Nominalism in the Philosophy of Mathematics

First published Mon Sep 16, 2013

Nominalism about mathematics (or mathematical nominalism) is the viewaccording to which either mathematical objects, relations, andstructures do not exist at all, or they do not exist as abstractobjects (they are neither located in space-time nor do they havecausal powers). In the latter case, some suitable concrete replacementfor mathematical objects is provided. Broadly speaking, there are twoforms of mathematical nominalism: those views that require thereformulation of mathematical (or scientific) theories in order toavoid the commitment to mathematical objects (e.g., Field 1980;Hellman 1989), and those views that do not reformulate mathematical orscientific theories and offer instead an account of how no commitmentto mathematical objects is involved when these theories are used(e.g., Azzouni 2004). Both forms of nominalism are examined, and theyare assessed in light of how they address five central problems in thephilosophy of mathematics (namely, problems dealing with theepistemology, the ontology, and the application of mathematics as wellas the use of a uniform semantics and the proviso that mathematicaland scientific theories be taken literally).

1. Two views about mathematics: nominalism and platonism

In ontological discussions about mathematics, two views areprominent. According to platonism, mathematical objects (as well asmathematical relations and structures) exist and are abstract; thatis, they are not located in space and time and have no causalconnection with us. Although this characterization ofabstractobjects is purely negative—indicating what suchobjectsare not—in the context of mathematics itcaptures the crucial features the objects in questions are supposed tohave. According to nominalism, mathematical objects (including,henceforth, mathematical relations and structures) do not exist, or atleast they need not be taken to exist for us to make sense ofmathematics. So, it is the nominalist's burden to show how tointerpret mathematics without the commitment to the existence ofmathematical objects. This is, in fact, a key feature of nominalism:those who defend the view need to show that it is possible to yield atleast as much explanatory work as the platonist obtains, but invokinga meager ontology. To achieve that, nominalists in the philosophy ofmathematics forge interconnections with metaphysics (whethermathematical objects do exist), epistemology (what kind of knowledgeof these entities we have), and philosophy of science (how to makesense of the successful application of mathematics in science withoutbeing committed to the existence of mathematical entities). Theseinterconnections are one of the sources of the variety of nominalistviews.

Despite the substantial differences between nominalism andplatonism, they have at least one feature in common: both come in manyforms. There are various versions of platonism in the philosophy ofmathematics: standard (or object-based) platonism (Gödel 1944,1947; Quine 1960), structuralism (Resnik 1997; Shapiro 1997), andfull-blooded platonism (Balaguer 1998), among other views. Similarly,there are also several versions of nominalism: fictionalism (Field1980, 1989), modal structuralism (Hellman 1989, 1996),constructibilism (Chihara 1990), the weaseling-away view (Melia 1995,2000), figuralism (Yablo 2001), deflationary nominalism (Azzouni2004), agnostic nominalism (Bueno 2008, 2009), and pretense views(Leng 2010), among others. Similarly to their platonist counterparts,the various nominalist proposals have different motivations, and facetheir own difficulties. These will be explored in turn. (A criticalsurvey of various nominalization strategies in mathematics can befound in Burgess and Rosen (1997). The authors address in detail boththe technical and philosophical issues raised by nominalism in thephilosophy of mathematics.)

Discussions about nominalism in the philosophy of mathematics inthe 20^th century started roughly with the work thatW. V. Quine and Nelson Goodman developed toward constructivenominalism (Goodman and Quine 1947). But, as Quine later pointed out,in the end it was indispensable to quantify over classes (Quine1960). As will become clear below, responses to this indispensabilityargument have generated a significant amount of work fornominalists. And it is the focus on the indispensability argument thatlargely distinguishes more recent nominalist views in the philosophyof mathematics, which I will focus on, from the nominalism developedin the early part of the 20^th century by the Polish schoolof logic (Simons 2010).

Mathematical nominalism is a form of anti-realism about abstractobjects. This is an independent issue from the traditional problem ofnominalism about universals. A universal, according to a widespreaduse, is something that can be instantiated by differententities. Since abstract objects are neither spatial nor temporal,they cannot be instantiated. Thus, mathematical nominalism andnominalism about universals are independent from one another (see theentry onnominalism inmetaphysics). It could be argued that certain sets encapsulate theinstantiation model, since a set of concrete objects can beinstantiated by such objects. But since thesame set cannotbe so instantiated, given that sets are individuated by their membersand as long as their members are different the resulting sets are notthe same, it is not clear that even these sets are instantiated. Iwill focus here on mathematical nominalism.

2. Five Problems

In contemporary philosophy of mathematics, nominalism has beenformulated in response to difficulties faced by platonism. But indeveloping their responses to platonism, nominalists also encounterdifficulties of their own. Five problems need to be addressed in thiscontext:

The epistemological problem of mathematics,
The problem of the application of mathematics,
The problem of uniform semantics,
The problem of taking mathematical discourse literally, and
The ontological problem.

Usually, problems (1) and (5) are considered as raisingdifficulties for platonism, whereas problems (2), (3), and (4) areoften taken as yielding difficulties for nominalism. (I will discussbelow to what extent such an assessment is accurate.) Each of theseproblems will be examined in turn.

2.1 The epistemological problem of mathematics

Given that platonism postulates the existence of mathematicalobjects, the question arises as to how we obtain knowledge aboutthem. The epistemological problem of mathematics is the problem ofexplaining the possibility of mathematical knowledge, given thatmathematical objects themselves do not seem to play any role ingenerating our mathematical beliefs (Field 1989).

This is taken to be a particular problem for platonism, since thisview postulates the existence of mathematical objects, and one wouldexpect such objects to play a role in the acquisition of mathematicalknowledge. After all, on the platonist view, such knowledge is aboutthe corresponding mathematical objects. However, despite varioussophisticated attempts by platonists, there is still considerablecontroversy as to how exactly this process should bearticulated. Should it be understood via mathematical intuition, bythe introduction of suitable mathematical principles and definitions,or does it require some form of abstraction?

In turn, the epistemological issue is far less problematic fornominalists, who are not committed to the existence of mathematicalobjects in the first place. They will have to explain other things,such as, how can the nominalist account for the difference between amathematician, who knows a significant amount of mathematics, and anon-mathematician, who does not? This difference, according to somenominalists, is based on empirical and logical knowledge—not onmathematical knowledge (Field 1989).

2.2 The problem of the application of mathematics

Mathematics is often successfully used in scientific theories. Howcan such a success be explained? Platonists allegedly have an answerto this problem. Given that mathematical objects exist and aresuccessfully referred to by our scientific theories, it is notsurprising that such theories are successful. Reference tomathematical objects is just part of the reference to those entitiesthat are indispensable to our best theories of the world. This framesthe problem of the application of mathematics in terms of theindispensability argument.

In fact, one of the main reasons for belief in the existence ofmathematical objects—some claim this is theonlynon-question begging reason (Field 1980)—is given by theindispensable use of mathematics in science. The crucial idea,originally put forward by W. V. Quine, and later articulated, in adifferent way, by Hilary Putnam, is that ontological commitment shouldbe restricted to just those entities that are indispensable to ourbest theories of the world (Quine 1960; Putnam 1971; Colyvan2001a). Mark Colyvan has formulated the argument in the followingterms:

(P1) We ought to be ontologically committed to all and only those entities that are indispensable to our best theories of the world.
(P2) Mathematical entities are indispensable to our best theories of the world.
Therefore, (C) we ought to be ontologically committed to mathematical entities.

The first premise relies crucially on Quine's criterion ofontological commitment. After regimenting our best theories of theworld in a first-order language, the ontological commitments of thesetheories can be read off as being the value of the existentiallyquantified variables. But how do we move from the ontologicalcommitments of a theory to what weought to be ontologicallycommitted to? This is the point where the first premise of theindispensability argument emerges. If we are dealing with our besttheories of the world, precisely those items that are indispensable tothese theories amount to what weought to be committedto. (Of course, a theory may quantify over more objects than thosethat are indispensable.) And by identifying the indispensablecomponents invoked in the explanation of various phenomena, and notingthat mathematical entities are among them, the platonist is then in aposition to make sense of the success of applied mathematics.

However, it turns out that whether the platonist can indeed explainthe success of the application of mathematics is, in fact,controversial. Given that mathematical objects are abstract, it isunclear why the postulation of such entities is helpful to understandthe success of applied mathematics. For the physical world—beingcomposed of objects located in space-time—is not constituted bythe entities postulated by the platonist. Hence, it is not clear whythe correct description of relations amongabstract(mathematical) entities is evenrelevant to understand thebehavior of concrete objects in the physical world involved in theapplication of mathematics. Just mentioning that the physicalworldinstantiates structures (or substructures) described ingeneral terms by various mathematical theories is not enough (see,e.g., Shapiro 1997). For there are infinitely many mathematicalstructures, and there is no way of uniquely determining which of themis actually instantiated—or even instantiated only inpart—in a finite region of the physical world. There is agenuine underdetermination here, given that the same physicalstructure in the world can be accommodated by very differentmathematical structures. For instance, quantum mechanical phenomenacan be characterized by group-theoretic structures (Weyl 1928) or bystructures emerging from the theory of Hilbert spaces (von Neumann1932). Mathematically, such structures are very different, but thereis no way of deciding between them empirically.

Despite the controversial nature of the platonist claim to be ableto explain the success of applied mathematics, to accommodate thatsuccess is often taken as a significant benefit of platonism. Lesscontroversially, the platonist is certainly able to describe the wayin which mathematical theories are actually used in scientificpractice without having to rewrite them. This is, as will become clearbelow, a significant benefit of the view.

Nominalism, in turn, faces the difficulty of having to explain thesuccessful use of mathematics in scientific theorizing. Since,according to the nominalist, mathematical objects do notexist—or, at least, are not taken to exist—it becomesunclear how referring to such entities can contribute to the empiricalsuccess of scientific theories. In particular, if it turns out thatreference to mathematical entities is indeed indispensable to our besttheories of the world, how can the nominalist deny the existence ofsuch entities? As we will see below, several nominalist views in thephilosophy of mathematics have emerged in response to the challengeraised by considerations based on the indispensability ofmathematics.

2.3 The problem of uniform semantics

One of the most significant features of platonism is the fact thatit allows us to adopt the same semantics for both mathematical andscientific discourse. Given the existence of mathematical objects,mathematical statements are true in the same way as scientificstatements are true. The only difference emerges from their respectivetruth makers: mathematical statements are true in virtue of abstract(mathematical) objects and relations among them, whereas scientificstatements are ultimately true in virtue of concrete objects and thecorresponding relations among such objects. This point is idealized inthat it assumes that, somehow, we can manage to distill the empiricalcontent of scientific statements independently of the contributionmade by the mathematics that is often used to express suchstatements. Platonists who defend the indispensability argument insistthat this is not possible to do (Quine 1960; Colyvan 2001a); even somenominalists concur (Azzouni 2011).

Moreover, as is typical in the application of mathematics, thereare alsomixed statements, which involve both terms referringto concrete objects and terms referring to abstract ones. Theplatonist has no trouble providing a unified semantics for suchstatements either—particularly if mathematical platonism isassociated with realism about science. In this case, the platonist canprovide a referential semantics throughout. Of course, the platonistabout mathematicsneed not be a realist aboutscience—although it's common to combine platonism and realism inthis way. In principle, the platonist could adopt some form ofanti-realism about science, such as constructive empiricism (vanFraassen 1980; Bueno 2009). As long as the form of anti-realismregarding science allows for a referential semantics (and many do),the platonist would have no trouble providing a unified semantics forboth mathematics and science (Benacerraf 1973).

It is not clear that the nominalist can deliver these benefits. Aswill become clear shortly, most versions of nominalism require asubstantial rewriting of mathematical language. As a result, adistinct semantics needs to be offered for that language in comparisonwith the semantics that is provided for scientific discourse.

2.4 The problem of taking mathematical discourse literally

A related benefit of platonism is that it allows one to takemathematical discourse literally, given that mathematical termsrefer. In particular, there is no change in the syntax of mathematicalstatements. So, when mathematicians claim that ‘There areinfinitely many prime numbers’, the platonist can take thatstatement literally as describing the existence of an infinitude ofprimes. On the platonist view, there are obvious truth-makers formathematical statements: mathematical objects and their correspondingproperties and relations (Benacerraf 1973).

We have here a major benefit of platonism. If one of the goals ofthe philosophy of mathematics is to provide understanding ofmathematics and mathematical practice, it is a significant advantagethat platonists are able to take the products of thatpractice—such as mathematical theories—literally and donot need to rewrite or reformulate them. After all, the platonist isthen in a position to examine mathematical theories as they areactually formulated in mathematical practice, rather than discuss aparallel discourse offered by various reconstructions of mathematicsgiven by those who avoid the commitment to mathematical objects (suchas the nominalists).

The inability to take mathematical discourse literally is indeed aproblem for nominalists, who typically need to rewrite the relevantmathematical theories. As will become clear below, it is common thatnominalization strategies for mathematics change either the syntax orthe semantics of mathematical statements. For instance, in the case ofmodal structuralism, modal operators are introduced to preserve verbalagreement with the platonist (Hellman 1989). The proposal is that eachmathematical statementS is translated into two modalstatements: (i) if there were structures of the suitablekind,S would be true in these structures, and (ii) it'spossible that there are such structures. As a result, both the syntaxand the semantics of mathematics are changed. In the case ofmathematical fictionalism, in order to preserve verbal agreement withthe platonist despite the denial of the existence of mathematicalobjects, fiction operators (such as, ‘According toarithmetic…’) are introduced (Field 1989). Once again,the resulting proposal changes the syntax (and, hence, the semantics)of mathematical discourse. This is a significant cost for theseviews.

2.5 The ontological problem

The ontological problem consists in specifying the nature of theobjects a philosophical conception of mathematics is ontologicallycommitted to. Can the nature of these objects be properly determined?Are the objects in question such that we simply lack good grounds tobelieve in their existence? Traditional forms of platonism have beencriticized for failing to offer an adequate solution to thisproblem. In response, some platonists have argued that the commitmentto mathematical objects is neither problematic nor mysterious (see,e.g., Hale and Wright 2001). Similarly, even though some nominalistsneed not be committed to mathematical objects, they may be committedto other entities that may also raise ontological concerns (such aspossibilia). The ontological problem is then the problem of assessingthe status of the ultimate commitments of the view.

Three nominalization strategies will be discussed below:mathematical fictionalism (Field 1980, 1989), modal structuralism(Hellman 1989, 1996), and deflationary nominalism (Azzouni 2004). Thefirst two reject the second premise of the indispensabilityargument. They provide ‘hard roads’ to nominalism (Colyvan2010), in the sense that the nominalist needs to develop the complex,demanding work of showing how quantification over mathematical objectscan be avoided in order to develop a suitable interpretation ofmathematics. The third strategy rejects the first premise of theargument, thus bypassing the need to argue for the dispensability ofmathematics (in fact, for the deflationary nominalist, mathematics isultimately indispensable). By reassessing Quine's criterion ofontological commitment, and indicating that quantification overcertain objects does not require their existence, this strategy yieldsan ‘easy road’ to nominalism.

Although this survey is clearly not exhaustive, since not everynominalist view available will be considered here, the three viewsdiscussed are representative: they occupy distinct points in thelogical space, and they have been explicitly developed to address thevarious problems just listed.

3. Mathematical Fictionalism

3.1 Central features of mathematical fictionalism

In a series of works, Hartry Field provided an ingenious strategyfor the nominalization of science (Field 1980, 1989). As opposed toplatonist views, in order to explain the usefulness of mathematics inscience, Field does not postulate the truth of mathematicaltheories. In his view, it is possible to explain successfulapplications of mathematics with no commitment to mathematicalobjects. Therefore, what he takes to be the main argument forplatonism, which relies on the (apparent) indispensability ofmathematics to science, is resisted. The nominalist nature of Field'saccount emerges from the fact that mathematical objects are notassumed to exist. Hence, mathematical theories are false. (Strictlyspeaking, Field notes, any existential mathematical statement isfalse, and any universal mathematical statement is vacuously true.) Bydevising a strategy that shows how to dispense with mathematicalobjects in the formulation of scientific theories, Field rejects theindispensability argument, and provides strong grounds for thearticulation of a nominalist stance.

Prima facie, it may sound counterintuitive to state that‘there are infinitely many prime numbers’ is false. But ifnumbers do not exist, that's the proper truth-value for that statement(assuming a standard semantics). In response to this concern, Field1989 introduces a fictional operator, in terms of which verbalagreement can be reached with the platonist. In the case at hand, onewould state: ‘According to arithmetic, there are infinitely manyprime numbers’, which is clearly true. Given the use of afictional operator, the resulting view is oftencalledmathematical fictionalism.

The nominalization strategy devised by the mathematicalfictionalist depends on two interrelated moves. The first is to changethe aim of mathematics, which is not taken to be truth, but somethingdifferent. On this view, the proper norm of mathematics, which willguide the nominalization program, isconservativeness. Amathematical theory is conservative if it is consistent with everyinternally consistent theory about the physical world, where suchtheories do not involve any reference to, nor quantification over,mathematical objects, such as sets, functions, numbers etc. (Field1989, p. 58). Conservativeness is stronger than consistency (since ifa theory is conservative, it is consistent, but not viceversa). However, conservativeness is not weaker than truth (Field1980, pp. 16–19; Field 1989, p. 59). So, Field is notcountenancing a weaker aim of mathematics, but onlyadifferent one.

It is precisely because mathematics is conservative that, despitebeing false, it can be useful. Of course, this usefulness is explainedwith no commitment to mathematical entities: mathematics is usefulbecause it shortens our derivations. After all, if a mathematicaltheoryM is conservative, then a nominalisticassertionA about the physical world (i.e. an assertion whichdoes not refer to mathematical objects) follows from a bodyNof such assertions andM only if it follows fromNalone. That is, provided we have a sufficiently rich body ofnominalistic assertions, the use of mathematics does not yield any newnominalistic consequences. Mathematics is only a useful instrument tohelp us in the derivations.

As a result, conservativeness can only be employed to do therequired job if we have nominalistic premises to start with (Field1989, p. 129). As Field points out, it is a confusion to argue againsthis view by claiming that if we add some bits of mathematics toabody of mathematical claims (not nominalistic ones), we mayobtain new consequences that could not be achieved otherwise (Field1989, p. 128). The restriction tonominalistic assertions iscrucial.

The second move of the mathematical fictionalist strategy is toprovide such nominalistic premises. Field has done that in oneimportant case: Newtonian gravitational theory. He elaborates on awork that has a respectable tradition: Hilbert's axiomatization ofgeometry (Hilbert 1971). What Hilbert provided was a syntheticformulation of geometry, which dispenses with metric concepts, andtherefore does not include any quantification over real numbers. Hisaxiomatization was based on concepts suchaspoint,betweenness,andcongruence. Intuitively speaking, we say that apointy isbetween the pointsx andzify is a point in the line-segment whose endpointsarex andz. Also intuitively, we say that theline-segmentxy iscongruent to theline-segmentzw if the distance from the pointx to thepointy is the same as that from the pointztow. After studying the formal properties of the resultingsystem, Hilbert proved a representation theorem. He showed, in astronger mathematical theory, that given a model of the axiom systemfor space he had put forward, there is a functiond from pairsof points onto non-negative real numbers such that the following‘homomorphism conditions’ are met:

xy is congruent tozw iffd(x,y) =d(z,w), for all pointsx,y,z, andw;
y is betweenx andz iffd(x,y) +d(y,z) =d(x,z), for all pointsx,y, andz.

As a result, if the functiond is taken to representdistance, we obtain the expected results about congruence andbetweenness. Thus, although we cannot talk about numbers in Hilbert'sgeometry (there are no such entities to quantify over), there is ametatheoretic result that associates assertions about distances withwhat can be said in the theory. Field calls such numericalclaimsabstract counterparts of purely geometric assertions,and they can be used to draw conclusions about purely geometricalclaims in a smoother way. Indeed, because of the representationtheorem, conclusions about space, statablewithout realnumbers, can be drawn far more easily than we could achieve by adeflationary proof from Hilbert's axioms. This illustrates Field'spoint that the usefulness of mathematics derives from shorteningderivations (Field 1980, pp. 24–29).

Roughly speaking, what Field established was how to extendHilbert's results about space to space-time. Similarly to Hilbert'sapproach, instead of formulating Newtonian laws in terms of numericalfunctors, Field showed that they can be recast in terms of comparativepredicates. For example, instead of adopting a functor such as‘the gravitational potential ofx’, which is takento have a numerical value, Field employed a comparative predicate suchas ‘the difference in gravitational potential betweenxandy is less than that betweenzandw’. Relying on a body of representation theorems(which plays the same role as Hilbert's representation theorem ingeometry), Field established how several numerical functors can be‘obtained’ from comparative predicates. But in order touse those theorems, he first showed how to formulate Newtoniannumerical laws (such as, Poisson's equation for the gravitationalfield) only in terms of comparative predicates. The result (Field1989, pp. 130–131) is the followingextended representationtheorem. LetN be a theory formulated only in terms ofcomparative predicates (with no recourse to numerical functors). Forany modelS ofN whose domain is constituted byspace-time regions, there exists:

A 1–1 spatio-temporal co-ordinate functionf (unique up to a generalized Galilean transformation) mapping the space-time ofS onto quadruples of real numbers;
A mass density functiong (unique up to a positive multiplicative transformation) mapping the space-time ofS onto an interval of non-negative real numbers; and
A gravitational potential functionh (unique up to a positive linear transformation) mapping the space-time onto an interval of real numbers.

Moreover, all these functions ‘preserve structure’, inthe sense that the comparative relations defined in terms of themcoincide with the comparative relations used inN. Furthermore,iff,g andh are taken as the denotation of theappropriate functors, the laws of Newtonian gravitational theory intheir functorial form hold.

Notice that, in quantifying over space-time regions, Field assumesa substantivalist view of space-time, according to which there arespace-time regions that are not fully occupied (Field 1980,pp. 34–36; Field 1989, pp. 171–180). Given this result,the mathematical fictionalist is allowed to draw nominalisticconclusions from premises involvingN plus a mathematicaltheoryT. After all, due to the conservativeness ofmathematics, such conclusions can be obtained independentlyofT. The role of the extended representation theorem is thento establish that, despite the lack of quantification overmathematical objects, precisely the same class of models is determinedby formulating Newtonian gravitational theory in terms of functors (asthe theory is usually expressed) or in terms of comparative predicates(as the mathematical fictionalist favors). Thus, the extendedrepresentation theorem ensures that the use of conservativeness ofmathematics together with suitable nominalistic claims (formulated viacomparative predicates) does not change the class of models of theoriginal theory: the same comparative relations are preserved. Hence,what Field provided is a nominalization strategy, and since it reducesontology, it seems a promising candidate for a nominaliststancevis-à-vis mathematics.

How should the mathematical fictionalist approach physicaltheories, such as perhaps string theory, that do not seem to be aboutconcrete observable objects? One possible response, assuming the lackof empirical import of such theories, is simply to reject that theyarephysical theories, and as such they are not the sorts oftheories for which the mathematical fictionalist needs to provide anominalistic counterpart. In other words, until the moment in whichsuch theories acquire the relevant empirical import, they need notworry the mathematical fictionalist. Theories of that sort would beclassified as part of the mathematics rather than the physics.

3.2 Metalogic and the formulation of conservativeness

But is mathematics conservative? In order to establish theconservativeness of mathematics, the mathematical fictionalist hasused metalogical results, such as the completeness and the compactnessof first-order logic (Field 1992, 1980, 1989). The issue then arisesas to whether the mathematical fictionalist canuse theseresults to develop the program.

At two crucial junctures, Field has made use of metalogicalresults: (a) in his reformulation of the notion of conservativeness innominalistically acceptable terms (Field 1989, pp. 119–120;Field 1991), and (b) in his nominalist proof of the conservativenessof set theory (Field 1992). These two outcomes are crucial for Field,since they establish the adequacy of conservativeness for themathematical fictionalist. For (a) settles that the latter canformulate that notion without violating nominalism, and (b) concludesthat conservativeness is a feature that mathematics actually has. Butif these two outcomes are not legitimate, Field's approach cannot getoff the ground. I will now consider whether these two uses ofmetalogical results are acceptable on nominalist grounds.

3.2.1 Conservativeness and the compactness theorem

Let me start with (a). The mathematical fictionalist has relied onthe compactness theorem to formulate the notionofconservativeness in an acceptable way, that is, withoutreference to mathematical entities. As noted above, conservativenessis defined in terms ofconsistency. But this notion isusually formulated either in semantic terms (as the existence of anappropriate model), or in proof-theoretic terms (in terms of suitableproofs). However, as Field acknowledges, these two formulations ofconsistency are platonist, since they depend on abstract objects(models and proofs), and therefore are not nominalisticallyacceptable.

The mathematical fictionalist way out is to avoid moving to themetalanguage in order to express the conservativeness ofmathematics. The idea is to state, in the object-language, the claimthat a given mathematical theory is conservative by introducing aprimitive notion oflogical consistency: ◊A. Thus,ifB is any sentence,B* is the result ofrestrictingB to non-mathematical entities,andM₁, …,M_n arethe axioms of a mathematical theoryM, the conservativenessofM can be expressed by the following schema (Field 1989,p. 120):

(C) If ◊B, then ◊(B* ∧M₁ ∧ … ∧M_n).

In other words, a mathematical theoryM is conservative ifit is consistent with every consistent theory about the physicalworldB*.

This assumes, of course, thatM is finitely axiomatized. Buthow can we apply (C) in the case of mathematical theories thatarenot finitely axiomatizable (such as Zermelo-Fraenkel settheory)? In this case, we cannot make the conjunction of all theaxioms of the theory, since there are infinitely many of them. Fieldhas addressed this issue, and he initially suggested that themathematical fictionalist could use substitutional quantification toexpress these infinite conjunctions (Field 1984). In a postscript tothe revised version of this essay (Field 1989, pp. 119–120), henotes that substitutional quantification can be avoided, provided thatthe mathematical and physical theories in question are expressed in alogic for whichcompactness holds. For in this case, theconsistency of the whole theory is reduced to the consistency of eachof its finite conjunctions.

There are, however, three problems with this move.

One concern about the use of substitutional quantification in this context involves the nature of substitutional instances. If the latter turn out to be abstract, which would be the case if such substitutional instances were not mere inscriptions, they are not available to the nominalist. If the substitutional instances are concrete, the nominalist needs to show that there are enough of them.
The verystatement of the compactness theorem involvesset-theoretic talk: letG be aset of formulas; if every finitesubset ofG is consistent, thenG is consistent. How can nominalists rely on a theorem whose very statement involves abstract entities? In order to use this theorem, an appropriate reformulation is required.
Let us grant that it is possible to reformulate this statement without referring to sets. Can then the nominalistuse the compactness theorem? As is well known, theproof of this theorem assumes set theory. The compactness theorem is usually presented as a corollary to the completeness theorem for first-order logic, whose proof assumes set theory (see, for example, Boolos and Jeffrey 1989, pp. 140–141). Alternatively, if the compactness result is to be proved directly, then one has to construct the appropriate model ofG—which again requires set theory. So, unless mathematical fictionalists are able to provide an appropriate nominalization strategy for set theory itself, they are not entitled to use this result. In other words, far more work is required before a Field-type nominalist is able to rely on metalogical results.

But maybe this criticism misses the whole point of Field'sprogram. As we saw, Field does not require that a mathematicaltheoryM be true for it to be used. Onlyitsconservativeness is demanded. So, ifM is added toa bodyB* ofnominalistic claims, no new nominalisticconclusion is obtained which was not obtained byB* alone. Inother words, what Field's strategy asks for is the formulation ofappropriatenominalistic bodies of claims to whichmathematics can be applied. The same point holds for metalogicalresults: provided that they are applied tonominalisticclaims, Field is fine.

The problem with this reply is that it involves the mathematicalfictionalist program in a circle. The fictionalist cannot rely ontheconservativeness of mathematics to justify the use of amathematical result (the compactness theorem) that is required fortheformulation of the notion of conservativeness itself. Forin doing that, the fictionalistassumes that the notion ofconservativeness is nominalistically acceptable, and this is exactlythe point in question. Recall that the motivation for Field to use thecompactness theorem was to reformulate conservativeness without havingto assume abstract entities (namely, those required by the semanticand the proof-theoretic accounts of consistency). Thus, at this point,the mathematical fictionalist cannot yet use the notion ofconservativeness; otherwise, the whole program would not get off theground. I conclude that, similarly to any other part of mathematics,metalogical results also need to be obtained nominalistically. Troublearises for nominalism otherwise.

3.2.2 Conservativeness and primitive modality

But perhaps the mathematical fictionalist has a way out. As we saw,Field spells out the notion of conservativeness in terms of aprimitive notion of logical consistency: ◊A. And he alsoindicates that this notion is related to the model-theoretic conceptof consistency—in particular, to the formulation of this conceptin von Neumann-Bernays-Gödel finitely axiomatizable set theory(NBG). This is done via two principles (Field 1989, p. 108):

(MTP^#) If ☐(NBG → there is a model for ‘A’), then ◊A
(ME^#) If ☐(NBG → there is no model for ‘A’), then ¬◊A.

I am following Field's terminology: ‘MTP^#’stands formodel-theoretic possibility, and‘ME^#’ formodel existence. The symbol‘^#’ indicates that, according to Field, theseprinciples are nominalistically acceptable. After all, they are modalsurrogates for the platonistic principles (Field 1989,pp. 103–109):

(MTP) If there is a model for ‘A’, then ◊A
(ME) If there is no model for ‘A’, then ¬◊A.

It may be argued that, by using these principles, the mathematicalfictionalist will be entitled to use the compactness theorem. First,one should try to state this theorem in a nominalistically acceptableway. Without worrying too much about details, let us grant, for thesake of argument, that the following characterization will do:

(Compact^#) If ¬◊T, then ∃_f A₁, …,A_n[¬◊(A₁ ∧ … ∧A_n)],

whereT is a theory and eachA_i, 1≤i ≤n, is a formula (an axiom ofT). Theexpression‘∃_f A₁…A_n’is to be read as ‘there arefinitely manyformulasA₁…A_n’. (Thisquantifier is not first-order. However, I am not going to press thepoint that the nominalist seems to need anon-first-orderquantifier to express a property typical offirst-orderlogic. This is only one of the worries we are leaving aside in thisformulation.) This version is parasitic on the following platonisticformulation of the compactness theorem:

(Compact) If there is no model forT, then ∃_f A₁, …,A_n such that there is no model for (A₁ ∧ … ∧A_n).

In order for mathematical fictionalists to be entitled to use thecompactness theorem, they will have to show that the nominalisticformulation (Compact^#) follows from the platonistic one(Compact). In this sense, if the latter is adequate, so is theformer. More accurately, what has to be shown is that(Compact^#) follows from amodal surrogate of(Compact). After all, since what is at issue is the legitimacy of thecompactness theorem on nominalist grounds, it would bequestion-begging to assume the full platonistic version from theoutset. As we will see, there are two ways to try to establish thisresult. Unfortunately, none of them works: both are formallyinadequate.

The two options start in the same way. Suppose that

(1) ¬◊T.

We have to establish that

(2) ∃_f A₁…A_n ¬◊(A₁∧…∧A_n).

It follows from (1) and (MTP^#) that

(3) ¬☐(NBG → there is a model for ‘T’),

and thus

(4) ◊(NBG ∧ there is no model for ‘T’).

Let us assume the modal surrogate for the compactness theorem:

(Compact^M) ☐(NBG → if there is no model for ‘T’, then ∃_f A₁…A_n such that there is no model for (A₁∧…∧A_n)).

Note that, since the modal surrogate is formulated in terms ofmodels (rather than in terms of the primitive modal operator), it isstill not what mathematical fictionalists need. What they need is(Compact^#), but one needs to show that they can get it. Atthis point, the options begin to diverge.

The first option consists in drawing from (4) and(Compact^M) that

(5) ◊(∃_f A₁…A_n such that there is no model for (A₁ ∧…∧A_n)).

There are, however, difficulties with this move. First, note that(5) is not equivalent to (2), which is the result to beachieved. Moreover, as opposed to (2), (5) is formulated inmodel-theoretic terms, since it incorporates a claim about thenonexistence of a certain model. And what is required is a similarstatement in terms of theprimitive notion of consistency. Inother words, we need the nominalistic counterpart of (5), rather than(5).

But (5) has a nice feature. It is a modalized formulation of theconsequent of (Compact). And since (5) only statesthepossibility that there is no model of a particular kind,it may be argued that itis nominalistically acceptable. (Aswill be examined below, modal structuralists advance a nominalizationstrategy exploring modality along these lines; see Hellman (1989).)Field, however, is skeptical about this move. On his view, modality isnot a general surrogate for ontology (Field 1989,pp. 252–268). And one of his worries is that by allowing theintroduction of modal operators, as a general nominalization strategy,we modalize away the physical content of the theory underconsideration. However, since metalogical claims are not expected tohave physical consequences, the worry need not arise here. At anyrate, given that (5) does not establish what needs to be established,it does not solve the problem.

The second option consists in moving to (5′) instead of(5):

(5′) ☐(NBG → ∃_f A₁…A_n such that there is no model for (A₁ ∧…∧A_n)).

Note that if (5′) were established, we would have settled thematter. After all, with a straightforward reworking of(ME^#) (namely, If ☐(NBG →∃_f A₁…A_nsuch that there is no model for(A₁∧…∧A_n)),then ¬◊(A₁∧…∧A_n)), it follows from(5′) and (ME^#) that

(2) ∃_f A₁…A_n ¬◊(A₁ ∧…∧A_n),

whichis the conclusion we need. The problem here is that(5′) doesnot follow from (4) and(Compact^M). Therefore, we cannot derive it.

Clearly, there may well be another option that establishes theintended conclusion. But, to say the least, the mathematicalfictionalist has to present it before being entitled to use metalogicresults. Until then, it is not clear that these results arenominalistically acceptable.

3.2.3 Metalogic and the proof of the conservativeness of set theory

I should now consider issue (b): Field's nominalistic proof of theconservativeness of set theory. Let us grant that the concept ofconservativeness has been formulated in some nominalisticallyacceptable way. If Field's proof were correct, he would have provedthat mathematics itself is conservative—as long as one assumesthe usual reductions of mathematics to set theory. How does Fieldprove the conservativeness of set theory? It is by an ingeniousargument, which adapts one of the Field's platonistic conservativenessproofs (Field 1980). For our present purposes, we need not examine thedetails of this argument, but simply note that at a crucial point thecompleteness of first-order logic is used to establish its conclusion(Field 1992, p. 118).

The problem with this move is that, even if mathematicalfictionalists formulate the statement of the completeness theoremwithout referring to mathematical entities, theproof of thistheorem assumes set theory (see, for instance, Boolos and Jeffrey1989, pp. 131–140). Therefore, fictionalists cannot use thetheorem without undermining their nominalism. After all, the point ofproviding a nominalistic proof of the conservativeness of set theoryis to show that, without recourse to platonist mathematics, themathematical fictionalist is able to establish that mathematics isconservative. Field has offered aplatonist argument for theconservativeness result (Field 1980)—an argument that explicitlyinvoked properties of set theory. The idea was to provideareductio of platonism: by using platonist mathematics,Field attempted to establish that mathematics was conservative and,thus, ultimately dispensable. In contrast with the earlier strategy,the goal was to provide a proof of the conservativeness of set theorythat a nominalist could accept. But since the nominalistic proofrelies on the completeness theorem, it is not at all clear that it isin fact nominalist. Mathematical fictionalists should first be able toprove the completeness resultwithout assuming settheory. Alternatively, they should provide a nominalization strategyfor set theory itself, which will then entitle them to use metalogicalresults.

But it may be argued that the mathematical fictionalist onlyrequires theconservativeness of the set theory in which thecompleteness theorem is proved. It should now be clear that this replyis entirely question begging, since the point at issue is exactlytoprove the conservativeness of set theory. Thus, thefictionalist cannot assume that this result is already established atthe metatheory.

In other words, without a broader nominalization strategy, whichallows set theory itself to be nominalized, it seems difficult to seehow mathematical fictionalists can use metalogical results as part oftheir program. The problem, however, is that it is not at all obviousthat, at least in the form articulated by Field, the mathematicalfictionalist program can be extended to set theory. For it onlyprovides a nominalization strategy forscientific theories,that is, for the use of mathematics inscience (e.g., inNewtonian gravitational theory). The approach doesn't address thenominalization ofmathematics itself.

In principle, one may object, this shouldn't be a problem. Afterall, the mathematical fictionalists’ motivation to develop theirapproach has focused on one issue: to overcome the indispensabilityargument—thus addressing the use of mathematicsinscience. And the overall strategy, as noted, has been toprovide nominalist counterparts to relevant scientific theories.

The problem with this objection, however, is that given the natureof Field's strategy, the task of nominalizing science cannot beachieved without also nominalizing set theory. Thus, what is needed isa more open-ended, broader nominalism: one that goes hand in hand notonly with science, but also with metalogic. As it stands, themathematical fictionalist approach still leaves a considerablegap.

3.3 Assessment: benefits and problems of mathematical fictionalism

3.3.1 The epistemological problem

Given that mathematical objects do not exist, on the mathematicalfictionalist perspective, the problem of how we can obtain knowledgeof them simply vanishes. But another problem emerges instead: what isit that distinguishes a mathematician (who knows a lot aboutmathematics) and a non-mathematician (who does not have suchknowledge)? The difference here (according to Field 1984) is not abouthaving or lacking mathematical knowledge, but rather it isaboutlogical knowledge: of knowing which mathematicaltheorems follow from certain mathematical principles, and which donot. The epistemological problem is then solved—as long as themathematical fictionalist provides an epistemologyforlogic.

In fact, what needs to be offered is ultimately an epistemologyformodality. After all, on Field's account, in order toavoid the platonist commitment to models or proofs, the concept oflogical consequence is understood in terms of the primitive modalconcept of logical possibility:A follows logicallyfromB as long as the conjunction ofB and the negationofA is impossible, that is, ¬◊(B ∧¬A).

However, how are such judgments of impossibility established? Underwhat conditions do we know that they hold? In simple cases, involvingstraightforward statements, to establish such judgments may beunproblematic. The problem emerges when more substantive statementsare invoked. In these cases, we seem to need a significant amountofmathematical information in order to be able to determinewhether the impossibilities in question really hold or not. Consider,for instance, the difficulty of establishing the independence of theaxiom of choice and the continuum hypotheses from the axioms ofZermelo-Fraenkel set theory. Significantly complex mathematical modelsneed to be constructed in this case, which rely on the development ofspecial mathematical techniques to build them. What is required fromthe mathematical fictionalist at this stage is the nominalization ofset theory itself—something that, as we saw, Field still owesus.

3.3.2 The problem of the application of mathematics

Similarly to the epistemological problem, the problem of theapplication of mathematics is partially solved by the mathematicalfictionalist. Field provides an account of the application ofmathematics that does not require the truth of mathematicaltheories. As we saw, this demands that mathematics be conservative inthe relevant sense. However, it is unclear whether Field hasestablished the conservativeness of mathematics, given his restrictiveway of introducing non-set-theoretic vocabulary into the axioms of settheory as part of his attempted proof of the conservativeness of settheory (Azzouni 2009b, p. 169, note 47; additional difficulties forthe mathematical fictionalist program can be found in Melia 1998,2000). Field was working with restricted ZFU, Zermelo-Fraenkel settheory with the axiom of choice modified to allowforUrelemente, objects that are not sets, butnotallowing for any non-set-theoretic vocabulary to appear in thecomprehension axioms, that is, replacement or separation (Field1980, p. 17). This is, however, a huge restriction, given that whenmathematics is actually applied, non-set-theoretic vocabulary, whentranslated into set-theoretic language, will have to appear in thecomprehension axioms. As formulated by Field, the proof failed toaddress the crucial case of actual applications of mathematics.

Moreover, it is also unclear whether the nominalization programadvanced by the mathematical fictionalist can be extended to otherscientific theories, such as quantum mechanics (Malament 1992). MarkBalaguer responded to this challenge by trying to nominalize quantummechanics along mathematical fictionalist lines (Balaguer1998). However, as argued by Bueno (Bueno 2003), Balaguer's strategyis incompatible with a number of interpretations of quantum mechanics,in particular with Bas van Fraassen's version of the modalinterpretation (van Fraassen 1991). And given that Balaguer's strategyinvokes physically real propensities, it is unclear whether it is evencompatible with nominalism. As a result, the nominalization of quantummechanics still remains a major problem for the mathematicalfictionalist.

But even if these difficulties can all be addressed, it is unclearthat the mathematical fictionalist has offered an account of theapplication of mathematics that allows us to make sense of howmathematical theories areactually applied. After all, thefictionalist account requires us to rewrite the relevant theories, byfinding suitable nominalistic versions for them. This leaves the issueof making sense of the actual process of the application ofmathematics entirely untouched, given that no such reformulations areever employed in actual scientific practice. Rather than engaging withactual features of the application process, the fictionalist creates aparallel discourse in an effort to provide a nominalist reconstructionof the use of mathematics in science. The reconstruction shows, atbest, that mathematical fictionalists need not worry about theapplication of mathematicsvis-à-vis increasing theirontology. But the problem still remains of whether they are in aposition to make sense of the actual use of mathematics inscience.This problem, which is crucial for a properunderstanding of mathematical practice, still remains.

A similar difficulty also emerges for Balaguer's version offictionalism (see the second half of Balaguer 1998). Balaguer relieson the possibility of distinguishing between the mathematical and thephysical contents of an applied mathematical theory: in particular,the truth of such a theory holds only in virtue of physical facts,with no contribution from mathematics. It is, however, controversialwhether the distinction between mathematical and physical content canbe characterized without implementing a Field-like nominalizationprogram. In this case, the same difficulties that the latter face alsocarry over to Balaguer's account (Colyvan 2010; Azzouni 2011).

Moreover, according to Azzouni (Azzouni 2009b), in order forscientists to use a scientific theory, they need toassertit. On his view, it is not enough for scientists simply to recognizethat a scientific theory is true (or exhibits some other theoreticalvirtue). It is required that they assert the theory. In particular,scientists would then need to assert a nominalistic theory. Theycannot simply contemplate such a theory; they need to be able toassert it as well (Azzouni 2009b, footnotes 31, 43, 53, and 55, andp. 171). Thus, nominalists who grant this point to Azzouni need toshow that scientists are in a position to assert the relevantnominalistic theories in order to address the issue of the applicationof mathematics.

3.3.3 Uniform semantics

In one respect, mathematical fictionalists offer a uniformsemantics for mathematical and scientific discourse, in anotherrespect, they don't. Initially, both types of discourse are assessedin the same way. Electrons and relations among them make certainquantum-mechanical statements true; in turn, mathematical objects andrelations amongthem make the corresponding mathematicalstatements true. It just happens that, as opposed to electrons on arealist interpretation of quantum mechanics, mathematical objects donot exist. Hence, as noted, existential mathematical statements, suchas ‘there are infinitely many prime numbers’, arefalse. Although the resulting truth-value assignments for existentialstatements conflict with those found in mathematical practice, atleast the same semantics is offered for mathematical and scientificlanguages.

In an attempt to agree with the truth-value assignments that areusually displayed in mathematical discourse, the mathematicalfictionalist introduces a fictional operator: ‘According tomathematical theoryM…’. Such an operator,however, changes the semantics of mathematical discourse. Applied to atrue mathematical statement, at least one that the platonistrecognizes as true, the result will be a true statement—evenaccording to the mathematical fictionalist. For instance, from bothplatonist and fictionalist perspectives, the statement‘according to arithmetic, there are infinitely many primenumbers’ comes out true. But, in this case, the mathematicalfictionalist can no longer offer a unified semantics for mathematicaland scientific languages, given that the latter does not involve theintroduction of fictional operators. Thus, whether mathematicalfictionalists are able to provide a uniform semantics ultimatelydepends on whether fictional operators are introduced or not.

3.3.4. Taking mathematics literally

An immediate consequence of the introduction of fiction operatorsis that mathematical discourse is no longer taken literally. As justnoted, without such operators, mathematical fictionalism producesnon-standard attributions of truth-values to mathematicalstatements. But with fiction operators in place, the syntax ofmathematical discourse is changed, and thus the latter cannot be takenliterally.

3.3.5 The ontological problem

The ontological problem—the problem of the acceptability ofthe ontological commitments made by the mathematicalfictionalist—is basically solved. No commitment to mathematicalobjects is, in principle, made. Although a primitive modal notion isintroduced, it has only a limited role in the nominalization ofmathematics: to allow for a nominalist formulation of the crucialconcept of conservativeness. As we saw, however, without a propernominalization of set theory itself, it is unclear whether themathematical fictionalist program ultimately succeeds.

4. Modal Structuralism

4.1 Central features of modal structuralism

Modal structuralism offers a program of interpretation ofmathematics which incorporates two features: (a) anemphasis onstructures as the main subject-matter of mathematics, and (b) acomplete elimination of reference to mathematical objects byinterpretingmathematics in terms of modal logic (as firstsuggested by Putnam (1967), and developed in Hellman (1989,1996)). Given these features, the resulting approach is calledamodal-structural interpretation (Hellman 1989,pp. vii–viii and 6–9).

The proposal is also supposed to meet two important requirements(Hellman 1989, pp. 2–6). The first is that mathematicalstatements should have truth-values, and thus‘instrumentalist’ readings are rejected from theoutset. The second is that: ‘a reasonable account should beforthcoming of how mathematics does in fact apply to the materialworld’ (Hellman 1989, p. 6). Thus, the applicability problemmust be examined.

In order to address these issues, the modal structuralist putsforward a general framework. The main idea is that althoughmathematics is concerned with the study of structures, this study canbe accomplished by focusing only onpossible structures, andnot actual ones. Thus, the modal interpretation is not committed toactual mathematical structures; there is no commitment to theirexistence as objects or to any objects that happen to‘constitute’ these structures. In this way, theontological commitment to them is avoided: the only claim is that thestructures in question arepossible. In order to articulatethis point, the modal-structural interpretation is formulated in asecond-order modal language based on S5. However, to preventcommitment to a set-theoretical characterization of the modaloperators, Hellman takes these operators as primitive (1989, pp. 17,and 20–23).

Two steps are taken. The first is to present an appropriatetranslation scheme in terms of which each ordinary mathematicalstatementS is taken as elliptical for a hypotheticalstatement, namely: thatSwould hold in a structure ofthe appropriate kind.

For example, if we are considering number-theoretic statements,such as those articulated in Peano arithmetic (PA, for short), thestructures we are concerned with are ‘progressions’ or‘ω-sequences’ satisfying PA's axioms. In this case,each particular statementS is to be (roughly) translatedas

☐∀X(X is an ω-sequence satisfying PA's axioms →S holds inX).

According to this statement, if there were ω-sequencessatisfying PA's axioms,S would hold in them. This isthehypothetical component of the modal-structuralinterpretation (for a detailed analysis and a precise formulation, see(Hellman 1989, pp. 16–24)). Thecategorical componentconstitutes the second step (Helman 1989, pp. 24–33). The ideais to assume that the structures of the appropriate kind are logicallypossible. In that case, we have that

◊∃X(X is an ω-sequence satisfying PA's axioms).

That is, it is logically possible that there are ω-sequencessatisfying PA's axioms. Following this approach, truth preservingtranslations of mathematical statements can be presented withoutontological costs, given that only thepossibility of thestructures in question is assumed.

The modal structuralist then indicates that the practice of theoremproving can be regained in this framework (roughly speaking, byapplying the translation scheme to each line of the original proof ofthe theorem under consideration). Moreover, by using the translationscheme and appropriate coding devices, one can argue that arithmetic,real analysis and, to a certain extent, even set theory are recoveredin a nominalist setting (Hellman 1989, pp. 16–33, 44–47,and 53–93). In particular, ‘by making use of codingdevices, virtually all the mathematics commonly encountered in currentphysical theories can be carried out within [real analysis]’(Hellman 1989, pp. 45–46). However, the issue of whether settheory has been nominalized in this way is, in fact,problematic—as the modal structuralist grants. After all, it isno obvious matter to establish even the possibility of the existenceof structures with inaccessibly many objects.

With the framework in place, the modal structuralist can thenconsider the applicability problem. The main idea is to adopt thehypothetical component as the basis for accommodating the applicationof mathematics. The relevant structures are those commonly used inparticular branches of science. Two considerations need to be made atthis point.

The first is the general form of applied mathematical statements(Hellman 1989, pp. 118–124). These statements involve threecrucial components: the structures that are used in appliedmathematics, the non-mathematical objects to which the mathematicalstructures are applied, and a statement of application that specifiesthe particular relations between the mathematical structures and thenon-mathematical objects. The relevant mathematical structures can beformulated in set theory. Let us call the set theory used in appliedcontextsZ. (This is second-order Zermelo set theory, which isfinitely axiomatizable; I'll denote the conjunction of the axiomsofZ by ∧Z.) The non-mathematical objects ofinterest in the context of application can be expressed inZasUrelemente, that is, as objects that are not sets. We willtake ‘U’ to be the statement that certainnon-mathematical objects of interest are includedasUrelemente in the structures ofZ. Finally,‘A’ is the statement of application, describing theparticular relations between the relevant mathematical structuresofZ and the non-mathematical objects describedinU. The particular relations involved depend on the case inquestion. We can now present the general form of an appliedmathematical statement (Hellman 1989, p. 119):

☐∀X∀f ((∧Z &U)^X (∈_f) →A).

In the antecedent ‘(∧Z&U)^X (∈_f)’is an abbreviation for the results from writing out the axioms ofZermelo set theory with all quantifiers relativized to the secondorder variableX, replacing each occurrence of the membershipsymbol ‘∈’ with the two place relation variable‘f’. According to the applied mathematicalstatement, if there were structures satisfying the conjunction of theaxioms of Zermelo set theoryZ including some non-mathematicalobjects referred to inU,A would hold in suchstructures. The application statementA expresses the relationsin questions, such as an isomorphism or a homomorphism between aphysical system and certain set theoretic structures. This is thehypothetical component interpreted to express which relations wouldhold between certain mathematical structures (formulated as structuresof ∧Z) and the entities studied in the world(theUrelemente).

The second consideration examines in more detail the relationshipsbetween the physical (or the material) objects studied and themathematical framework. These are the “syntheticdetermination” relations (Hellman 1989, pp. 124–135). Morespecifically, we have to determine which relations amongnon-mathematical objects can be taken, in the antecedent of an appliedmathematical statement, as the basis for specifying “the actualmaterial situation” (Hellman 1989, p. 129). The modalstructuralist proposal is to consider the models of a comprehensivetheoryT′. This theory embraces and links the vocabularyof theapplied mathematical theory (T) and thesynthetic vocabulary (S) in question, which intuitively fixesthe actual material situation. It is assumed thatT determines,up to isomorphism, a particular kind of mathematical structure(containing, for example,Z), and thatT′ is anextension ofT. In that case, a proposed “syntheticbasis” will be adequate if the following condition holds:

Leta be the class of (mathematically) standard models ofT′, and letV denote the full vocabulary ofT′: thenS determinesV ina iff for any two modelsm andm′ ina, and any bijectionf between their domains, iff is anS isomorphism, it is also aV isomorphism. (Hellman 1989, p. 132.)

The introduction of isomorphism in this context comes, of course,from the need to accommodate the preservation of structure between the(applied) mathematical part of the domain under study and thenon-mathematical part. This holds in the crucial case in which thepreservation of the synthetic properties and relations(S-isomorphism) byf leads to the preservation of theanalytic applied mathematical relations (V-isomorphism) of theoverall theoryT′. It should be noted that the‘synthetic’ structure is not meant to‘capture’ thefull structure of the mathematicaltheory in question, but only itsapplied part. (Recall thatHellman started with anapplied mathematicaltheoryT.)

This can be illustrated with a simple example. Suppose thatfinitely many physical objects display a linear order. We can describethis by defining a function from those objects to an initial segmentof the natural numbers. What is required by the modal structuralist'ssynthetic determination condition is that the physical ordering amongthe objects alone captures this function and the description it offersof the objects. It isnot claimed that thefullnatural number structure is thus captured. This example also providesan illustration of the applied mathematical statement mentionedabove. TheUrelemente (objects that are not sets) are thephysical objects in question, the relevant mathematical relation isisomorphism, and the mathematical structure is a segment of naturalnumbers with their usual linear order.

On the modal structural conception, mathematics is applied byestablishing an appropriate isomorphism between (parts of)mathematical structures and those structures that represent thematerial situation. This procedure is justified, since suchisomorphism establishes the structural equivalence between the(relevant parts of the) mathematical and the non-mathematicallevels.

However, this proposal faces two difficulties. The first concernsthe ontological status of the structural equivalence between the(applied) mathematical and the non-mathematical domains. On whatgrounds can we claim that the structures under consideration aremathematically the same if some of them concern ‘material’objects? Of course, given that the structural equivalence isestablished by an isomorphism the material objects are alreadyformulated in structural terms—this means that some mathematicshas already been applied to the domain in question. In other words, inorder to be able to represent the applicability of mathematics,Hellman assumes that some mathematics has already been applied. Thismeans that a purely mathematical characterization of the applicabilityof mathematics (via structure preservation) is inherentlyincomplete. Thefirst step in the application, namely themathematical modeling of the material domain, is not, and cannot, beaccommodated, since no isomorphism is involved there. Indeed, giventhat by hypothesis the domain isnot articulated inmathematical terms, no isomorphism isdefined there.

It may be argued that the modal structural account does not requirean isomorphism between (applied) mathematical structures and thosedescribing the material situation. The account only requires anisomorphism between two standard models of the overalltheoryT′, which links the mathematical theoryTand the descriptionS of the material domain. In reply, notethat this only moves the difficulty one level up. In orderforT′ to extend the applied mathematical theoryTand to provide a link betweenT and the material situation, amodel ofT′ will have to be, in particular, a model ofbothT andS. Thus, if the modal structuralist'ssynthetic determination claim is satisfied, an isomorphism between twomodels ofT′ will determine an isomorphism between themodels ofS and those ofT. In this way, an isomorphismbetween structures describing the material situation and those arisingfrom applied mathematics is still required.

The second difficulty addresses the epistemological status of theclaim that there is a structural equivalence between the mathematicaland the non-mathematical domains. On what grounds do weknowthat such equivalence holds? Someone may say that the equivalence isnormatively imposed in order for the application process to get offthe ground. But this suggestion leads to a dilemma. Either it is justassumed that we know that the equivalence holds, and theepistemological question is begged (given that the grounds for thisare in question), or it is assumed that we donot know thatthe equivalence holds—and that is why we have toimposethe condition—in which case the latter is clearlygroundless. However, it may be argued that there is no problem here,since we establish the isomorphism by examining the physical theoriesof the material objects under consideration. But the problem is thatin order to formulate these physical theories we typically usemathematics. And the issue is precisely to explain this use, that is,to provide some understanding of the grounds in terms of which we cometo know that the relevant mathematical structures are isomorphic tothe physical ones.

The main point underlying these considerations has been stressedoften enough (although in a different context): isomorphism does notseem to be an appropriate condition for capturing the relation betweenmathematical structures and the world (see, e.g., da Costa and French2003). There is, of course, a correct intuition underlying the use ofisomorphism at this level, and this relates to the idea of justifyingthe application of mathematics: the isomorphism does guarantee thatapplied mathematical structuresS and the structuresMwhich represent the material situation are mathematically thesame. The problem is that isomorphism-based characterizations tend tobe unrealistically strong. They require that some mathematics hasalready been applied to the material situation, and that we haveknowledge of the structural equivalence betweenSandM. What is needed is a framework in which the relationbetween the relevant structures is weaker than isomorphism, but whichstill supports the applicability, albeit in a less demanding way(e.g., Bueno, French and Ladyman 2002).

4.2 Assessment: benefits and problems of modal structuralism

4.2.1 The epistemological problem

The modal structuralist solves partially the epistemologicalproblem for mathematics. Assuming that the modal-structuraltranslation scheme works for set theory, modal structuralists need notexplain how we can have knowledge of the existence of mathematicalobjects, relations or structures—given the lack of commitment tothese entities. However, they still need to explain our knowledge ofthe possibility of the relevant structures, since the translationscheme commits them to such possibility.

One worry that emerges here is that, in the case of substantivemathematical structures (such as those invoked in set theory),knowledge of the possibility of such structures may require knowledgeof substantial parts of mathematics. For instance, in order to knowthat the structures formulated in Zermelo set theory are possible,presumably we need to know that the theory itself is consistent. Butthe consistency of the theory can only be established in anothertheory, whose consistency, in turn, also needs to beestablished—and we face a regress. It would be arbitrary simplyto assume the consistency of the theories in question, given that ifsuch theories turn out to be in fact inconsistent, given classicallogic, everything could be proved in them.

Of course, these considerations do not establish that the modalstructuralist cannot develop an epistemology for mathematics. Theyjust suggest that further developments on the epistemological frontseem to be called for in order to address more fully theepistemological problem for mathematics.

4.2.2 The problem of the application of mathematics

Similarly, the problem of the application of mathematics ispartially solved by the modal structuralist. After all, a framework tointerpret the use of mathematics in science is provided, and in termsof this framework the application of mathematics can be accommodatedwithout the commitment to the existence of the correspondingobjects.

One concern that emerges (besides those already mentioned at theend of section 4.1 above) is that, similarly to what happens tomathematical fictionalism, the proposed framework does not allow us tomake sense of actual uses of the application of mathematics. Ratherthan explaining how mathematics is in fact applied in scientificpractice, the modal-structural framework is advanced in order toregiment that practice and dispense with the commitment tomathematical entities. But even if the framework succeeds at thelatter task, thus allowing the modal structuralist to avoid therelevant commitment, the issue of how to make sense of the waymathematics is actually used in scientific contexts stillremains. Providing a translation scheme into a nominalistic languagedoes not address this issue. A significant aspect of mathematicalpractice is then left unaccounted for.

The status of the indispensability argument within themodal-structural interpretation is quite unique. On the one hand, theconclusion of the argument is undermined (if the proposed translationscheme goes through), since commitment to the existence ofmathematical objects can be avoided. On the other hand, a revisedversion of the indispensability argument can be used to motivate thetranslation into the modal language, thus emphasizing theindispensable role played by the primitive modal notions introduced bythe modal structuralist. The idea is to change the argument's secondpremise, insisting thatmodal-structural translations ofmathematical theories are indispensable to our best theories of theworld, and concluding that we ought to be ontologically committed tothe possibility of the corresponding structures. In this sense, modalstructuralists can invoke the indispensability argument in support ofthe translation scheme they favor and, hence, the possibility of therelevant structures, which are referred to in the conclusion of therevised argument. But rather than supporting the existence ofmathematical objects, the argument would only support commitment tomodal-structural translations of mathematical theories and thepossibility of mathematical structures.

4.2.3 Uniform semantics

With the introduction of modal operators and the proposedtranslation scheme, the modal structuralist is unable to provide auniform semantics for scientific and mathematical theories. Only thelatter, as opposed to the former, requires such operators. In fact,Field has argued that if modal operators were invoked in theformulation of scientific theories, not only their mathematicalcontent, but also their physical content would be nominalized (Field1989). After all, in that case, instead of asserting that somephysical situation is actually the case, the theory would only statethe possibility that this is so.

One strategy to avoid this difficulty (of losing the physicalcontent of a scientific theory due to the use of modal operators) isto employ an actuality operator. By properly placing this operatorwithin the scope of the modal operators, it is possible to undo thenominalization of the physical content in question (Friedman2005). Without the introduction of the actuality operator, or somerelated maneuver, it is unclear that the modal structuralist would bein a position to preserve the physical content of the scientifictheory in question.

But the introduction of an actuality operator in this contextrequires the distinction between nominalist and mathematicalcontent. (That such a distinction cannot be drawn at all is argued inAzzouni 2011.) Otherwise, there is no guarantee that the applicationof the actuality operator will not yield more than what is physicallyreal.

However, even with the introduction of such an operator, therewould still be a significant difference, on the modal-structuraltranslation scheme, between the semantics for mathematical andscientific discourse. For the former, as opposed to the latter, doesnot invoke such an operator. The result is that modal structuralismdoes not seem to be able to provide a uniform semantics formathematical and scientific language.

4.2.4 Taking mathematics literally

Given the need for introducing modal operators, the modalstructuralist does not take mathematical discourse literally. In fact,it may be argued, this is the whole point of the view! Takenliterally, mathematical discourse seems to be committed to abstractobjects and structures—a commitment that the modal structuralistclearly aims to avoid.

However, the point still stands that, in order to block suchcommitment, a parallel discourse to actual mathematical practice isoffered. The discourse is ‘parallel’ given thatmathematical practice typically does not invoke the modal operatorsintroduced by the modal structuralist. For those who aim to understandmathematical discourse as it is used in the practice of mathematics,and who try to identify suitable features of that practice thatprevent commitment to mathematical entities, the proposed translationwill make the realization of that goal particularly difficult.

4.2.5 The ontological problem

The modal structuralist has solved, in part, the ontologicalproblem. No commitment to mathematical objects or structures seems tobe needed to implement the proposed translation scheme. The mainconcern emerges from the introduction of modal operators. But as themodal structuralist emphasizes, these operators do not presuppose apossible-worlds semantics: they are introduced as primitive terms.

However, since the modal translation of mathematical axioms istaken to be true, the question arises as to what makes suchstatementstrue. For instance, when it is asserted that‘it is possible that there are structures satisfying the axiomsof Peano Arithmetic’, what is responsible for the truth of suchstatement? Clearly, the modal structuralist will not ground thepossibility in question on the actual truth of the Peano axioms, forthis move, on a reasonable interpretation, would requireplatonism. Nor will the modal structuralist support the relevantpossibility on the basis of the existence of a consistency proof forthe Peano axioms. After all, any such proof is an abstract object, andto invoke it at the foundation of modal structuralism clearlythreatens the coherence of the overall view. Furthermore, to invoke amodalized version of such a consistency proof would beg the question,since it assumes that the use of modal operators is alreadyjustified. Ultimately, what is needed to solve properly theontological problem is a suitable account of modal discourse.

5. Deflationary Nominalism

5.1 Central features of deflationary nominalism

According to the deflationary nominalist, it is perfectlyconsistent to insist that mathematical theories are indispensable toscience, to assert that mathematical and scientific theories are true,and to deny that mathematical objects exist. I am calling the view‘deflationary nominalism’ given that it demands veryminimal commitments to make sense of mathematics (Azzouni 2004), itadvances a deflationary view of truth (Azzouni 2004, 2006), andadvocates a direct formulation of mathematical theories, withoutrequiring that they be reconstructed or rewritten (Azzouni 1994,2004).

Deflationary nominalism offers an ‘easy road’ tonominalism, which does not require any form of reformulation ofmathematical discourse, while granting the indispensability ofmathematics. Despite the fact that quantification over mathematicalobjects and relations is indispensable to our best theories of theworld, this fact offers no reason to believe in the existence of thecorresponding entities. This is because, as Jody Azzouni points out,two kinds of commitment should be distinguished:quantifiercommitment andontological commitment (Azzouni 1997; 2004,p. 127 and pp. 49–122). We incur a quantifier commitmentwhenever our theories imply existentially quantified statements. Butexistential quantification, Azzouni insists, is not sufficient forontological commitment. After all, we often quantify over objects wehave no reason to believe exist, such as fictional entities.

To incur anontological commitment—that is, to becommitted to the existence of a given object—a criterion forwhat exists needs to be satisfied. There are, of course, variouspossible criteria for what exists (such as causal efficacy,observability, possibility of detection, and so on). But the criterionAzzouni favors, and he takes it to be the one that has beencollectively adopted, is ontological independence (2004, p. 99). Whatexist are the things that are ontologically independent of ourlinguistic practices and psychological processes. The point is that ifwe have just made something up through our linguistic practices orpsychological processes, there's no need for us to be committed to theexistence of the corresponding object. And typically, we would resistany such commitment.

Do psychological processes themselves exist, according to theontologically independence criterion? It may be argued that mostpsychological processes do exist, at least those we undergo ratherthan those we make up. After all, the motivation underlying theindependence criterion is that those things we just made up verballyor psychologically do not exist. Having a headache or believing thatthere is a laptop computer in front of me now are psychologicalprocesses that I did not make up. Therefore, it seems that at leastthese kinds of psychological processes do exist. In contrast,imaginings, desires, and hopes are processes we make up, and thus theydo not exist. However, the underlying motivation for the criterionseems to diverge, in these cases, from what is entailed by thecriterion's actual formulation. For the criterion insists on theontological independence of “our linguistic practices andpsychological processes”. Since headaches and beliefs arepsychological processes themselves, presumably they arenotontologically independent of psychological processes. Hence, they donot exist. This means that if the criterion is applied as stated, nopsychological process exists. For similar reasons, novels, mentalcontents, and institutions do not exist either, since they are allabstract objects dependent on our linguistic practices andpsychological processes, according to the deflationary nominalist(Azzouni 2010a, 2012).

Quine, of course, identifies quantifier and ontologicalcommitments, at least in the crucial case of the objects that areindispensable to our best theories of the world. Such objects arethose that cannot be eliminated through paraphrase and over which wehave to quantify when we regiment the relevant theories (usingfirst-order logic). According to Quine's criterion, these areprecisely the objects we are ontologically committed to. Azzouniinsists that we should resist this identification. Even if the objectsin our best theories are indispensable, even if we quantify over them,this is not sufficient for us to be ontologically committed tothem. After all, the objects we quantify over might beontologicallydependent on us—on our linguisticpractices or psychological processes—and thus we might have justmade them up. But, in this case, clearly there is no reason to becommitted to their existence. However, for those objectsthatare ontologically independent of us, wearecommitted to their existence.

As it turns out, on Azzouni's view, mathematical objects areontologically dependent on our linguistic practices and psychologicalprocesses. And so, even though they may be indispensable to our besttheories of the world, we are not ontologically committed tothem. Hence, deflationary nominalism is indeed a form ofnominalism.

But in what sense do mathematical objects depend on our linguisticpractices and psychological processes? In the sense that the sheerpostulation of certain principles is enough for mathematical practice:‘A mathematical subject with its accompanying posits can becreatedex nihilo by simply writing down a set of axioms’(Azzouni 2004, p. 127). The only additional constraint that sheerpostulation has to meet, in practice, is that mathematicians shouldfind the resulting mathematics interesting. That is, the consequencesthat follow from the relevant mathematical principles shouldn't beobvious, and they should be computationally tractable. Thus, giventhat sheer postulation is (basically) enough in mathematics,mathematical objects have no epistemic burdens. Such objects, or‘posits’, are calledultrathin (Azzouni 2004,p. 127).

The same move that the deflationary nominalist makes to distinguishontological commitment from quantifier commitment is also used todistinguish ontological commitment toFs from assertingthetruth of ‘There areFs’. Althoughmathematical theories used in science are (taken to be) true, this isnot sufficient to commit us to the existence of the objects thesetheories are supposedly about. After all, according to thedeflationary nominalist, it may betrue that thereareFs, but to be ontologically committed toFs, acriterion for what exists needs to be satisfied. As Azzouni pointsout:

I take true mathematical statements as literallytrue; I forgo attempts to show that such literally true mathematical statements arenot indispensable to empirical science, and yet, nonetheless, I can describe mathematical terms as referring to nothing at all. Without Quine's criterion to corrupt them, existential statements are innocent of ontology. (Azzouni 2004, pp. 4–5.)

On the deflationary nominalist picture, ontological commitment isnot signaled in any special way in natural (or even formal)language. We just don't read off the ontological commitment ofscientific doctrines (even if they were suitably regimented). Afterall, without Quine's criterion of ontological commitment, neitherquantification over a given object (in a first-order language) norformulation of true claims about such an object entails the existenceof the latter.

In his 1994 book, Azzouni did not commit himself to nominalism, onthe grounds that nominalists typically require a reconstruction ofmathematical language—something that, as discussed above, isindeed the case with both mathematical fictionalism (Field 1989) andmodal structuralism (Hellman 1989). However, no such reconstructionwas implemented, or needed, in the proposal advanced by Azzouni(Azzouni 1994). The fact that mathematical objects play no role in howmathematical truths are known clearly expresses a nominalistattitude—an attitude that Azzouni explicitly endorsed in(Azzouni 2004).

The deflationary nominalist proposal nicely expresses a view thatshould be taken seriously. And as opposed to other versions ofnominalism, it has the significant benefit of aiming to takemathematical discourse literally.

5.2 Assessment: benefits of deflationary nominalism and a problem

Of the nominalist views discussed in this essay, deflationarynominalism is the view that comes closest to solving (or, in somecases, dissolving) the five problems that have been used to assessnominalist proposals. With the possible exceptions of the issue oftaking mathematical language literally and the ontological problem,all of the remaining problems are explicitly and successfullyaddressed. I will discuss each of them in turn.

5.2.1 The epistemological problem dissolved

How can the deflationary nominalist explain the possibility ofmathematical knowledge, given the abstract nature of mathematicalobjects? On this version of nominalism, this problemvanishes. Mathematical knowledge is ultimately obtained from whatfollows from mathematical principles. Given that mathematical objectsdo not exist, they play no role in how mathematical results are known(Azzouni 1994). What is required is that the relevant mathematicalresult be established via a proof. Proofs are the source ofmathematical knowledge.

It might be argued that certain mathematical statements are knownwithout the corresponding proof. Consider the Gödel sentenceinvoked in the proof of Gödel's incompleteness theorem: thesentence is true, but it cannot be proved in the system underconsideration (if the latter is consistent). Do we have knowledge ofthe Gödel sentence? Clearly we do, despite the fact that thesentence is not derivable in the system in question. As a result, theknowledge involved here is of a different sort than the onearticulated in terms of what can be proved in a given system.

In my view, the deflationary nominalist has no problem making senseof our knowledge of the Gödel sentence. It is an intuitive sortof knowledge, which emerges from what the sentence in questionstates. All that is required in order to know that the sentence istrue is to properly understand it. But that's not how mathematicalresults are typically established: they need to be proved.

According to Azzouni, we know the Gödel sentence as long as weare able to embed the syntactically incomplete system (such as Peanoarithmetic) in a stronger system in which the truth predicate for theoriginal system occurs and in which the Gödel sentence is proved(Azzouni 1994, pp. 134–135; Azzouni 2006, p. 89, note 38, lastparagraph, and pp. 161–162).

Clearly, the account does not turn mathematical knowledge intosomething easy to obtain, given that, normally, it is nostraightforward matter to determine whether some result follows from agiven group of axioms. Part of the difficulty emerges from the factthat the logical consequences of a non-trivial group of axioms areoften not transparent: significant work is required to establish suchconsequences. This is as it should be, given the non-trivial nature ofmathematical knowledge.

5.2.2 Dissolving the problem of the application of mathematics

The deflationary nominalist offers various considerations to theeffect that there is no genuine philosophical problem in the successof applied mathematics (Azzouni 2000). Once particular attention isgiven to implicational opacity—our inability to see, before aproof is offered, the consequences of various mathematicalstatements—much of the alleged surprise in the successfulapplication of mathematics should vanish. Ultimately, the so-calledproblem of the application of mathematics—of understanding howit is possible that mathematics can be successfully applied to thephysical world should—is an artificially designed issue ratherthan a genuine problem.

Colyvan defends the opposing view (Colyvan 2001b), insisting thatthe application of mathematics to science does present a genuineproblem. In particular, he argues that two major philosophicalaccounts of mathematics, Field's mathematical fictionalism and Quine'splatonist realism, are unable to explain the problem. Thus, heconcludes that the problem cuts across the realism/anti-realism debatein the philosophy of mathematics. The deflationary nominalist wouldinsist that what is ultimately at issue—implicationalopacity—is not a special problem, even though to the extent thatit is a problem, it is one that is equally faced by realists andanti-realists about mathematics.

This does not mean that the application of mathematics is astraightforward matter. Clearly, it is not. But the difficultiesinvolved in the successful application of mathematics do not raise aspecial philosophical problem, particularly as soon as the issue ofimplicational opacity is acknowledged—an issue that is common toboth pure and applied mathematics.

The issue of understanding the way in which mathematics in factgets applied is something that the deflationary nominalist explicitlyaddresses, carefully examining the central features and thelimitations of different models of the application of mathematics(see, in particular, the second part of (Azzouni 2004)).

5.2.3 Uniform semantics

The deflationary nominalist, as noted above, is not committed tooffering a reconstruction, or any kind of reformulation, ofmathematical theories. (The exception here is the case of inconsistentmathematical or scientific theories, which according to thedeflationary nominalist, ideally are regimented as consistentfirst-order theories.) No special semantics is required to make senseof mathematics: the same semantics that is used in the case ofscientific theories is invoked for mathematical theories. It may seemthat the uniform semantics requirement is thus satisfied. But thesituation is more complicated.

It may be argued that the deflationary nominalist needs to providethe semantics for the existential and universal claims in mathematics,science and ordinary language. After all, it does sound puzzling tostate: “It is true that there are numbers, but numbers do notexist”. What is such semantics? The deflationary nominalist willrespond by noting that this semantics is precisely the standardsemantics of classical logic, with the familiar conditions for theexistential and universal quantifiers, but without the assumption thatsuch quantifiers are ontologically committing. The fact that noontological import is assigned to the quantifiers does not changetheir semantics. After all, the metalanguage in which the semantics isdeveloped already has universal and existential quantifiers, and thesequantifiers need not be interpreted as providing ontologicalcommitment any more than the object language quantifiers do. As aresult, the same semantics is used throughout.

It may be argued that the deflationary nominalist needs tointroduce the distinction between ontologically serious (orontologically committing) uses of the quantifiers and ontologicallyinnocent (or ontologically non-committing) uses. If so, this wouldpresumably require a different semantics for these quantifiers. Inresponse, the deflationary nominalist will deny the need for suchdistinction. In order to mark ontological commitment, an existencepredicate, which expresses ontological independence, is used. Thosethings that are ontologically independent of us (that is, of ourlinguistic practices and psychological processes) are those to whichwe are ontologically committed. The mark of ontological commitment isnot made at the level of the quantifiers, but via the existencepredicate.

This means, however, that even though the semantics is uniformthroughout the sciences, mathematics and ordinary language,deflationary nominalism requires the introduction of the existencepredicate. But, at least on the surface, this predicate does not seemto have a counterpart in the way language is used in these domains. Itis the same semantics throughout, but the formalization of thediscourse requires an extended language to accommodate the existencepredicate. As a result, the uniformity of the semantics comes with thecost of the introduction of a special predicate into the language tomark ontological commitment for formalization.

Perhaps the deflationary nominalist will respond by arguing thatthe existence predicate is already part of the language, maybeimplicitly via contextual and rhetorical factors (Azzouni 2007,Section III; Azzouni 2004, Chapter 5). What would be needed then isevidence for such a claim, and an indication of how exactly thepredicate is in fact found in scientific, mathematical and ordinarycontexts. Consider, for instance, the sentences:

(S) There is no set of all sets.
(P) Perfectly frictionless planes do not exist.
(M) Mice exist; talking mice don't.

Presumably, in all of these cases the existence predicate isused. As a result, the sentences could be formalized as follows:

(S) ∀x(Sx → ¬Ex), where ‘S’ is (for simplicity) the predicate ‘set of all sets’, and ‘E’ is the existence predicate.
(P) ∀x(Px → ¬Ex), where ‘P’ is (for simplicity) the predicate ‘perfectly frictionless plane’, and ‘E’ is the existence predicate.
(M) ∃x(Mx ∧Ex) ∧ ∀x((Mx ∧Tx) → ¬Ex), where ‘M’ is the predicate ‘mice’, ‘T’ is the predicate ‘to talk’, and ‘E’ is the existence predicate.

In all of these cases, the formalization requires some change inthe logical form of the natural language sentences in order tointroduce the existence predicate. And that is arguably a cost for theview. After all, in these cases, mathematical, scientific and ordinarylanguages do not seem to be taken literally—a topic to which Iturn now.

5.2.4 Taking mathematical language literally

We saw that with the introduction of the existence predicate it isnot clear that the deflationary nominalist is in fact able to takemathematical language literally. After all, some reconstruction ofthat language seems to be needed. It should be granted that the levelof reconstruction involved is significantly less than what is found inthe other versions of nominalism discussed above. As opposed to them,the deflationary nominalist is able to accommodate significant aspectsof mathematical practice without the need for creating a full paralleldiscourse (in particular, no operators, modal or fictional, need to beintroduced). However, some level of reconstruction is still needed toaccommodate the existence predicate, which then compromises thedeflationary nominalist's capacity to take mathematical languageliterally.

A related concern is that the deflationary nominalist introduces anon-standard notion of reference that does not presuppose theexistence of the objects that are referred to (Bueno and Zalta2005). This move goes hand in hand with the understanding of thequantifiers as not being ontologically committing, and it does seem tolimit the deflationary nominalist's capacity to take mathematicallanguage literally. After all, a special use of ‘refers’is needed to accommodate the claim that “‘a’refers tob, butb does not exist”. Thedeflationary nominalist, however, resists this charge (Azzouni 2009a,2010a, 2010b).

5.2.5 The ontological problem

The ontological problem is also dissolved by the deflationarynominalism. Clearly, deflationary nominalism has no commitment eitherto mathematical objects or to a modal ontology of any kind (includingpossible worlds, abstract entities as proxy for possible worlds, orother forms of replacement for the expression of modal claims). Thedeflationary nominalists not only avoid the commitment to mathematicalobjects, they also claim that such objects have no propertieswhatsoever. This means that the deflationary nominalist's ontology isextremely minimal: only concrete objects are ultimatelyassumed—objects that are ontologically independent of ourpsychological processes and linguistic practices. In particular, nodomain of nonexistent objects is posited nor a realm of genuineproperties of such objects. By ‘genuine properties’ I meanthose properties that hold only in virtue of what the objects inquestion are, and not as the result of some external relations toother objects. For example, although Sherlock Holmes does not exist,he has the property of being thought of by me as I write thissentence. This is not, however, a genuine property of Sherlock Holmesin the intended sense.

Deflationary nominalism is not a form of Meinongianism (Azzouni2010a). Although the ontology of deflationary nominalism is notsignificantly different from that of the Meinongian, the ideology ofthe two views—at least assuming a particular, traditionalinterpretation of Meinongianism—is importantly different. Thedeflationary nominalist is not committed to any subsisting objects, incontrast to what is often claimed to be a distinctive feature ofMeinongianism.

It is not clear to me, however, that this traditional reading ofMeinongianism is correct. If we consider the subsisting objects asthose that are abstract, and if we take only concrete objects asexisting, the resulting picture ideologically is not significantlydifferent from the one favored by the deflationary nominalist. Still,the deflationary nominalists distance themselves from Meinongianism(Azzouni 2010a).

With the meager ontological commitments, the deflationarynominalist fares very well on the ontological front. One source ofconcern is how meager the deflationary nominalist's ontologyultimately is. For instance, platonists would insist that mathematicalobjects are ontologically independent of our psychological processesand linguistic practices, and—using the criterion of ontologicalcommitment offered by the deflationary nominalist—they wouldinsist that these objectsdo exist. Similarly, modal realists(such as Lewis 1986) would also argue that possible worlds areontologically independent of us in the relevant sense, thus concludingthat these objects also exist. Deflationary nominalists will try toresist these conclusions. But unless their arguments are successful atthis point, the concern remains that the deflationary nominalist mayhave a significantly more robust ontology—given the proposedcriterion of ontological commitment—than advertised.

It may be argued that deflationary nominalists are changing therules of the debate. They state that mathematicians derive statementsof the form “There areFs”, but insist that theobjects in question do not exist, given that quantifier commitment andontological commitment should be distinguished. This strategy isfundamentally different from those found in the nominalist proposalsdiscussed before. It amounts to the denial of the first premise of theindispensability argument (“We ought to be ontologicallycommitted to all and only those entities that are indispensable to ourbest theories of the world”). Even though quantification overmathematical entities is indispensable to our best theories of theworld (thus, the deflationary nominalist accepts the second premise ofthe argument), this fact does not entail that these entitiesexist. After all, we can quantify over objects that do not exist,given the rejection of the indispensability argument's firstpremise.

But are deflationary nominalists really changing the rules of thedebate? If Quine's criterion of ontological commitment provides suchrules, then they are. But why should we grant that Quine's criterionplay such a role? Deflationary nominalists challenge this deeply heldassumption in ontological debates. And by doing so, they pave the wayfor the development of a distinctive form of nominalism in thephilosophy of mathematics.

Bibliography

Azzouni, J., 1994,Metaphysical Myths, Mathematical Practice:The Ontology and Epistemology of the Exact Sciences, Cambridge:Cambridge University Press.
–––, 1997, “Applied Mathematics,Existential Commitment and the Quine-Putnam IndispensabilityThesis”,Philosophia Mathematica, 5:193–209.
–––, 2000, “Applying Mathematics: AnAttempt to Design a Philosophical Problem”,The Monist,83: 209–227.
–––, 2004,Deflating Existential Consequence:A Case for Nominalism, New York: Oxford University Press.
–––, 2006,Tracking Reason: Proof,Consequence, and Truth, New York: Oxford University Press.
–––, 2007, “Ontological Commitment in theVernacular”,Noûs, 41: 204–226.
–––, 2009a, “Emptyde re Attitudesabout Numbers”,Philosophia Mathematica, 17:163–188.
––– 2009b, “Evading Truth Commitments: TheProblem Reanalyzed”,Logique et Analyse, 206:139–176.
–––, 2010a,Talking about Nothing: Numbers,Hallucinations, and Fictions, New York: Oxford UniversityPress.
––– 2010b, “Ontology and the Word‘Exist’: Uneasy Relations”,PhilosophiaMathematica, 18: 74–101.
–––, 2011, “Nominalistic Content”,in Cellucci, Grosholtz, and Ippoliti (eds.) 2011,pp. 33–51.
–––, 2012, “Taking the Easy Road Out ofDodge”,Mind, 121: 951–965. (Paper published in2013.)
Balaguer, M., 1998,Platonism and Anti-Platonism inMathematics, New York: Oxford University Press.
Benacerraf, P., 1973, “Mathematical Truth”,Journalof Philosophy, 70: 661–679.
Benacerraf, P., and Putnam, H. (eds.), 1983,Philosophy ofMathematics: Selected Readings, second edition, Cambridge:Cambridge University Press.
Boolos, G., and Jeffrey, R., 1989,Computability andLogic, third edition, Cambridge: Cambridge UniversityPress.
Bueno, O., 2003, “Is It Possible to Nominalize QuantumMechanics?”,Philosophy of Science, 70:1424–1436.
–––, 2008, “Nominalism and MathematicalIntuition”,Protosociology, 25: 89–107.
–––, 2009, “MathematicalFictionalism”, in Bueno and Linnebo (eds.) 2009,pp. 59–79.
Bueno, O., French, S., and Ladyman, J., 2002, “OnRepresenting the Relationship between the Mathematical and theEmpirical”,Philosophy of Science, 69:497–518.
Bueno, O., and Linnebo, Ø. (eds.), 2009,New Waves inPhilosophy of Mathematics, Hampshire: Palgrave MacMillan.
Bueno, O., and Zalta, E., 2005, “A Nominalist's Dilemma andits Solution”,Philosophia Mathematica, 13:294–307.
Burgess, J., and Rosen, G., 1997,A Subject With No Object:Strategies for Nominalistic Interpretation of Mathematics, Oxford:Clarendon Press.
Cellucci, C., Grosholtz, E., and Ippoliti, E. (eds.),2011,Logic and Knowledge, Newcastle: Cambridge ScholarsPublishing.
Chihara, C.S., 1990,Constructibility and MathematicalExistence, Oxford: Clarendon Press.
Colyvan, M., 2001a,The Indispensability ofMathematics, New York: Oxford University Press.
–––, 2001b, “The Miracle of AppliedMathematics”,Synthese, 127: 265–277.
–––, 2010, “There's No Easy Road toNominalism”,Mind, 119: 285–306.
da Costa, N., and French, S., 2003,Science and PartialTruth, New York: Oxford University Press.
Field, H., 1980,Science without Numbers: A Defense ofNominalism, Princeton, N.J.: Princeton University Press.
–––, 1984, “Is Mathematical Knowledge JustLogical Knowledge?”,Philosophical Review, 93:509–552; reprinted with a postscript and some changes inField 1989, pp. 79–124.
–––, 1989,Realism, Mathematics andModality, Oxford: Basil Blackwell.
–––, 1991, “Metalogic andModality”,Philosophical Studies, 62: 1–22.
–––, 1992, “A Nominalistic Proof of theConservativeness of Set Theory”,Journal of PhilosophicalLogic, 21: 111–123.
Friedman, J., 2005, “Modal Platonism: An Easy Way to AvoidOntological Commitment to Abstract Entities”,Journal ofPhilosophical Logic, 34: 227–273.
Gödel, K., 1944, “Russell's Mathematical Logic”,in Schilpp (ed.) 1944, pp. 125–153; reprinted in Benacerraf andPutnam (eds.) 1983, pp. 447–469.
–––, 1947, “What Is Cantor's ContinuumProblem?”,American Mathematical Monthly, 54:515–525; revised and expanded version in Benacerraf and Putnam(eds.) 1983, pp. 470–485.
Goodman, N., and Quine, W. V., 1947, “Steps Toward aConstructive Nominalism”,Journal of Symbolic Logic, 12:105–122.
Hale, B., and Wright, C., 2001,The Reason's Proper Study:Essays Towards a Neo-Fregean Philosophy of Mathematics, Oxford:Clarendon Press.
Hellman, G., 1989,Mathematics without Numbers: Towards aModal-Structural Interpretation, Oxford: Clarendon Press.
–––, 1996, “Structuralism withoutStructures”,Philosophia Mathematica, 4:100–123.
Hilbert, D., 1971,Foundations of Geometry, La Salle: OpenCourt; English translation of the tenth German edition, published in1968.
Leng, M., 2010,Mathematics and Reality, New York: OxfordUniversity Press.
Lewis, D., 1986,On the Plurality of Worlds, Oxford:Blackwell.
Malament, D., 1982, “Review of Field,Science withoutNumbers”,Journal of Philosophy, 79:523–534.
Melia, J., 1995, “On What There'sNot”,Analysis, 55: 223–229.
–––, 1998, “Field's Programme: SomeInterference”,Analysis, 58: 63–71.
–––, 2000, “Weaseling Away theIndispensability Argument”,Mind, 109:455–479.
Putnam, H., 1967, “Mathematics without Foundations”,The Journal of Philosophy, 64(1): 5–22; reprinted inBenacerraf and Hilary Putnam 1983: 295–312.
–––, 1971,Philosophy of Logic, New York:Harper and Row; reprinted in Putnam 1979, pp. 323–357.
–––, 1979,Mathematics, Matter andMethod (Philosophical Papers: Volume 1), second edition,Cambridge: Cambridge University Press.
Quine, W. V., 1960,Word and Object, Cambridge, Mass.: TheMIT Press.
Resnik, M., 1997,Mathematics as a Science ofPatterns, Oxford: Clarendon Press.
Schilpp, P.A. (ed.), 1944,The Philosophy of BertrandRussell (The Library of Living Philosophers), Chicago: NorthwesternUniversity.
Shapiro, S., 1997,Philosophy of Mathematics: Structure andOntology, New York: Oxford University Press.
Simons, P., 2010,Philosophy and Logic in Central Europe fromBolzano to Tarski, Dordrecht: Kluwer.
van Fraassen, B.C., 1980,The Scientific Image, Oxford:Clarendon Press.
–––, 1991,Quantum Mechanics: An EmpiricistView, Oxford: Clarendon Press.
von Neumann, J., 1932,Mathematical Foundations of QuantumMechanics, English translation, Princeton: PrincetonUniversity Press, 1955.
Weyl, H., 1928,Gruppentheorie un Quantenmechanik, Leipzig:S. S. Hirzel; English translation,The Theory of Groups and QuantumMechanics, H.P. Roberton (trans.), New York: Dover, 1931.
Yablo, S., 2001, “Go Figure: A Path throughFictionalism”,Midwest Studies in Philosophy, 25:72–102.

Academic Tools

How to cite this entry.
Preview the PDF version of this entry at theFriends of the SEP Society.
Look up topics and thinkers related to this entry at the Internet Philosophy Ontology Project (InPhO).
Enhanced bibliography for this entryatPhilPapers, with links to its database.

Other Internet Resources

FOM: Foundations of Mathematics mailing list

Acknowledgments

My thanks go to two anonymous referees for their helpful commentson earlier versions of this entry. Their suggestions led tosignificant improvements. My thanks are also due to Jody Azzouni, UriNodelman, and Ed Zalta for all of their comments and help.

Open access to the SEP is made possible by a world-wide funding initiative.
The Encyclopedia Now Needs Your Support
Please Read How You Can Help Keep the Encyclopedia Free

Browse

About

Support SEP

Mirror Sites

View this site from another server:

USA (Main Site)Philosophy, Stanford University

Info about mirror sites

Library of Congress Catalog Data: ISSN 1095-5054

	How to cite this entry.
	Preview the PDF version of this entry at theFriends of the SEP Society.
	Look up topics and thinkers related to this entry at the Internet Philosophy Ontology Project (InPhO).
	Enhanced bibliography for this entryatPhilPapers, with links to its database.

Movatterモバイル変換