Abstract Singular Terms And Thin Reference

Abstract Singular Terms and Thin Reference
by
GEORGE DUKE
Deakin
Abstract: The prevailing approach to the problem of the ontological status of mathematical entities
such as numbers and sets is to ask in what sense it is legitimate to ascribe a reference to abstract
singular terms; those expressions of our language which, taken at face value, denote abstract objects.
On the basis of this approach, neo-Fregean Abstractionists such as Hale and Wright have argued that
abstract singular terms may be taken to effect genuine reference towards objects, whereas nominal-
ists such as Field have asserted that these apparent ontological commitments should not be taken at
face value. In this article I argue for an intermediate position which upholds the legitimacy of
ascribing a reference to abstract singular terms in an attenuated sense relative to the more robust
ascription of reference applicable to names denoting concrete entities. In so doing I seek to clear up
some confusions regarding the ramifications of such a thin notion of reference for ontological claims
about mathematical objects.
Keywords: reference, abstract singular terms, ontology, abstract objects
THE PREVAILING APPROACH TO the problem of the ontological status of mathemati-
cal entities such as numbers and sets is to ask in what sense it is legitimate to ascribe
a reference to abstract singular terms; those expressions of our language which,
taken at face value, denote abstract objects. On the basis of this approach, neo-
Fregean Abstractionists such as Hale and Wright (Wright, 1983; Hale, 1987; Hale
and Wright, 2001, 2009) have argued that abstract singular terms may be taken to
effect genuine reference towards objects, whereas nominalists such as Field (1980,
1984) have asserted that these apparent ontological commitments should not be
taken at face value. In this article I argue for an intermediate position which
upholds the legitimacy of ascribing a reference to abstract singular terms in an
attenuated sense relative to the more robust ascription of reference applicable to
names denoting concrete entities. In so doing I seek to clear up some confusions
regarding the ramifications of such a thin notion of reference for ontological claims
about mathematical objects. The article is divided into three sections. Section I
briefly surveys some key assumptions of the current literature on abstract singular
terms. In section II, I critically examine the thin theories of reference proposed by
Dummett and Linnebo. These theories have been referred to as “tolerant reduction-
ist” because they suggest that we should tolerate the attribution of a semantic role
to abstract singular terms featuring in true statements but also allow for a meta-
semantic reduction of our ontological commitment to abstract objects. In the final
section of the article I argue that the case for this position can be strengthened by
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THEORIA, 2012, 78, 276–292
doi:10.1111/j.1755-2567.2012.01143.x
© 2012 Stiftelsen Theoria. Published by Blackwell Publishing, 9600 Garsington Road, Oxford OX4 2DQ, UK,
and 350 Main Street, Malden, MA 02148, USA.

a clarification of the links between reference, existential quantification and onto-
logical commitment. This clarification explains why reference to abstract objects is
legitimate without this entailing that such objects exist in the strong mind and
language independent sense of traditional Platonism.
I
The problem of abstract objects is generally framed as an investigation into whether
abstract singular terms genuinely refer to existent objects.1
Three key assumptions
of this approach are that:
i. the ontological categorization of entities is dependent upon a prior logical
categorization of expressions in accordance with a broadly Fregean syntax;
ii. the ontology of a theory is to be sought in the values of the variables that it
employs; and
iii. existential quantification and reference are intimately connected: that “n”
refers to n iff “there is an x such that x = n” is true.
In order to understand what is at stake in debates about the reference of abstract
singular terms, it is worth briefly considering these assumptions in turn.2
The thesis that an ontological categorization of entities is dependent upon a prior
logical categorization of expressions (i) suggests that our experience of reality as
divided up into objects, properties, relations etc. is determined by the logical-
syntactic categories we employ. In his break with the traditional interpretation of
logic, Frege replaced Aristotle’s distinction between universals and particulars with
the mathematical concepts of function and argument and their corresponding
ontological categories of concept and object. The notion of object at play here is
perfectly general; objects are regarded as the referents of singular terms that
compose the domain of quantification. Insofar as Frege’s logic applies to all that
can be thought [1884, §14], his notion of an object is thus a formal one, encom-
passing what we call abstract objects as well as observable concrete objects located
in space-time. According to assumption (i), however, it is not only that an object is
conceived as the possible referent of a singular term, a property as the non-
linguistic correlate of a predicate etc., but that the categories of linguistic expres-
1 Charles Parsons (2008, p. 1) captures the standard contemporary meaning of the term “abstract object”
when he characterizes it as referring to objects that are “not located in space and time” and which do not
“stand in causal relations.” Mathematical entities such as numbers, geometrical shapes and sets are thus
paradigmatic abstract objects.
2 Although the framework for all three assumptions is Fregean, it is significant that the joint paper by
Nelson Goodman and W. V. O. Quine, “Steps towards a Constructive Nominalism” (1947), is the earliest
frequently cited paper which frames the problem of “abstract” entities in terms of assumption (ii).
ABSTRACT SINGULAR TERMS AND THIN REFERENCE 277
© 2012 Stiftelsen Theoria.

sion we employ are attributed logical or explanatory priority over the correlative
ontological categories. Hale and Wright, building on Dummett’s interpretation of
Frege, have characterized this claim as the “syntactic priority principle” or the
“priority of syntactic over ontological categories” (Wright, 1983, p. 30).
The notion of ontological commitment (assumption (ii)) can be thought of as
relating either to theories or people or both. To say that a theory T embodies a
commitment to Fs is to say that the truth of T requires that the world contain Fs.
To say that a person is committed to Fs is to say that they accept as true a theory
that embodies a commitment to Fs. In terms of the existence of objects of a
particular kind, Quine’s theory of ontological commitment accordingly states that
“a theory is committed to those and only those entities to which the bound vari-
ables of the theory must be capable of referring in order that the affirmations
made in the theory be true” (Quine, 1948, pp. 13–14). For abstract objects like
numbers, then, the key question is often regarded as whether quantification over
such objects is indispensable for our best natural scientific theories. The Quine-
Putnam indispensability argument thus suggests the need to admit numbers into
our ontology, despite our inability to get into causal interactions with such pur-
ported objects.
The key insight behind assumption (iii) can perhaps be best expressed by the
claim that the verification of the existence of a referent for a term N is a verifi-
cation of a statement of the form: ($x) (x = N). This way of framing the link
between reference and existential quantification, however, leaves many questions
open, not least of which is how we are to determine whether there really exists a
referent for a term when the mode of verification here suggested seems circular
insofar as existential quantification is used as a criterion for existence. The above
formulation also raises concerns about inscrutability and ontological relativity. I
will return to these points in the third section of this article; for here it is enough
to register that the link suggested in (iii) is not as unproblematic as is sometimes
assumed.
The combination of assumptions (i–iii) suggests that the problem of abstract
objects is one of determining whether our employment of abstract singular terms,
those expressions of our language which at the surface grammatical level appear to
denote abstract objects, is ontologically committing, i.e., whether we should regard
such expressions as genuinely referring to such objects. A number of responses are
possible, but within the framework under consideration these may generally be
characterized as neo-Fregean, nominalist or intermediate (sometimes referred to as
anti-nominalist).
Frege’s abstraction principles allow for a concise illustration of these divergent
stances on the referential status of abstract singular terms. If we start from the
assumption that the right hand side (RHS) of the Fregean abstraction principle for
directions
278 GEORGE DUKE

Dir D D iff //
: a b a b
( ) = ( ) 3
features terms referring to concrete objects (parallel lines), then three interpreta-
tions of the referential commitments of the terms on the LHS seem possible.
According to the austere nominalist interpretation, favoured by Field (1984), either
the LHS does not have the syntactic and semantic structure that it appears to have
on surface level, or the LHS is not actually equivalent to the RHS. The robust
interpretation, favoured by Hale and Wright, claims that the LHS has precisely the
semantic structure that its surface syntactic form suggests; the abstract singular
terms for directions formed by applying the relevant functional expression effect
objectual reference (Hale and Wright, 2001, p. 202). The intermediate interpreta-
tion, suggested by Dummett’s work, would amount to the view that the LHS has
semantically significant structure, “but that the singular terms involved have refer-
ence only in the sense of having a semantic role, and lack reference realistically
construed as a relation to an external object” (Hale and Wright, 2001, p. 202). This
intermediate interpretation asserts, in other words, that a “thin” notion of reference
is applicable to abstract singular terms; whereby “thin” designates a contrast with
the “thick” notion of reference applicable in the case of concrete singular terms,
whose referents may be demonstratively identified as constituents of external
reality.
From the austere perspective, it could be argued that while the expressions on the
LHS may be said to have a use, they lack reference altogether, because they simply
provide us with a way of rewriting what we already know on the basis of the RHS.
If we allow that an abstraction principle such as Dir at least appears to possess a
truth-value, however, then we require a form of error-theory to explain why the
terms on the LHS do not refer and are not ontologically committing. As Wright
suggests, however, basic forms of inference allowing us to move from the attribu-
tion of a property to a direction [F(D(a))] to existential generalization [($x)Fx]
seem to presuppose that terms for directions in this context genuinely refer to
objects. The denial that the D-terms genuinely refer, would divest the context
containing them of any truth-value, insofar as we accept that only identity state-
ments featuring terms with reference on both sides have the potential to be true or
false.
The essential claim of neo-Fregeanism is that abstraction principles of the form
( )( ) ,
∀ ∀ ∑( ) = ∑( ) ↔ ( )
( )
a b a b E a b
4
3 Read “The direction of line a is identical to the direction of line b iff line a is parallel to line b”.
4 Where a and b are variables of a given type, S is a term-forming operator denoting a function from
items of the given type to objects in the range of the first-order variables, and E is an equivalence over items
of the given type (Hale and Wright, 2009, p. 178).

serve as stipulative implicit definitions of the S-operator and in so doing also of the
new kind of term introduced by means of it with the corresponding sortal concept
(Hale and Wright, 2009, p. 179). According to Hale and Wright (2009, p. 179),
abstraction principles thereby allow us to overcome Benacerraf-inspired concerns
about our epistemic access to abstract objects by giving an account of the “truth
conditions of S identities as coincident with those of a kind of statement we already
understand”. We can therefore exploit this prior understanding so as to establish our
knowledge of the referents of the S-terms, referents whose status as existent objects
is guaranteed by the truth of the identity statements by means of which we gain
access to them.
I do not have time to consider the neo-Fregean thesis in detail here, except to note
that recent remarks on minimalism by Hale and Wright (2009, p. 207) suggest a
shift away from the robust Platonism upholding the strong mind and language
independence of abstract objects suggested by some of their earlier rhetoric towards
a more deflationary account of abstract singular terms that has affinities with the
thin theory of reference here outlined. What is important for my present purpose,
however, is Hale and Wright’s contention that it is not coherent to uphold an
intermediate position between the austere and the robust interpretations of the LHS
of Fregean abstraction principles. If we accept that “the truth of appropriate sen-
tential contexts containing what is, by syntactic criteria, a singular term is sufficient
to take care, so to speak, of its reference” (Wright, 1983, p. 24), then it is indeed
difficult to see why it would be illegitimate to claim that the relevant terms
genuinely refer to abstract objects. As we shall see in the next section, Dummett’s
rejection of a robust interpretation of the capacity of contextual explanations to
proffer a reference for abstract singular terms is ultimately predicated on the
nominalist demand that we should always be able to be “shown” the referent of a
term for it to be legitimately denoting. While Hale and Wright’s critique of Dum-
mett’s relapse into a demonstrative model of identifying knowledge for abstract
singular terms seems plausible, however, this does not, contrary to appearances,
speak against the case for a thin theory of reference for such terms.
II
In his early work Dummett held to a view of the referential purport of abstract
singular terms very similar to that later developed in more detail by Hale and
Wright. It is worth engaging in a brief exposition of the considerations that moti-
vated Dummett’s shift towards an intermediate account of the reference of abstract
singular terms in his middle and later period, insofar as this points us in the
direction of arguments which suggest that a thin theory of reference for abstract
singular terms is the most compelling option for all participants in the debate.
280 GEORGE DUKE

In his early critiques of the reductive nominalism of Goodman and Quine,
Dummett claims that Frege’s great insight was that our ontological commitment
to a range of entities should be based on an account of how expressions standing
for those entities function in sentential contexts. For the early Dummett, if a term
genuinely fulfils the syntactical function of a proper name in sentences, some of
which are true, then we have not only fixed the sense, but also the reference, of that
proper name (Dummett, 1956, p. 40). As a result, we do not, as the nominalist
suggests, have a right to demand the possibility of an ostensive encounter with an
extra-linguistic correlate of an expression for that expression to be considered
legitimately referential. Insofar as the numeral “7” operates as a singular term by
syntactic criteria in the statement “7 is prime”, and it is possible to determine such
a statement as true, then we are justified as regarding the numeral as effecting
reference to an object. Reductive nominalism, which seeks to translate all state-
ments about abstract entities into statements about concrete entities, Dummett says,
“boils down to nothing but a simple-minded materialism” (1955, p. 33) predicated
on a misunderstanding of the extent to which even our apprehension of concrete
objects is dependent upon the employment of linguistic criteria of identity.
Dummett, however, later revisited his claim that it is sufficient to guarantee a
term a reference that we have established its syntactic credentials and capacity to
feature in true sentences. In the case of an abstract singular term, the determination
of the truth or falsity of a sentence in which it occurs does not involve “an
identification of an object as the referent of the term” (Dummett, 1978, p. xlii),
where this referent is considered as part of “external” reality. As a thesis about
sense, the context principle still serves as a corrective against the nominalist
superstition that all talk of abstract objects is to be regarded with suspicion, but it
is, Dummett now argues, “a great deal more dubious” when interpreted as a thesis
about reference (1978, p. xlii). Frege’s doctrine that it is only in the context of a
sentence that a word has a meaning can accordingly no longer “be used to give a
knock-down demonstration of the absurdity of a suspicious attitude to abstract
objects” (1978, p. xlii).
Dummett’s approach to the problem of abstract objects in his later work is to
investigate the applicability to abstract singular terms of a model for the meaning
of proper names “in the more usual sense” (1973, p. 671). By proper names “in the
more usual sense”, Dummett intends names denoting concrete objects, which,
given an associated criterion of identity, can be picked out by an ostensive gesture
accompanied by the use of a demonstrative. Dummett maintains that the nominalist
is confused in asserting that numbers do not exist when doing philosophy, whilst
asserting that there is a perfect number between 7 and 30 when doing mathematics
(1973, p. 497). In the case of abstract singular terms, however, the theory of
meaning operative cannot be “construed after a realistic model” (1973, p. 508).
When we make mathematical statements featuring numerals, for example, there is

no analogue to the process of identifying the bearer of a name to that which occurs
when we seek to determine the truth conditions of sentences containing names for
medium-sized physical objects. Dummett insists that, whilst it would be excessive
to deny numerals a reference altogether on this basis, at no point in the explanation
of the truth-conditions of sentences in which numerals occur is an appeal to the
identification of abstract objects necessary. Accordingly, reference may only be
ascribed to abstract singular terms “as a facon de parler” (Dummett, 1973, p. 508).
Dummett’s later position seems less an intermediate position than a sophisti-
cated variant of nominalism. On one plausible interpretation, his contention is that
although we cannot attribute an extra-linguistic reference to abstract singular terms,
the attribution of an intra-linguistic reference is philosophically harmless. Yet
because Dummett fails to articulate clearly what attributing intra-linguistic refer-
ence to abstract singular terms actually entails, Hale and Wright (Wright, 1983;
Hale, 1987; Hale and Wright, 2001) have attempted a reconstruction based on other
Dummettian commitments. Dummett distinguishes two ingredients in the notion of
reference as it applies to proper names: the identification of the referent of a name
with its bearer, and the notion of semantic role (or the contribution of an expression
to the truth-value of a sentence) (Dummett, 1973, pp. 190–191). Dummett contends
that our reluctance to accept Frege’s claim that incomplete expressions are refer-
ential is due to the tendency to focus upon the name/bearer prototype and neglect
the notion of semantic role. The ascription of reference to predicates and other
incomplete expressions in this “thin” sense is legitimate, Dummett argues, because
they play an indispensable role in determining the truth conditions of sentences in
which they occur (1973, p. 211). Hale and Wright’s conjecture is that Dummett
extends these views on the reference of incomplete expressions to abstract singular
terms.
Dummett (1991a) argues for a “tolerant reductionist” position that conforms to
some extent with Hale and Wright’s interpretation. The tolerant reductionist admits
the intelligibility of statements such as “there is a prime that divides 49”. He
recognizes, moreover, “that ‘31’ refers to an object” can be construed, untenden-
tiously, as simply the equivalent, in the formal mode, of “there is such a number as
31”, and hence as uncontroversially true (Dummett, 1991a, p. 191). This suggests
a notion of reference for abstract singular terms that is similar to the redundancy
theory of truth. Dummett is, however, more circumspect about the positive com-
ponent of his proposal. The thesis that reference to abstract objects is acceptable
once we adopt a redundancy theory of truth and reference is weaker than the
assertion that it is legitimate to ascribe such terms a semantic role. Moreover, in
direct opposition to Hale and Wright’s more robust interpretation, Dummett denies
that we can take sentences featuring abstract singular terms as having just the
semantic structure that they appear to have, insofar as the notion of reference is not
semantically operative in contextual definitions such as Dir and N=. This is to say
282 GEORGE DUKE

that an explanation of our understanding of a sentence containing an abstract
singular term is mediated by our understanding of a sentence not containing that
term; the notion of the reference of that term, as determined by its sense, thus plays
no role in our grasp of what determines it as true or false (Dummett, 1991a, p. 193).
As Hale and Wright suggest, from the perspective of the theory of meaning, this
amounts to the claim that the semantic structure of the LHS of the equivalences
cannot be taken at face value.
The “tolerance” of Dummett’s position appears to consist in its acceptance of
the claims of the practicing mathematician to refer to mathematical objects. His
“reductionism”, in contrast, asserts the much stronger thesis that reference to
abstract objects is ultimately a function of language and that an account of such
reference should be based on a model of meaning for concrete singular terms
regarded as referring to demonstratively identifiable constituents of “external”
reality. Indeed, the thesis that the notion of reference as employed in contextual
definitions is “semantically idle” does not seem prima facie incommensurable with
a thoroughgoing reductionism according to which “the mere possibility of contex-
tually defining the direction-operator shows that there are no such things as direc-
tions” (Dummett, 1991a, p. 196). Dummett here suggests that the contextual
explanation of Dir, precisely insofar as it presupposes the epistemological priority
of statements about concrete objects (parallel lines), offers significant weight to
reductionism.5
The real issue, however, is the suggestion that we need to be shown
the bearer of a name for it to be considered legitimately referential. Such a position
would appear to be highly problematic, insofar as it renders mysterious not only our
understanding of abstract objects, but also the vast majority of concrete objects,
which are not objects of direct acquaintance. As Hale (1987, p. 170) has argued,
Dummett’s later work suggests a fixation with “the demonstration-based concep-
tion of identifying knowledge” generally associated with nominalism.
It may be argued in Dummett’s defence that his position is a “tolerant” form of
reductionism precisely insofar as it allows for a “thin” language-internal notion of
reference repudiated by the nominalist. When we consider in more detail Dum-
mett’s linkage of the “thin” notion of reference with redundancy theories of truth,
however, the limits of his “tolerance” are manifest. According to Dummett (1991a,
p. 195), a “thin” notion of reference is one according to which “ ‘the direction of a’
refers to something” is indisputably true, because it reduces to “the line a has a
5 This perhaps explains Dummett”s (1991a, p. 191) cryptic remark that his intermediate view is “perhaps
one more austere than that which Wright has in mind.” In “What is Mathematics About?” (1991b, p. 435),
Dummett suggests that a sophisticated contemporary variant of nominalism, of the kind represented in the
work of Hartry Field, offers “a new strategy for resolving the problem of mathematical objects.” Whilst he
rejects Field’s “superstitious” blanket denial of abstract entities, the later Dummett is sympathetic towards
a sophisticated form of nominalism which characterizes mathematical truth in terms of logical relations.

direction”, and “ ‘the direction of a refers to the direction of a’ is a claim that is
trivially true, because it reduces to ‘the direction of a is the direction of a’.” On this
basis, Dummett (1991a, p. 196) claims that the context principle as employed in
Grundlagen is “strictly analogous to the redundancy theory of truth”. This sugges-
tion that a thin notion of reference is one in which the notion of reference is
legitimate but “redundant”, suggests that the best way to make sense of the “tol-
erance” of Dummett’s position is in fact in terms of its recognition of the distinction
between statements about mathematical objects from the perspectives of the
object- and meta-language. Such a distinction is only a minor concession to a more
thoroughgoing reductionism, insofar as it allows us to quantify over objects in the
object-language whilst suggesting that they are either dubious or superfluous from
the perspective of the meta-language.
Øystein Linnebo’s “meta-ontological minimalism” concerning abstract singular
terms is instructive in this context. Linnebo, like Dummett, seeks a middle way
between mathematical Platonism and nominalism by arguing for the legitimacy of
ascribing abstract singular terms a semantic role in sentences.6
Where Linnebo’s
position supplements that of Dummett is in its more explicit appeal to the distinc-
tion between semantics, understood as a theory of how the truth of sentences is
determined by the semantic values of their components, and meta-semantics, which
describes how this process takes place with reference to objects of different kinds,
i.e., which provides an account of the relevant model of meaning. Linnebo places
the reductionist aspect of his account of abstract singular terms entirely at the
meta-semantic level or the level of explanation concerned with what is required for
the reference relation to obtain between a term and its semantic value.
Linnebo’s application of the distinction between semantics and meta-semantics
supplements Dummett’s intermediate theory of abstract entities by demonstrating
the sense in which it is legitimate to grant sentences featuring abstract singular
terms a truth-value. Objects of a given kind (such as numbers) are described as
“light-weight” if the sentences concerning them admit of a meta-semantic reduc-
tion to sentences not containing them. Dummett (2007, p. 794) interprets this as
asserting, “in my own terminology, that the conception of reference to such objects
is thin if the explanation of the use of the sense of sentences containing terms for
them does not make use of the notion of reference to them”. Here the meta-
semantic analysis operates at the explanatory level of the senses of expressions and
6 Linnebo refers to his intermediate position on abstract objects as “anti-nominalist” insofar as it allows
for reference to mathematical objects at the level of the object-language whilst denying their mind or
language independence. If mathematical Platonism can be defined as the conjunction of the theses that (i)
there are mathematical objects; (ii) mathematical objects are abstract; and (iii) mathematical objects are
independent of intelligent agents and their language, thought and practices (Linnebo, 2009), then anti-
nominalism is an attempt to maintain the first two theses while dismissing the third as both misleading and
mysterious.
284 GEORGE DUKE

sentences, whereas the semantic level works at the level of reference by giving a
compositional account of how the semantic values of expressions go towards
determining the semantic value of larger linguistic units, such as sentences, of
which they are a part. This does not resolve the question, however, as to whether
objects that are “light-weight” in the sense that the sentences concerning them are
easily amenable to meta-semantic reduction should be regarded as genuine objects
of reference at all.
Linnebo freely grants that his meta-ontological minimalism is suggestive of a
form of reductionism, whilst seeking, like Dummett, to avoid the extremes of an
intolerant nominalism. As Linnebo’s own account of truth-value realism suggests,
however, it would also appear possible to integrate his views concerning the
semantic values of abstract singular terms with a sophisticated variant of nominal-
ism. By making a distinction between “the language LM in which mathematicians
make their claims and the language LP in which nominalists and other philosophers
make theirs” (Linnebo, 2009), the nominalist can assert that the statement “there
are prime numbers between 10 and 20” is true whilst simultaneously arguing that
there are no numbers. This is because the nominalist’s statement about prime
numbers is made in LM whereas the nominalist’s statement that there are no
numbers is made in LP. As a result, it would appear that that the nominalist’s
assertion regarding the (non) existence of numbers is cogent provided that the
sentence about primes “is translated non-homophonically from LM into LP”
(Linnebo, 2009).
Linnebo’s claim, then, that the reference of numbers is “exhausted” by their
numerical presentation suggests a position amenable to the nominalist, who may
assert that the meta-semantic analysis is precisely a demonstration of the non-
existence of numbers. In order to provide a theory of reference for mathematical
objects that embodies a genuinely intermediate position between Platonism and
nominalism, it would seem necessary to explain how numerical singular terms
can refer to objects that are more than “mere shadows of syntax” (Wright, 1992,
pp. 181–182) without thereby rendering knowledge of them mysterious. In this
context it is worth investigating whether the form of meta-semantic analysis
carried out by Dummett and Linnebo can be prosecuted in a way that does not
prejudice the question of the referential status of abstract singular terms by privi-
leging the demonstration-based conception of identifying knowledge. Such an
account is congruent with the claim that the reference of abstract singular terms
is solely determined through our linguistic practices, by contrast with terms for
concrete objects, where “how the world is”, independently of our thought and
talk, is constitutive of the semantic value of sentences containing them. The
missing requirement for an adequate theory of reference for abstract singular
terms is a more precise explanation of the link between reference to objects and
ontology.

III
Extant intermediate positions on abstract objects fall short of providing an adequate
explanation of a “thin” notion of reference for abstract singular terms. Both Dum-
mett’s “tolerant reductionism” and Linnebo’s “meta-ontological minimalism”
suggest that we may attribute abstract singular terms a semantic role in determining
the truth conditions of mathematical statements whilst also allowing for a meta-
semantic reduction of the ontological status of abstract objects. Although Dummett
and Linnebo’s distinction between semantics and meta-semantics and appeal to a
deflationary account of reference and truth explains the reductionist aspect of their
account, however, it does not adequately explain its tolerance. This explanatory gap
renders the claim that abstract singular terms play a semantic role opaque and
makes it hard to differentiate the intermediate position from a sophisticated nomi-
nalism. In the closing section of this article, I argue that a clarification of the
relationship between reference, existential quantification and ontological commit-
ment both motivates and justifies a tolerant attitude towards abstract objects. Once
we acknowledge that abstract singular terms can play a semantic role in true
mathematical statements, without this necessarily entailing the existence of
such objects in a robust Platonist sense, then the way is open to uphold both the
reductionist and the tolerant components of the thin theory.
The explanatory gap in extant intermediate accounts of abstract singular terms is
best brought out by considering a simplified version of the neo-Fregean objection
discussed in section I. A statement such as “58 + 67 = 125” is true according to
basic arithmetic. Our scientific theories also quantify over numbers like 67 and 58.
If we reject the nominalist claim that numerals like “58” and “67” are empty names,
then we seem committed to the view that numerals refer to objects. But if
“58 + 67 = 125” contains numerals that effect objectual reference then we are
committed to the existence of numbers as objects based on standard assumptions
concerning the link between objectual quantification and ontological commitment.
On what grounds, then, should we take on the complexities associated with a thin
theory of reference rather than simply asserting, with the neo-Fregean, that suc-
cessful objectual reference to mathematical objects secures their existence in a
robust sense?
The key presupposition of this argument is that objectual reference in true
sentences is sufficient to get us to the existence of objects in a robust mind and
language independent sense. This is to say that the neo-Fregean’s argument
assumes a strong interpretation of assumption (iii) discussed in section one.
Assumption (iii) states that:
iii. existential quantification and reference are intimately connected: “n” refers
to n iff “there is an x such that x = n” is true.
286 GEORGE DUKE

If we assume a robust ontological interpretation of the existential quantifier, then
assumption (iii) could be taken to imply that ontological commitment to the
existence of a kind of object follows without further argument from a successful
act of referring in a sentential context. If such an assumption cannot be granted,
however, then the tolerant reductionist claim that the attribution of a reference in the
sense of a semantic role to abstract singular terms can be harmonized with onto-
logical reductionism at the meta-semantic level appears well motivated.
An historical digression is very instructive at this point. The early Frege’s
account of reference does not have the direct ontological import suggested by a
strong interpretation of assumption (iii), which is a Quinean innovation.7
Frege
notes in Über Sinn und Bedeutung (1892, pp. 31–32) that in order to justify
speaking of the Bedeutung of a sign (e.g., a proper name), it is enough to point out
that our intention in speaking or thinking is to refer to such an object. In the context
of a formal language the existence of the relevant objects designated by a particular
sign is a presupposition of our discourse (Frege, 1892, p. 32). Passages such as this
suggest that Frege’s concern is not with an ontological justification of the existence
of certain entities on the basis of his formula language, but rather with the con-
struction of a formal language that can represent objects, concepts, etc., that can
already be assumed to be ontologically legitimate and an explication of the con-
cepts that are required to justify the use of such a language from a semantic
perspective.
Quine’s doctrine of ontological commitment suggests that we are committed to
those entities we objectually quantify over in our best theories and that to quantify
over an object is to secure reference to it. It can also be argued, however, that our
ontological commitments are determined by our intention to talk about a certain
range of objects (to refer to such objects) as if such objects really exist. This line of
thought has been well captured by Azzouni (2004, p. 55) in a nominalist context
when he asserts that we are best placed to read ontological commitments from
semantic conditions if we have already smuggled in the ontological commitments
that we want to read off. From this perspective our commitment to objects of a
certain kind existing involves a substantive metaphysical decision regarding criteria
for existence. One could then deny that mathematical objects meet criteria of, say,
strong mind and language independence, without this entailing the illegitimacy of
attributing abstract singular terms a role in the determination of the truth of
mathematical statements.
7 Although the more ontologically-oriented approach derives in large part from Russell, who sets out
from the problem of the existence of mathematical objects. For Russell objectivity is to be explained in
terms of objects. Russellian propositions and “terms” are therefore notions with immediate ontological
significance. Such a position contrasts with Frege’s emphasis upon the importance of the context principle,
according to which commitment to objects is explained in terms of the truth of statements.

As suggested in section I, the notion of ontological commitment is usually
thought of as relating either to theories or people or both. Yet the view that theories
are ontologically committed to entities in complete independence of the commit-
ments of people is inherently problematic.8
It makes sense to say that a theory T
embodies a commitment to Fs if the truth of T requires that the world contain Fs,
but only in the context of the fact that theories are things that humans formulate to
explain the world. So to say that a person, or community of rational linguistic
agents, is committed to Fs is to say that they accept as true a theory that embodies
a commitment to Fs.9
As indispensability arguments indirectly indicate, however,
such commitment suggests nothing in itself about the need for a direct ostensive
encounter with the bearers of the terms of the theory. So what are we to make of the
notion of ontological commitment in this context where we take ontology to be an
account of the “furniture” of the world? In fact, talk of the “furniture” of the world
is misleading where we are concerned with mathematical truth insofar as it sug-
gests that we are concerned with the constituents of external reality.
For Quine (1953, p. 131) the notion of ontological commitment belongs to the
theory of reference in the sense that to say “a given existential quantification
presupposes objects of a given kind is simply to say that the open sentence which
follows the quantifier is true of some objects of that kind and none not of that kind”.
Such a proposal seems to demand that we interpret the existential quantifier as
objectual rather than substitutional.10
Nonetheless the claim that the objectual
quantifier quantifies over a domain of mind and language independent objects is a
postulate and is not guaranteed by the semantic condition for the quantifier. Even if
it has been determined that “there are Fs” commits us to Fs, and that “there are Fs”
is true, we cannot necessarily conclude that there really are Fs (see Brogaard, 2008;
Eklund, 2010). In order to get to this further conclusion we need to adopt a stance
whereby objectual reference is sufficient for existence. The conclusion we should
draw is that there we should adopt a more cautious attitude towards the move from
reference to ontology in the robust sense of telling us what there “really” is.
The intelligibility of regarding Quine’s doctrine of ontological commitment as a
method that allows us to limn the ultimate structure of reality is already placed in
question by Quine’s own doctrines of the inscrutability of reference and his account
of ontological relativity. Matti Eklund draws a helpful contrast here between
8 Arguments by Cartwright (1954) and Parsons (1967) suggest why we should be reluctant to regard
ontological commitment as extensional and referentially transparent. This provides support for an account
of reference and ontological commitment based on speaker-intentions.
9 Some of Quine’s own formulations are suggestive of this, i.e., “we commit ourselves to an ontology
containing Pegasus when we say Pegasus is” (Quine, 1948, p. 8; emphasis added).
10 Where objectual quantification asserts that “$xFx” is true iff there is at least one object in the range
of the variables that satisfies “Fx” and substitutional quantification asserts that “$xFx” is true iff “Fx” is
true for some substitution instance substituting “t” for “x”, where “t” is any closed term in the language.
288 GEORGE DUKE

ontology as a transcendent and an immanent enterprise.11
If we assume that the
ontological commitments of a theory are assessed in the meta-language, then
transcendent ontology sets out from the assumption that it can describe the ultimate
constituents of reality by keeping the meta-language distinct from the object-
language. Such a project seems severely compromised by considerations of the
inscrutability of reference and ontological relativity put forward by Quine him-
self.12
Immanent ontology, by contrast, presupposes the meta-language to contain
the object language and thus restricts an assessment of ontological commitment to
“the enterprise of figuring out which ‘there are’ sentences are true” (Eklund, 2010).
A “thin” theory of reference for abstract singular terms is best understood in the
context of immanent ontology, namely as an account of which mathematical sen-
tences are true and assertable, rather than as an account of whether numbers exist
in a transcendent sense. The justification for adopting a tolerant attitude to math-
ematical objects is accordingly that the attribution to abstract singular terms of a
reference in the sense of semantic role is needed to explain what makes our
mathematical statements true and assertable. The question of whether mathematical
terms such as numerals refer is not the question as to whether such objects truly
exist, but rather whether we succeed in talking about them in a way which makes
our everyday or scientific statements intelligible, i.e., true or false, or at least
subject to justification and proof.13
As Strawson (1950) says in his polemic against
Russell, we should keep in mind that it is not that expressions refer as such but that
people use expressions to refer. The failure to remember this insight leads to the
confusion of questions about whether we can successfully refer to objects – refer-
ence to which determines whether sentences are true or false – and whether such
objects truly exist.
In the case of abstract singular terms, we use such terms to refer to objects that
are not objects of direct sensory experience or part of the causal flux. In this sense
the identification of the bearer of an abstract singular term must be regarded as not
playing a key part in the determination of the truth or falsity of a sentence
containing that term. But statements such as “58 + 67 = 125” nonetheless come out
true or false. So, if we accept the full generality of Fregean assumption (i) about
what it is to be an object, namely the correlate of a singular term, we are justified
in saying that people can use abstract singular terms to refer to mathematical
objects and that such terms play a semantic role in determining the truth conditions
11 There are notable similarities between Eklund’s distinction and Carnap’s (1950) differentiation of
internal and external questions.
12 I here assume familiarity with the relevant arguments for inscrutability of reference and ontological
relativity. See Quine (1960, 1968), Davidson (1979) and Putnam (1980).
13 Rayo (2007) demonstrates two different implementations of a structure which are ontologically
committing and ontologically innocent respectively can prove equally successful as tools for understanding
arithmetical truth.

of statements. We are not justified, however, in concluding from this that math-
ematical objects possess a strong mind and language independent existence in the
sense demanded by the traditional Platonist.14
The tolerance of the tolerant reductionist position consists in recognition that, in
the case of abstract singular terms, where the model of demonstrative identification
is not operative, we must be content to accept the legitimate referential status of
such terms just if the objects in the domain of discourse go towards determining the
truth-values of sentences containing the terms. Our reference to such objects is thin
because it does not involve contact with extra-linguistic reality but is no less
legitimately referential for that. Such a notion of reference is “attenuated” in
relation to objects of direct perceptual acquaintance, yet what led Dummett’s
intermediate theory of abstract objects astray is the assumption that a model of
meaning for concrete singular terms should serve as a paradigm for the case of
abstract singular terms. Hidden within this assumption is an implicit commitment
to a demonstration-based conception of identifying knowledge, and a resultant
suspicion of the specific claims to intelligibility and objectivity of the mathematical
realm.
As suggested above, however, the question of whether we should be construc-
tivists or Platonists regarding the existence of abstract objects is secondary to the
question of whether and how abstract singular terms refer. Such terms do in fact
refer, because we use them to successfully identify objects and because they feature
in true statements of mathematical and logical theories. The kind of reference in
question may be considered attenuated relative to the kind of reference whereby we
use a word to pick out an object in our immediate environment, but for all this our
words hook onto objects of reference. Of course in itself this allows us to regard
mathematical objects as mere fictions, but does not in itself provide a compelling
case that such objects are constructions or fictions, because the questions as to
whether we can successfully refer to a certain kind of object and whether it is
possible to provide a meta-ontological reduction of that kind of object are separate.
In the final section of this article I have argued that a tolerant attitude towards
mathematical objects is motivated and justified by the gap between reference and
14 Nonetheless, the approach to the reference of abstract singular terms advocated here does not
necessarily conflict with the neo-Fregean approach. As Eklund (2010) has suggested, the attempt to
prosecute a semantic approach to ontology via the syntactic priority thesis is legitimate on the condition
that such an approach is based on the assumption that the meta-language contains the object language.
Field, of course, has suggested that quantification over numbers is not necessary for science. Yet this does
not mean that the mathematician could fail to refer to numbers when he talks about them or that he is
uncommitted to considering numerals as referring to objects from an inferential immanent point of view.
One could be a constructivist whilst still believing that such construction only describes human access to
mathematical entities that already exist and one could be a Platonist while still acknowledging that from
the perspective of finite humans our only access to mathematical truth is through methods of symbolic
proof.
290 GEORGE DUKE

robust ontological commitment. If we understand reference as a relation between
our words and what we use our words to talk about, then reference to mathematical
objects is not only possible but a precondition of mathematical truth. Of course, a
meta-semantic reduction of the ontological status of mathematical entities is pos-
sible, but this proves neither that we do not successfully refer to mathematical
objects nor that there is no mathematical reality that transcends our thought and
talk. Ontological reduction proves that we can explain away our ontological com-
mitment to such entities, not that such entities do not exist in the Platonic heaven.
All that we can know with certainty in this context is that we can use abstract
singular terms in sentences that are true or false given our construction or adoption
of a particular system or model of objects. This is all we need to know for reference
to mathematical objects, which is after all a function of our linguistic practice, to be
successful from both an epistemic and a pragmatic point of view.
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292 GEORGE DUKE

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