## 1. Two views about mathematics: nominalism and platonism


In ontological discussions about mathematics, two views are
prominent. According to platonism, mathematical objects (as well as
mathematical relations and structures) exist and are abstract; that
is, they are not located in space and time and have no causal
connection 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 crucial features the objects in questions are supposed to
have. According to nominalism, mathematical objects (including,
henceforth, mathematical relations and structures) do not exist, or at
least they need not be taken to exist for us to make sense of
mathematics. So, it is the nominalist's burden to show how to
interpret mathematics without the commitment to the existence of
mathematical objects. This is, in fact, a key feature of nominalism:
those who defend the view need to show that it is possible to yield at
least as much explanatory work as the platonist obtains, but invoking
a meager ontology. To achieve that, nominalists in the philosophy of
mathematics forge interconnections with metaphysics (whether
mathematical objects do exist), epistemology (what kind of knowledge
of these entities we have), and philosophy of science (how to make
sense of the successful application of mathematics 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 least one feature in common: both come in many
forms. There are various versions of platonism in the philosophy of
mathematics: standard (or object-based) platonism (Godel 1944,
1947; Quine 1960), structuralism (Resnik 1997; Shapiro 1997), and
full-blooded platonism (Balaguer 1998), among other views. Similarly,
there are also several versions of nominalism: 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 (Bueno 2008, 2009), and pretense views
(Leng 2010), among others. Similarly to their platonist counterparts,
the various nominalist proposals have different motivations, and face
their own difficulties. These will be explored in turn. (A critical
survey of various nominalization strategies in mathematics can be
found in Burgess and Rosen (1997). The authors address in detail both
the technical and philosophical issues raised by nominalism in the
philosophy of mathematics.)


Discussions about nominalism in the philosophy of mathematics in
the 20th century started roughly with the work that
W. V. Quine and Nelson Goodman developed toward constructive
nominalism (Goodman and Quine 1947). But, as Quine later pointed out,
in the end it was indispensable to quantify over classes (Quine
1960). As will become clear below, responses to this indispensability
argument have generated a significant amount of work for
nominalists. And it is the focus on the indispensability argument that
largely distinguishes more recent nominalist views in the philosophy
of mathematics, which I will focus on, from the nominalism developed
in the early part of the 20th century by the Polish school
of logic (Simons 2010).


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


## 2. Five Problems


In contemporary philosophy of mathematics, nominalism has been
formulated in response to difficulties faced by platonism. But in
developing their responses to platonism, nominalists also encounter
difficulties of their own. Five problems need to be addressed in this
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, and
5. The ontological problem.


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


### 2.1 The epistemological problem of mathematics


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


This is taken to be a particular problem for platonism, since this
view postulates the existence of mathematical objects, and one would
expect such objects to play a role in the acquisition of mathematical
knowledge. After all, on the platonist view, such knowledge is about
the corresponding mathematical objects. However, despite various
sophisticated attempts by platonists, there is still considerable
controversy as to how exactly this process should be
articulated. Should it be understood via mathematical intuition, by
the introduction of suitable mathematical principles and definitions,
or does it require some form of abstraction?


In turn, the epistemological issue is far less problematic for
nominalists, who are not committed to the existence of mathematical
objects in the first place. They will have to explain other things,
such as, how can the nominalist account for the difference between a
mathematician, who knows a significant amount of mathematics, and a
non-mathematician, who does not? This difference, according to some
nominalists, is based on empirical and logical 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 be explained? Platonists allegedly have an answer
to this problem. Given that mathematical objects exist and are
successfully referred to by our scientific theories, it is not
surprising that such theories are successful. Reference to
mathematical objects is just part of the reference to those entities
that are indispensable to our best theories of the world. This frames
the problem of the application of mathematics in terms of the
indispensability argument.


In fact, one of the main reasons for belief in the existence of
mathematical objects--some claim 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 ontological commitment should
be restricted to just those entities that are indispensable to our
best theories of the world (Quine 1960; Putnam 1971; Colyvan
2001a). Mark Colyvan has formulated the argument in the following
terms:



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


The first premise relies crucially on Quine's criterion of
ontological commitment. After regimenting our best theories of the
world in a first-order language, the ontological commitments of these
theories can be read off as being the value of the existentially
quantified variables. But how do we move from the ontological
commitments of a theory to what we *ought* to be ontologically
committed to? This is the point where the first premise of the
indispensability argument emerges. If we are dealing with our best
theories of the world, precisely those items that are indispensable to
these theories amount to what we *ought* to be committed
to. (Of course, a theory may quantify over more objects than those
that are indispensable.) And by identifying the indispensable
components invoked in the explanation 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 the application of mathematics is, in fact,
controversial. Given that mathematical objects are abstract, it is
unclear why the postulation of such entities is helpful to understand
the success of applied mathematics. For the physical world--being
composed of objects located 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 physical world involved in the
application of mathematics. Just mentioning that the physical
world *instantiates* structures (or substructures) described in
general terms by various mathematical theories is not enough (see,
e.g., Shapiro 1997). For there are infinitely many mathematical
structures, and there is no way of uniquely determining which of them
is actually instantiated--or even instantiated only in
part--in a finite region of the physical world. There is a
genuine underdetermination here, given that the same physical
structure in 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 between them empirically.


Despite the controversial nature of the platonist claim to be able
to explain the success of applied mathematics, to accommodate that
success is often taken as a significant benefit of platonism. Less
controversially, the platonist is certainly able to describe the way
in which mathematical theories are actually used in scientific
practice without having to rewrite them. 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 of mathematics in scientific theorizing. Since,
according to the nominalist, mathematical objects do not
exist--or, at least, are not taken to exist--it becomes
unclear how referring to such entities can contribute to the empirical
success of scientific theories. In particular, if it turns out that
reference to mathematical entities is indeed indispensable to our best
theories of the world, how can the nominalist deny the existence of
such entities? As we will see below, several nominalist views in the
philosophy of mathematics have emerged in response to the challenge
raised by considerations based on the indispensability of
mathematics.


### 2.3 The problem of uniform semantics


One of the most significant features of platonism is the fact that
it allows us to adopt the same semantics for both mathematical and
scientific discourse. Given the existence of mathematical objects,
mathematical statements are true in the same way as scientific
statements are true. The only difference emerges from their respective
truth makers: mathematical statements are true in virtue of abstract
(mathematical) objects and relations among them, whereas scientific
statements are ultimately true in virtue of concrete objects and the
corresponding relations among such objects. This point is idealized in
that it assumes that, somehow, we can manage to distill the empirical
content of scientific statements independently of the contribution
made by the mathematics that is often used to express such
statements. Platonists who defend the indispensability argument insist
that this 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 this case, the platonist can
provide a referential semantics throughout. Of course, the platonist
about 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
both mathematics 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 a result, a
distinct semantics needs to be offered for that language in comparison
with the semantics that is provided for scientific discourse.



### 2.4 The problem of taking mathematical discourse literally


A related benefit of platonism is that it allows one to take
mathematical discourse literally, given that mathematical terms
refer. In particular, there is no change in the syntax of mathematical
statements. So, when mathematicians claim that 'There are
infinitely many prime numbers', the platonist can take that
statement literally as describing the existence of an infinitude of
primes. On the platonist view, there are obvious truth-makers for
mathematical statements: mathematical objects and their corresponding
properties and relations (Benacerraf 1973).


We have here a major benefit of platonism. If one of the goals of
the philosophy of mathematics is to provide understanding of
mathematics and mathematical practice, it is a significant advantage
that platonists are able to take the products of that
practice--such as mathematical 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
actually formulated in mathematical practice, rather than discuss a
parallel discourse offered by various reconstructions of mathematics
given by those who avoid the commitment to mathematical 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 clear below, it is common that
nominalization strategies for mathematics change either the syntax or
the semantics of mathematical statements. For instance, in the case of
modal structuralism, modal operators are introduced to preserve verbal
agreement with the platonist (Hellman 1989). The proposal is that each
mathematical statement *S* is translated into two modal
statements: (i) if there were structures of the suitable
kind, *S* would be true in these structures, and (ii) it's
possible that there are such structures. As a result, both the syntax
and the semantics of mathematics are changed. In the case of
mathematical fictionalism, in order to preserve verbal agreement with
the platonist despite the denial of the existence of mathematical
objects, fiction operators (such as, 'According to
arithmetic...') are introduced (Field 1989). Once again,
the resulting proposal changes the syntax (and, hence, the semantics)
of mathematical discourse. This is a significant cost for these
views.


### 2.5 The ontological problem


The ontological problem consists in specifying the nature of the
objects a philosophical conception of mathematics is ontologically
committed to. Can the nature of these objects be properly determined?
Are the objects in question such that we simply lack good grounds to
believe in their existence? Traditional forms of platonism have been
criticized for failing to offer an adequate solution to this
problem. In response, some platonists have argued that the 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 to mathematical objects, they may be committed
to other entities that may also raise ontological concerns (such as
possibilia). The ontological problem is then the problem of assessing
the status of the ultimate commitments of the view.


Three nominalization strategies will be discussed below:
mathematical fictionalism (Field 1980, 1989), modal structuralism
(Hellman 1989, 1996), and deflationary nominalism (Azzouni 2004). The
first two reject the second premise of the indispensability
argument. They provide 'hard roads' to nominalism (Colyvan
2010), in the sense that the nominalist needs to develop the complex,
demanding work of showing how quantification over mathematical objects
can be avoided in order to develop a suitable interpretation of
mathematics. The third strategy rejects the first premise of the
argument, thus bypassing the need to argue for the dispensability of
mathematics (in fact, for the deflationary nominalist, mathematics is
ultimately indispensable). By reassessing Quine's criterion of
ontological commitment, and indicating that quantification over
certain objects does not require their existence, this strategy yields
an 'easy road' to nominalism.


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


## 3. Mathematical Fictionalism


### 3.1 Central features of mathematical fictionalism


In a series of works, Hartry Field provided an ingenious strategy
for the nominalization of science (Field 1980, 1989). As opposed to
platonist views, in order to explain the usefulness 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 with no commitment to mathematical
objects. Therefore, what he takes to be the main argument for
platonism, which relies on the (apparent) indispensability of
mathematics to science, is resisted. The nominalist nature of Field's
account emerges from the fact that mathematical objects are not
assumed to exist. Hence, mathematical theories are false. (Strictly
speaking, Field notes, any existential mathematical statement is
false, and any universal mathematical statement is vacuously true.) By
devising a strategy that shows how to dispense with mathematical
objects in the formulation of scientific theories, Field rejects the
indispensability argument, and provides strong grounds for the
articulation of a nominalist stance.


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


The nominalization strategy devised by the mathematical
fictionalist depends on two interrelated moves. The first is to change
the aim of mathematics, which is not taken to be truth, but something
different. On this view, the proper norm of mathematics, which will
guide the nominalization program, is *conservativeness*. A
mathematical theory is conservative if it is consistent with every
internally consistent theory about the physical world, 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 is consistent, but not vice
versa). However, conservativeness is not weaker than truth (Field
1980, pp. 16-19; Field 1989, p. 59). So, Field is not
countenancing a weaker aim of mathematics, but only
a *different* one.


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


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


The second move of the mathematical fictionalist strategy is to
provide such nominalistic premises. Field has done that in one
important case: Newtonian gravitational theory. He elaborates on a
work that has a respectable tradition: Hilbert's axiomatization of
geometry (Hilbert 1971). What Hilbert provided was a synthetic
formulation of geometry, which dispenses with metric concepts, and
therefore does not include any quantification over real numbers. His
axiomatization was based on concepts such
as *point*, *betweenness*,
and *congruence*. Intuitively speaking, we say that a
point *y* is *between* the points *x* and *z*
if *y* is a point in the line-segment whose endpoints
are *x* and *z*. Also intuitively, we say that the
line-segment *xy* is *congruent* to the
line-segment *zw* if the distance from the point *x* to the
point *y* is the same as that from the point *z*
to *w*. After studying the formal properties of the resulting
system, Hilbert proved a representation theorem. He showed, in a
stronger mathematical 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 the following
'homomorphism conditions' are met:


1. *xy* is congruent to *zw* iff *d*(*x*, *y*) = *d*(*z*,*w*), for all points *x*, *y*, *z*, and *w*;
2. *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 results about congruence and
betweenness. Thus, although we cannot talk about numbers in Hilbert's
geometry (there are no such entities to quantify over), there is a
metatheoretic result that associates assertions about distances with
what can be said in the theory. Field calls such numerical
claims *abstract counterparts* of purely geometric assertions,
and they can be used to draw conclusions about purely geometrical
claims in a smoother way. Indeed, because of the representation
theorem, conclusions about space, statable *without* real
numbers, can be drawn far more easily than we could achieve by a
deflationary proof from Hilbert's axioms. This illustrates Field's
point that the usefulness of mathematics derives from shortening
derivations (Field 1980, pp. 24-29).


Roughly speaking, what Field established was how to extend
Hilbert's results about space to space-time. Similarly to Hilbert's
approach, instead of formulating Newtonian laws in terms of numerical
functors, Field showed that they can be recast in terms of comparative
predicates. For example, instead of adopting a functor such as
'the gravitational potential of *x*', 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*
and *w*'. Relying on a body of representation theorems
(which plays the same role as Hilbert's representation theorem in
geometry), Field established how several numerical functors can be
'obtained' from comparative predicates. But in order to
use those theorems, he first showed how to formulate Newtonian
numerical laws (such as, Poisson's equation for the gravitational
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 formulated only in terms of
comparative predicates (with no recourse to numerical functors). For
any model *S* of *N* whose domain is constituted by
space-time regions, there exists:


1. A 1-1 spatio-temporal co-ordinate function *f* (unique up to a generalized Galilean transformation) mapping the space-time of *S* onto quadruples of real numbers;
2. 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
3. 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 comparative relations 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 laws of Newtonian gravitational theory in
their functorial form hold.


Notice that, in quantifying over space-time regions, Field assumes
a substantivalist view of space-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 mathematical fictionalist is allowed to draw nominalistic
conclusions from premises involving *N* plus a mathematical
theory *T*. After all, due to the conservativeness of
mathematics, such conclusions can be obtained independently
of *T*. The role of the extended representation theorem is then
to establish that, despite the lack of quantification over
mathematical objects, precisely the same class of models is determined
by formulating Newtonian gravitational theory in terms of functors (as
the theory is usually expressed) or in terms of comparative predicates
(as the mathematical fictionalist favors). Thus, the extended
representation theorem ensures that the use of conservativeness of
mathematics together with suitable nominalistic claims (formulated via
comparative predicates) does not change the 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, it seems a promising candidate for a nominalist
stance *vis-a-vis* mathematics.


How should the mathematical fictionalist approach physical
theories, such as perhaps string theory, that do not seem to be about
concrete observable objects? One possible response, 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 the mathematical fictionalist needs to provide a
nominalistic counterpart. In other words, until the moment in which
such theories acquire the relevant empirical import, they need not
worry the mathematical fictionalist. Theories of that sort would be
classified as part of the mathematics rather than the physics.


### 3.2 Metalogic and the formulation of conservativeness


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


At two crucial junctures, Field has made use of metalogical
results: (a) in his reformulation of 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 adequacy of conservativeness for the
mathematical fictionalist. For (a) settles that the latter can
formulate that notion without violating nominalism, and (b) concludes
that conservativeness is a feature that mathematics actually has. But
if these two outcomes are not legitimate, Field's approach cannot get
off the ground. I will now consider whether these 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 theorem to formulate the notion
of *conservativeness* in an acceptable way, that is, without
reference to mathematical entities. As noted above, conservativeness
is defined in terms of *consistency*. But this notion is
usually formulated either in semantic terms (as the existence of an
appropriate model), or in proof-theoretic terms (in terms of suitable
proofs). However, as Field acknowledges, these two formulations of
consistency are platonist, since they depend on abstract objects
(models and proofs), and therefore are not nominalistically
acceptable.


The mathematical fictionalist way out is to avoid moving to the
metalanguage in order to express 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
primitive notion of *logical consistency*: **A*. Thus,
if *B* is any sentence, *B*\* is the result of
restricting *B* to non-mathematical entities,
and *M*1, ..., *M**n* 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*\*
>  [?] *M*1 [?] ...
>  [?] *M**n*).
> 
> 
> 


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


This assumes, of course, that *M* is finitely axiomatized. But
how can we apply (C) in the case of mathematical theories that
are *not* finitely axiomatizable (such as Zermelo-Fraenkel set
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 initially suggested that the
mathematical fictionalist could use substitutional quantification to
express these infinite conjunctions (Field 1984). In a postscript to
the revised version of this essay (Field 1989, pp. 119-120), he
notes that substitutional quantification can be avoided, provided that
the mathematical and physical theories in question are expressed in a
logic for which *compactness* holds. For in this case, the
consistency of the whole theory is reduced to the consistency of each
of its finite conjunctions.


There are, however, three problems with this move.


1. One concern about the use of substitutional quantification in this context involves the nature of substitutional instances. If the latter turn out to be abstract, which would be the case if such substitutional instances were not mere inscriptions, they are not available to the nominalist. If the substitutional instances are concrete, the nominalist needs to show that there are enough of them.
2. The very *statement* of the compactness theorem involves *set-theoretic* talk: let *G* be a *set* of formulas; if every finite *subset* of *G* is consistent, then *G* is consistent. How can nominalists rely on a theorem whose very statement involves abstract entities? In order to use this theorem, an appropriate reformulation is required.
3. 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 *proof* of this theorem assumes set theory. The compactness theorem is usually presented as a corollary to the completeness theorem for first-order logic, whose proof assumes set theory (see, for example, Boolos and Jeffrey 1989, pp. 140-141). Alternatively, if the compactness result is to be proved directly, then one has to construct the appropriate model of *G*--which again requires set theory. So, unless mathematical fictionalists are able to provide an appropriate nominalization strategy for set theory itself, they are not entitled to use this result. In other words, far more work is required before a Field-type nominalist is able to rely on metalogical results.


But maybe this criticism misses the whole point of Field's
program. As we saw, Field does not require that a mathematical
theory *M* be true for it to be used. Only
its *conservativeness* is 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's strategy asks for is the formulation of
appropriate *nominalistic* bodies of claims to which
mathematics can be applied. The same point holds for metalogical
results: provided that they are applied to *nominalistic*
claims, Field is fine.


The problem with this reply is that it involves the mathematical
fictionalist program in a circle. The fictionalist cannot rely on
the *conservativeness* of mathematics to justify the use of a
mathematical result (the compactness theorem) that is required for
the *formulation* of the notion of conservativeness itself. For
in doing that, the fictionalist *assumes* that the notion of
conservativeness is nominalistically acceptable, and this is exactly
the point in question. Recall that the motivation for Field to use the
compactness theorem was to reformulate conservativeness without having
to assume abstract entities (namely, those required by the semantic
and the proof-theoretic accounts of consistency). Thus, at this point,
the mathematical fictionalist cannot yet use the notion of
conservativeness; otherwise, the whole program would not get off the
ground. I conclude that, similarly to any other part of mathematics,
metalogical results also need to be obtained nominalistically. 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 the notion of conservativeness in terms of a
primitive notion of logical consistency: **A*. And he also
indicates that this notion is related to the model-theoretic concept
of consistency--in particular, to the formulation of this concept
in von Neumann-Bernays-Godel finitely axiomatizable set theory
(NBG). This is done via two principles (Field 1989, p. 108):



> 
> (MTP#) If (NBG - there is a model for
>  '*A*'), then **A*
> 
> 
> (ME#) If (NBG - there is no model for
>  '*A*'), then !**A*.
> 
> 
> 


I am following Field's terminology: 'MTP#'
stands for *model-theoretic possibility*, and
'ME#' for *model existence*. The symbol
'#' indicates that, according to Field, these
principles are nominalistically acceptable. After all, they are modal
surrogates for the platonistic principles (Field 1989,
pp. 103-109):



> 
> (MTP) If there is a model for '*A*', then
>  **A*
> 
> 
> (ME) If there is no model for '*A*', then
>  !**A*.
> 
> 
> 


It may be argued that, by using these principles, the mathematical
fictionalist will be entitled 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, for the
sake of argument, that the following characterization will do:



> 
> (Compact#) If !**T*, then
>  [?]*f* *A*1,
>  ..., *A**n*[!*(*A*1
>  [?] ... [?] *A**n*)],
> 
> 
> 


where *T* is a theory and each *A**i*, 1
[?] *i* [?] *n*, is a formula (an axiom of *T*). The
expression
'[?]*f* *A*1...*A**n*'
is to be read as 'there are *finitely many*
formulas *A*1...*A**n*'. (This
quantifier is not first-order. However, I am not going to press the
point that the nominalist seems to need a *non*-first-order
quantifier to express a property typical of *first-order*
logic. This is only one of the worries we are leaving aside in this
formulation.) This version is parasitic on the following platonistic
formulation of the compactness theorem:



> 
> (Compact) If there is no model for *T*, then
>  [?]*f* *A*1,
>  ..., *A**n* such that there is no model
>  for (*A*1 [?] ...
>  [?] *A**n*).
> 
> 
> 


In order for mathematical fictionalists to be entitled to use the
compactness theorem, they will have to show that the nominalistic
formulation (Compact#) follows from the platonistic one
(Compact). In this sense, if the latter is adequate, so is the
former. More accurately, 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 on nominalist grounds, it would be
question-begging to assume the full platonistic version from 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* *A*1...*A**n*
>  !*(*A*1[?]...[?]*A**n*).
> 
> 
> 


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



> 
> (3) !(NBG - there is a model for
>  '*T*'),
> 
> 
> 


and thus



> 
> (4) *(NBG [?] there is no model for
>  '*T*').
> 
> 
> 


Let us assume the modal surrogate for the compactness theorem:



> 
> (CompactM) (NBG - if there is no model for
>  '*T*', then
>  [?]*f* *A*1...*A**n*
>  such that there is no model for
>  (*A*1[?]...[?]*A**n*)).
> 
> 
> 


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


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



> 
> (5)
>  *([?]*f* *A*1...*A**n*
>  such that there is no model for (*A*1
>  [?]...[?] *A**n*)).
> 
> 
> 


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 may be argued that it *is* nominalistically acceptable. (As
will be examined below, modal structuralists advance a nominalization
strategy exploring modality along these lines; see Hellman (1989).)
Field, however, is skeptical about this move. On his view, modality is
not a general surrogate for ontology (Field 1989,
pp. 252-268). And one of his worries is that by allowing the
introduction of modal operators, as a general nominalization strategy,
we modalize away the physical content of the theory under
consideration. However, since metalogical claims are not expected to
have physical consequences, the worry need not arise here. At any
rate, given that (5) does not establish what needs to be established,
it does not solve the problem.


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



> 
> (5') (NBG -
>  [?]*f* *A*1...*A**n*
>  such that there is no model for (*A*1
>  [?]...[?] *A**n*)).
> 
> 
> 


Note that if (5') were established, we would have settled the
matter. After all, with a straightforward reworking of
(ME#) (namely, If (NBG -
[?]*f* *A*1...*A**n*
such that there is no model for
(*A*1[?]...[?]*A**n*)),
then !*(*A*1
[?]...[?] *A**n*)), it follows from
(5') and (ME#) that



> 
> (2)
>  [?]*f* *A*1...*A**n*
>  !*(*A*1
>  [?]...[?] *A**n*),
> 
> 
> 


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, to say the least, the mathematical
fictionalist has to present it before being entitled to use metalogic
results. Until then, it is not clear that these results are
nominalistically acceptable.


#### 3.2.3 Metalogic and the proof of the conservativeness of set theory


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


The problem with this move is that, even if mathematical
fictionalists formulate the statement of the completeness theorem
without referring to mathematical entities, the *proof* of this
theorem assumes set theory (see, for instance, Boolos and Jeffrey
1989, pp. 131-140). Therefore, fictionalists cannot use the
theorem without undermining their nominalism. After all, the point of
providing a nominalistic proof of the conservativeness of set theory
is to show that, without recourse to platonist mathematics, the
mathematical fictionalist is able to establish that mathematics is
conservative. Field has offered a *platonist* argument for the
conservativeness result (Field 1980)--an argument that explicitly
invoked properties of set theory. The idea was to provide
a *reductio* of platonism: by using platonist mathematics,
Field attempted to establish that mathematics was conservative and,
thus, ultimately dispensable. In contrast with the earlier strategy,
the goal was to provide a proof of the conservativeness of set theory
that a nominalist could accept. But since the nominalistic proof
relies on the completeness theorem, it is not at all clear that it is
in fact nominalist. Mathematical fictionalists should first be able to
prove the completeness result *without* assuming set
theory. Alternatively, they should provide a nominalization strategy
for set theory itself, which will then entitle them to use metalogical
results.


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


In other words, without a broader nominalization strategy, which
allows set theory itself to be nominalized, it seems difficult to see
how mathematical fictionalists can use metalogical results as part of
their program. The problem, however, is that it is not at all obvious
that, at least in the form articulated by Field, the mathematical
fictionalist program can be extended 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). The approach doesn't address the
nominalization of *mathematics* itself.


In principle, one may object, this shouldn't be a problem. After
all, the mathematical fictionalists' motivation to develop their
approach has focused on one issue: to overcome the indispensability
argument--thus addressing the use of mathematics
in *science*. And the overall strategy, as noted, has been to
provide nominalist counterparts to relevant scientific theories.


The problem with this objection, however, is that given the nature
of Field's strategy, the task 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 in hand not
only with science, but also with metalogic. As it stands, the
mathematical fictionalist 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 another problem emerges instead: what is
it that distinguishes a mathematician (who knows a lot about
mathematics) and a non-mathematician (who does not have such
knowledge)? The difference here (according to Field 1984) is not about
having or lacking mathematical knowledge, but rather it is
about *logical* knowledge: of knowing which mathematical
theorems follow from certain mathematical principles, and which do
not. The epistemological problem is then solved--as long as the
mathematical fictionalist provides an epistemology
for *logic*.


In fact, what needs to be offered is ultimately an epistemology
for *modality*. After all, on Field's account, in order to
avoid the platonist commitment to models or proofs, the concept of
logical consequence is understood in terms of the primitive modal
concept of logical possibility: *A* follows logically
from *B* as long as the conjunction of *B* and the negation
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 establish such judgments may be
unproblematic. The problem emerges when more substantive statements
are invoked. In these cases, we seem to need a significant amount
of *mathematical* information in order to be able to determine
whether the impossibilities in question really hold or not. Consider,
for instance, the difficulty of establishing the independence of the
axiom of choice and the continuum hypotheses from the axioms of
Zermelo-Fraenkel set theory. Significantly complex mathematical models
need to be constructed in this case, which rely on the development of
special mathematical techniques to build them. What is required from
the mathematical fictionalist at this stage is the nominalization 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 is partially solved by the mathematical
fictionalist. Field provides an account of the application of
mathematics that does not require the truth of mathematical
theories. As we saw, this demands that mathematics be conservative in
the relevant sense. However, it is unclear whether Field has
established the conservativeness of mathematics, given his restrictive
way of introducing non-set-theoretic vocabulary into the axioms of set
theory as part 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 in Melia 1998,
2000). Field was working with restricted ZFU, Zermelo-Fraenkel set
theory with the axiom of choice modified to allow
for *Urelemente*, objects that are not sets, but *not
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, when
translated into set-theoretic language, will have to appear in the
comprehension axioms. As formulated by Field, the proof failed to
address the crucial case of actual applications of mathematics.


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


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


A similar difficulty also emerges for Balaguer's version of
fictionalism (see the second half of Balaguer 1998). Balaguer relies
on the possibility of distinguishing between the mathematical 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 from mathematics. It is, however, controversial
whether the distinction between mathematical and physical content can
be characterized without implementing a Field-like nominalization
program. In this case, the same difficulties that the latter face also
carry over to Balaguer's account (Colyvan 2010; Azzouni 2011).


Moreover, according to Azzouni (Azzouni 2009b), in order for
scientists to use a scientific theory, they need to *assert*
it. On his view, it is not enough for scientists simply to recognize
that a scientific theory is true (or exhibits some other theoretical
virtue). It is required that they assert the theory. In particular,
scientists would then need to assert a nominalistic theory. They
cannot simply contemplate such a theory; they need to be able to
assert 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 to assert the relevant
nominalistic theories in order to address the issue of the application
of mathematics.


#### 3.3.3 Uniform semantics


In one respect, mathematical fictionalists offer a uniform
semantics for mathematical and scientific discourse, in another
respect, they don't. Initially, both types of discourse are assessed
in the same way. Electrons and relations among them make certain
quantum-mechanical statements true; in turn, mathematical objects and
relations among *them* make the corresponding mathematical
statements true. It just happens that, as opposed to electrons on a
realist interpretation of quantum mechanics, mathematical objects do
not exist. Hence, as noted, existential mathematical statements, such
as 'there are infinitely many prime numbers', are
false. Although the resulting truth-value assignments for existential
statements conflict with those found in mathematical practice, at
least the same semantics is offered for mathematical and scientific
languages.


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


#### 3.3.4. Taking mathematics literally


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


#### 3.3.5 The ontological problem


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



## 4. Modal Structuralism


### 4.1 Central features of modal structuralism


Modal structuralism offers a program of interpretation of
mathematics which incorporates two features: (a) an *emphasis on
structures* as the main subject-matter of mathematics, and (b) a
complete elimination of reference to mathematical objects by
interpreting *mathematics in terms of modal logic* (as first
suggested by Putnam (1967), and developed in 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 reasonable account should be
forthcoming of how mathematics does in fact apply to the material
world' (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, this study 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 is no 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 only claim is that the
structures in question are *possible*. In order to articulate
this point, the modal-structural interpretation is formulated in a
second-order modal language based on S5. However, to prevent
commitment to a set-theoretical characterization of the modal
operators, 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 of which each ordinary mathematical
statement *S* is taken as elliptical for a hypothetical
statement, namely: that *S* *would hold* in a structure of
the appropriate kind.


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



> 
> [?]*X*(*X* is an o-sequence
>  satisfying PA's axioms - *S* holds in *X*).
> 
> 
> 


According to this statement, if there were o-sequences
satisfying PA's axioms, *S* would hold in them. This is
the *hypothetical component* of the modal-structural
interpretation (for a detailed analysis and a precise formulation, see
(Hellman 1989, pp. 16-24)). The *categorical component*
constitutes the second step (Helman 1989, pp. 24-33). The idea
is to assume that the structures of the appropriate kind are logically
possible. In that case, we have that



> 
> *[?]*X*(*X* is an o-sequence satisfying
>  PA's axioms).
> 
> 
> 


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


The modal structuralist then indicates that the practice of theorem
proving can be regained in this framework (roughly speaking, by
applying the translation scheme to each line of the original proof of
the theorem under consideration). Moreover, by using the translation
scheme and appropriate coding devices, one can argue that arithmetic,
real analysis and, to a 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 the mathematics 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 has been 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
of structures with inaccessibly many objects.


With the framework in place, the modal structuralist can then
consider the applicability problem. The main idea is to adopt the
hypothetical component as the basis for accommodating the application
of mathematics. The relevant structures are those commonly used in
particular branches of science. Two considerations need to be made at
this 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 in applied
mathematics, the non-mathematical objects to which the mathematical
structures are applied, and a statement of application that specifies
the particular relations between the mathematical structures and the
non-mathematical objects. The relevant mathematical structures can be
formulated in set theory. Let us call the set theory used in applied
contexts *Z*. (This is second-order Zermelo set theory, which is
finitely axiomatizable; I'll denote the conjunction of the axioms
of *Z* by [?]*Z*.) The non-mathematical objects of
interest in the context of application can be expressed in *Z*
as *Urelemente*, that is, as objects that are not sets. We will
take '*U*' to be the statement that certain
non-mathematical objects of interest are included
as *Urelemente* in the structures of *Z*. Finally,
'*A*' is the statement of application, describing the
particular relations between the relevant mathematical structures
of *Z* and the non-mathematical objects described
in *U*. The particular relations involved depend on the case in
question. We can now present the general form of an applied
mathematical statement (Hellman 1989, p. 119):



> 
> [?]*X*[?]*f* (([?]*Z*
>  & *U*)*X* ([?]*f*)
>  - *A*).
> 
> 
> 


In the antecedent '([?]*Z*
& *U*)*X* ([?]*f*)'
is an abbreviation for the results from writing out the axioms 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 relation variable
'*f*'. According to the applied mathematical
statement, if there were structures satisfying 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 application statement *A* expresses the relations
in questions, such as an isomorphism or a homomorphism between a
physical system and certain set theoretic structures. This is the
hypothetical component interpreted to express which relations would
hold between certain mathematical structures (formulated as structures
of [?]Z) and the entities studied in the world
(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 "synthetic
determination" relations (Hellman 1989, pp. 124-135). More
specifically, we have to determine which relations among
non-mathematical objects can be taken, in the antecedent of an applied
mathematical statement, as the basis for specifying "the actual
material situation" (Hellman 1989, p. 129). The modal
structuralist proposal is to consider the models of a comprehensive
theory *T*'. This theory embraces and links the vocabulary
of the *applied* mathematical theory (*T*) and the
synthetic vocabulary (*S*) in question, which intuitively fixes
the actual material situation. It is assumed that *T* determines,
up to isomorphism, a particular kind of mathematical structure
(containing, for example, *Z*), and that *T*' is an
extension of *T*. In that case, a proposed "synthetic
basis" will be adequate if the following condition holds:



> 
> Let *a* be the class of (mathematically) standard models
>  of *T*', and let *V* denote the full vocabulary
>  of *T*': then *S* determines *V* in *a*
>  iff for any two models *m* and *m*' in *a*, and
>  any bijection *f* between their domains, if *f* is
>  an *S* isomorphism, it is also a *V* isomorphism. (Hellman
>  1989, p. 132.)
> 
> 
> 


The introduction of isomorphism in this context comes, of course,
from the need to accommodate the preservation of structure between the
(applied) mathematical part of the domain under study and the
non-mathematical part. This holds in the crucial case in which the
preservation of the synthetic properties and relations
(*S*-isomorphism) by *f* leads to the preservation of the
analytic applied mathematical relations (*V*-isomorphism) of the
overall theory *T*'. It should be noted that the
'synthetic' structure is not meant to
'capture' the *full* structure of the mathematical
theory in question, but only its *applied* part. (Recall that
Hellman started with an *applied* mathematical
theory *T*.)


This can be illustrated with a simple example. Suppose that
finitely many physical objects display a linear order. We can describe
this by defining a function from those objects to an initial segment
of the natural numbers. What is required by the modal structuralist's
synthetic determination condition is that the physical ordering among
the objects alone captures this function and the description it offers
of the objects. It is *not* claimed that the *full*
natural number structure is thus captured. This example also provides
an illustration of the applied mathematical statement mentioned
above. The *Urelemente* (objects that are not sets) are the
physical objects in question, the relevant mathematical relation is
isomorphism, and the mathematical structure is a segment of natural
numbers with their usual linear order.


On the modal structural conception, mathematics is applied by
establishing an appropriate isomorphism between (parts of)
mathematical structures and those structures that represent the
material situation. This procedure is justified, since such
isomorphism establishes the structural 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 of the structural equivalence between the
(applied) mathematical and the non-mathematical domains. On what
grounds can we claim that the structures under consideration are
mathematically the same if some of them concern 'material'
objects? Of course, given that the structural equivalence is
established by an isomorphism the material objects are already
formulated in structural terms--this means that some mathematics
has already been applied to the domain in question. In other words, in
order to be able to represent the applicability of mathematics,
Hellman assumes that some mathematics has already been applied. This
means that a purely mathematical characterization of the applicability
of mathematics (via structure preservation) is inherently
incomplete. The *first* step in the application, namely the
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 *not* articulated in
mathematical terms, no isomorphism is *defined* there.


It may be argued that the modal structural account does not require
an isomorphism between (applied) mathematical structures and those
describing the material situation. The account only requires an
isomorphism between two standard models of the overall
theory *T*', which links the mathematical theory *T*
and the description *S* of the material domain. In reply, note
that this only moves the difficulty one level up. In order
for *T*' to extend the applied mathematical theory *T*
and to provide a link between *T* and the material situation, a
model of *T*' will have to be, in particular, a model of
both *T* and *S*. Thus, if the modal structuralist's
synthetic determination claim is satisfied, an isomorphism between two
models of *T*' will determine an isomorphism between the
models of *S* and those of *T*. In this 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 a structural equivalence between the mathematical
and the non-mathematical domains. On what grounds do we *know*
that such equivalence holds? Someone may say that the equivalence is
normatively imposed in order for the application process to get off
the ground. But this suggestion leads to a dilemma. Either it is just
assumed that we know that the equivalence holds, and the
epistemological question is begged (given that the grounds for 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 clearly
groundless. However, it may be argued that there is no problem here,
since we establish the isomorphism by examining the physical theories
of the material objects under consideration. But the problem is that
in order to formulate these physical theories we typically use
mathematics. And the issue is precisely to explain this use, that is,
to provide some understanding of the grounds in terms of which we come
to know that the relevant mathematical structures are isomorphic to
the physical ones.


The main point underlying these considerations has been stressed
often enough (although in a different context): isomorphism does not
seem to be an appropriate condition for capturing the relation between
mathematical structures and the world (see, e.g., da Costa and French
2003). There is, of course, a correct intuition underlying the use of
isomorphism at this level, and this relates to the idea of justifying
the application of mathematics: the isomorphism does guarantee that
applied mathematical structures *S* and the structures *M*
which represent the material situation are mathematically the
same. The problem is that isomorphism-based characterizations tend to
be unrealistically strong. They require that some mathematics has
already been applied to the material situation, and that we have
knowledge of the structural equivalence between *S*
and *M*. What is needed is a framework in which the relation
between the relevant structures is weaker than isomorphism, 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, modal structuralists need not
explain how we can have knowledge of the existence of mathematical
objects, relations or structures--given the lack of commitment to
these entities. However, they still need to explain our knowledge of
the possibility of the relevant structures, 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 may require knowledge
of substantial parts of mathematics. For instance, in order to know
that the structures formulated in Zermelo set theory are possible,
presumably we need to know that the theory itself is consistent. But
the consistency of the theory can only be established in another
theory, whose consistency, in turn, also needs to be
established--and we face a regress. 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 develop an epistemology for mathematics. They
just suggest that further developments on the epistemological front
seem to be called for in order to address more fully the
epistemological 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 modal structuralist. After all, a framework to
interpret the use of mathematics in science is provided, and in terms
of this framework the application of mathematics can be accommodated
without the commitment to the existence of the corresponding
objects.


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


The status of the indispensability argument within the
modal-structural interpretation is quite unique. On the one hand, the
conclusion of the argument is undermined (if the proposed translation
scheme goes through), since commitment to the existence of
mathematical objects can be avoided. On the other hand, a revised
version of the indispensability argument can be used to motivate the
translation into the modal language, thus emphasizing the
indispensable role played by the primitive modal notions introduced by
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 best theories of the
world, and concluding that we ought to be ontologically committed to
the possibility of the corresponding structures. In this sense, modal
structuralists can invoke the 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 the
revised argument. But rather than supporting the existence of
mathematical objects, the argument would only support commitment to
modal-structural translations of mathematical theories and the
possibility of mathematical structures.


#### 4.2.3 Uniform semantics


With the introduction of modal operators and the proposed
translation scheme, the modal structuralist is unable to provide a
uniform semantics for scientific and mathematical theories. Only the
latter, as opposed to the former, requires such operators. In fact,
Field has 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 is actually 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 theory due to the use of modal operators) is
to employ an actuality operator. By properly placing this operator
within the scope of the modal operators, it is possible to undo the
nominalization of the physical content in question (Friedman
2005). Without the introduction of the actuality operator, or some
related maneuver, it is unclear that the modal structuralist would be
in a position to preserve the physical content of the scientific
theory in question.


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


However, even with the introduction of such an operator, there
would still be a significant difference, on the modal-structural
translation scheme, between the semantics for mathematical and
scientific discourse. For the former, as opposed to the latter, does
not invoke such an operator. The result is that modal structuralism
does not seem to be able to provide 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 the view! Taken
literally, mathematical discourse seems to be committed to abstract
objects and 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 discourse to actual mathematical practice is
offered. The discourse is 'parallel' given that
mathematical practice typically does not invoke the modal operators
introduced by the modal structuralist. For those who aim to understand
mathematical discourse as it is used in the practice of mathematics,
and who try to identify suitable features of that practice that
prevent commitment to mathematical entities, the proposed translation
will make the realization of that goal particularly difficult.


#### 4.2.5 The ontological problem


The modal structuralist has solved, in part, the ontological
problem. No commitment to mathematical objects or structures seems to
be needed to implement the proposed translation 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, the question 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 is responsible for the truth of such
statement? Clearly, the modal structuralist will not ground the
possibility in question on the actual truth of the Peano axioms, for
this move, on a reasonable interpretation, would require
platonism. Nor will the modal structuralist support the relevant
possibility on the basis of the existence of a consistency proof for
the Peano axioms. After all, any such proof is an abstract object, and
to invoke it at the foundation of modal structuralism clearly
threatens the coherence of the overall view. Furthermore, to invoke a
modalized version of such a consistency proof would beg the question,
since it assumes that the use of modal operators is already
justified. Ultimately, what is needed to solve properly the
ontological problem is a suitable account of modal discourse.



## 5. Deflationary Nominalism


### 5.1 Central features of deflationary nominalism


According to the deflationary nominalist, it is perfectly
consistent to insist that mathematical theories are indispensable to
science, to assert that mathematical and scientific theories are true,
and to deny that mathematical objects exist. I am calling the view
'deflationary nominalism' given that it demands very
minimal commitments to make sense of mathematics (Azzouni 2004), it
advances a deflationary view of truth (Azzouni 2004, 2006), and
advocates a direct formulation of mathematical theories, without
requiring that they be reconstructed or rewritten (Azzouni 1994,
2004).


Deflationary nominalism offers an 'easy road' to
nominalism, which does not require any form of reformulation of
mathematical discourse, while granting the indispensability of
mathematics. Despite the fact that quantification over mathematical
objects and relations is indispensable to our best theories of the
world, this fact offers no reason to believe in the existence of the
corresponding entities. This is because, as Jody Azzouni points out,
two kinds of commitment should be distinguished: *quantifier*
commitment and *ontological* commitment (Azzouni 1997; 2004,
p. 127 and pp. 49-122). We incur a quantifier commitment
whenever our theories imply existentially quantified statements. But
existential quantification, Azzouni insists, is not sufficient for
ontological commitment. After all, we often quantify over objects we
have no reason to believe exist, such as fictional entities.


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


Do psychological processes themselves exist, according to the
ontologically independence criterion? It may be argued that most
psychological processes do exist, at least those we undergo rather
than those we make up. After all, the motivation underlying the
independence criterion is that those things we just made up verbally
or psychologically do not exist. Having a headache or believing that
there is a laptop computer in front of me now are psychological
processes that I did not make up. Therefore, it seems that at least
these kinds of psychological processes do exist. In contrast,
imaginings, desires, and hopes are processes we make up, and thus they
do not exist. However, the underlying motivation for the criterion
seems to diverge, in these cases, from what is entailed by the
criterion's actual formulation. For the criterion insists on the
ontological independence of "our linguistic practices and
psychological processes". Since headaches and beliefs are
psychological processes themselves, presumably they are *not*
ontologically independent of psychological processes. Hence, they do
not exist. This means that if the criterion is applied as stated, no
psychological process exists. For similar reasons, novels, mental
contents, and institutions do not exist either, since they are all
abstract objects dependent on our linguistic practices and
psychological processes, according to the deflationary nominalist
(Azzouni 2010a, 2012).


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


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


But in what sense do mathematical objects depend on our linguistic
practices and psychological processes? In the sense that the sheer
postulation of certain principles is enough for mathematical practice:
'A mathematical subject with its accompanying posits can 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 that mathematicians should
find the resulting mathematics interesting. That is, the consequences
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. Such objects, or
'posits', are called *ultrathin* (Azzouni 2004,
p. 127).


The same move that the deflationary nominalist makes to distinguish
ontological commitment from quantifier commitment is also used to
distinguish ontological commitment to *F*s from asserting
the *truth* of 'There are *F*s'. Although
mathematical theories used in science are (taken to be) true, this is
not sufficient to commit us to the existence of the objects these
theories are supposedly about. After all, according to the
deflationary nominalist, it may be *true* that there
are *F*s, but to be ontologically committed to *F*s, 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 show that such literally true mathematical
>  statements are *not* indispensable to empirical science, and
>  yet, nonetheless, I can describe mathematical terms as referring to
>  nothing at all. Without Quine's criterion to corrupt them,
>  existential statements are innocent of ontology. (Azzouni 2004,
>  pp. 4-5.)
> 
> 
> 


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


In his 1994 book, Azzouni did not commit himself to nominalism, on
the grounds that nominalists 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
was implemented, or needed, in the proposal advanced by Azzouni
(Azzouni 1994). The fact that mathematical objects play no role in how
mathematical truths are known clearly expresses a nominalist
attitude--an attitude that Azzouni explicitly endorsed in
(Azzouni 2004).


The deflationary nominalist proposal nicely expresses a view that
should be taken seriously. And as opposed to other versions of
nominalism, it has the significant benefit of aiming 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 that comes closest to solving (or, in some
cases, dissolving) the five problems that have been used to assess
nominalist proposals. With the possible exceptions of the issue of
taking mathematical language literally and the ontological problem,
all of the remaining problems are 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, this problem
vanishes. Mathematical knowledge is ultimately obtained from what
follows from mathematical principles. Given that mathematical objects
do not exist, they play no role in how 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
mathematical knowledge.


It might be argued that certain mathematical statements are known
without the corresponding proof. Consider the Godel sentence
invoked in the proof of Godel's incompleteness theorem: the
sentence is true, but it cannot be proved in the system under
consideration (if the latter is consistent). Do we have knowledge of
the Godel 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 in terms of what can be proved in a given system.


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


According to Azzouni, we know the Godel sentence as long as we
are able to embed the syntactically incomplete system (such as Peano
arithmetic) in a stronger system in which the truth predicate for the
original system occurs and in which the Godel 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 result follows from a
given group of axioms. Part of the difficulty emerges from the fact
that the logical consequences of a non-trivial group of axioms are
often not transparent: significant work 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 no genuine philosophical problem in the success
of applied mathematics (Azzouni 2000). Once particular attention is
given to implicational opacity--our inability to see, before a
proof is offered, the consequences of various mathematical
statements--much of the alleged surprise in the successful
application of mathematics should vanish. Ultimately, the so-called
problem of the application of mathematics--of understanding how
it is possible that mathematics can be successfully applied to the
physical world should--is an artificially designed issue rather
than a genuine problem.


Colyvan defends the opposing view (Colyvan 2001b), insisting that
the application of mathematics to science does present a genuine
problem. In particular, he argues that two major philosophical
accounts of mathematics, Field's mathematical fictionalism and Quine's
platonist realism, are unable to explain the problem. Thus, he
concludes that the problem cuts across the realism/anti-realism debate
in the philosophy of mathematics. The deflationary 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 is equally 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 not raise a
special philosophical problem, particularly as soon as the issue of
implicational opacity 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 something that the deflationary nominalist explicitly
addresses, carefully examining the central features and the
limitations of different models of the application of mathematics
(see, in particular, the second part of (Azzouni 2004)).


#### 5.2.3 Uniform semantics


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


It may be argued that the deflationary nominalist needs to provide
the semantics for the existential and universal claims in mathematics,
science and ordinary language. After all, it does 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 this semantics is precisely the standard
semantics of classical logic, with the familiar conditions for the
existential and universal quantifiers, but without the assumption that
such quantifiers are ontologically committing. The fact that no
ontological import is assigned to the quantifiers does not change
their semantics. After all, the metalanguage in which the semantics is
developed already has universal and existential quantifiers, and these
quantifiers need not be interpreted as providing ontological
commitment any more than the object 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 between ontologically serious (or
ontologically committing) uses of the quantifiers and ontologically
innocent (or ontologically non-committing) uses. If so, this would
presumably require a different semantics for these quantifiers. In
response, the deflationary nominalist will deny the need for such
distinction. In order to mark ontological commitment, an existence
predicate, which expresses ontological independence, is used. Those
things that are ontologically independent of us (that is, of our
linguistic practices and psychological processes) are those to which
we are ontologically committed. The mark of ontological commitment is
not made at the level of the quantifiers, but via the existence
predicate.


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


Perhaps the deflationary nominalist will respond by arguing that
the existence predicate is already part of the language, maybe
implicitly via contextual and rhetorical factors (Azzouni 2007,
Section III; Azzouni 2004, Chapter 5). What would be needed then is
evidence for such a claim, and an indication of how exactly the
predicate is in fact found in scientific, 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 sentences could be formalized as follows:



> 
> (S) [?]*x*(*S**x* -
>  !*E**x*), where '*S*' is (for
>  simplicity) the predicate 'set of all sets', and
>  '*E*' is the existence predicate.
> 
> 
> (P) [?]*x*(*P**x* -
>  !*E**x*), where '*P*' is (for
>  simplicity) the predicate 'perfectly frictionless
>  plane', and '*E*' is the existence
>  predicate.
> 
> 
> (M) [?]*x*(*M**x* [?] *E**x*)
>  [?] [?]*x*((*M**x* [?] *T**x*)
>  - !*E**x*), 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 the natural language sentences in order to
introduce the existence predicate. And that is arguably a cost for the
view. After all, in these cases, mathematical, scientific and ordinary
languages 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 the deflationary 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 level
of reconstruction involved is significantly less than what is found in
the other versions of nominalism discussed above. As opposed to them,
the deflationary nominalist is able to accommodate significant aspects
of mathematical practice without the need for creating a full parallel
discourse (in particular, no operators, modal or fictional, need to be
introduced). However, some level of reconstruction is still needed to
accommodate the existence predicate, which then compromises the
deflationary nominalist's capacity to take mathematical language
literally.


A related concern is that the deflationary nominalist introduces a
non-standard notion of reference that does not presuppose the
existence of the objects that are referred to (Bueno and Zalta
2005). This move goes hand in hand with the understanding of the
quantifiers as not being ontologically committing, and it does seem to
limit the deflationary nominalist's capacity to take mathematical
language literally. After all, a special use of 'refers'
is needed to accommodate the claim that "'*a*'
refers to *b*, but *b* does not exist". The
deflationary nominalist, however, resists this charge (Azzouni 2009a,
2010a, 2010b).


#### 5.2.5 The ontological problem


The ontological problem is also dissolved by the deflationary
nominalism. Clearly, deflationary nominalism has no commitment either
to mathematical objects or to a modal ontology of any kind (including
possible worlds, abstract entities as proxy for possible worlds, or
other forms of replacement for the expression of modal claims). The
deflationary nominalists not only avoid the commitment to mathematical
objects, they also claim that such objects have no properties
whatsoever. This means that the deflationary nominalist's ontology is
extremely minimal: only concrete objects are ultimately
assumed--objects that are ontologically independent of our
psychological processes and linguistic practices. In particular, no
domain of nonexistent objects is posited nor a realm of genuine
properties 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 to
other objects. For example, although Sherlock Holmes does not exist,
he has the property of being thought of by me as I write this
sentence. This is not, however, a genuine property of Sherlock Holmes
in the intended sense.


Deflationary nominalism is not a form of Meinongianism (Azzouni
2010a). Although the ontology of deflationary nominalism is not
significantly different from that of the Meinongian, the ideology of
the two views--at least assuming a particular, traditional
interpretation of Meinongianism--is importantly different. The
deflationary nominalist is not committed to any subsisting objects, in
contrast to what is often claimed to be a distinctive feature of
Meinongianism.


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


With the meager ontological commitments, the deflationary
nominalist fares very well on the ontological front. One source of
concern is how meager the deflationary nominalist's ontology
ultimately is. For instance, platonists would insist that mathematical
objects are ontologically independent of our psychological processes
and linguistic practices, and--using the criterion of ontological
commitment offered by the deflationary nominalist--they would
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 relevant sense, thus concluding
that these objects also exist. Deflationary nominalists will try to
resist these conclusions. But unless their arguments are successful at
this point, the concern remains 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. They state that mathematicians derive statements
of the form "There are *F*s", but insist that the
objects in question do not exist, given that quantifier commitment and
ontological commitment should be distinguished. This strategy is
fundamentally different from those found in the nominalist proposals
discussed before. It amounts to the denial of the first premise of the
indispensability argument ("We ought to be ontologically
committed to all and only those entities that are indispensable to our
best theories of the world"). Even though quantification over
mathematical entities is indispensable to our best theories of the
world (thus, the deflationary nominalist accepts the second premise of
the argument), this fact does not entail that these entities
exist. After all, we can quantify over objects that do not 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 criterion of ontological commitment provides such
rules, then they are. But why should we grant that Quine's criterion
play such a role? Deflationary nominalists challenge this deeply held
assumption in ontological debates. And by doing so, they pave the way
for the development of a distinctive form of nominalism in the
philosophy of mathematics.