Nominalism in the Philosophy of Mathematics (Stanford ... · in the philosophy of mathematics forge interconnections with metaphysics (whether mathematical objects do exist), epistemology

9/16/13 2:21 PMNominalism in the Philosophy of Mathematics (Stanford Encyclopedia of Philosophy)

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Nominalism in the Philosophy ofMathematicsFirst published Mon Sep 16, 2013

Nominalism about mathematics (or mathematical nominalism) is the view according towhich either mathematical objects, relations, and structures do not exist at all, or they donot exist as abstract objects (they are neither located in space-time nor do they have causalpowers). In the latter case, some suitable concrete replacement for mathematical objects isprovided. Broadly speaking, there are two forms of mathematical nominalism: those viewsthat require the reformulation of mathematical (or scientific) theories in order to avoid thecommitment to mathematical objects (e.g., Field 1980; Hellman 1989), and those viewsthat do not reformulate mathematical or scientific theories and offer instead an account ofhow no commitment to mathematical objects is involved when these theories are used (e.g.,Azzouni 2004). Both forms of nominalism are examined, and they are assessed in light ofhow they address five central problems in the philosophy of mathematics (namely,problems dealing with the epistemology, the ontology, and the application of mathematicsas well as the use of a uniform semantics and the proviso that mathematical and scientifictheories be taken literally).

1. Two views about mathematics: nominalism and platonism2. Five Problems

2.1 The epistemological problem of mathematics2.2 The problem of the application of mathematics2.3 The problem of uniform semantics2.4 The problem of taking mathematical discourse literally2.5 The ontological problem

3. Mathematical Fictionalism3.1 Central features of mathematical fictionalism3.2 Metalogic and the formulation of conservativeness3.3 Assessment: benefits and problems of mathematical fictionalism

4. Modal Structuralism

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4.1 Central features of modal structuralism4.2 Assessment: benefits and problems of modal structuralism

5. Deflationary Nominalism5.1 Central features of deflationary nominalism5.2 Assessment: benefits of deflationary nominalism and a problem

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1. Two views about mathematics: nominalism andplatonismIn ontological discussions about mathematics, two views are prominent. According toplatonism, mathematical objects (as well as mathematical relations and structures) existand are abstract; that is, they are not located in space and time and have no causalconnection with us. Although this characterization of abstract objects is purely negative—indicating what such objects are not—in the context of mathematics it captures the crucialfeatures the objects in questions are supposed to have. According to nominalism,mathematical objects (including, henceforth, mathematical relations and structures) do notexist, or at least they need not be taken to exist for us to make sense of mathematics. So, itis the nominalist's burden to show how to interpret mathematics without the commitment tothe existence of mathematical objects. This is, in fact, a key feature of nominalism: thosewho defend the view need to show that it is possible to yield at least as much explanatorywork as the platonist obtains, but invoking a meager ontology. To achieve that, nominalistsin the philosophy of mathematics forge interconnections with metaphysics (whethermathematical objects do exist), epistemology (what kind of knowledge of these entities wehave), and philosophy of science (how to make sense of the successful application ofmathematics in science without being committed to the existence of mathematical entities).These interconnections are one of the sources of the variety of nominalist views.

Despite the substantial differences between nominalism and platonism, they have at leastone feature in common: both come in many forms. There are various versions of platonismin the philosophy of mathematics: standard (or object-based) platonism (Gödel 1944, 1947;Quine 1960), structuralism (Resnik 1997; Shapiro 1997), and full-blooded platonism(Balaguer 1998), among other views. Similarly, there are also several versions ofnominalism: fictionalism (Field 1980, 1989), modal structuralism (Hellman 1989, 1996),constructibilism (Chihara 1990), the weaseling-away view (Melia 1995, 2000), figuralism(Yablo 2001), deflationary nominalism (Azzouni 2004), agnostic nominalism (Bueno2008, 2009), and pretense views (Leng 2010), among others. Similarly to their platonist

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counterparts, the various nominalist proposals have different motivations, and face theirown difficulties. These will be explored in turn. (A critical survey of variousnominalization strategies in mathematics can be found in Burgess and Rosen (1997). Theauthors address in detail both the technical and philosophical issues raised by nominalismin the philosophy of mathematics.)

Discussions about nominalism in the philosophy of mathematics in the 20th century startedroughly with the work that W. V. Quine and Nelson Goodman developed towardconstructive nominalism (Goodman and Quine 1947). But, as Quine later pointed out, inthe end it was indispensable to quantify over classes (Quine 1960). As will become clearbelow, responses to this indispensability argument have generated a significant amount ofwork for nominalists. And it is the focus on the indispensability argument that largelydistinguishes more recent nominalist views in the philosophy of mathematics, which I willfocus on, from the nominalism developed in the early part of the 20th century by the Polishschool of logic (Simons 2010).

Mathematical nominalism is a form of anti-realism about abstract objects. This is anindependent issue from the traditional problem of nominalism about universals. Auniversal, according to a widespread use, is something that can be instantiated by differententities. Since abstract objects are neither spatial nor temporal, they cannot be instantiated.Thus, mathematical nominalism and nominalism about universals are independent fromone another (see the entry on nominalism in metaphysics). It could be argued that certainsets encapsulate the instantiation model, since a set of concrete objects can be instantiatedby such objects. But since the same set cannot be so instantiated, given that sets areindividuated by their members and as long as their members are different the resulting setsare not the same, it is not clear that even these sets are instantiated. I will focus here onmathematical nominalism.

2. Five ProblemsIn contemporary philosophy of mathematics, nominalism has been formulated in responseto difficulties faced by platonism. But in developing their responses to platonism,nominalists also encounter difficulties of their own. Five problems need to be addressed inthis context:

1. The epistemological problem of mathematics,2. The problem of the application of mathematics,3. The problem of uniform semantics,4. The problem of taking mathematical discourse literally, and5. The ontological problem.

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Usually, problems (1) and (5) are considered as raising difficulties for platonism, whereasproblems (2), (3), and (4) are often taken as yielding difficulties for nominalism. (I willdiscuss below to what extent such an assessment is accurate.) Each of these problems willbe examined in turn.

2.1 The epistemological problem of mathematics

Given that platonism postulates the existence of mathematical objects, the question arisesas to how we obtain knowledge about them. The epistemological problem of mathematicsis the problem of explaining the possibility of mathematical knowledge, given thatmathematical objects themselves do not seem to play any role in generating ourmathematical beliefs (Field 1989).

This is taken to be a particular problem for platonism, since this view postulates theexistence of mathematical objects, and one would expect such objects to play a role in theacquisition of mathematical knowledge. After all, on the platonist view, such knowledge isabout the corresponding mathematical objects. However, despite various sophisticatedattempts by platonists, there is still considerable controversy as to how exactly this processshould be articulated. Should it be understood via mathematical intuition, by theintroduction of suitable mathematical principles and definitions, or does it require someform of abstraction?

In turn, the epistemological issue is far less problematic for nominalists, who are notcommitted to the existence of mathematical objects in the first place. They will have toexplain other things, such as, how can the nominalist account for the difference between amathematician, who knows a significant amount of mathematics, and a non-mathematician,who does not? This difference, according to some nominalists, is based on empirical andlogical knowledge—not on mathematical knowledge (Field 1989).

2.2 The problem of the application of mathematics

Mathematics is often successfully used in scientific theories. How can such a success beexplained? Platonists allegedly have an answer to this problem. Given that mathematicalobjects exist and are successfully referred to by our scientific theories, it is not surprisingthat such theories are successful. Reference to mathematical objects is just part of thereference to those entities that are indispensable to our best theories of the world. Thisframes the problem of the application of mathematics in terms of the indispensabilityargument.

In fact, one of the main reasons for belief in the existence of mathematical objects—someclaim this is the only non-question begging reason (Field 1980)—is given by the



indispensable use of mathematics in science. The crucial idea, originally put forward by W.V. Quine, and later articulated, in a different way, by Hilary Putnam, is that ontologicalcommitment should be restricted to just those entities that are indispensable to our besttheories of the world (Quine 1960; Putnam 1971; Colyvan 2001a). Mark Colyvan hasformulated the argument in the following terms:

(P1) We ought to be ontologically committed to all and only those entities thatare 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 mathematicalentities.

The first premise relies crucially on Quine's criterion of ontological commitment. Afterregimenting our best theories of the world in a first-order language, the ontologicalcommitments of these theories can be read off as being the value of the existentiallyquantified variables. But how do we move from the ontological commitments of a theory towhat we ought to be ontologically committed to? This is the point where the first premiseof the indispensability argument emerges. If we are dealing with our best theories of theworld, precisely those items that are indispensable to these theories amount to what weought to be committed to. (Of course, a theory may quantify over more objects than thosethat are indispensable.) And by identifying the indispensable components invoked in theexplanation of various phenomena, and noting that mathematical entities are among them,the platonist is then in a position to make sense of the success of applied mathematics.

However, it turns out that whether the platonist can indeed explain the success of theapplication of mathematics is, in fact, controversial. Given that mathematical objects areabstract, it is unclear why the postulation of such entities is helpful to understand thesuccess of applied mathematics. For the physical world—being composed of objectslocated in space-time—is not constituted by the entities postulated by the platonist. Hence,it is not clear why the correct description of relations among abstract (mathematical)entities is even relevant to understand the behavior of concrete objects in the physicalworld involved in the application of mathematics. Just mentioning that the physical worldinstantiates structures (or substructures) described in general terms by variousmathematical theories is not enough (see, e.g., Shapiro 1997). For there are infinitely manymathematical structures, and there is no way of uniquely determining which of them isactually instantiated—or even instantiated only in part—in a finite region of the physicalworld. There is a genuine underdetermination here, given that the same physical structurein the world can be accommodated by very different mathematical structures. For instance,quantum mechanical phenomena can be characterized by group-theoretic structures (Weyl



1928) or by structures emerging from the theory of Hilbert spaces (von Neumann 1932).Mathematically, such structures are very different, but there is no way of deciding betweenthem empirically.

Despite the controversial nature of the platonist claim to be able to explain the success ofapplied mathematics, to accommodate that success is often taken as a significant benefit ofplatonism. Less controversially, the platonist is certainly able to describe the way in whichmathematical theories are actually used in scientific practice without having to rewritethem. This is, as will become clear below, a significant benefit of the view.

Nominalism, in turn, faces the difficulty of having to explain the successful use ofmathematics in scientific theorizing. Since, according to the nominalist, mathematicalobjects do not exist—or, at least, are not taken to exist—it becomes unclear how referringto such entities can contribute to the empirical success of scientific theories. In particular, ifit turns out that reference to mathematical entities is indeed indispensable to our besttheories of the world, how can the nominalist deny the existence of such entities? As wewill see below, several nominalist views in the philosophy of mathematics have emerged inresponse to the challenge raised 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 that it allows us to adopt thesame semantics for both mathematical and scientific discourse. Given the existence ofmathematical objects, mathematical statements are true in the same way as scientificstatements are true. The only difference emerges from their respective truth makers:mathematical statements are true in virtue of abstract (mathematical) objects and relationsamong them, whereas scientific statements are ultimately true in virtue of concrete objectsand the corresponding relations among such objects. This point is idealized in that itassumes that, somehow, we can manage to distill the empirical content of scientificstatements independently of the contribution made by the mathematics that is often used toexpress such statements. Platonists who defend the indispensability argument insist thatthis is not possible to do (Quine 1960; Colyvan 2001a); even some nominalists concur(Azzouni 2011).

Moreover, as is typical in the application of mathematics, there are also mixed statements,which involve both terms referring to concrete objects and terms referring to abstract ones.The platonist has no trouble providing a unified semantics for such statements either—particularly if mathematical platonism is associated with realism about science. In thiscase, the platonist can provide a referential semantics throughout. Of course, the platonistabout mathematics need not be a realist about science—although it's common to combine



platonism and realism in this way. In principle, the platonist could adopt some form of anti-realism about science, such as constructive empiricism (van Fraassen 1980; Bueno 2009).As long as the form of anti-realism regarding science allows for a referential semantics(and many do), the platonist would have no trouble providing a unified semantics for bothmathematics and science (Benacerraf 1973).

It is not clear that the nominalist can deliver these benefits. As will become clear shortly,most versions of nominalism require a substantial rewriting of mathematical language. As aresult, a distinct semantics needs to be offered for that language in comparison with thesemantics 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 take mathematical discourse literally,given that mathematical terms refer. In particular, there is no change in the syntax ofmathematical statements. So, when mathematicians claim that ‘There are infinitely manyprime numbers’, the platonist can take that statement literally as describing the existence ofan infinitude of primes. On the platonist view, there are obvious truth-makers formathematical statements: mathematical objects and their corresponding properties andrelations (Benacerraf 1973).

We have here a major benefit of platonism. If one of the goals of the philosophy ofmathematics is to provide understanding of mathematics and mathematical practice, it is asignificant advantage that platonists are able to take the products of that practice—such asmathematical theories—literally and do not need to rewrite or reformulate them. After all,the platonist is then in a position to examine mathematical theories as they are actuallyformulated in mathematical practice, rather than discuss a parallel discourse offered byvarious reconstructions of mathematics given by those who avoid the commitment tomathematical objects (such as the nominalists).

The inability to take mathematical discourse literally is indeed a problem for nominalists,who typically need to rewrite the relevant mathematical theories. As will become clearbelow, it is common that nominalization strategies for mathematics change either thesyntax or the semantics of mathematical statements. For instance, in the case of modalstructuralism, modal operators are introduced to preserve verbal agreement with theplatonist (Hellman 1989). The proposal is that each mathematical statement S is translatedinto two modal statements: (i) if there were structures of the suitable kind, S would be truein these structures, and (ii) it's possible that there are such structures. As a result, both thesyntax and the semantics of mathematics are changed. In the case of mathematicalfictionalism, in order to preserve verbal agreement with the platonist despite the denial ofthe existence of mathematical objects, fiction operators (such as, ‘According to



arithmetic…’) are introduced (Field 1989). Once again, the resulting proposal changes thesyntax (and, hence, the semantics) of mathematical discourse. This is a significant cost forthese views.

2.5 The ontological problem

The ontological problem consists in specifying the nature of the objects a philosophicalconception of mathematics is ontologically committed to. Can the nature of these objectsbe properly determined? Are the objects in question such that we simply lack good groundsto believe in their existence? Traditional forms of platonism have been criticized for failingto offer an adequate solution to this problem. In response, some platonists have argued thatthe commitment to mathematical objects is neither problematic nor mysterious (see, e.g.,Hale and Wright 2001). Similarly, even though some nominalists need not be committed tomathematical objects, they may be committed to other entities that may also raiseontological concerns (such as possibilia). The ontological problem is then the problem ofassessing the status of the ultimate commitments of the view.

Three nominalization strategies will be discussed below: mathematical fictionalism (Field1980, 1989), modal structuralism (Hellman 1989, 1996), and deflationary nominalism(Azzouni 2004). The first two reject the second premise of the indispensability argument.They provide ‘hard roads’ to nominalism (Colyvan 2010), in the sense that the nominalistneeds to develop the complex, demanding work of showing how quantification overmathematical objects can be avoided in order to develop a suitable interpretation ofmathematics. The third strategy rejects the first premise of the argument, thus bypassingthe need to argue for the dispensability of mathematics (in fact, for the deflationarynominalist, mathematics is ultimately indispensable). By reassessing Quine's criterion ofontological commitment, and indicating that quantification over certain objects does notrequire their existence, this strategy yields an ‘easy road’ to nominalism.

Although this survey is clearly not exhaustive, since not every nominalist view availablewill be considered here, the three views discussed are representative: they occupy distinctpoints in the logical space, and they have been explicitly developed to address the variousproblems just listed.

3. Mathematical Fictionalism3.1 Central features of mathematical fictionalism

In a series of works, Hartry Field provided an ingenious strategy for the nominalization ofscience (Field 1980, 1989). As opposed to platonist views, in order to explain theusefulness of mathematics in science, Field does not postulate the truth of mathematical



theories. In his view, it is possible to explain successful applications of mathematics withno commitment to mathematical objects. Therefore, what he takes to be the main argumentfor platonism, which relies on the (apparent) indispensability of mathematics to science, isresisted. The nominalist nature of Field's account emerges from the fact that mathematicalobjects are not assumed to exist. Hence, mathematical theories are false. (Strictly speaking,Field notes, any existential mathematical statement is false, and any universal mathematicalstatement is vacuously true.) By devising a strategy that shows how to dispense withmathematical objects in the formulation of scientific theories, Field rejects theindispensability argument, and provides strong grounds for the articulation of a nominaliststance.

Prima facie, it may sound counterintuitive to state that ‘there are infinitely many primenumbers’ is false. But if numbers do not exist, that's the proper truth-value for thatstatement (assuming a standard semantics). In response to this concern, Field 1989introduces a fictional operator, in terms of which verbal agreement can be reached with theplatonist. In the case at hand, one would state: ‘According to arithmetic, there are infinitelymany prime numbers’, which is clearly true. Given the use of a fictional operator, theresulting view is often called mathematical fictionalism.

The nominalization strategy devised by the mathematical fictionalist depends on twointerrelated moves. The first is to change the aim of mathematics, which is not taken to betruth, but something different. On this view, the proper norm of mathematics, which willguide the nominalization program, is conservativeness. A mathematical theory isconservative if it is consistent with every internally consistent theory about the physicalworld, where such theories do not involve any reference to, nor quantification over,mathematical objects, such as sets, functions, numbers etc. (Field 1989, p. 58).Conservativeness is stronger than consistency (since if a theory is conservative, it isconsistent, but not vice versa). However, conservativeness is not weaker than truth (Field1980, pp. 16–19; Field 1989, p. 59). So, Field is not countenancing a weaker aim ofmathematics, but only a different one.

It is precisely because mathematics is conservative that, despite being false, it can beuseful. Of course, this usefulness is explained with no commitment to mathematicalentities: mathematics is useful because it shortens our derivations. After all, if amathematical theory M is conservative, then a nominalistic assertion A about the physicalworld (i.e. an assertion which does not refer to mathematical objects) follows from a bodyN of such assertions and M only if follows from N alone. That is, provided we have asufficiently rich body of nominalistic assertions, the use of mathematics does not yield anynew nominalistic consequences. Mathematics is only a useful instrument to help us in thederivations.



As a result, conservativeness can only be employed to do the required job if we havenominalistic premises to start with (Field 1989, p. 129). As Field points out, it is aconfusion to argue against his view by claiming that if we add some bits of mathematics toa body of mathematical claims (not nominalistic ones), we may obtain new consequencesthat could not be achieved otherwise (Field 1989, p. 128). The restriction to nominalisticassertions is crucial.

The second move of the mathematical fictionalist strategy is to provide such nominalisticpremises. Field has done that in one important case: Newtonian gravitational theory. Heelaborates on a work that has a respectable tradition: Hilbert's axiomatization of geometry(Hilbert 1971). What Hilbert provided was a synthetic formulation of geometry, whichdispenses with metric concepts, and therefore does not include any quantification over realnumbers. His axiomatization was based on concepts such as point, betweenness, andcongruence. Intuitively speaking, we say that a point y is between the points x and z if y is apoint in the line-segment whose endpoints are x and z. Also intuitively, we say that theline-segment xy is congruent to the line-segment zw if the distance from the point x to thepoint y is the same as that from the point z to w. After studying the formal properties of theresulting system, Hilbert proved a representation theorem. He showed, in a strongermathematical theory, that given a model of the axiom system for space he had put forward,there is a function d from pairs of points onto non-negative real numbers such that thefollowing ‘homomorphism conditions’ are met:

i. xy is congruent to zw iff d(x, y) = d(z,w), for all points x, y, z, and w;ii. y is between x and z iff d(x, y) + d(y, z) = d(x, z), for all points x, y, and z.

As a result, if the function d is taken to represent distance, we obtain the expected resultsabout congruence and betweenness. Thus, although we cannot talk about numbers inHilbert's geometry (there are no such entities to quantify over), there is a metatheoreticresult that associates assertions about distances with what can be said in the theory. Fieldcalls such numerical claims abstract counterparts of purely geometric assertions, and theycan be used to draw conclusions about purely geometrical claims in a smoother way.Indeed, because of the representation theorem, conclusions about space, statable withoutreal numbers, can be drawn far more easily than we could achieve by a deflationary prooffrom Hilbert's axioms. This illustrates Field's point that the usefulness of mathematicsderives from shortening derivations (Field 1980, pp. 24–29).

Roughly speaking, what Field established was how to extend Hilbert's results about spaceto space-time. Similarly to Hilbert's approach, instead of formulating Newtonian laws interms of numerical functors, 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 taken to have a numerical value, Field employed a comparative predicate such



as ‘the difference in gravitational potential between x and y is less than that between z andw’. Relying on a body of representation theorems (which plays the same role as Hilbert'srepresentation theorem in geometry), Field established how several numerical functors canbe ‘obtained’ from comparative predicates. But in order to use those theorems, he firstshowed how to formulate Newtonian numerical laws (such as, Poisson's equation for thegravitational field) only in terms of comparative predicates. The result (Field 1989, pp.130–131) is the following extended representation theorem. Let N be a theory formulatedonly in terms of comparative predicates (with no recourse to numerical functors). For anymodel S of N whose domain is constituted by space-time regions, there exists:

i. A 1–1 spatio-temporal co-ordinate function f (unique up to a generalized Galileantransformation) mapping the space-time of S onto quadruples of real numbers;

ii. A mass density function g (unique up to a positive multiplicative transformation)mapping the space-time of S onto an interval of non-negative real numbers; and

iii. A gravitational potential function h (unique up to a positive linear transformation)mapping the space-time onto an interval of real numbers.

Moreover, all these functions ‘preserve structure’, in the sense that the comparativerelations defined in terms of them coincide with the comparative relations used in N.Furthermore, if f, g and h are taken as the denotation of the appropriate functors, the lawsof Newtonian gravitational theory in their functorial form hold.

Notice that, in quantifying over space-time regions, Field assumes a substantivalist view ofspace-time, according to which there are space-time regions that are not fully occupied(Field 1980, pp. 34–36; Field 1989, pp. 171–180). Given this result, the mathematicalfictionalist is allowed to draw nominalistic conclusions from premises involving N plus amathematical theory T. After all, due to the conservativeness of mathematics, suchconclusions can be obtained independently of T. The role of the extended representationtheorem is then to establish that, despite the lack of quantification over mathematicalobjects, precisely the same class of models is determined by formulating Newtoniangravitational theory in terms of functors (as the theory is usually expressed) or in terms ofcomparative predicates (as the mathematical fictionalist favors). Thus, the extendedrepresentation theorem ensures that the use of conservativeness of mathematics togetherwith suitable nominalistic claims (formulated via comparative predicates) does not changethe class of models of the original theory: the same comparative relations are preserved.Hence, what Field provided is a nominalization strategy, and since it reduces ontology, itseems a promising candidate for a nominalist stance vis-à-vis mathematics.

How should the mathematical fictionalist approach physical theories, such as perhapsstring theory, that do not seem to be about concrete observable objects? One possibleresponse, assuming the lack of empirical import of such theories, is simply to reject that



they are physical theories, and as such they are not the sorts of theories for which themathematical fictionalist needs to provide a nominalistic counterpart. In other words, untilthe moment in which such theories acquire the relevant empirical import, they need notworry the mathematical fictionalist. Theories of that sort would be classified as part of themathematics rather than the physics.

3.2 Metalogic and the formulation of conservativeness

But is mathematics conservative? In order to establish the conservativeness ofmathematics, the mathematical fictionalist has used metalogical results, such as thecompleteness and the compactness of first-order logic (Field 1992, 1980, 1989). The issuethen arises as to whether the mathematical fictionalist can use these results to develop theprogram.

At two crucial junctures, Field has made use of metalogical results: (a) in his reformulationof the notion of conservativeness in nominalistically acceptable terms (Field 1989, pp.119–120; Field 1991), and (b) in his nominalist proof of the conservativeness of set theory(Field 1992). These two outcomes are crucial for Field, since they establish the adequacyof conservativeness for the mathematical fictionalist. For (a) settles that the latter canformulate that notion without violating nominalism, and (b) concludes thatconservativeness is a feature that mathematics actually has. But if these two outcomes arenot legitimate, Field's approach cannot get off the ground. I will now consider whetherthese two uses of metalogical results are acceptable on nominalist grounds.

3.2.1 Conservativeness and the compactness theorem

Let me start with (a). The mathematical fictionalist has relied on the compactness theoremto formulate the notion of conservativeness in an acceptable way, that is, without referenceto mathematical entities. As noted above, conservativeness is defined in terms ofconsistency. But this notion is usually formulated either in semantic terms (as the existenceof an appropriate model), or in proof-theoretic terms (in terms of suitable proofs).However, as Field acknowledges, these two formulations of consistency are platonist, sincethey depend on abstract objects (models and proofs), and therefore are not nominalisticallyacceptable.

The mathematical fictionalist way out is to avoid moving to the metalanguage in order toexpress the conservativeness of mathematics. The idea is to state, in the object-language,the claim that a given mathematical theory is conservative by introducing a primitivenotion of logical consistency: ◊A. Thus, if B is any sentence, B* is the result of restricting Bto non-mathematical entities, and M1, …, Mn are the axioms of a mathematical theory M,the conservativeness of M can be expressed by the following schema (Field 1989, p. 120):



(C) If ◊B, then ◊(B* ∧ M1 ∧ … ∧ Mn).

In other words, a mathematical theory M is conservative if it is consistent with everyconsistent theory about the physical world B*.

This assumes, of course, that M is finitely axiomatized. But how can we apply (C) in thecase of mathematical theories that are not finitely axiomatizable (such as Zermelo-Fraenkelset theory)? In this case, we cannot make the conjunction of all the axioms of the theory,since there are infinitely many of them. Field has addressed this issue, and he initiallysuggested that the mathematical fictionalist could use substitutional quantification toexpress these infinite conjunctions (Field 1984). In a postscript to the revised version ofthis essay (Field 1989, pp. 119–120), he notes that substitutional quantification can beavoided, provided that the mathematical and physical theories in question are expressed ina logic for which compactness holds. For in this case, the consistency of the whole theoryis reduced to the consistency of each of its finite conjunctions.

There are, however, three problems with this move.

i. One concern about the use of substitutional quantification in this context involves thenature of substitutional instances. If the latter turn out to be abstract, which would bethe case if such substitutional instances were not mere inscriptions, they are notavailable to the nominalist. If the substitutional instances are concrete, the nominalistneeds to show that there are enough of them.

ii. The very statement of the compactness theorem involves set-theoretic talk: let G be aset of formulas; if every finite subset of G is consistent, then G is consistent. Howcan nominalists rely on a theorem whose very statement involves abstract entities? Inorder to use this theorem, an appropriate reformulation is required.

iii. Let us grant that it is possible to reformulate this statement without referring to sets.Can then the nominalist use the compactness theorem? As is well known, the proofof this theorem assumes set theory. The compactness theorem is usually presented asa corollary to the completeness theorem for first-order logic, whose proof assumes settheory (see, for example, Boolos and Jeffrey 1989, pp. 140–141). Alternatively, if thecompactness result is to be proved directly, then one has to construct the appropriatemodel of G—which again requires set theory. So, unless mathematical fictionalistsare able to provide an appropriate nominalization strategy for set theory itself, theyare not entitled to use this result. In other words, far more work is required before aField-type nominalist is able to rely on metalogical results.

But maybe this criticism misses the whole point of Field's program. As we saw, Field doesnot require that a mathematical theory M be true for it to be used. Only its conservativenessis demanded. So, if M is added to a body B* of nominalistic claims, no new nominalistic



conclusion is obtained which was not obtained by B* alone. In other words, what Field'sstrategy asks for is the formulation of appropriate nominalistic bodies of claims to whichmathematics can be applied. The same point holds for metalogical results: provided thatthey are applied to nominalistic claims, Field is fine.

The problem with this reply is that it involves the mathematical fictionalist program in acircle. The fictionalist cannot rely on the conservativeness of mathematics to justify the useof a mathematical result (the compactness theorem) that is required for the formulation ofthe notion of conservativeness itself. For in doing that, the fictionalist assumes that thenotion of conservativeness is nominalistically acceptable, and this is exactly the point inquestion. Recall that the motivation for Field to use the compactness theorem was toreformulate conservativeness without having to assume abstract entities (namely, thoserequired by the semantic and the proof-theoretic accounts of consistency). Thus, at thispoint, the mathematical fictionalist cannot yet use the notion of conservativeness;otherwise, the whole program would not get off the ground. I conclude that, similarly toany other part of mathematics, metalogical results also need to be obtainednominalistically. Trouble arises 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 thenotion of conservativeness in terms of a primitive notion of logical consistency: ◊A. And healso indicates that this notion is related to the model-theoretic concept of consistency—inparticular, to the formulation of this concept in von Neumann-Bernays-Gödel finitelyaxiomatizable 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 for model-theoretic possibility, and‘ME#’ for model existence. The symbol ‘#’ indicates that, according to Field, theseprinciples are nominalistically acceptable. After all, they are modal surrogates for theplatonistic 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 mathematical fictionalist will beentitled to use the compactness theorem. First, one should try to state this theorem in a



nominalistically acceptable way. Without worrying too much about details, let us grant, forthe sake of argument, that the following characterization will do:

(Compact#) If ¬◊T, then ∃f A1, …, An[¬◊(A1 ∧ … ∧ An)],

where T is a theory and each Ai, 1 ≤ i ≤ n, is a formula (an axiom of T). The expression‘∃f A1…An’ is to be read as ‘there are finitely many formulas A1…An’. (This quantifier isnot first-order. However, I am not going to press the point that the nominalist seems toneed a non-first-order quantifier to express a property typical of first-order logic. This isonly one of the worries we are leaving aside in this formulation.) This version is parasiticon the following platonistic formulation of the compactness theorem:

(Compact) If there is no model for T, then ∃f A1, …, An such that there is nomodel for (A1 ∧ … ∧ An).

In order for mathematical fictionalists to be entitled to use the compactness theorem, theywill have to show that the nominalistic formulation (Compact#) follows from theplatonistic one (Compact). In this sense, if the latter is adequate, so is the former. Moreaccurately, what has to be shown is that (Compact#) follows from a modal surrogate of(Compact). After all, since what is at issue is the legitimacy of the compactness theorem onnominalist grounds, it would be question-begging to assume the full platonistic versionfrom the outset. As we will see, there are two ways to try to establish this result.Unfortunately, none of them works: both are formally inadequate.

The two options start in the same way. Suppose that

(1) ¬◊T.

We have to establish that

(2) ∃f A1…An ¬◊(A1∧…∧An).

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:

(CompactM) ☐(NBG → if there is no model for ‘T’, then ∃f A1…An such that



there is no model for (A1∧…∧An)).

Note that, since the modal surrogate is formulated in terms of models (rather than in termsof the primitive modal operator), it is still not what mathematical fictionalists need. Whatthey need is (Compact#), but one needs to show that they can get it. At this point, theoptions begin to diverge.

The first option consists in drawing from (4) and (CompactM) that

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

There are, however, difficulties with this move. First, note that (5) is not equivalent to (2),which is the result to be achieved. Moreover, as opposed to (2), (5) is formulated in model-theoretic terms, since it incorporates a claim about the nonexistence of a certain model.And what is required is a similar statement in terms of the primitive notion of consistency.In other words, we need the nominalistic counterpart of (5), rather than (5).

But (5) has a nice feature. It is a modalized formulation of the consequent of (Compact).And since (5) only states the possibility that there is no model of a particular kind, it maybe argued that it is nominalistically acceptable. (As will be examined below, modalstructuralists advance a nominalization strategy exploring modality along these lines; seeHellman (1989).) Field, however, is skeptical about this move. On his view, modality is nota general surrogate for ontology (Field 1989, pp. 252–268). And one of his worries is thatby allowing the introduction of modal operators, as a general nominalization strategy, wemodalize away the physical content of the theory under consideration. However, sincemetalogical claims are not expected to have physical consequences, the worry need notarise here. At any rate, given that (5) does not establish what needs to be established, itdoes not solve the problem.

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

(5$) ☐(NBG → ∃f A1…An such that there is no model for (A1 ∧…∧ An)).

Note that if (5$) were established, we would have settled the matter. After all, with astraightforward reworking of (ME#) (namely, If ☐(NBG → ∃f A1…An such that there is nomodel for (A1∧…∧An)), then ¬◊(A1 ∧…∧ An)), it follows from (5$) and (ME#) that

(2) ∃f A1…An ¬◊(A1 ∧…∧ An),

which is the conclusion we need. The problem here is that (5$) does not follow from (4) and(CompactM). Therefore, we cannot derive it.



Clearly, there may well be another option that establishes the intended conclusion. But, tosay the least, the mathematical fictionalist has to present it before being entitled to usemetalogic results. Until then, it is not clear that these results are nominalisticallyacceptable.

3.2.3 Metalogic and the proof of the conservativeness of set theory

I should now consider issue (b): Field's nominalistic proof of the conservativeness of settheory. Let us grant that the concept of conservativeness has been formulated in somenominalistically acceptable way. If Field's proof were correct, he would have proved thatmathematics itself is conservative—as long as one assumes the usual reductions ofmathematics to set theory. How does Field prove the conservativeness of set theory? It isby an ingenious argument, which adapts one of the Field's platonistic conservativenessproofs (Field 1980). For our present purposes, we need not examine the details of thisargument, but simply note that at a crucial point the completeness of first-order logic isused to establish its conclusion (Field 1992, p. 118).

The problem with this move is that, even if mathematical fictionalists formulate thestatement of the completeness theorem without referring to mathematical entities, the proofof this theorem assumes set theory (see, for instance, Boolos and Jeffrey 1989, pp. 131–140). Therefore, fictionalists cannot use the theorem without undermining theirnominalism. After all, the point of providing a nominalistic proof of the conservativenessof set theory is to show that, without recourse to platonist mathematics, the mathematicalfictionalist is able to establish that mathematics is conservative. Field has offered aplatonist argument for the conservativeness result (Field 1980)—an argument thatexplicitly invoked properties of set theory. The idea was to provide a reductio of platonism:by using platonist mathematics, Field attempted to establish that mathematics wasconservative and, thus, ultimately dispensable. In contrast with the earlier strategy, the goalwas to provide a proof of the conservativeness of set theory that a nominalist could accept.But since the nominalistic proof relies on the completeness theorem, it is not at all clearthat it is in fact nominalist. Mathematical fictionalists should first be able to prove thecompleteness result without assuming set theory. Alternatively, they should provide anominalization strategy for set theory itself, which will then entitle them to use metalogicalresults.

But it may be argued that the mathematical fictionalist only requires the conservativenessof the set theory in which the completeness theorem is proved. It should now be clear thatthis reply is entirely question begging, since the point at issue is exactly to prove theconservativeness of set theory. Thus, the fictionalist cannot assume that this result isalready established at the metatheory.



In other words, without a broader nominalization strategy, which allows set theory itself tobe nominalized, it seems difficult to see how mathematical fictionalists can use metalogicalresults as part of their program. The problem, however, is that it is not at all obvious that, atleast in the form articulated by Field, the mathematical fictionalist program can beextended to set theory. For it only provides a nominalization strategy for scientific theories,that is, for the use of mathematics in science (e.g., in Newtonian gravitational theory). Theapproach doesn't address the nominalization of mathematics itself.

In principle, one may object, this shouldn't be a problem. After all, the mathematicalfictionalists’ motivation to develop their approach has focused on one issue: to overcomethe indispensability argument—thus addressing the use of mathematics in science. And theoverall strategy, as noted, has been to provide nominalist counterparts to relevant scientifictheories.

The problem with this objection, however, is that given the nature of Field's strategy, thetask of nominalizing science cannot be achieved without also nominalizing set theory.Thus, what is needed is a more open-ended, broader nominalism: one that goes hand inhand not only with science, but also with metalogic. As it stands, the mathematicalfictionalist approach still leaves a considerable gap.

3.3 Assessment: benefits and problems of mathematical fictionalism

3.3.1 The epistemological problem

Given that mathematical objects do not exist, on the mathematical fictionalist perspective,the problem of how we can obtain knowledge of them simply vanishes. But anotherproblem emerges instead: what is it that distinguishes a mathematician (who knows a lotabout mathematics) and a non-mathematician (who does not have such knowledge)? Thedifference here (according to Field 1984) is not about having or lacking mathematicalknowledge, but rather it is about logical knowledge: of knowing which mathematicaltheorems follow from certain mathematical principles, and which do not. Theepistemological problem is then solved—as long as the mathematical fictionalist providesan epistemology for logic.

In fact, what needs to be offered is ultimately an epistemology for modality. After all, onField's account, in order to avoid the platonist commitment to models or proofs, theconcept of logical consequence is understood in terms of the primitive modal concept oflogical possibility: A follows logically from B as long as the conjunction of B and thenegation of A is impossible, that is, ¬◊(B ∧ ¬A).

However, how are such judgments of impossibility established? Under what conditions do



we know that they hold? In simple cases, involving straightforward statements, to establishsuch judgments may be unproblematic. The problem emerges when more substantivestatements are invoked. In these cases, we seem to need a significant amount ofmathematical information in order to be able to determine whether the impossibilities inquestion really hold or not. Consider, for instance, the difficulty of establishing theindependence of the axiom of choice and the continuum hypotheses from the axioms ofZermelo-Fraenkel set theory. Significantly complex mathematical models need to beconstructed in this case, which rely on the development of special mathematical techniquesto build them. What is required from the mathematical fictionalist at this stage is thenominalization of set theory itself—something that, as we saw, Field still owes us.

3.3.2 The problem of the application of mathematics

Similarly to the epistemological problem, the problem of the application of mathematics ispartially solved by the mathematical fictionalist. Field provides an account of theapplication of mathematics that does not require the truth of mathematical theories. As wesaw, this demands that mathematics be conservative in the relevant sense. However, it isunclear whether Field has established the conservativeness of mathematics, given hisrestrictive way of introducing non-set-theoretic vocabulary into the axioms of set theory aspart of his attempted proof of the conservativeness of set theory (Azzouni 2009b, p. 169,note 47; additional difficulties for the mathematical fictionalist program can be found inMelia 1998, 2000). Field was working with restricted ZFU, Zermelo-Fraenkel set theorywith the axiom of choice modified to allow for Urelemente, objects that are not sets, butnot allowing for any non-set-theoretic vocabulary to appear in the comprehension axioms,that is, replacement or separation (Field 1980, p. 17). This is, however, a huge restriction,given that when mathematics is actually applied, non-set-theoretic vocabulary, whentranslated into set-theoretic language, will have to appear in the comprehension axioms. Asformulated by Field, the proof failed to address the crucial case of actual applications ofmathematics.

Moreover, it is also unclear whether the nominalization program advanced by themathematical fictionalist can be extended to other scientific theories, such as quantummechanics (Malament 1992). Mark Balaguer responded to this challenge by trying tonominalize quantum mechanics along mathematical fictionalist lines (Balaguer 1998).However, as argued by Bueno (Bueno 2003), Balaguer's strategy is incompatible with anumber of interpretations of quantum mechanics, in particular with Bas van Fraassen'sversion of the modal interpretation (van Fraassen 1991). And given that Balaguer's strategyinvokes physically real propensities, it is unclear whether it is even compatible withnominalism. As a result, the nominalization of quantum mechanics still remains a majorproblem for the mathematical fictionalist.



But even if these difficulties can all be addressed, it is unclear that the mathematicalfictionalist has offered an account of the application of mathematics that allows us to makesense of how mathematical theories are actually applied. After all, the fictionalist accountrequires us to rewrite the relevant theories, by finding suitable nominalistic versions forthem. This leaves the issue of making sense of the actual process of the application ofmathematics entirely untouched, given that no such reformulations are ever employed inactual scientific practice. Rather than engaging with actual features of the applicationprocess, the fictionalist creates a parallel discourse in an effort to provide a nominalistreconstruction of the use of mathematics in science. The reconstruction shows, at best, thatmathematical fictionalists need not worry about the application of mathematics vis-à-visincreasing their ontology. But the problem still remains of whether they are in a position tomake sense of the actual use of mathematics in science. This problem, which is crucial fora proper understanding of mathematical practice, still remains.

A similar difficulty also emerges for Balaguer's version of fictionalism (see the second halfof Balaguer 1998). Balaguer relies on the possibility of distinguishing between themathematical and the physical contents of an applied mathematical theory: in particular,the truth of such a theory holds only in virtue of physical facts, with no contribution frommathematics. It is, however, controversial whether the distinction between mathematicaland physical content can be characterized without implementing a Field-likenominalization program. In this case, the same difficulties that the latter face also carryover to Balaguer's account (Colyvan 2010; Azzouni 2011).

Moreover, according to Azzouni (Azzouni 2009b), in order for scientists to use a scientifictheory, they need to assert it. On his view, it is not enough for scientists simply torecognize that a scientific theory is true (or exhibits some other theoretical virtue). It isrequired that they assert the theory. In particular, scientists would then need to assert anominalistic theory. They cannot simply contemplate such a theory; they need to be able toassert it as well (Azzouni 2009b, footnotes 31, 43, 53, and 55, and p. 171). Thus,nominalists who grant this point to Azzouni need to show that scientists are in a position toassert the relevant nominalistic theories in order to address the issue of the application ofmathematics.

3.3.3 Uniform semantics

In one respect, mathematical fictionalists offer a uniform semantics for mathematical andscientific discourse, in another respect, they don't. Initially, both types of discourse areassessed in the same way. Electrons and relations among them make certain quantum-mechanical statements true; in turn, mathematical objects and relations among them makethe corresponding mathematical statements true. It just happens that, as opposed toelectrons on a realist interpretation of quantum mechanics, mathematical objects do not



exist. Hence, as noted, existential mathematical statements, such as ‘there are infinitelymany prime numbers’, are false. Although the resulting truth-value assignments forexistential statements conflict with those found in mathematical practice, at least the samesemantics is offered for mathematical and scientific languages.

In an attempt to agree with the truth-value assignments that are usually displayed inmathematical discourse, the mathematical fictionalist introduces a fictional operator:‘According to mathematical theory M…’. Such an operator, however, changes thesemantics of mathematical discourse. Applied to a true mathematical statement, at least onethat the platonist recognizes as true, the result will be a true statement—even according tothe mathematical fictionalist. For instance, from both platonist and fictionalist perspectives,the statement ‘according to arithmetic, there are infinitely many prime numbers’ comes outtrue. But, in this case, the mathematical fictionalist can no longer offer a unified semanticsfor mathematical and scientific languages, given that the latter does not involve theintroduction of fictional operators. Thus, whether mathematical fictionalists are able toprovide a uniform semantics ultimately depends on whether fictional operators areintroduced or not.

3.3.4. Taking mathematics literally

An immediate consequence of the introduction of fiction operators is that mathematicaldiscourse is no longer taken literally. As just noted, without such operators, mathematicalfictionalism produces non-standard attributions of truth-values to mathematical statements.But with fiction operators in place, the syntax of mathematical discourse is changed, andthus the latter cannot be taken literally.

3.3.5 The ontological problem

The ontological problem—the problem of the acceptability of the ontological commitmentsmade by the mathematical fictionalist—is basically solved. No commitment tomathematical objects is, in principle, made. Although a primitive modal notion isintroduced, it has only a limited role in the nominalization of mathematics: to allow for anominalist formulation of the crucial concept of conservativeness. As we saw, however,without a proper nominalization of set theory itself, it is unclear whether the mathematicalfictionalist program ultimately succeeds.

4. Modal Structuralism4.1 Central features of modal structuralism



Modal structuralism offers a program of interpretation of mathematics which incorporatestwo features: (a) an emphasis on structures as the main subject-matter of mathematics, and(b) a complete elimination of reference to mathematical objects by interpretingmathematics in terms of modal logic (as first suggested by Putnam (1967), and developedin Hellman (1989, 1996)). Given these features, the resulting approach is called a modal-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 mathematical statements should have truth-values, and thus‘instrumentalist’ readings are rejected from the outset. The second is that: ‘a reasonableaccount should be forthcoming of how mathematics does in fact apply to the materialworld’ (Hellman 1989, p. 6). Thus, the applicability problem must be examined.

In order to address these issues, the modal structuralist puts forward a general framework.The main idea is that although mathematics is concerned with the study of structures, thisstudy can be accomplished by focusing only on possible structures, and not actual ones.Thus, the modal interpretation is not committed to actual mathematical structures; there isno commitment to their existence as objects or to any objects that happen to ‘constitute’these structures. In this way, the ontological commitment to them is avoided: the onlyclaim is that the structures in question are possible. In order to articulate this point, themodal-structural interpretation is formulated in a second-order modal language based onS5. However, to prevent commitment 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 appropriate translation scheme in terms ofwhich each ordinary mathematical statement S is taken as elliptical for a hypotheticalstatement, namely: that S would hold in a structure of the appropriate kind.

For example, if we are considering number-theoretic statements, such as those articulatedin Peano arithmetic (PA, for short), the structures we are concerned with are ‘progressions’or ‘ω-sequences’ satisfying PA's axioms. In this case, each particular statement S is to be(roughly) translated as

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

According to this statement, if there were ω-sequences satisfying PA's axioms, S wouldhold in them. This is the hypothetical component of the modal-structural interpretation (fora detailed analysis and a precise formulation, see (Hellman 1989, pp. 16–24)). Thecategorical component constitutes the second step (Helman 1989, pp. 24–33). The idea isto assume that the structures of the appropriate kind are logically possible. In that case, wehave that



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

That is, it is logically possible that there are ω-sequences satisfying PA's axioms.Following this approach, truth preserving translations of mathematical statements can bepresented without ontological costs, given that only the possibility of the structures inquestion is assumed.

The modal structuralist then indicates that the practice of theorem proving can be regainedin this framework (roughly speaking, by applying the translation scheme to each line of theoriginal proof of the theorem under consideration). Moreover, by using the translationscheme and appropriate coding devices, one can argue that arithmetic, real analysis and, toa certain extent, even set theory are recovered in a nominalist setting (Hellman 1989, pp.16–33, 44–47, and 53–93). In particular, ‘by making use of coding devices, virtually all themathematics commonly encountered in current physical theories can be carried out within[real analysis]’ (Hellman 1989, pp. 45–46). However, the issue of whether set theory hasbeen nominalized in this way is, in fact, problematic—as the modal structuralist grants.After all, it is no obvious matter to establish even the possibility of the existence ofstructures with inaccessibly many objects.

With the framework in place, the modal structuralist can then consider the applicabilityproblem. The main idea is to adopt the hypothetical component as the basis foraccommodating the application of mathematics. The relevant structures are thosecommonly used in particular 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 three crucial components: the structures that are used inapplied mathematics, the non-mathematical objects to which the mathematical structuresare applied, and a statement of application that specifies the particular relations betweenthe mathematical structures and the non-mathematical objects. The relevant mathematicalstructures can be formulated in set theory. Let us call the set theory used in appliedcontexts Z. (This is second-order Zermelo set theory, which is finitely axiomatizable; I'lldenote the conjunction of the axioms of Z by ∧Z.) The non-mathematical objects of interestin the context of application can be expressed in Z as Urelemente, that is, as objects that arenot sets. We will take ‘U’ to be the statement that certain non-mathematical objects ofinterest are included as Urelemente in the structures of Z. Finally, ‘A’ is the statement ofapplication, describing the particular relations between the relevant mathematical structuresof Z and the non-mathematical objects described in U. The particular relations involveddepend on the case in question. 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 theaxioms of Zermelo set theory with all quantifiers relativized to the second order variable X,replacing each occurrence of the membership symbol ‘∈’ with the two place relationvariable ‘f’. According to the applied mathematical statement, if there were structuressatisfying the conjunction of the axioms of Zermelo set theory Z including some non-mathematical objects referred to in U, A would hold in such structures. The applicationstatement A expresses the relations in questions, such as an isomorphism or ahomomorphism between a physical system and certain set theoretic structures. This is thehypothetical component interpreted to express which relations would hold between certainmathematical structures (formulated as structures of ∧Z) and the entities studied in theworld (the Urelemente).

The second consideration examines in more detail the relationships between the physical(or the material) objects studied and the mathematical framework. These are the “syntheticdetermination” relations (Hellman 1989, pp. 124–135). More specifically, we have todetermine which relations among non-mathematical objects can be taken, in the antecedentof an applied mathematical statement, as the basis for specifying “the actual materialsituation” (Hellman 1989, p. 129). The modal structuralist proposal is to consider themodels of a comprehensive theory T$. This theory embraces and links the vocabulary of theapplied mathematical theory (T) and the synthetic vocabulary (S) in question, whichintuitively fixes the actual material situation. It is assumed that T determines, up toisomorphism, a particular kind of mathematical structure (containing, for example, Z), andthat T$ is an extension of T. In that case, a proposed “synthetic basis” will be adequate if thefollowing condition holds:

Let a be the class of (mathematically) standard models of T$, and let V denotethe full vocabulary of T$: then S determines V in a iff for any two models m andm$ in a, and any bijection f between their domains, if f is an S isomorphism, it isalso a V isomorphism. (Hellman 1989, p. 132.)

The introduction of isomorphism in this context comes, of course, from the need toaccommodate the preservation of structure between the (applied) mathematical part of thedomain under study and the non-mathematical part. This holds in the crucial case in whichthe preservation of the synthetic properties and relations (S-isomorphism) by f leads to thepreservation of the analytic applied mathematical relations (V-isomorphism) of the overalltheory T$. It should be noted that the ‘synthetic’ structure is not meant to ‘capture’ the fullstructure of the mathematical theory in question, but only its applied part. (Recall thatHellman started with an applied mathematical theory T.)



This can be illustrated with a simple example. Suppose that finitely many physical objectsdisplay a linear order. We can describe this by defining a function from those objects to aninitial segment of the natural numbers. What is required by the modal structuralist'ssynthetic determination condition is that the physical ordering among the objects alonecaptures this function and the description it offers of the objects. It is not claimed that thefull natural number structure is thus captured. This example also provides an illustration ofthe applied mathematical statement mentioned above. The Urelemente (objects that are notsets) are the physical objects in question, the relevant mathematical relation isisomorphism, and the mathematical structure is a segment of natural numbers with theirusual linear order.

On the modal structural conception, mathematics is applied by establishing an appropriateisomorphism between (parts of) mathematical structures and those structures that representthe material situation. This procedure is justified, since such isomorphism establishes thestructural equivalence between the (relevant parts of the) mathematical and the non-mathematical levels.

However, this proposal faces two difficulties. The first concerns the ontological status ofthe structural equivalence between the (applied) mathematical and the non-mathematicaldomains. On what grounds can we claim that the structures under consideration aremathematically the same if some of them concern ‘material’ objects? Of course, given thatthe structural equivalence is established by an isomorphism the material objects are alreadyformulated in structural terms—this means that some mathematics has already been appliedto the domain in question. In other words, in order to be able to represent the applicabilityof mathematics, Hellman assumes that some mathematics has already been applied. Thismeans that a purely mathematical characterization of the applicability of mathematics (viastructure preservation) is inherently incomplete. The first step in the application, namelythe mathematical modeling of the material domain, is not, and cannot, be accommodated,since no isomorphism is involved there. Indeed, given that by hypothesis the domain is notarticulated in mathematical terms, no isomorphism is defined there.

It may be argued that the modal structural account does not require an isomorphismbetween (applied) mathematical structures and those describing the material situation. Theaccount only requires an isomorphism between two standard models of the overall theoryT$, which links the mathematical theory T and the description S of the material domain. Inreply, note that this only moves the difficulty one level up. In order for T$ to extend theapplied mathematical theory T and to provide a link between T and the material situation, amodel of T$ will have to be, in particular, a model of both T and S. Thus, if the modalstructuralist's synthetic determination claim is satisfied, an isomorphism between twomodels of T$ will determine an isomorphism between the models of S and those of T. Inthis way, an isomorphism between structures describing the material situation and those



arising from applied mathematics is still required.

The second difficulty addresses the epistemological status of the claim that there is astructural equivalence between the mathematical and the non-mathematical domains. Onwhat grounds do we know that such equivalence holds? Someone may say that theequivalence is normatively imposed in order for the application process to get off theground. But this suggestion leads to a dilemma. Either it is just assumed that we know thatthe equivalence holds, and the epistemological question is begged (given that the groundsfor this are in question), or it is assumed that we do not know that the equivalence holds—and that is why we have to impose the condition—in which case the latter is clearlygroundless. However, it may be argued that there is no problem here, since we establish theisomorphism by examining the physical theories of the material objects underconsideration. But the problem is that in order to formulate these physical theories wetypically use mathematics. And the issue is precisely to explain this use, that is, to providesome understanding of the grounds in terms of which we come to know that the relevantmathematical structures are isomorphic to the physical ones.

The main point underlying these considerations has been stressed often enough (althoughin a different context): isomorphism does not seem to be an appropriate condition forcapturing the relation between mathematical structures and the world (see, e.g., da Costaand French 2003). There is, of course, a correct intuition underlying the use ofisomorphism at this level, and this relates to the idea of justifying the application ofmathematics: the isomorphism does guarantee that applied mathematical structures S andthe structures M which represent the material situation are mathematically the same. Theproblem is that isomorphism-based characterizations tend to be unrealistically strong. Theyrequire that some mathematics has already been applied to the material situation, and thatwe have knowledge of the structural equivalence between S and M. What is needed is aframework in which the relation between the relevant structures is weaker thanisomorphism, but which still 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 epistemological problem for mathematics.Assuming that the modal-structural translation scheme works for set theory, modalstructuralists need not explain how we can have knowledge of the existence ofmathematical objects, relations or structures—given the lack of commitment to theseentities. However, they still need to explain our knowledge of the possibility of the relevantstructures, since the translation scheme commits them to such possibility.



One worry that emerges here is that, in the case of substantive mathematical structures(such as those invoked in set theory), knowledge of the possibility of such structures mayrequire knowledge of substantial parts of mathematics. For instance, in order to know thatthe structures formulated in Zermelo set theory are possible, presumably we need to knowthat the theory itself is consistent. But the consistency of the theory can only be establishedin another theory, whose consistency, in turn, also needs to be established—and we face aregress. It would be arbitrary simply to assume the consistency of the theories in question,given that if such theories turn out to be in fact inconsistent, given classical logic,everything could be proved in them.

Of course, these considerations do not establish that the modal structuralist cannot developan epistemology for mathematics. They just suggest that further developments on theepistemological front seem 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 is partially solved by the modalstructuralist. After all, a framework to interpret the use of mathematics in science isprovided, and in terms of this framework the application of mathematics can beaccommodated without the commitment to the existence of the corresponding objects.

One concern that emerges (besides those already mentioned at the end of section 4.1above) is that, similarly to what happens to mathematical fictionalism, the proposedframework does not allow us to make sense of actual uses of the application ofmathematics. Rather than explaining how mathematics is in fact applied in scientificpractice, the modal-structural framework is advanced in order to regiment that practice anddispense with the commitment to mathematical entities. But even if the frameworksucceeds at the latter task, thus allowing the modal structuralist to avoid the relevantcommitment, the issue of how to make sense of the way mathematics is actually used inscientific contexts still remains. Providing a translation scheme into a nominalisticlanguage does not address this issue. A significant aspect of mathematical practice is thenleft unaccounted for.

The status of the indispensability argument within the modal-structural interpretation isquite unique. On the one hand, the conclusion of the argument is undermined (if theproposed translation scheme goes through), since commitment to the existence ofmathematical objects can be avoided. On the other hand, a revised version of theindispensability argument can be used to motivate the translation into the modal language,thus emphasizing the indispensable role played by the primitive modal notions introducedby the modal structuralist. The idea is to change the argument's second premise, insisting



that modal-structural translations of mathematical theories are indispensable to our besttheories of the world, and concluding that we ought to be ontologically committed to thepossibility of the corresponding structures. In this sense, modal structuralists can invokethe indispensability argument in support of the translation scheme they favor and, hence,the possibility of the relevant structures, which are referred to in the conclusion of therevised argument. But rather than supporting the existence of mathematical objects, theargument would only support commitment to modal-structural translations of mathematicaltheories and the possibility of mathematical structures.


With the introduction of modal operators and the proposed translation scheme, the modalstructuralist is unable to provide a uniform semantics for scientific and mathematicaltheories. Only the latter, as opposed to the former, requires such operators. In fact, Fieldhas argued that if modal operators were invoked in the formulation of scientific theories,not only their mathematical content, but also their physical content would be nominalized(Field 1989). After all, in that case, instead of asserting that some physical situation isactually the case, the theory would only state the possibility that this is so.

One strategy to avoid this difficulty (of losing the physical content of a scientific theorydue to the use of modal operators) is to employ an actuality operator. By properly placingthis operator within the scope of the modal operators, it is possible to undo thenominalization of the physical content in question (Friedman 2005). Without theintroduction of the actuality operator, or some related maneuver, it is unclear that themodal structuralist would be in a position to preserve the physical content of the scientifictheory in question.

But the introduction of an actuality operator in this context requires the distinction betweennominalist and mathematical content. (That such a distinction cannot be drawn at all isargued in Azzouni 2011.) Otherwise, there is no guarantee that the application of theactuality operator will not yield more than what is physically real.

However, even with the introduction of such an operator, there would still be a significantdifference, on the modal-structural translation scheme, between the semantics formathematical and scientific discourse. For the former, as opposed to the latter, does notinvoke such an operator. The result is that modal structuralism does not seem to be able toprovide a uniform semantics for mathematical and scientific language.

4.2.4 Taking mathematics literally

Given the need for introducing modal operators, the modal structuralist does not take



mathematical discourse literally. In fact, it may be argued, this is the whole point of theview! Taken literally, mathematical discourse seems to be committed to abstract objectsand structures—a commitment that the modal structuralist clearly aims to avoid.

However, the point still stands that, in order to block such commitment, a parallel discourseto actual mathematical practice is offered. The discourse is ‘parallel’ given thatmathematical practice typically does not invoke the modal operators introduced by themodal structuralist. For those who aim to understand mathematical discourse as it is used inthe practice of mathematics, and who try to identify suitable features of that practice thatprevent commitment to mathematical entities, the proposed translation will make therealization of that goal particularly difficult.


The modal structuralist has solved, in part, the ontological problem. No commitment tomathematical objects or structures seems to be needed to implement the proposedtranslation scheme. The main concern emerges from the introduction of modal operators.But as the modal structuralist emphasizes, these operators do not presuppose a possible-worlds semantics: they are introduced as primitive terms.

However, since the modal translation of mathematical axioms is taken to be true, thequestion arises as to what makes such statements true. For instance, when it is asserted that‘it is possible that there are structures satisfying the axioms of Peano Arithmetic’, what isresponsible for the truth of such statement? Clearly, the modal structuralist will not groundthe possibility in question on the actual truth of the Peano axioms, for this move, on areasonable interpretation, would require platonism. Nor will the modal structuralist supportthe relevant possibility on the basis of the existence of a consistency proof for the Peanoaxioms. After all, any such proof is an abstract object, and to invoke it at the foundation ofmodal structuralism clearly threatens the coherence of the overall view. Furthermore, toinvoke a modalized version of such a consistency proof would beg the question, since itassumes that the use of modal operators is already justified. Ultimately, what is needed tosolve properly the ontological problem is a suitable account of modal discourse.

5. Deflationary Nominalism5.1 Central features of deflationary nominalism

According to the deflationary nominalist, it is perfectly consistent to insist thatmathematical theories are indispensable to science, to assert that mathematical andscientific theories are true, and to deny that mathematical objects exist. I am calling theview ‘deflationary nominalism’ given that it demands very minimal commitments to make



sense of mathematics (Azzouni 2004), it advances a deflationary view of truth (Azzouni2004, 2006), and advocates a direct formulation of mathematical theories, withoutrequiring that they be reconstructed or rewritten (Azzouni 1994, 2004).

Deflationary nominalism offers an ‘easy road’ to nominalism, which does not require anyform of reformulation of mathematical discourse, while granting the indispensability ofmathematics. Despite the fact that quantification over mathematical objects and relations isindispensable to our best theories of the world, this fact offers no reason to believe in theexistence of the corresponding entities. This is because, as Jody Azzouni points out, twokinds of commitment should be distinguished: quantifier commitment and ontologicalcommitment (Azzouni 1997; 2004, p. 127 and pp. 49–122). We incur a quantifiercommitment whenever our theories imply existentially quantified statements. Butexistential quantification, Azzouni insists, is not sufficient for ontological commitment.After all, we often quantify over objects we have no reason to believe exist, such asfictional entities.

To incur an ontological commitment—that is, to be committed to the existence of a givenobject—a criterion for what exists needs to be satisfied. There are, of course, variouspossible criteria for what exists (such as causal efficacy, observability, possibility ofdetection, and so on). But the criterion Azzouni favors, and he takes it to be the one thathas been collectively adopted, is ontological independence (2004, p. 99). What exist are thethings that are ontologically independent of our linguistic practices and psychologicalprocesses. The point is that if we have just made something up through our linguisticpractices or psychological processes, there's no need for us to be committed to theexistence of the corresponding object. And typically, we would resist any suchcommitment.

Do psychological processes themselves exist, according to the ontologically independencecriterion? It may be argued that most psychological processes do exist, at least those weundergo rather than those we make up. After all, the motivation underlying theindependence criterion is that those things we just made up verbally or psychologically donot exist. Having a headache or believing that there is a laptop computer in front of menow are psychological processes that I did not make up. Therefore, it seems that at leastthese kinds of psychological processes do exist. In contrast, imaginings, desires, and hopesare processes we make up, and thus they do not exist. However, the underlying motivationfor the criterion seems to diverge, in these cases, from what is entailed by the criterion'sactual formulation. For the criterion insists on the ontological independence of “ourlinguistic practices and psychological processes”. Since headaches and beliefs arepsychological processes themselves, presumably they are not ontologically independent ofpsychological processes. Hence, they do not exist. This means that if the criterion isapplied as stated, no psychological process exists. For similar reasons, novels, mental



contents, and institutions do not exist either, since they are all abstract objects dependenton our linguistic practices and psychological processes, according to the deflationarynominalist (Azzouni 2010a, 2012).

Quine, of course, identifies quantifier and ontological commitments, at least in the crucialcase of the objects that are indispensable to our best theories of the world. Such objects arethose that cannot be eliminated through paraphrase and over which we have to quantifywhen we regiment the relevant theories (using first-order logic). According to Quine'scriterion, these are precisely the objects we are ontologically committed to. Azzouni insiststhat we should resist this identification. Even if the objects in our best theories areindispensable, even if we quantify over them, this is not sufficient for us to be ontologicallycommitted to them. After all, the objects we quantify over might be ontologicallydependent on us—on our linguistic practices or psychological processes—and thus wemight have just made them up. But, in this case, clearly there is no reason to be committedto their existence. However, for those objects that are ontologically independent of us, weare committed to their existence.

As it turns out, on Azzouni's view, mathematical objects are ontologically dependent onour linguistic practices and psychological processes. And so, even though they may beindispensable to our best theories of the world, we are not ontologically committed tothem. Hence, deflationary nominalism is indeed a form of nominalism.

But in what sense do mathematical objects depend on our linguistic practices andpsychological processes? In the sense that the sheer postulation of certain principles isenough for mathematical practice: ‘A mathematical subject with its accompanying positscan be created ex nihilo by simply writing down a set of axioms’ (Azzouni 2004, p. 127).The only additional constraint that sheer postulation has to meet, in practice, is thatmathematicians should find the resulting mathematics interesting. That is, theconsequences that follow from the relevant mathematical principles shouldn't be obvious,and they should be computationally tractable. Thus, given that sheer postulation is(basically) enough in mathematics, mathematical objects have no epistemic burdens. Suchobjects, or ‘posits’, are called ultrathin (Azzouni 2004, p. 127).

The same move that the deflationary nominalist makes to distinguish ontologicalcommitment from quantifier commitment is also used to distinguish ontologicalcommitment to Fs from asserting the truth of ‘There are Fs’. Although mathematicaltheories used in science are (taken to be) true, this is not sufficient to commit us to theexistence of the objects these theories are supposedly about. After all, according to thedeflationary nominalist, it may be true that there are Fs, but to be ontologically committedto Fs, a criterion for what exists needs to be satisfied. As Azzouni points out:



I take true mathematical statements as literally true; I forgo attempts to showthat such literally true mathematical statements are not indispensable toempirical science, and yet, nonetheless, I can describe mathematical terms asreferring to nothing at all. Without Quine's criterion to corrupt them, existentialstatements are innocent of ontology. (Azzouni 2004, pp. 4–5.)

On the deflationary nominalist picture, ontological commitment is not signaled in anyspecial way in natural (or even formal) language. We just don't read off the ontologicalcommitment of scientific doctrines (even if they were suitably regimented). After all,without Quine's criterion of ontological commitment, neither quantification over a givenobject (in a first-order language) nor formulation of true claims about such an object entailsthe existence of the latter.

In his 1994 book, Azzouni did not commit himself to nominalism, on the grounds thatnominalists typically require a reconstruction of mathematical language—something that,as discussed above, is indeed the case with both mathematical fictionalism (Field 1989)and modal structuralism (Hellman 1989). However, no such reconstruction wasimplemented, or needed, in the proposal advanced by Azzouni (Azzouni 1994). The factthat mathematical objects play no role in how mathematical truths are known clearlyexpresses a nominalist attitude—an attitude that Azzouni explicitly endorsed in (Azzouni2004).

The deflationary nominalist proposal nicely expresses a view that should be takenseriously. And as opposed to other versions of nominalism, it has the significant benefit ofaiming to take mathematical discourse literally.

5.2 Assessment: benefits of deflationary nominalism and a problem

Of the nominalist views discussed in this essay, deflationary nominalism is the view thatcomes closest to solving (or, in some cases, dissolving) the five problems that have beenused to assess nominalist proposals. With the possible exceptions of the issue of takingmathematical language literally and the ontological problem, all of the remaining problemsare explicitly and successfully addressed. I will discuss each of them in turn.

5.2.1 The epistemological problem dissolved

How can the deflationary nominalist explain the possibility of mathematical knowledge,given the abstract nature of mathematical objects? On this version of nominalism, thisproblem vanishes. Mathematical knowledge is ultimately obtained from what follows frommathematical principles. Given that mathematical objects do not exist, they play no role inhow mathematical results are known (Azzouni 1994). What is required is that the relevant



mathematical result be established via a proof. Proofs are the source of mathematicalknowledge.

It might be argued that certain mathematical statements are known without thecorresponding proof. Consider the Gödel sentence invoked in the proof of Gödel'sincompleteness theorem: the sentence is true, but it cannot be proved in the system underconsideration (if the latter is consistent). Do we have knowledge of the Gödel sentence?Clearly we do, despite the fact that the sentence is not derivable in the system in question.As a result, the knowledge involved here is of a different sort than the one articulated interms of what can be proved in a given system.

In my view, the deflationary nominalist has no problem making sense of our knowledge ofthe Gödel sentence. It is an intuitive sort of knowledge, which emerges from what thesentence in question states. All that is required in order to know that the sentence is true isto properly understand it. But that's not how mathematical results are typically established:they need to be proved.

According to Azzouni, we know the Gödel sentence as long as we are able to embed thesyntactically incomplete system (such as Peano arithmetic) in a stronger system in whichthe truth predicate for the original system occurs and in which the Gödel sentence is proved(Azzouni 1994, pp. 134–135; Azzouni 2006, p. 89, note 38, last paragraph, and pp. 161–162).

Clearly, the account does not turn mathematical knowledge into something easy to obtain,given that, normally, it is no straightforward matter to determine whether some resultfollows from a given group of axioms. Part of the difficulty emerges from the fact that thelogical consequences of a non-trivial group of axioms are often not transparent: significantwork is required to establish such consequences. This is as it should be, given the non-trivial nature of mathematical knowledge.

5.2.2 Dissolving the problem of the application of mathematics

The deflationary nominalist offers various considerations to the effect that there is nogenuine philosophical problem in the success of applied mathematics (Azzouni 2000).Once particular attention is given to implicational opacity—our inability to see, before aproof is offered, the consequences of various mathematical statements—much of thealleged surprise in the successful application of mathematics should vanish. Ultimately, theso-called problem of the application of mathematics—of understanding how it is possiblethat mathematics can be successfully applied to the physical world should—is anartificially designed issue rather than a genuine problem.



Colyvan defends the opposing view (Colyvan 2001b), insisting that the application ofmathematics to science does present a genuine problem. In particular, he argues that twomajor philosophical accounts of mathematics, Field's mathematical fictionalism andQuine's platonist realism, are unable to explain the problem. Thus, he concludes that theproblem cuts across the realism/anti-realism debate in the philosophy of mathematics. Thedeflationary nominalist would insist that what is ultimately at issue—implicational opacity—is not a special problem, even though to the extent that it is a problem, it is one that isequally faced by realists and anti-realists about mathematics.

This does not mean that the application of mathematics is a straightforward matter. Clearly,it is not. But the difficulties involved in the successful application of mathematics do notraise a special philosophical problem, particularly as soon as the issue of implicationalopacity is acknowledged—an issue that is common to both pure and applied mathematics.

The issue of understanding the way in which mathematics in fact gets applied is somethingthat the deflationary nominalist explicitly addresses, carefully examining the centralfeatures and the limitations of different models of the application of mathematics (see, inparticular, the second part of (Azzouni 2004)).


The deflationary nominalist, as noted above, is not committed to offering a reconstruction,or any kind of reformulation, of mathematical theories. (The exception here is the case ofinconsistent mathematical or scientific theories, which according to the deflationarynominalist, ideally are regimented as consistent first-order theories.) No special semanticsis required to make sense of mathematics: the same semantics that is used in the case ofscientific theories is invoked for mathematical theories. It may seem that the uniformsemantics requirement is thus satisfied. But the situation is more complicated.

It may be argued that the deflationary nominalist needs to provide the semantics for theexistential and universal claims in mathematics, science and ordinary language. After all, itdoes sound puzzling to state: “It is true that there are numbers, but numbers do not exist”.What is such semantics? The deflationary nominalist will respond by noting that thissemantics is precisely the standard semantics of classical logic, with the familiar conditionsfor the existential and universal quantifiers, but without the assumption that suchquantifiers are ontologically committing. The fact that no ontological import is assigned tothe quantifiers does not change their semantics. After all, the metalanguage in which thesemantics is developed already has universal and existential quantifiers, and thesequantifiers need not be interpreted as providing ontological commitment any more than theobject language quantifiers do. As a result, the same semantics is used throughout.



It may be argued that the deflationary nominalist needs to introduce the distinction betweenontologically serious (or ontologically committing) uses of the quantifiers andontologically innocent (or ontologically non-committing) uses. If so, this wouldpresumably require a different semantics for these quantifiers. In response, the deflationarynominalist will deny the need for such distinction. In order to mark ontologicalcommitment, an existence predicate, which expresses ontological independence, is used.Those things that are ontologically independent of us (that is, of our linguistic practices andpsychological processes) are those to which we are ontologically committed. The mark ofontological commitment is not made at the level of the quantifiers, but via the existencepredicate.

This means, however, that even though the semantics is uniform throughout the sciences,mathematics and ordinary language, deflationary nominalism requires the introduction ofthe existence predicate. But, at least on the surface, this predicate does not seem to have acounterpart in the way language is used in these domains. It is the same semanticsthroughout, but the formalization of the discourse requires an extended language toaccommodate the existence predicate. As a result, the uniformity of the semantics comeswith the cost of the introduction of a special predicate into the language to markontological commitment for formalization.

Perhaps the deflationary nominalist will respond by arguing that the existence predicate isalready part of the language, maybe implicitly 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 the predicate is in fact found inscientific, mathematical and ordinary contexts. 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 is used. As a result, the sentencescould 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 ‘perfectlyfrictionless 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 in the logical form of thenatural language sentences in order to introduce the existence predicate. And that isarguably a cost for the view. After all, in these cases, mathematical, scientific and ordinarylanguages do not seem to be taken literally—a topic to which I turn now.

5.2.4 Taking mathematical language literally

We saw that with the introduction of the existence predicate it is not clear that thedeflationary nominalist is in fact able to take mathematical language literally. After all,some reconstruction of that language seems to be needed. It should be granted that the levelof reconstruction involved is significantly less than what is found in the other versions ofnominalism discussed above. As opposed to them, the deflationary nominalist is able toaccommodate significant aspects of mathematical practice without the need for creating afull parallel discourse (in particular, no operators, modal or fictional, need to beintroduced). However, some level of reconstruction is still needed to accommodate theexistence predicate, which then compromises the deflationary nominalist's capacity to takemathematical language literally.

A related concern is that the deflationary nominalist introduces a non-standard notion ofreference that does not presuppose the existence of the objects that are referred to (Buenoand Zalta 2005). This move goes hand in hand with the understanding of the quantifiers asnot being ontologically committing, and it does seem to limit the deflationary nominalist'scapacity to take mathematical language literally. After all, a special use of ‘refers’ isneeded to accommodate the claim that “‘a’ refers to b, but b does not exist”. Thedeflationary nominalist, however, resists this charge (Azzouni 2009a, 2010a, 2010b).


The ontological problem is also dissolved by the deflationary nominalism. Clearly,deflationary nominalism has no commitment either to mathematical objects or to a modalontology of any kind (including possible worlds, abstract entities as proxy for possibleworlds, or other forms of replacement for the expression of modal claims). Thedeflationary nominalists not only avoid the commitment to mathematical objects, they alsoclaim that such objects have no properties whatsoever. This means that the deflationarynominalist's ontology is extremely minimal: only concrete objects are ultimately assumed—objects that are ontologically independent of our psychological processes and linguisticpractices. In particular, no domain of nonexistent objects is posited nor a realm of genuineproperties of such objects. By ‘genuine properties’ I mean those properties that hold only in



virtue of what the objects in question are, and not as the result of some external relations toother objects. For example, although Sherlock Holmes does not exist, he has the propertyof being thought of by me as I write this sentence. This is not, however, a genuine propertyof Sherlock Holmes in the intended sense.

Deflationary nominalism is not a form of Meinongianism (Azzouni 2010a). Although theontology of deflationary nominalism is not significantly different from that of theMeinongian, the ideology of the two views—at least assuming a particular, traditionalinterpretation of Meinongianism—is importantly different. The deflationary nominalist isnot committed to any subsisting objects, in contrast to what is often claimed to be adistinctive feature of Meinongianism.

It is not clear to me, however, that this traditional reading of Meinongianism is correct. Ifwe consider the subsisting objects as those that are abstract, and if we take only concreteobjects as existing, the resulting picture ideologically is not significantly different from theone favored by the deflationary nominalist. Still, the deflationary nominalists distancethemselves from Meinongianism (Azzouni 2010a).

With the meager ontological commitments, the deflationary nominalist fares very well onthe ontological front. One source of concern is how meager the deflationary nominalist'sontology ultimately is. For instance, platonists would insist that mathematical objects areontologically independent of our psychological processes and linguistic practices, and—using the criterion of ontological commitment offered by the deflationary nominalist—theywould insist that these objects do exist. Similarly, modal realists (such as Lewis 1986)would also argue that possible worlds are ontologically independent of us in the relevantsense, thus concluding that these objects also exist. Deflationary nominalists will try toresist these conclusions. But unless their arguments are successful at this point, the concernremains that the deflationary nominalist may have a significantly more robust ontology—given the proposed criterion of ontological commitment—than advertised.

It may be argued that deflationary nominalists are changing the rules of the debate. Theystate that mathematicians derive statements of the form “There are Fs”, but insist that theobjects in question do not exist, given that quantifier commitment and ontologicalcommitment should be distinguished. This strategy is fundamentally different from thosefound in the nominalist proposals discussed before. It amounts to the denial of the firstpremise of the indispensability argument (“We ought to be ontologically committed to alland only those entities that are indispensable to our best theories of the world”). Eventhough quantification over mathematical entities is indispensable to our best theories of theworld (thus, the deflationary nominalist accepts the second premise of the argument), thisfact does not entail that these entities exist. After all, we can quantify over objects that donot exist, given the rejection of the indispensability argument's first premise.



But are deflationary nominalists really changing the rules of the debate? If Quine's criterionof ontological commitment provides such rules, then they are. But why should we grantthat Quine's criterion play such a role? Deflationary nominalists challenge this deeply heldassumption in ontological debates. And by doing so, they pave the way for thedevelopment of a distinctive form of nominalism in the philosophy of mathematics.

BibliographyAzzouni, J., 1994, Metaphysical Myths, Mathematical Practice: The Ontology andEpistemology of the Exact Sciences. Cambridge: Cambridge University Press.–––, 1997, “Applied Mathematics, Existential Commitment and the Quine-PutnamIndispensability Thesis”, Philosophia Mathematica 5, pp. 193–209.–––, 2000, “Applying Mathematics: An Attempt to Design a Philosophical Problem”,The Monist 83, pp. 209–227.–––, 2004, Deflating Existential Consequence: A Case for Nominalism. New York:Oxford University Press.–––, 2006, Tracking Reason: Proof, Consequence, and Truth. New York: OxfordUniversity Press.–––, 2007, “Ontological Commitment in the Vernacular”, Noûs 41, pp. 204–226.–––, 2009a, “Empty de re Attitudes about Numbers”, Philosophia Mathematica 17,pp. 163–188.––– 2009b, “Evading Truth Commitments: The Problem Reanalyzed”, Logique etAnalyse 206, pp. 139–176.–––, 2010a, Talking about Nothing: Numbers, Hallucinations, and Fictions. NewYork: Oxford University Press.––– 2010b, “Ontology and the Word ‘Exist’: Uneasy Relations”, PhilosophiaMathematica 18, pp. 74–101.–––, 2011, “Nominalistic Content”, in Cellucci, Grosholtz, and Ippoliti (eds.) 2011,pp. 33–51.–––, 2012, “Taking the Easy Road Out of Dodge”, Mind 121, pp. 951–965. (Paperpublished in 2013.)Balaguer, M., 1998, Platonism and Anti-Platonism in Mathematics. New York:Oxford University Press.Bueno, O., 2003, “Is It Possible to Nominalize Quantum Mechanics?”, Philosophy ofScience 70, pp. 1424–1436.–––, 2008, “Nominalism and Mathematical Intuition”, Protosociology 25, pp. 89–107.–––, 2009, “Mathematical Fictionalism”, in Bueno and Linnebo (eds.) 2009, pp. 59–79.Bueno, O., French, S., and Ladyman, J., 2002, “On Representing the Relationship



between the Mathematical and the Empirical”, Philosophy of Science 69, pp. 497–518.Bueno, O., and Linnebo, Ø. (eds.), 2009, New Waves in Philosophy of Mathematics.Hampshire: Palgrave MacMillan.Bueno, O., and Zalta, E., 2005, “A Nominalist's Dilemma and its Solution”,Philosophia Mathematica 13, pp. 294–307.Benacerraf, P., 1973, “Mathematical Truth”, Journal of Philosophy 70, pp. 661–679.Benacerraf, P., and Putnam, H. (eds.), 1983, Philosophy of Mathematics: SelectedReadings. (Second edition.) Cambridge: Cambridge University Press.Boolos, G., and Jeffrey, R., 1989, Computability and Logic. (Third edition.)Cambridge: Cambridge University Press.Burgess, J., and Rosen, G., 1997, A Subject With No Object: Strategies forNominalistic Interpretation of Mathematics. Oxford: Clarendon Press.Cellucci, C., Grosholtz, E., and Ippoliti, E. (eds.), 2011, Logic and Knowledge.Newcastle: Cambridge Scholars Publishing.Chihara, C.S., 1990, Constructibility and Mathematical Existence. Oxford:Clarendon Press.Colyvan, M., 2001a, The Indispensability of Mathematics. New York: OxfordUniversity Press.–––, 2001b, “The Miracle of Applied Mathematics”, Synthese 127, pp. 265–277.–––, 2010, “There's No Easy Road to Nominalism”, Mind 119, pp. 285–306.da Costa, N., and French, S., 2003, Science and Partial Truth. New York: OxfordUniversity Press.Field, H., 1980, Science without Numbers: A Defense of Nominalism. Princeton, N.J.:Princeton University Press.–––, 1984, “Is Mathematical Knowledge Just Logical Knowledge?”, PhilosophicalReview 93, pp. 509–552. (Reprinted with a postscript and some changes in Field1989, pp. 79–124.)–––, 1989, Realism, Mathematics and Modality. Oxford: Basil Blackwell.–––, 1991, “Metalogic and Modality”, Philosophical Studies 62, pp. 1–22.–––, 1992, “A Nominalistic Proof of the Conservativeness of Set Theory”, Journal ofPhilosophical Logic 21, pp. 111–123.Friedman, J., 2005, “Modal Platonism: An Easy Way to Avoid OntologicalCommitment to Abstract Entities”, Journal of Philosophical Logic 34, pp. 227–273.Gödel, K., 1944, “Russell's Mathematical Logic”, in Schilpp (ed.) 1944, pp. 125–153.(Reprinted in Benacerraf and Putnam (eds.) 1983, pp. 447–469.)–––, 1947, “What Is Cantor's Continuum Problem?”, American MathematicalMonthly 54, pp. 515–525. (A revised and expanded version can be found inBenacerraf and Putnam (eds.) 1983, pp. 470–485.)Goodman, N., and Quine, W. V., 1947, “Steps Toward a Constructive Nominalism”,



Journal of Symbolic Logic 12, pp. 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 a Modal-StructuralInterpretation. Oxford: Clarendon Press.–––, 1996, “Structuralism without Structures”, Philosophia Mathematica 4, pp. 100–123.Hilbert, D., 1971, Foundations of Geometry, La Salle: Open Court. (Englishtranslation of the tenth German edition, published in 1968.)Leng, M., 2010, Mathematics and Reality. New York: Oxford University Press.Lewis, D., 1986, On the Plurality of Worlds. Oxford: Blackwell.Malament, D., 1982, “Review of Field, Science without Numbers”, Journal ofPhilosophy 79: 523–534.Melia, J., 1995, “On What There's Not”, Analysis 55, pp. 223–229.–––, 1998, “Field's Programme: Some Interference”, Analysis 58, pp. 63–71.–––, 2000, “Weaseling Away the Indispensability Argument”, Mind 109, pp. 455–479.Putnam, H., 1971, Philosophy of Logic. New York: Harper and Row. (Reprinted inPutnam 1979, pp. 323–357.)–––, 1979, Mathematics, Matter and Method. Philosophical Papers, volume 1.(Second edition.) Cambridge: Cambridge University Press.Quine, W. V., 1960, Word and Object. Cambridge, Mass.: The MIT Press.Resnik, M., 1997, Mathematics as a Science of Patterns. Oxford: Clarendon Press.Schilpp, P.A. (ed.), 1944, The Philosophy of Bertrand Russell. The Library of LivingPhilosophers. Chicago: Northwestern University.Shapiro, S., 1997, Philosophy of Mathematics: Structure and Ontology. New York:Oxford University Press.Simons, P., 2010, Philosophy and Logic in Central Europe from Bolzano to Tarski.Dordrecht: Kluwer.van Fraassen, B.C., 1980, The Scientific Image. Oxford: Clarendon Press.–––, 1991, Quantum Mechanics: An Empiricist View. Oxford: Clarendon Press.von Neumann, J., 1932, Mathematical Foundations of Quantum Mechanics. (Englishtranslation, 1955.) Princeton: Princeton University Press.Weyl, H., 1928, Gruppentheorie un Quantenmechanik. Leipzig: S. S. Hirzel.[English translation, 1931, The Theory of Groups and Quantum Mechanics, H.P.Roberton (trans.), New York: Dover.]Yablo, S., 2001, “Go Figure: A Path through Fictionalism”, Midwest Studies inPhilosophy 25, pp. 72–102.

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Acknowledgements

My thanks go to two anonymous referees for their helpful comments on earlier versions ofthis entry. Their suggestions led to significant improvements. My thanks are also due toJody Azzouni, Uri Nodelman, and Ed Zalta for all of their comments and help.

Copyright © 2013 by Otávio Bueno <[email protected]>

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