamp and I did some years ago but setting aside its sub-stitutional features. Two sets of considerations in addition to the substitutional considerations just outlinedmotivate the revised semantics for quantification that I am about to present. The firstconcerns the mathematician's freedom to introduce existence assumptions, which seemsanalogous to the freedom of an author to introduce objects in a work of fiction. GeorgCantor [I8831 wrote that \"the very essence of mathematics is its freedom.\" David Hilbert[I9801 saw consistency as the only constraint on mathematical freedom: \"[Ilf the arbi-trarily given axioms do not contradict one another with all their consequences, then theyare true and the things defined by the axioms exist. This for me is the criterion of truthand existence.\" Henri PoincarC similarly maintained that \"A mathematical entity existsprovided there is no contradiction implied in its definition, either in itself, or with theproposition previously admitted\" [PoincarC, 19521; \"In mathematics the word exist canhave only one meaning; it signifies exemption from contradiction.\" This suggests that ex-istence, in mathematics as in fiction, is tied to possibility. Something of a certain kindexists if it is possible to find or construct a thing of that kind. The second motivation stems from thinking of mathematical objects and domains as

380 Daniel Bonevacconstructed. We might think of fictional entities as introduced, not all at once, but as awork of fiction (or a series of works of fiction, such as Doyle's Sherlock Holmes sto-ries or Whedon's B u f i the Vampire Slayer and Angel episodes) unfolds. Similarly, wemight think of mathematical objects and mathematical domains as constructed over timeor, more generally, over a series of creative mathematical acts. We can assert that a math-ematical entity of a kind exists if and only if it is possible to construct an object of thatkind. This might suggest that exists means something quite different in mathematics fromwhat it means in ordinary contexts. Poincare [1952], indeed, maintained that \"the word'existence' has not the same meaning when it refers to a mathematical entity as when itrefers to a material object.\" This flies in the face of Benacerraf's semantic continuity re-quirement, that language used in both mathematical and nonmathematical contexts mustbe given a uniform semantics covering both. What follows attempts to do just that: pro-vide a uniform semantics for mathematical and nonmathematical language that explainsthe difference in existence criteria not in terms of meaning but in terms of ontology -interms, that is, of the kinds of objects and models under consideration.7.3 DejinitionsThe semantics I shall present follows the pattern of Kripke semantics for intuitionisticlogic [Kripke, 196.51. A Kripke model ff for language L is a quadruple < K, 5 ,D, It>,where < K, 9 is a poset, D is a monotonic function from K to inhabited sets, calleddomains, and It is a relation from K to the set of atomic formulas of L' = L U D(K)(=,...{D(k) : k E K}) such that (a) k It Rn(d1...dn) d, E D(k) for I 5 j 5 n (existence), and(b) k IF Rn(d dn) and k 5 k' k' IF Rn(d1...dn) (persistence). We may extend It to all formulas by inductive clauses for compound formulas in sev-eral ways. One familiar method is intuitionistic: k ItiA & B o k ItiA and k ItiB k IFi A V B o k ItiA or k IFi B k ItiA + B a Vk' 2 k if k' ItiA then k' It, B k I F i ~ A o V k f > kWiA k Iti3xA(x) a 3d 6 D(k) k IF; A(d) k IFi VxA(x) G Vk' 2 kVd E D(kf) k' ItiA(d) Another is classical: k It,.A & B @ k It,.A and k IF,. B k It,.A v B o k It, A or k IF,. B k It,.A -+ B a k W,. A or k It,.B kt,7A@kWCA k It,.3xA(x) a 3 d E D(k) k Il-, A(d) k Ik,.VxA(x) a V d D~(k) klF,A(d) For now, however, I want to focus on another possibility. Suppose we take seriouslythe thought that mathematical objects, like fictional objects, are constructed -are charac-terized in stages of creative acts, as Kripke models for intuitionistic logic seem to reflect.

We might be tempted to adopt intuitionistic logic as that appropriate to mathematical andfictional reasoning alike. Yet there is something odd about intuitionistic logic in this con-nection, something that the thought of the objects of the domain as constructed does notitself justify or explain. Intuitionistic logic treats the quantifiers asymmetrically. It iswell-known that, in intuitionism, one cannot define the quantifiers as duals of each other.In fact, one could not d o so even if negation were given a classical rather than intuitionis-tic analysis. The Kripke semantics brings out the reason why: the universal quantifier isessentially forward-looking, taking into account future stages of construction, while theexistential quantifier is not. The existential quantifier, one might say, ranges over con-structed objects, while the universal quantifier ranges over constructible objects. Nothingabout taking the domain as constructed in stages seems to require that. Intuitionistic logic,in short, is not Quinean: to be cannot be construed as being the value of a variable, forbeing the value of a variable has no univocal meaning. Intuitionistic logic rests on two independent theses expressing quite different kinds ofconstructivism. One is the metaphysical thesis that the domain consists of constructedobjects; the other, the epistemological thesis that only constructive proofs can justifyexistence statements. The former seems compatible with fictionalism, and perhaps evento be entailed by it. The latter, however, has no obvious link to fictionalism, and in factseems inconsistent with the freedom of existential assertion that Cantor, PoincarC, Hilbert,and Balaguer have found central to mathematical practice. To be clear about this, we need to distinguish the perspective of the author of a workof fiction from the perspective of someone talking about the fiction -and, similarly, theperspective of the creative mathematician form the perspective of someone talking aboutthe mathematical theory. What justifies me in saying that there are vampires with souls inthe universe of Bufi the Vampire Slayer is Josh Whedon's construction or, better, stipula-tion of them. What justifies Josh Whedon in saying so is an entirely different matter, andseems to be nothing more than logical possibility. Just so, what justifies me in saying thatevery Banach space has a norm is Banach's stipulation, but what justified Stefan Banachin saying so is something else, and again seems to be nothing more than logical possi-bility. (Perhaps, as Vaihinger suggests, not even logical possibility is required; we coulddevelop a version of the theory using a paraconsistent logic, by, for example, thinking ofthe semantics as relating sentences to truth values.) In thinking of mathematics as in some sense fictional, it is important to focus on theperspective of the author or more generally creator of a fictional work rather than the per-spective of the reader or viewer. The latter perspectives are derivative. I am justified insaying that there are ensouled vampires in the Buffyverse by virtue of Whedon's stipu-lation of them. I am justified in saying that all Banach spaces have norms by virtue ofBanach's stipulation of them. In short, my epistemological perspective with respect tothe universe of BuRy or Banach spaces is derivative. It requires no epistemic contact withvampires or Banach spaces. Given the stipulations of Whedon and Banach, it seems fairlyeasy to explain my knowledge of the relevant domains; I have contact not with the objectsbut with the stipulations. The philosophically interesting question concerns their justification in making thosestipulations in the first place. If they, as the authors of the relevant fictions, were indeed

382 Daniel Bonevacfree, their acts of creation evidently required nothing like a constructive existence proof.One might be tempted to think that the author's act of creation is itself a construction ofthe kind demanded in intuitionism and other forms of constructive mathematics. But thatwould be a mistake. The author is not at all like \"the constructive mathematician [who]must be presented with an algorithm that constructs the object x before he will recognizethat x exists\" [Bridges and Richman, 19871. The author can stipulate whatever he or shepleases. For fictionalist purposes, then, it makes sense to isolate the thought that mathematicalobjects and domains are constructed -and the related thought that existence in mathe-matics amounts to constructibility -from the further thought that existence claims can bejustified only by the completion of certain kinds of constructions. Kripke semantics lendsitself naturally to constructivism in the metaphysical sense. To free it from an insistenceon constructive proof, however, and to bring it into line with Quine's understanding ofthe role of quantification as ranging over a domain, we might reasonably adapt the intu-itionistic truth clauses to treat the quantifiers symmetrically, as ranging over constructibleobjects: k It A&B ok It A and k It B kItAvBokII-AorkItB k I t A + B o Vk' 1 k i f k ' It A thenk' ll- B k I t ~ A o V k ' > kk F A k It 3xA(x) o 3k' 2 k 3 d E D ( k f ) k' II- A(d) k It- V x A ( x )o Vk' 2 kVd E D ( k f ) k' I t A ( d ) As usual, say that A is valid at k in f f o k It A, and that A is valid in f f o Vk E K k It-A (written f f II- A). C II- A o V f fi f B It- B for all B E C then B It A. A is valid (It-A ) iff0 I!- A. Let f f k be < Kt,<',D', Itf>, where K' = {k' : k' 2 k ) , <'=<k',ID' = D 1 K', andIt'=It 1 (K'x the atomic formulas of L*). It is easy to show the following: k IF A e 411, It' A o k It' A. Validity in f f is just validity in B's bottom node, if there is one. If A is built up fromjust disjunction and conjunction - the connectives receiving the same truth conditionsin all three logics so far defined -then It A o Iti A o It, A. Monotonicity fails forformulas containing existential quantifiers. That is, in intuitionistic logic, k Iti A andk' 2 k imply k' It A. That holds in the logic I have defined only for formulas without3. On such formulas, it agrees entirely with intuitionistic logic. In general, however, it isweaker. Every valid formula is valid in intuitionistic logic. The reverse, however, fails. Some intuitionistically valid formulas that fail: 2. ( 3 x A & 3 y B ) + 3x3y(A & B) (where x does not occur in B and y does not occur in A) 3. VxVy(AV B) + (VxA v VyB)(where x does not occur in B and y does not occur in A)