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Reference to Numbers in Natural Language
Friederike Moltmann
IHPST (Paris 1/CNRS/ENS)
fmoltmann@univ-paris1.fr
The question whether numbers are objects and if so how such objects can be understood is a
central question in the philosophy of mathematics. Often in addressing this question
philosophers have made appeal to natural language, either to clarify our intuitions about
numbers in general or to obtain supporting evidence for a particular philosophical view about
numbers. Frege is a good case in point. For Frege numbers are objects because there are
singular terms that stand for them, and not just singular terms in some formal language, but in
natural language in particular. Thus, Frege thought that both noun phrases like the number of
planets and simple numerals like eight as in (1) are singular terms referring to numbers as
abstract objects:
(1) The number of planets is eight.
Thus, it appears that natural language gives straightforward support for numbers having the
status of abstract objects. The view that natural language allows pervasive reference to
numbers as objects with terms like numerals and the number of planets is in fact not only
Frege’s, but is the most common view about the semantic role of numbers in natural language.
In this paper I will argue that view about natural language is fundamentally mistaken. The
number of planets, I like to show, while it is a referential term, is not a term referring to a
number. It does not refer to an abstract object, but rather to what I will call a number trope,
the concrete instantiation of a ‘number property’ in a plurality, namely the instantiation of the
property of being eight in the plurality of the planets.
Simple numerals like eight, I will argue, are not really referential terms at all, even though
they may occur in apparently referential position with predicates with which referential terms
can also occur and fulfil other standard (Fregean or Neofregean) criteria for referential terms.
A range of new linguistic data shows that numerals behave in a number of respects differently
from terms like the number eight, which are truly referential terms. Numerals rather are quasireferential terms, on a par with certain other kinds of expressions such as clausal complements
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(given the view on which that-clauses are not referential terms). As a quasi-referential term, a
numeral has the syntactic status of a full noun phrase, but it retains the semantic value it has
when acting as a determiner or noun modifier (eight planets, the eight planets). While this
view coincides with that of Hofweber (2005), the motivations are new and unlike in Hofweber
(2005), numerals are sharply distinguished semantically from explicit number-referring terms
like the number of eight as well as terms like the number of planets.
The data about numerals give strong indications of an ‘adjectival strategy’ being
operative in natural language, the view on which number terms in arithmetical sentences do
not have the status of objects of references, but rather make contributions to generalizations
about ordinary (and possible) objects.
Natural language at the same time makes use of the ‘substantival strategy’: noun
phrases like the number eight act as truly referential terms, referring to numbers as objects of
some sort. However, these objects have an obviously derived status, going along with the
compositional semantics of such noun phrases: number here sets up an intensional context in
which eight occurs nonreferentially, number itself then is not a mere sortal but has a reifying
function, enabling reference to an object whose nature can be read off, in the number-specific
way, from the role of the semantic value of the numeral in arithmetical contexts. Thus, natural
language indicates also a fictionalist treatment of numbers as objects, on which numbers are
on a par with fictional characters as objects of reference
This paper not only argues against the Fregean view according to which there is just one
kind of singular term in natural language, having the function of standing for an object; it also
undermines the linguistic motivation for Frege’s specific view of numbers of objects. For
Frege, the form of number-referring terms in natural language was indicative of the nature of
numbers as objects. Frege assumed that in the number of planets, the number of expresses a
function that applies to a concept (denoted by planets) and yields a number acting as an
object. We will see that natural language reflects a very different view of the nature of
numbers. Natural language is in fact indicative of an older notion of abstraction, where
numbers (or number-like entities) are the ‘abstractions’ from concrete pluralities. In addition,
natural language supports the adjectival strategy and a version of a fictionalist account of
‘pure numbers’ as objects of reference.
1. The number of planets and number tropes
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1.1. The number of planets as a referential, but not a number-referring term
Let me call terms like the number of planets ‘ the number of-terms’. At least since Frege
(1884), it has been taken for granted that terms like the number of planets are referential terms
referring to numbers. It was Frege’s view that since the number of-terms are referential terms,
they must have the function of standing for an object (Frege’s context principle), and since
Frege thought that only numbers could be the right objects of reference, numbers are objects.
I will argue that in many (though not all) contexts, the number of planets has indeed the
status of a referential term, but it does not refer to a number as an abstract object or to what I
will later call a ‘pure number’. Rather it refers to what I call a ‘number trope’, a particularized
property (or rather relation) which is the instantiation of a ‘number property’ (or rather
number relation, as we will see) in a plurality of entities. For example, the number of planets
will refer to the instantiation of the number property ‘eight’ in the plurality of the planets.
There is a range of semantic evidence that indicates that noun phrases of the sort the
number of planets (the number of-terms for short) do not refer to numbers as abstract objects.
First of all, it is easy to see that Frege’s example (1) cannot be a statement of identity.
Substituting the simple numeral eight in (1) by an explicit number-referring term results in
unacceptability:1
(2) * The number of planets is the number eight.
But if (1) is not an identity statement, what is its logical form? I will later argue that (1) is
neither an identity statement nor a subject-predicate sentence, but rather is of a third sort,
namely what linguists call a pseudocleft or specificational sentence, a sentence where (at least
on one view) the subject specifies a question and the postcopula NP an answer. Then, the
number of planets in (1) in fact does not act as a referential term and thus does not have the
function of standing for an object. The same holds for eight: eight in that sentence does not
act as a referential term standing for an object either (I will later argue that eight in general
does not have that function though it acts as a singular term, but lets us set that aside for the
moment). A common analysis of specificational sentence, which I will later adopt, is that the
subject has the function of specifying a question and the expression in postcopula position
that of specifying an answer. On such an analysis, (1) says in fact ‘the answer to the question
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of how many planets there are is ‘eight’’, rather than ‘the object that is the number of planets
is identical to eight’. I will return to this analysis of (1) later.
The number of-terms, however, clearly occur as referential terms in a range of contexts,
and I will now focus on those. For example, in contexts such as (3a), the number of women
satisfies any tests of referentiality. In particular, it here occurs as subject of a sentence whose
predicate generally acts as a predicate of individuals, just as in (3b):
(3) a. The number of women is small.
b. The number eight is small.
Let me call terms like the number eight explicit ‘number-referring terms’. Explicit numberreferring terms and the number of-terms display a range of semantic differences with various
classes of predicates as well as in other respects. These differences are evidence that the two
kinds of terms refer to fundamentally different kinds of entities: the number of-terms refer to
number tropes; by contrast, explicit number-referring terms refer abstract objects, to ‘ pure
numbers’.
1. 2. Predicates
Most importantly, the number of-terms and explicit number-referring terms differ in the range
of predicates they accept or in the readings they display with particular kinds of predicates.
There are a number of predicates that are perfectly acceptable with the number of-terms,
but not explicit number-referring terms. These predicates characteristically are possible also
with the plural which follows the number of. One such predicate is the comparative predicate
more than:2
(4) a. The number of women is more than the number of men
b. The women are more than the men.
c. * The number eight is more than the number six.
Another predicate is the nonspecific comparative predicate exceed, as in (5a). With the
plural, this predicate asks for the modifier in number, as in (5b):
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(5) a. The number of the women exceeds the number of the men.
b. The women exceed the men (in number).
c.* The number fifty exceeds the number forty.
In general, also one-place predicates of comparative measurement display the same pattern
with the number of-terms, the corresponding plurals, and explicit number-referring terms.
Examples of such predicates are significant, negligible, high, and low:
(6) a. The number of animals is significant / negligible.
b. The animals are significant / negligible in number.
c. ? The number 10 is significant / negligible.
(7) a. The number of deaths is high / low.
b. The deaths are high / low in number.
c. ??? The number ten is high / low.
The closeness of the referents of the number of-terms to the associated plurality also shows
in the readings such terms yield with other evaluative predicates: they do not yield the kinds
of readings expected when evaluative and comparative predicates apply to abstract objects,
but rather readings that yield an evaluation of the plurality in just one particular respect,
namely in regard to how many they are:
(8) a. The number of women is unusual.
b. The number fifty is unusual.
(9) a. John compared the number fifty to the number forty.
b. John compared the number of women to the number of men.
(8a) has quite a different reading from (8b), and (9a) from (9b). In fact, the readings (8a)
and (9a) display can be made transparent by the near-equivalence with a sentence just about
the plurality such as (9a) and (9b), with a modifier specifying the respect ‘in number’, that is,
the respect of just ‘how many they are’:3
(10) a. The women are unusual in regard to their number.
b. John compared the women to the men in number / in regard to their number.
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To summarize, unlike pure numbers, the entities that the number of-terms refer to share
certain kinds of properties with the corresponding plurality to which the number of applies.
These are precisely the properties that can be attributed to the plurality when in addition using
the modifier ‘in number’. Those properties are just the kinds of properties the plurality has
when viewed only as ‘how many it consist in’, that is, when focusing just on how many
entities make the plurality up. This gives the first indication of what kinds of entities the
number of-terms refer to: they are entities that are pluralities ‘reduced to’ just one of their
aspects, namely how many objects they consist in.
2.3.3. Properties of concrete objects
There are further properties that show that referents of the number of-terms, unlike pure
numbers are entities that are close to the associated plurality. These properties indicate that as
long as the plurality consists of concrete entities, the referent of the number of-terms also
qualifies as concrete.
Given common criteria, what qualifies an entity as concrete rather than abstract is its
ability to act as an object of perception, as an argument of causal relations, and to be spatiotemporally located. It is easy to see that as long as the plurality in question consists of
concrete entities, perceptual and causal predicates make sense with the number of-terms,
though not with explicit number-referring terms:4
perceptual predicates:
(11) John noticed the number of the women / * the number fifty.
causal predicates:
(12) The number of the women / * The number fifty caused Mary consternation.
The number of terms thus refer to entities that have two characteristics:
[1] They share those properties with the corresponding plurality that can be attributed to the
plurality with the addition of the modifier specifying the respect of ‘how many they are’.
[2] They have properties qualifying them as concrete as long as the corresponding plurality
consists of concrete entities.
There is one kind of entity that fits just these two roles, and this is a certain kind of trope,
namely what I will call a number trope. A trope is a particularized property, a concrete
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manifestation of a property in an individual (the bearer of the trope). A trope, one can also
say, is an entity that attends to just one property of a particular individual and disregards all
others.4’ A number trope is a trope whose bearer is a plurality and which attends to just the
numerical aspect of the plurality, namely just to how many entities the plurality consists in. It
disregards all qualitative aspects of those individuals. A number trope is the instantiation of a
property being so-and-so-many in a plurality. For example, the trope that the number of
planets refers to will be the concrete manifestation of the property of being eight in the
plurality of the planets.
A number trope differs from standard examples of tropes, such as Socrates’ wisdom or
the redness of the apple, in that it is purely quantitative. Psychologically speaking, it involves
‘abstracting’ from all the qualitative respects of a plurality and focussing just on how many it
consists in. Ontologically speaking, a number trope is just like the underlying plurality except
that it shares only those properties of the plurality that pertain to how many entities it consists
in. Nonetheless number tropes belong to the same ontological category as qualitative tropes
such as Socrates’ wisdom or the redness of the apple.
There is in fact a nice syntactic indication for number tropes and qualitative tropes
belonging to the same category, and that is the impossibility of extraction of a complement. In
general a complement of an (attributively used) definite noun phrase can be extracted from
the complements position, as in (13):
(13) a. Whom did John read the biography of e?
b. What did John read the first chapter of e?
c. What house did John recognize the street number of e?
Interestingly, wh-extraction is impossible from noun phrases that describe tropes, including
those describing number tropes:
(14) a. * What did John notice the wisdom of e?
b. * What did John see the redness of e?
c. * What did John notice the number of e?
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Whatever the syntactic reason for the unacceptability of extraction in (14), it is clear that
trope-denoting NPs pattern the same, as opposed to various other kinds of definite NPs, as in
(13).5
Number tropes have still other kinds of properties than those discussed so far. In particular,
number tropes display a range of mathematical properties. But first let us focus on the
conception of number tropes itself and the semantics of number trope terms.
For the semantics of number trope terms, it is important to distinguish a relational noun
number as it occurs in the number of-terms from the homophonic nonrelational noun as it
occurs in explicit number-referring terms such as the number eight.6
There are two fundamentally different approaches to plurals, which will go along with two
different ways of conceiving of number tropes. On the first approach, plurals are taken to refer
to single entities which are collections of some sort, for example mereological sums or sets.
Let me call this the reference to a plurality view. On such a view, the number of planets
would be the instantiation of the property of having eight members (that is, atomic parts) of
the plurality that consists of all the planets. The alternative to taking pluralities to be single
objects of reference is to analyse plural terms as referring plurally to various individuals at
once (Boolos 1984, Yi 1999, 2995, 2006, Oliver/Smiley 2004). Let me call this the plural
reference view. On the plural reference view, eight would not be a one-place predicate of
entities that are pluralities, but rather a multigrade predicate taking an in principle unlimited
number of individuals as arguments. Eight will express the relation that holds among any
eight individuals just in case they are distinct (Bigelow 1988, Yi 1999).
The two views of plural terms, reference to a plurality and plural reference, correspond to
two different ways of conceiving of number tropes. On the first conception, number tropes
have a single plurality as their bearer and are instances of a property of pluralities. On the
second view, number tropes have one or more bearers and instantiate a relation.
While the reference to a plurality view is the more common one in linguistics, the plural
reference view has been pursued more recently with renewed interest by a number of
philosophers and logicians. There are various arguments against the reference to a plurality
view, which I will not go into.7 I just like to mention just one problem for the view, which also
manifest itself with number trope terms. The problem is that if plural terms were to refer to
just a plurality of some sort, the term should be substitutable by a singular term referring to
just that entity. But with many predicates such a substitution is impossible, resulting either in
unacceptability or in a different reading of the predicate, as in (15), (16):
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(15) a. John compared his children.
b. ?? John compared the set (sum / collection) of his children.
(16) a. John counted his children.
b. ?? John counted the set (sum / collection) of his children.
Unlike with standard ‘category mistakes’ or semantic selectional requirements, no semantic
type shift or coercion can make (15b) and (15b) acceptable with the reading that is available
in (16a) and (16a). This is one major argument in favour of plural reference. On the plural
reference view, his children and the set of children differ fundamentally in their semantic
function: his children refers to several children at once, whereas the set of his children refers
to a single entity. As a consequence the plural term will involve a different kind of predicate
than the singular term: the plural term requires a predicate that allows an unlimited number of
arguments for one of its positions; the singular term, by contrast, will just need an argument
position for single objects (Oliver/Smiley 2004).
The substitution problem manifests itself in just the same way with terms referring to
number tropes. Thus replacing the plural as in (17a) by a singular collective NP results in
unacceptability, on the intended reading:
(17) a. the number of children
b. ?? the number of the set (sum / collection) of children
Number trope terms are formed with the unspecific functional relational noun number.
Number in trope-referring terms expresses a multigrade function mapping some n individuals
to the trope that is the instantiation of being n in those individuals:8
(18) For entities d1, ..., dn, [number]w, i (d1, ..., dn) = f(R, d1, ..., dn)) for the number relation R
such that R(d1, ..., dn).
Here ‘f’ stands for the function mapping a property or a relation and an individual or several
individuals to the instantiation of the property or relation in the individual or the individuals.
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The number of-terms may also refer to what appears an entity that has variable
manifestations as number tropes, namely at different times, as in (19), and in different
possible circumstances, as in (20):
(19) a. The number of students has increased.
b. The number of students was the same then as it is now.
(20) The number of students might have been greater than it is.
The number of students in (19a) and (19b) does not refer to a single number trope, but rather
to a function-like entity, characterized by a function f mapping a world w and a time t to a
manifestations that is a number trope, i.e. f(w, i) = [the number of the students]w, i.
Some predicates, such as those in (19-20), apply to the function-like entity, whereas
others, those considered sofar, seem to apply just to the number tropes that act as
manifestations. This actually does not mean that number trope terms need to be considered
ambiguous between referring to number tropes and the corresponding function-like entity.
Perhaps number trope terms always refer to the function-like entity, rather than a single
number trope. Some predicates (those that seem to apply to the number trope itself) then
apply to the functional entity by applying (with their ‘primary’ meaning) to a particular
manifestation at a time in a world.9
One potential semantic problem with number trope terms in English is that the number of
is actually not followed by a standard plural term, a definite plural NP, but rather by a bare
(that is, determinerless) plural. While there are different views about the semantic function of
bare plurals, it is generally agreed that bare plurals can act as kind-referring terms, as in (21a),
and perhaps also in (21b) (Carlson 1977):
(21) a. Lions are rare.
b. John has killed lions.
Kind-referring terms can however themselves be considered plurally referring terms,
referring plurally to the various instances in the various possible circumstances, so that rare
would be a multigrade predicate. There are the same sorts of arguments for bare plurals
plurally referring to the various possible instances of the kind, rather than referring to a single
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entity that is a kind. Thus, for example replacement of lions by a singular description such as
the kind of lions is unacceptable (on a plausible reading).
In certain contexts such as after the number of a bare plural will refer only to the instances
of an actualized kind, that is, to the instances of the kind when restricted to the actual
circumstances. In that case, a kind-referring term will refer to exactly the same entities as a
definite plural term. Note that in other languages ‘the number of’ can be followed by a
definite plural only (or a specific indefinite), for example in German (die Anzahl der Planeten
/ * von Planeten ‘the number of the planets / of planets)’.
The semantics of the number of women can thus be given as follows where []w, c is the
restriction of the plurally referring term women to the actual circumstances:
(22) [the number of planets]w, c = f(R, [the planets]w, c), where R is the number relation that
holds of [planets]w, c
There is one potential problem for the number trope analysis of the number of terms and
that is cases when the relevant plurality is empty, as in (23):
(23) The number of students this year is zero.
But (23) is in fact a specificational sentence. That is, the subject here has the function of
specifying the question ‘How many students are there?’ and the numeral in postcopula
position that of giving an answer.
Another apparent problem are identity statements as in (24):
(24) The number of women is exactly the same as the number of men.
There is good evidence, however, that the expression the same as in (24) expresses not
numerical identity, but rather qualitative identity, that is, exact similarity among tropes. First,
the same as clearly expresses exact similarity with other trope-referring terms:
(25) a. John’s excitement today is exactly the same as John’s excitement yesterday.
b. John’s irritation is exactly the same as Mary’s.
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The second piece of evidence that the same as in (24) does not express numerical identity
is that (26) becomes rather unacceptable once the same as is replaced by the is of identity:
(26) ?? The number of women is the number of men.
In fact, (26) sound false, just as does (27), with terms referring to qualitative tropes:
(27) John’s excitement today is John’s excitement yesterday.
Also identity statements such as (28a), which would have to express numerical identity (or
perhaps ‘relative identity’), are bad. Such statements are of course fine with ordinary
descriptions as in (28b):
(28) a. ?? The number of women and the number of men are the same number.
b. The carpenter and the professor are the same person.
Number tropes belong to a greater class of more ‘abstract’ tropes: they belong to the class
of quantitative tropes, tropes that disregard any qualitative features of their bearer and focus
entirely on a particular quantitative aspect (see also Campbell’s instances of ‘quantities’,
Campbell (1990, p. 83pp.)). Other quantitative tropes are John’s height, Mary’s age, Bill’s
weight, the extent of Mary’s anger, and the degree of Bill’s happiness. Criteria such as the
applicability of predicates for concrete objects show straightforwardly that those terms do not
refer to purely abstract objects, but to tropes as well.10
That the number of-terms refer to tropes rather than purely abstract objects is not surprising
semantically. It fits naturally with an independently plausible generalization about ‘abstract’
nouns. It appears that whenever an ‘abstract’ relational noun is used with a definite
complement or specifier, the resulting term will not refer to an abstract object, but rather to a
trope. Other cases in point are the form of the car, the relation between John and Mary, the
color the apple, the difference between John and Bill, and the shape of the object That those
terms refer to tropes can easily be seen by using criteria for trope reference such as the
applicability of predicates for concrete entities (for other criteria for qualitative tropes see
Moltmann 2007).
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1.2. Mathematical properties
Number tropes have not only the kinds of properties that are characteristic of tropes in
general. They also have certain kinds of mathematical properties. Though they do not share
the full range of mathematical properties that pure numbers can have, that is, the referents of
explicit number-referring terms. I will argue that the more limited range of properties that
number tropes may have (in contrast to pure numbers) follows from the nature of number
tropes, as tropes that have concrete pluralities as bearers.
Let us first look at predicates that classify numbers according to their mathematical
properties. Predicates such as even, uneven, finite and infinite are possible both with number
tropes and with pure numbers:
(29) a. Mary was puzzled by the uneven / even number of guests.
b. Given the only finite number of possibilities, ...
c. John pointed out the infinite number of possibilities.
There are other predicates, however, that are acceptable only with pure numbers but not
number tropes. They include natural, rational, and real:
(30) * the natural / rational / real number of women
Furthermore, many mathematical operations are inapplicable to number tropes. These
include one-place operations such as the successor function as well as some two-place
functions such as product and smallest common denominator:
(31) a. * the successor / predecessor / root / exponent of the number of planets
b. * the product / smallest common denominator of the number of men and the number
of women
By contrast, the two-place functor sum as well as plus is applicable to number tropes:
(32) a. the sum of the number of men and the number of women
b. The number of children plus the number of adults is more than a hundred.
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What distinguishes the mathematical predicates or functors that are applicable to number
tropes from those that are not? The answer to this question can be obtained by reflecting on
the kinds of mathematical properties concrete pluralities can have and the kinds of operations
that can apply to them.
First of all, there is a sense in which pluralities can be even or uneven: to see whether a
plurality is even or uneven, it just needs to be checked whether or not the plurality can be
divided into two equal subpluralities. Similarly, in order to see whether a plurality is finite or
infinite it simply needs to be seen whether or not a 1-1 mapping can be established from the
elements of the plurality onto themselves. A number trope will then be even, uneven, finite or
infinite simply because the plurality that is its bearer is. Let us then state the following
generalization: a mathematical predicate is applicable to one or more number tropes just in
case its application conditions consist in hypothetical operations on the pluralities that are the
bearers of the number tropes. Such a condition also explains the applicability of the functor
sum: the sum operation is applicable to two number tropes because it can be defined in terms
of an operation on the two pluralities that are the bearers of the number tropes, as in (32):
(33) Addition of Number Tropes
For two number tropes t and t’, sum(t, t’) = f(R, d1, ..., dn) with R a number relation
and{d1, ..., dn} = {e1, ..., em}  {e’1, ..., e’k}, where t = f(R1, e’1, ..., e’m),
t’ = f(R2, e’1, ..., e’m} for number relations R1 and R2, provided
{e1, ..., em}  {e’1, ..., e’k} =  .
Why isn’t the successor function applicable to number tropes? The reason is simply that
the successor function cannot be viewed as an operation on concrete pluralities: the successor
function as a function applying to a concrete plurality would require adding an entity to the
plurality. However, given a ‘normal’ universe, there is not just one single object that could be
added that there are many choices as to what object could be added to the plurality to yield its
successor. Thus, no uniqueness is guaranteed, which means as an operation on pluralities, the
successor function just is not a function. Therefore, the successor function is inapplicable to
number tropes too. Similar considerations rule out the predecessor, root, and exponent
function as operations on number tropes.
Thus we can state the condition arithmetical operations on number tropes as follows:
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(33’) Condition on arithmetical properties of and functions on number tropes
a. If P is an n-place arithmetical property of number tropes, then for some n-place
property of pluralities Q, for any number tropes t1, ..., tn: Q(p1, ..., pn) iff
P(t1, ..., tn) for the bearers p1, ..., pn of t1, ..., tn.
b. If f is an n-place function on number tropes, then for some n-place function on
pluralities g, for any number tropes t1, ..., tn : g(p1, ..., pn) = f(t1, ..., tn) for the bearers
p1, ..., pn of t1, ..., tn.
Here p1, p2, .. are used as plural variables standing for several objects at once (Yi 2005, 2006).
What about the predicates natural, rational, and real? These are technical predicates that
already at the outset are defined just for the domain of all numbers, rather than just the natural
numbers. They will thus not be applicable to number tropes, which are outside the domain of
their application.11
The possibility of some mathematical properties and functions to be applicable to number
tropes on the basis of operations on concrete pluralities is also reflected in the acceptability of
descriptions of agent-related mathematical operations on number tropes:
(34) a. John added the number of children to the number of adults, and found there were
too many people to fit into the bus.
b. John subtracted the number of children from the number of invited guests.
Addition as a mathematical operation performed by an agent, as in (34a), is possible with
number tropes for the same reason as addition as a mathematical function. What matters is
that the operation as an operation on number tropes is definable in terms of an operation on
the underlying pluralities. This does not necessarily mean that when John added the number
of children to the number of adults he first mentally put together the plurality of children with
the plurality of adults and then counted the result. However, it means that if he obtained the
correct result, he might as well have obtained it by operating on the concrete pluralities first.
Subtraction of a number trope t from a number trope t’ as in (34b) is possible just in case
the plurality that is the bearer of t’ includes the plurality that is the bearer of t. If that is not the
case, subtraction applied to number tropes is not possible:
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(35) ?? John subtracted the number of planets from the number of invited guests.
Division of one number trope by another is, as expected, strange when it involves two
number tropes, as in (36a); though it is acceptable when the second (but not the first) term is a
numeral, as seen in (36b) and (36c):
(36) a. ?? John divided the number of invited guests by the number of the planets.
b. John divided the number of invited guests by two.
c. ?? John divided eighteen by the number of invited guests.
Divide by two is a complex predicate that involves an arithmetical operation definable as an
operation on a plurality. By contrast divide eighteen by is not such a predicate: eighteen is not
associated with a particular plurality that a division could target, and the plurality of a number
trope is not something by which it could be divided.
Interestingly, multiplication with number tropes is available, though. Two particularly
natural examples are those below:
(37) a. John doubled the number of invited guests.
b. Three times the number of children can fit into the bus.
Those examples, crucially, involve number tropes both as point of departure and as the result
of the multiplication. In (37a), John’s act of ‘doubling’ consists not just in a mathematical
operation, but in the replacement of one number trope (the number of invited guests at time t)
by another (the number of invited guests at t’). In (37a), the doubling of the number trope may
consist in adding as many names as there already are on the list of invited guests. Also (37b)
does not just describe a mathematical operation of multiplication of the number of children by
three, but rather compares the actual number of children to a hypothetical number trope whose
bearer consists in a maximal number of children that fit into the bus. (37b), that is, compares
the actual number of children to a hypothetical number trope with three times as many
children as bearers.
How is multiplication of a particular number trope by three achieved such that it results
again in a particular, if hypothetical, number trope? It is helpful to look at the actual linguistic
form for expressing number trope multiplication. With multiplication applied to number
17
tropes the occurrence of the expression times is crucial: in (37b), times first applies to the
number trope term the number of children; then the numeral three applies to times the number
of children. The expression times in general appears to have the function of ‘generating’
quantitative copies of tropes (or events), that is, of mapping a trope (or event) t to copies of t
that are quantitatively (or temporally) equivalent to t. More precisely times maps a particular
trope t to quantitative units of measurement of tropes for which t acts as the standard. This is
illustrated well also in (38), with a qualitative trope:
(38) John has three times the ability of Mary.
In (38), times maps a trope t that is the trope of Mary’s ability to ‘concatenations’ of
quantitative copies of t. (38) thus says that John has a trope that ‘measures’ three quantitative
copies of Mary’s ability. Multiplication of tropes thus proceeds by measurement, and the units
of measurement are quantitative copies of tropes. There are two different kinds of quantitative
units of tropes: first, those that are obtained by a cognitive operation of copying a particular
trope that serves as a standard, as in (38), and second, ‘copies’ of number tropes, which are
inherently quantitative.
The use of times for multiplication of number tropes gives further support for the view that
the number of-terms do in fact refer to number tropes: number tropes require an additional
operation of abstracting measuring units, quantitative copies of number tropes, in order to
allow for multiplication.12
Arithmetical operations thus are possible with number tropes just in case they can be
defined as operations on the underlying pluralities, possibly involving measurement based on
quantitative copies of tropes. It is then expected that ‘mixed operations’ involving both
number tropes and pure numbers are excluded. This is indeed the case:
(39) a. * John subtracted the number ten from the number of children.
b. * John added the number twenty to the number of children.
Arithmetical operations are possible on number tropes only in sofar as they are derivative of
operations on the underlying pluralities, and pure numbers have no pluralities as bearers.
Number tropes can have only those mathematical properties that are derivative of
operations on the underlying pluralities. They share the same properties with pure numbers. In
18
addition, though, number tropes have empirical properties tied to the particular nature of their
bearers. This difference in the range of properties number tropes and pure numbers may have
also shows in the way general property-related expressions are understood with number trope
terms and explicit number-referring terms. Such expressions include investigate, property,
and behaviour:
(40) a. John investigated the number 888.
b. John investigated the number of women.
(41) a. the properties / behavior of the number 8 ( mathematical properties)
b. the properties / behaviour of the number of women ( other properties)
Whereas (40a) can only mean that John investigated the mathematical properties of 888,
(40b) implies that John’s investigation was also an empirical one regarding the women in
question, namely how many women there were. Similarly, whereas (41a) can only refer to the
mathematical properties or the mathematical behaviour of a number, (41b) also refers to
nonmathematical, empirical properties or behaviour of the plurality of women.
Natural language thus displays reference to numbers tropes in a rather pervasive form and
number tropes participate in arithmetical operations as long as they can be reduced to
operations on the corresponding pluralities. Using number tropes fit well within a nominalist
approach to mathematical entities: number tropes are simply particularized relations among
particular objects. Number tropes are thus neither universals nor are they abstract objects, at
least as long as they have concrete objects as bearers. If Mayberry (2000, Chapt. 2) is correct,
number tropes also match well ancient views of numbers, in particular that of Aristotle
(Metaphysics M). For Aristotle, numbers were not special abstract objects, but concrete
pluralities viewed ‘in the manner appropriate to mathematicians’, pluralities consisting of
normal objects but viewed only ‘qua single countable things’.
Number tropes are concrete particulars that approximate natural numbers. However, a 1-1
correspondence exists only between natural numbers and classes of number tropes, namely
classes of exactly similar number tropes. Thus, 2 corresponds to the class of number tropes
that are instantiations of the property xy[x ≠ y] in some plurality. Number tropes are
‘obtained by’ abstraction from particular pluralities (as an ontological, not psychological
operation). The result is still a particular, but one that establishes a new resemblance class of
particulars which corresponds to a natural number.
19
One question to ask then is, does natural language allow for reference to such a
resemblance class, as a way of referring to a ‘pure number’? The most plausible object of
reference would then be a kind, a kind of number tropes. While natural language does allow
reference to kinds with terms like lions and kinds of tropes in particular with terms like
wisdom (Moltmann 2003c), at least English does not seem to have a device to refer to a kind
of number trope.
The same holds for numbers viewed as kinds of pluralities of ordinary objects. Numerals,
one might think, refer to kinds of pluralities, so that two would refer to the kind whose
instances are all the pluralities of two things (as proposed in Mayberry 2000). However,
numerals just do not act like kind-referring terms, like lions or wisdom. (42a) cannot mean
‘pluralities of two things are rare’, and (42b) cannot mean ‘John killed some plurality of two
things’ (two here is rather followed by a silent discourse-related anaphoric noun):
(42) a. ?? Two is rare.
b. ?? John killed two.
There is another kind of abstraction conceivable by which a natural number could be
obtained from a plurality of particulars, one that would require abstraction also from the
particuliary of the particulars, so that the result would be a plurality of ‘pure units’. This
would be Cantorian abstraction (see Fine 1998 for a reconstruction of this notion). Setting
aside the problems and limits of Cantorian abstraction, one might ask the question whether
natural language makes in fact use of Cantorian abstraction to refer to pure numbers. Given
Fine’s reconstruction, we should then find reference to number tropes with arbitrary
pluralities as bearers. Natural language does not seem to provide any such construction that
would amount to reference to a natural number. The closest one might get is a construction
like the number of any three children (where any three children should stand for a arbitrary
plurality of three objects), but this does not get close to what the number three refers to. The
strategy that natural language displays to get ‘pure numbers’ rather is from a quasi-referential
use of numerals, as we will see.
1.3. Apparent identity statements
Let us now turn to the problem of apparent identity statements like (1), repeated below:
20
(1) The number of planets is eight.
As mentioned earlier, this statement cannot be an identity statement involving two numberreferring terms. This is quite clear from the unacceptability of the sentences below:
(43) a. * The number of planets is the number eight.
b. * Which number is the number of the planets?
c. * The number of the planets is the same number as eight.
One obvious alternative analysis of (1) to that of an identity statement is that of a subjectpredicate sentence, with the subject referring to a trope and the numeral acting as a predicate
of tropes. Thus, eight as a predicate of tropes would be true of a trope t in case t is the
instantiation of the eight-relation in a plurality. However, there is one strong argument against
the subject-predicate analysis of (1), and that is that subject-predicate sentences generally do
not allow for inversion, illustrated in (44) (Heycock/Koch 1990), whereas (1) does, as seen in
(45):
(44) a. John is honest.
b. * Honest is John.
(45) Eight is the number of planets.
There is in fact a third kind of sentence besides identity statements and subject-predicate
sentences for which (1) is a candidate and that is a specificational or pseudocleft sentence
(Higgins 1973, Heycock/Krock 1990, den Dikken et al. 2000, Schlenker 2003). A
specificational sentence typically involves a wh question or question-like expression in
subject position and a not necessarily referential expression in postcopula position. A typical
example is:
(46) What John did is kiss Mary.
One important analysis of specificational sentences takes them to express a relation
between a question and an answer (den Dikken et al. 2000, Schlenker 2003).13 The answer
21
may of course consist in the content of a nonreferential expression with a complete answer
being a completion of that expression as a full sentence.
Crucially, specificational sentences allow for inversion:
(47) Kiss Mary is what John did.
(46) illustrates the most important type of a specificational sentence, in which the subject
is a wh-phrase and thus arguably an indirect question. However, there are also specificational
sentences with a definite NP as subject, such as:
(48) The biggest problem is John.
Here the subject can be taken to act as a concealed question, a non-interrogative expression
whose meaning, though, is question-like. That is, the biggest problem stands for a question of
the sort ‘what is the biggest problem?’. It is thus justified to take the subject in (1) to be a
concealed question as well, that is, the subject will have as its denotation a question or
question-like entity of the sort ‘how many planets are there?’.
There are different views of how to conceive of questions, which should not concern us
here in detail. To mention just one such view, Romero (2005) argues that the subjects of
specificational sentences and indirect questions denote functions from worlds to denotations.
Thus, the subject of (48) would denote the function mapping a world to the entity that in that
world (in the context in question) is the biggest problem, and the subject of (1) would denote
the function mapping a world to the entity that in that world provides the answer to the
question of how many planets there are. The value of the ‘question function’ need not be an
object, it may just be the semantic value of the constituent that provides the incomplete
answer to the corresponding question, thus, in the case of (1), the semantic value of a numeral
(as a nonreferential expression).14
Hofweber (2005a) gives a very different account of (1), which also avoids taking numbers
to be objects of reference. Hofweber argues that (1) is derived from (49) by a syntactic
operation of focus movement, just as (50a) would be derived from (50b):
(49) There are eight planets.
(50) a. The number of bagels I have is five.
22
b. I have five bagels.
More generally, Hofweber maintains that the number of generally has the function of a placeholder for a focus-moved numeral.
There does not seem to be any independent syntactic evidence for the alleged focusmovement, however, and in fact the kind of movement required is very implausible: to derive
(1) from (49) would require inversion of there and eight planets, lowering of the numeral to
postcopula position, and spelling out the remaining trace as the number of -- all operations
that would not be acceptable on current syntactic theory and have no parallel elsewhere. In
fact, Hofweber does not give any syntactic justification or exact description of the required
syntactic operations at all. Another problem for Hofweber’s view is that the number of-terms
can occur in any contexts other than specificational sentences (as discussed in the preceding
section), contexts in which the number of could not possibly act as a place-holder for a focusmoved numeral.
2. Numerals as Quasi-Referential Terms
Let me now turn to simple numerals. Simple numerals obviously occur primarily as
determiners of noun phrases, as in (51a), and secondarily as (apparent) referential terms in
(51b):
(51) a. Eight women were invited.
b. Eight is divisible by two.
Eight in (51b) has the (apparent) status of a referential term both because of its occurrence in
the subject position of a sentence and because it occurs with a predicate that ordinarily applies
to the referents of referential terms (such as the number eight).
The common view about occurrences of numerals as in (51a, b) is that they act as ordinary
referential terms referring to numbers as abstract objects. Standard tests for referential terms
customarily used by philosophers seem to support this view, such as Frege’s criterion of being
able to occur at both sides of the identity symbol and the possibility of replacing the numeral
by a quantifier such as something (cf. Hale 1987):15
23
(52) a. The sum of two and six is eight.
b. If eight is divisible by two, then there is something that is divisible by two.
However, these criteria are not conclusive as to the semantic role of such terms. There are
further data and criteria that distinguish among apparent referential terms. In particular, there
are criteria that distinguish between explicit number-referring terms and simple numerals, to
the effect that simple numerals should not be regarded as ordinary referential terms, but rather
as ‘quasireferential’.
2.1. Mathematical and nonmathematical properties
With a number of predicates simple numerals and explicit number-referring terms seem to be
interchangeable, for example with those below:
(53) a. Four is divisible by two.
b. The number four is divisible by the number two.
(54) a. John added two to four.
b. John added the number two to the number four.
However, at closer inspection, it turns out that simple numerals and explicit number-referring
terms do not at all share the same range of predicates. There are three classes of predicates
that are of interest:
1. nonmathematical predicates
2. mathematical predicates
3. predicates describing agent-related mathematical operations
2.1.1. Nonmathematical predicates
I will start with nonmathematical predicates, since with them the difference is particularly
striking. The difference is most apparent with relative clause constructions, as below:16
(55) a. ?? Twelve, which interests me a lot, is an important number in religious and cultural
contexts.
24
b. The number twelve, which interests me a lot, is an important number in religious and
cultural contexts.
(56) a. ?? Twelve, which I like to write my dissertation about, is an important number in
religious and cultural contexts.
b. The number twelve, which I like to write my dissertation about, is an important number
in religious and cultural contexts.
(57) a. ?? twelve, which I thought about a lot, ...
b. the number twelve, which I thought about a lot, ....
The same construction with the relative clauses modifying a simple numeral is fine when
the predicate expresses a purely mathematical property:
(58) a. Twelve, which is divisible by two, is not a prime number.
b. Twelve, which is smaller than fifteen, is greater than ten.
If the sortal number occurs in the predicate, though, non-mathematical predicates with
simple numerals are acceptable:
(59) a. twelve, which is a number that interests me a lot, ...
b. twelve, which is a number I like to write my dissertation about, ....
c. twelve, which is a number I thought about a lot, ....
The difference also shows in the following constructions:
(60) a. ?? Twelve is what I write my dissertation about.
b. Twelve is a number I like to write my dissertation about.
(61) a. ?? I like to write my dissertation about twelve.
b. I like to write my dissertation about the number twelve.
What characterizes predicates like interest me, write about, or think about? They are
intentional object-oriented predicates and thus express relations outside of the mathematical
context. It seems in fact safe to generalize that in general nonmathematical predicates are
excluded with simple numerals. What about mathematical predicates? Are the same
25
mathematical predicates possible with simple numerals and explicit number-referring terms?
Let us have a closer look at how arithmetical statements are ordinarily expressed in natural
language.
2.1.2. Mathematical properties
Before turning to arithmetical statements themselves, let us first attend to one particular
number-related predicate, namely the verb count, a verb that distinguishes between simple
numerals and explicit number-referring terms. Count allows only simple numerals but not
explicit number-referring terms as objects, at least not on the relevant reading:
(62) a. Every day, John has to count the visitors. Today he counted ten, yesterday he counted
two, before yesterday he counted zero.
b. # John counted the number ten.
In (62a) count takes simple numeral as objects and displays an intensional reading: John may
have counted ten, even if there were only nine individuals he tried to count.16’ The reading of
count in (62a) is quite different from that in (62b). If when counting the visitors, John counted
ten, he did not count the number ten. If John counted the number ten, then (if he counted
correctly) he must have counted one, rather than ten.
Count also has an intransitive use, and here again only the simple numeral is acceptable:
(63) a. One day, the visitors counted one thousand.
b. * One day, the visitors counted the number one thousand.
Why does count require simple numerals and does not allow for explicit number-referring
terms on the relevant reading? An explanation is straightforwardly obtained by reflecting on
the lexical meaning of count. Count as an accomplishment verb as in (62a) describes an
action that results in the attribution of a plural property of a plurality, as in ‘there are ten
visitors’; counting as a ‘state’ as in (63a) consists in a plurality having a plural property of the
relevant sort. The simple numeral expresses such a property (a multigrade relation), but not
the explicit number-referring term. Counting a plurality does not require assigning a number
26
as an object to it, but only attributing a plural property. It is certainly the latter sense of
counting that natural language makes use of. Thus, it is the particular lexical meaning of the
predicate that is responsible for why the predicate only takes simple numerals, but not explicit
number-referring terms.
Let us now turn to other mathematical contexts. In general, one-place mathematical
predicates as well as comparative mathematical predicates are shared by simple numerals and
explicit number-referring terms:
(64) a. Ten is divisible by five / is uneven / is smaller than twelve.
b. The number ten is divisible by five / is uneven / is smaller than twelve.
Also mathematical one-place functors are applicable both to simple numerals and explicit
number-referring terms:
(65) a. the root / successor / predecessor of four
b. the root / successor / predecessor of the number four
However, with sentences describing mathematical operations on two or more numbers
noticeable differences among the two kinds of terms appear. Thus, whereas (66a) is the
natural way of expressing addition, (66b) is very strange:
(66) a. Two and two is four.
b. ?? The number two and the number two is the number four.
The examples below display similar if less striking contrasts:
(67) a. Two plus two is four.
b. ? The number two plus the number two is the number four.
(68) a. Two times two is four.
b. ? The number two times the number two is the number four.
(69) a. Four minus two is two.
b. ? The number four minus the number two is the number two.
27
Note that the contrast holds in the same way with other kinds of explicit number terms,
without the numeral:17
(67) c. ?? The first number and the second number is the third number.
(68) c.? The first number times the second number is the third number.
(69) c. ? The first number minus the second number is the third number.
The way mathematical formulae like ‘2 + 2 = 4’ are expressed in natural language thus does
not seem to correspond to the syntax of the mathematical formula: here ‘2’ and ‘4’ are
singular terms that stand for numbers as objects, ‘+’ is a two-place functor and ‘=’ the twoplace predicate expressing identity among objects. In natural language ‘and’ and even ‘plus’
do not seem to act as functors taking singular terms and is does not in this context seem to
express identity among objects. Yet, syntactically, two plus two occurs as a normal subject,
and is as a predicate taking four as its complement. Do arithmetical statements in natural
language therefore have a fundamentally different semantics from that of the corresponding
mathematical formula? The answer is negative: there is also a view within the philosophy of
mathematics according to which arithmetical formulae do not presuppose numbers acting as
objects and numerals do not act as terms referring numbers as objects, namely ‘the Adjectival
Strategy’.
2.2. The Adjectival Strategy
Let us set aside one-place mathematical properties and functors, and focus on why
descriptions of mathematical operations prefer simple numerals and why nonmathematical
predicates are impossible with simple numerals.
A straightforward explanation is available once it is accepted that numerals in referential
position are not in fact number-referring terms, but rather retain the meaning they have when
occurring as determiners.
This view has been proposed for linguistic reasons by Hofweber (2005). Hofweber’s
motivation was to account for the general linguistic fact that numerals occur both as
determiners or adjectival modifiers and as singular terms. Thus Hofweber proposed that in
arithmetical statements like (70a) the numeral retains its meaning as a quantifier, and the
sentence itself is a generic sentences, with a meaning as in the paraphrase in (70b):
28
(70) a. Two and two is four.
b. Two things and two things are four things.
Hofweber argues that numerals undergo ‘cognitive type shift’: they retain their meaning as
(generalized) quantifiers, but syntactically they become singular terms, for the purpose of
facilitating arithmetical calculation.
The view that numerals do not semantically act as a singular terms referring to numbers
has also been explored for purely philosophical reasons by philosophers of mathematics such
as Bostock (1974), Gottlieb (1980) and Hodes (1984, 1990), and has become known as the
Adjectival Strategy, following Dummett (1995). The Adjectival Strategy analyses must make
use of modality to account for a domain that is too small to make the relevant sentences true
or rather false. Thus, (70a) is best paraphrased as below:
(71) a. If there are (were) two things and two other things, then there would be four things.
Using plural logic, (71a) can be formalized as in (71b), where ‘xx’ and ‘yy’ are plural
variables possibly standing for several individuals at once, and ‘≤’ symbolizes the ‘is / are
some of’ relation:18
(71) b. (xxyy(R2(xx) & R2(yy) & z(z ≤ xx & z ≤ yy)  xxyy(R2(xx) & R2(yy)
& z(z < xx & z < yy)  zz(R4(ww) & xx ≤ ww & yy ≤ ww) & z ( z ≤ ww
&  z ≤ xx &  z < yy))))
That is: ‘in any world in which there are two things and a different two things, for any two
things and a different two things there were two things, there are four things consisting of just
them’.
Numerals will then take as their semantic values plural properties. To formalize (70a) as
(71b), and together with the copula is needs to treated as a semantic unit which corresponds to
the following three-place relation among number relations:
(72) trans(and, is) = R’R’’R’’’[ (xxyy(R2(xx) & R2(yy) & z(z ≤ xx & z ≤ yy) 
xxyy(R’(xx) & R’’(yy) & z(z ≤ xx & z ≤ yy) zz(R’’(ww) & xx ≤ ww &
29
yy ≤ ww) & z ( z ≤ ww &  z ≤ xx &  z ≤ yy))))]
Both and and is in (72) are thus treated as syncategorematic expressions.
Also one-place functors can be accounted for that way. For example successor can be
defined syncategorematically as follows:
(73) trans(successor of t) = yxx[Rt(xx) &  y ≤ xx]
That is, the successor of the relation that corresponds to the term t is the relation that holds of
y and a plurality of xs in case R holds of the xs and y is not among the xs.
The generalization presented in the previous section adds strong linguistic evidence for
the Adjectival Strategy: the adjectival strategy obviously can apply only to mathematical
predicates and functors and not nonmathematical ones, such as intentional object-related
predicates. But how can one make sense of the adjectival strategy from a linguistic point of
view? Hodes (1990) has shown how a formal language of arithmetic can be interpreted within
the adjectival strategy, so that number terms do not designate numbers, but ‘encode’
quantifiers. Just the same in the context of natural language, it is possible to take noun phrases
that fulfil a range of criteria for ‘singular terms’ to not refer to objects, but rather to express
concepts, a generalized quantifier, or a multigrade relation among individuals, or in fact to
just have a syncategorematic function. The predicate then will not express a property of
objects, but rather as a syncategorematic expression will act together with the contribution of
the noun phrase to yield the overall meaning of the sentence. Thus, the predicate will have a
different meaning than when it occurs with referential noun phrases. As we have seen, is with
simple numerals has a different meaning than when it occurs with a referential noun phrase as
subject, and so for and, which with numerals will have a different meaning than when it
occurs with referential noun phrases. Natural language predicates and connectives that can
occur in arithmetical contexts thus are generally polysemous and display a special meaning
when they are used to make arithmetical statements.
Explicit number-referring terms are truly referential NPs referring to objects. Thus, there is
no reason to expect that they should be acceptable with predicates that require nonreferential
subjects or objects. One question then is: why are at least some mathematical predicates
acceptable with explicit number-referring terms? More important even is the question, given
the ‘presence’ of the adjectival strategy in natural language semantics, what is the status of the
30
pure numbers that explicit number-referring terms make reference to? I will turn to those
questions shortly.
Certain kinds of quantifiers can replace numerals in referential position, namely quantifiers
like something or the same thing:
(74) a. John added something to something else, namely he added ten to twenty
b. John added something to the number of children: he added two.
c. John counted the same thing as Mary when he counted the visitors, namely twenty
three.
Is this evidence against the adjectival strategy and in favour of numbers acting as objects? The
answer must be negative. Something is a very special quantifier. It is wellknown as being able
to replace various kinds of nonreferential complements, for examples predicates, as in (75a)
(Moltmann 2003a), and plural and mass NPs, as in (75b):
(75) a. John is wise.
b. John is something.
(76) a. John ate something, namely the apples.
b. John drank something, namely red wine.
The possibility of replacing simple numerals by quantifiers like something is no evidence for
simple numerals acting as referential terms. Rather, more plausibly, the quantifier either has a
substitutional or quasi-substitutional status, its role is just that of enabling inferences
(Hofweber 2005b), or it acts as a nominalising quantifier, introducing new entities into the
semantic structure of the sentence that would not have been present in the absence of the
nominalising quantifier (Moltmann 2003a). A discussion of these options will go beyond the
scope of this paper though.
3. Explicit number-referring terms
It appears that the linguistic structure and the linguistic properties of explicit propertyreferring terms are quite revealing as regards the nature of pure numbers as referents of
explicit number-referring terms.
31
The picture suggested by the data so far is that at some primary level, that of basic
arithmetical operations with natural numbers, the adjectival strategy is present and numbers in
fact do not occur as objects; at a next level, the adjectival strategy as well as reference to
numbers is permitted, the level of certain one-place mathematical properties as well as agentrelated mathematical operations. Finally, at a third level, that of nonmathematical properties,
the adjectival strategy is not available anymore, but only reference to numbers as objects.
This picture is strongly suggestive of a fictionalist account of numbers. The parallels with
fictional characters are strong, given a view of fictional entities such as that of Kripke (1973),
Searle (1979), or van Inwagen’s (2000). On that view, there is only pretend reference within
the story, where nuclear properties are attributed to an individual the author pretends to refer
to. But reference to a fictional character takes place as soon as ‘extranuclear’ properties are
predicated of the individual described in the story, properties from outside the context of the
story. While in purely mathematical contexts, given the adjectival strategy, there is neither
reference nor pretend reference, mathematical properties certainly side with nuclear properties
on the nuclear – extranuclear distinction. Nonmathematical properties, by contrast, side with
extranuclear properties, and thus they require reference to numbers as objects. Numbers as
objects of reference thus enable the attribution of nonmathematical predicates, just like
fictional characters as objects of reference enabled the attribution of extranuclear properties.
Numbers also can have at least certain mathematical properties, namely certain one-place
properties as well as agent-related properties. In the case of fictional characters, they do not
‘have’ the properties attributed to them in the story (otherwise conflicts may arise with certain
extranuclear properties they may have), but rather they ‘hold’ them, as van Inwagen (2000)
has argued.
Fictional characters depend entirely for their existence and identity on the story and its
origins. However, this does not mean fictional characters themselves are ‘created’. Rather, as
Schiffer (1996) has argued, once the story exist in a world, the fictional character exists there
as well, whether or not anyone conceived of it or refers to it. Fictional characters thus are
language-created, but language-independent objects (Schiffer 1996). The use of a referential
term making reference to a fictional character does not bring it into existence, but simply
enables epistemic access to it. The same can be said about a plausible fictionalist account of
numbers: once there are the mathematical contexts with an adjectival strategy being operative,
natural numbers as objects can be read off those contexts. The use of explicit numberreferring terms simply enables reference to them.
32
The two challenges this account faces is a clarification and a justification of the strategy by
which a context, fictional or mathematical, individuates objects of the relevant sort. Once
such a procedure is justified, the role of the referential term is simply to pick out the objects
so defined.
The parallel to fictional characters is also linguistically plausible. The following
constructions are of exactly the same sort:
(77) a. the fictional character Hamlet
b. the number ten
In this construction, a sortal (fictional character or number) as head nominal is followed by a
nonreferential expression, a fictional name or simple numeral. Given the general view of the
introduction of entities on the basis of fictional or mathematical contexts just outlined, the
semantics of this construction is evident: the sortal here does not just have the function of
characterizing the kind to which an entity belongs (with its particular identity conditions), but
rather serves to characterize the strategy on the basis of which an entity can be ‘read off’ a
nonreferential expression in a context (such as that of a fiction or of arithmetics). The sortal
fictional character characterises one such strategy; the sortal number another.
The construction has even greater generality. The following are also examples of it:
(78) a. the name Hamlet
b. the person John
c. the numeral eight
d. the color green
In (78a) and (78c) the strategy for introducing an object on the basis of a nonreferential
expression is yet a different one. Here the object introduced is an expression and clearly here
the sortal name or numeral creates a hyperintensional context: what follows it is simply the
‘presentation’ of an expression. In (78b), the sortal person just takes into account the referent
of the expression that follows it. In (78d), again a non-referential expression (an adjective)
follows the sortal color. Color here introduces an abstract object on the basis of the adjectival,
nonreferential use of green (see also Dummett 1995). The noun phrases in (77) and (78) can
thus be called reifying terms.
33
That the sortal in reifying terms sets up an intensional and in fact hyperintensional context
is clear also from the fact that substitution of the expression following the sortal by a
coreferential expression is impossible (* the number that number,* the number what I wrote
down, * the number what I added to two,* the numeral that expression,* the numeral that
numeral, * the color the complementary color of red).
In reifying terms, it is the sortal, the reifying strategy it characterizes, that guarantees
uniqueness. Thus, the definite determiner is obligatory (* a number four, * a person John, * a
color nuance green), and is best seen as having a ‘pleonastic’ function (being indicative of a
semantic condition already expressed otherwise in the construction).
Returning now to explicit number-referring terms, the question is, why do their referents
have certain mathematical properties but not others? That they can be involved in agentrelated mathematical operations is straightforwardly explained: agent-related mathematical
operations involve both the intentionality of actions (and thus a non-mathematical aspect) and
a purely mathematical function; it is the former that licences explicit number-referring terms,
and the latter must be what makes also simple numerals acceptable. Licensing of simple
numerals comes from the arithmetic aspect, licensing of explicit number-referring term from
the intentional aspect.
What about predicates like even or infinite which are also possible with the two kinds of
terms? Both properties are generally defined in terms of a mathematical operation, requiring
simple numerals. But since their content is derivative on that operation, the definition can
equally well be given for pure numbers: the pure number n has a property P just in case the
numeral corresponding to n plays such and such a role in a particular mathematical operation
from which P is derived.
Thus, number objects will not systematically inherit all the properties attributed to the
corresponding numeral; rather, like fictional characters, they have properties of their own
many of which are defined by using the corresponding numeral. Other properties are simply
supervenient on the role the content of the numeral plays in various possible mathematical
contexts and in various mathematical and nonmathematical uses of it.
2.3. Further linguistic evidence for the nonreferential status of numerals
There is also syntactic linguistic evidence for the nonreferential status of simple numerals.
One piece of such evidence comes from the choice among two kinds of relative pronouns in
34
German. In German there are two kinds of relative pronouns. The first kind, lets call it ‘wpronouns’, consists in the pronoun was. Was occurs with sortal-free quantifiers as in alles,
was; nichts, was; vieles, was (‘everything that’, ‘nothing that’, ‘many things that’). The
second kind, lets call it ‘d-pronouns’, consists in das, der, die (singular feminine), and die
(plural). D-pronouns cannot occur with sortal free quantifiers, alles, vieles, nichts, but must, it
appears, modify an NP with a sortal as head, as in der Mann, der;, (‘the man who’), die Frau,
die (‘the woman who’); das Kind, das (‘the child that’); die Leute, die (‘the people that’), or
else a proper name as in Hans, der (‘John, who’) What is most interesting in the present
context is that the two kinds of relative pronouns show a systematic difference with respect to
referential and quasi-referential expressions. Predicative complements generally are not held
to be referential: they do not refer to a property that acts as an argument of the embedding
copula verb, but rather express a property that will contribute to a complex predicate. Only wpronouns can be used with predicative complements. By contrast, explicit property-referring
terms (with the sortal property) require d-pronouns:
(79) a. Hand wurde weise, was / *das Maria bereits war.
‘John became wise, which Mary already was.’
b. Hans hat die Eigenschaft, weise zu sein, die / *was Maria auch hat.
‘John has the property of being wise, which Mary has too.’
It has frequently been argued that also that-clauses are not referential terms (Moltmann 2003b
and references therein). In fact, the distinction between the two kinds of relative pronouns in
German sharply distinguish simple that-clause-complements from truly referential NPs like
the proposition that S, or the fact that S:
(80) a. Hans glaubt, das es regnet, was / * das Maria auch glaubt.
‘John believes that it will rain, which Mary believes too.’
b. Hans glaubt die Proposition, dass es regnet, die / * was Maria auch glaubt.
‘John believes the proposition that it will rain, which Mary believes too.’
(81) a. Hans hat erwahnt, dass die Sonne schien, was / * das Maria nicht erwahnt hatte.
‘John has mentioned that the sun is shining, which Mary has not mentioned.’
b. Hans hat die Tatsache bemerkt, dass die Sonne schien, die / * was Maria nicht
erwaehnt hatte.
35
‘John has mentioned the fact that the sun is shining, which Mary has not mentioned.’
Now the crucial observation is that simple numerals can only go with w-pronouns, not
with d-pronouns, as opposed to explicit number-referring terms, which require d-pronouns:
(82) a. zwoelf, was / * das / *? die eine Zahl ist, die mich sehr interessiert, …
‘twelve, which is a number that interests me a lot, …’
b. zwoelf, was / * das / * die durch zwei teilbar ist, …
‘twelve, which is divisible by two, …
c. die Zahl zwoelf, die / * was durch zwei teilbar ist, …
‘the number twelve, which divisible by two, …’
By this test, simple numerals are not treated as referential. Rather they have the same nonreferential status as predicative complements, as well as, arguably, that-clauses.19
German provides another test for the non-referentiality of simple numerals, and that is
support of plural anaphora (for some reason the data are much less clear in English). The
generalization is that conjoined simple numerals generally do not support plural anaphora, as
opposed to conjoined explicit number-referring terms:
(83) a. Hans addierte zehn und zwanzig. Maria addierte ?? sie / ok diese Zahlen auch.
‘John added ten and twenty. Mary added them / those numbers too.’
b. Hans addierte die Zahlen zehn und zwanzig. Maria addierte sie auch.
‘John added the numbers ten and twenty. Mary added them too.’
(84) a. Hans notierte zehn und zwanzig. Maria notierte ?? sie / dies Zahlen auch.
‘ John wrote down ten and twenty. Mary wrote them / those numbers down too.’
b. Hans notierte die Zahlen zehn und zwanzig. Maria notierte sie auch.
‘John wrote down the numbers ten and twenty. Mary wrote them down too.’
Again a number sortal in the predicate has the same effect as a number sortal in an explicit
number-referring term, rendering a plural anaphor acceptable:
(85) a. Drei und fuenf sind beides Primzahlen. Sie sind nicht durch zwei teilbar.
‘Three and five are both prime numbers. They are not divisible by two.’
36
b. Drei und fuenf sind nicht durch zwei teilbar. Sie sind beide(s) Primzahlen.
‘Three and five are not divisible by two. ? They are both prime numbers.’
Plural anaphora (in German) obviously must refer to pluralities of objects, which are not what
conjunctions of non-referential terms (without subsequent reification) can introduce. (Why the
generalization seems much less strong in English than in German remains to be explained.)
The two linguistic criteria from German indicating the quasi-referential status of numerals
and the referential status of explicit number-referring terms also show the quasi-referential
status of ‘simple’ quotations (‘John’) and the referential status of explicit expression-referring
terms (the name John). Moreover, they show the quasi-referential status of simple color words
(green) and the referential status of explicit color-referring terms (the color green).
Simple quotations pattern just like simple numerals with respect to the choice of relative
pronouns and support of plural anaphora, whereas explicit expression-referring terms go with
explicit number-referring terms (and here the data are just as clear in English):
(86) a. ‘Maria’, was / * der der Name dieser Frau ist…
‘Mary’, which is the name of this woman….
b. der Name ‘Maria’, der der Name dieser Frau ist….
(87) a. Bill wrote down ‘Mary’ and ‘Sue’.
b. ?? John wrote them down too.
(88) a. Bill wrote down the names ‘Mary’ and ‘Sue’.
b. Sue wrote them down too.
Color adjectives that are used as ‘bare’ nominalizations, as in (92), pattern with simple
numerals, whereas explicit colour-referring terms, with the sortal colour, as in (93), go with
explicit number-referring terms:
(89) a. Green is my favourite colour.
b. Green is nicer than blue
(90) a. The colour green is my favourite colour.
b. The colour green is nicer than the colour blue.
The relevant data are given below:
37
(91) a. Gruen, was / * das meine Liebingsfarbe ist, ....
‘Green, which is my favourite colour, ...
b. Gruen und Rot stehen Anna am besten. * Sie / ok Die Farben standen ihr schon immer
gut.
(92) a. Die Farbe Gruen, die / * was meine Lieblingsfarbe ist ...
‘the colour green, which is my favourite color, ...
b. Die Farbe Gruen und die Farbe Rot stehen ihr am besten. Sie standen ihr schon immer
gut.
‘The color green and the color red suit her best. They have always suited her.’
The conclusion is obvious: colour words in referential position, even though they seem to
stand for abstract entities of some sort, do not yet act as referential terms, but have a quasireferential status. It is only with the sortal colour that a referential term is formed, referring to
an abstract object obtained in some way from the conditions of use of the adjective that
follows the sortal. I will leave with these indications of the generality of the phenomenon. It is
a task for future work to make precise how the difference is to be understood between a
expression as an object of reference and an expression occurring in a hyperintensional context
and the difference between a color object and the semantic value of a bare color adjective
nominalization.
5. Conclusion
In this paper, the approach was to look more closely at the linguistic data involving reference
to natural numbers in natural language, and the result was a treatment of numbers that differs
greatly from what philosophers usually take for granted: natural language primarily allows for
reference to number tropes and makes use of the adjectival strategy for quasi-referential
occurrences of numerals. It also, with a more derivative construction, supports a certain kind
of fictionalist account of numbers as objects of reference.
The paper also has pointed at other new directions that are of independent interest. First, it
has shown the importance of reference to tropes in the semantics of natural language. Second,
it has introduced a notion of quasi-referentiality which has much wider applications than just
to number-referring terms. Quasi-referential terms require not only a revision of the standard
38
semantics of ‘singular terms’ (or rather the equivalent category in natural language), but also
of lexical meaning. Predicates that occur with quasi-referential terms cannot just express a
property of objects that applies to the semantic value of the term, but rather they will need to
interact in more complex ways with the semantic contribution of the quasi-referential term.
The reconception of lexical semantics that is required may in some ways coincide with
more recent developments in lexical theory, in particular the theory of the Generative Lexicon
(Pustejovsky 1995). Here predicates do not come with one given meaning that is a relation
among objects, but show considerable flexibility of what their actual contribution to the
meaning of the sentence may be. Also, noun phrases that have a quasi-referential status seem
to be admitted, namely singular noun phrases that stand for what is called ‘dot-objects’,
concepts that may correspond to objects of distinct categories. Thus, dinner in (93) does not
stand for a single object that is both an event and consists of food, but rather it stands for a
combination of two concepts that may be developed into one or the other object, depending
on what predicate is applied:
(93) Dinner was delicious and lasted for three hours.
Acknowledgments
I would like to thank audiences at the IHPST, the universities of St Andrews (Arché), Geneva,
Kyoto, Palermo, and Geneva as well as Bob Fiengo, Kit Fine, Oleg Prosorov, and Ted Sider
for stimulating discussions.
Notes
This is despite Frege’s own claim to the contrary (Frege 1884). The example is equally
unacceptable in German. In fact also Frege’s other German example below, with a numeral
containing a definite determiner is unacceptable:
1
(1) ?? Die Anzahl der Planeten ist die Acht.
‘The number of planets is the eight.’
In other languages, for example German, also the opposite of more, weniger ‘less than’,
patterns that way:
2
(1) a. Die Anzahl der Frauen ist weniger als die Anzahl der Maenner.
‘The number of the women is less than the number of the men.’
b. The women are less than the men.
‘Die Frauen sind weniger als die Maenner.’
c. ?? Die Zahl fuenfzig ist weniger als die Zahl vierzig.
39
‘The number fifty is less than the number forty.’
(2) OK The number fifty is less than the number forty.
The fact that English the number ten is less than the number twenty is acceptable should best
be traced to a lexical idiosyncracy of less than.
3
Some comparative predicates, though, are unacceptable with plurals and are acceptable with
the number of-terms and explicit number-referring terms:
(1) a. The number fifty is greater than / is smaller than the number forty.
b. The number of the women is greater than / is smaller than the number of the men.
4
Temporal predicates that are natural with the number of-terms are harder to find. The
example below might be a case in point:
(1) a. The number of participants stays the same as long as the participants do not change.
b. ? The number fifty stays the same.
However, (1a) involves a case of reference to an entity with variable manifestations at
different times and worlds, as discussed below in the text.
4’
The term ‘trope’ is from Williams (1953). Other references on tropes are Bacon (1995),
Campbell (1990), Lowe (2006), and Woltersdorff (1970). Tropes already played an important
role as accidents or modes in ancient (Aristotle) and medieval philosophy.
A number trope is the instantiation not of an accidental property in an entity (as with John’s
happiness or Socrates’ wisdom), but rather of an essential property, if pluralities are
considered objects: a plurality considered as an object has its members and thus the number of
its members essentially. There is another view of pluralities, though, which I will adopt later.
On that view, a plurality is not ‘one, but ‘many’. More precisely, a plural term such as the
planets is a term not referring to a single collective entity, but rather refers plurally to several
entities.
5
6
The distinction is reflected in some languages, for example German. In German, Zahl is used
relationally or nonrelationally. By contrast Anzahl is used only relationally (though in fact for
my ears entirely deviant uses are found in Frege 1884). This means in particular Anzahl is
limited to the number of-terms and does not allow for the plural:
(1) a. Die Zahl / Anzahl der Planeten ist acht.
‘The number of the planets is eight’.
b. die Zahl / * Anzahl acht
‘the number eight’
c. diese Zahl / * diese Anzahl
‘this number’
d. Er addierte die Zahlen / * die Anzahlen.
‘He added the number / * the numbers (relational).
The number of planets will thus refer to the concrete manifestation of being eight in the
various planets.
40
7
See for example Boolos (1884) and Yi (2005, 2006) for discussion.
8
The plural reference view has the additional advantage of allowing for a uniform meaning of
numerals, namely as relational predicates expressing multigrade relations. They will have that
meaning in contexts such as (1a), as well as (1b) (see also Bigelow 1988):
(1) a. Five students received a prize.
b. The five students resemble each other.
9
Of course, number can also occur as just part of a quantifier, as in (1):
(1) A great number of women were arrested.
is no reference to a number trope in this case is indicated by the obligatory plural agreement
rather than singular agreement. The same holds for German Anzahl, as in (2b):
(2) a. * A great number of women was arrested.
b. Eine grosse Anzahl von Frauen wurden / * wurde festgenommen.
10
Note that other quantitative tropes show the same behaviour with predicates expressing
comparison:
(1) a. John’s age exceeds Mary’s age.
b. John exceeds Mary in age.
c. ?? Ten years exceeds five years.
(2) a. John’s height exceeds Mary’s height.
b. John exceeds Mary in height.
c. ?? Two meters exceed 150 cm.
11
In fact, it is not clear that predicates like natural here have a meaning that is independent of
the content of number: This number is natural is quite bad. This indicates that natural number
does not express a property compositionally obtained from natural and number, but is
idiomatic. Then natural (with the relevant meaning) could not possibly apply to things other
than numbers anyway.
12
There is in principle a different way in which multiplication could be defined on the basis of
particular pluralities, namely by making use of quantification over higher-order pluralities.
Mayberry (2000), who also pursues a reduction of arithmetical operations to operations on
pluralities, proposes hat ‘2 x 3 = 6’ be analysed as ‘two (nonoverlapping) pluralities of three
(distinct) entities is six entities’. The present account avoids quantification over pluralities by
taking two in ‘two times the number of N’ to range over number tropes that ‘measure’ two
copies of the number trope ‘the number of N’. Measurement of tropes thus replaces higherorder plural quantification.
13
For an alternative analysis involver higher-order equations see Jacobson (1994) and Sharvit
(1999).
14
This also corresponds roughly to Karttunen’s (1978) view of questions.
41
15
Hale’s (1987) criterion involving quantifiers is, actually, considerably more complex.
16
For some reason the difference is less apparent in main clauses:
(1) a. Twelve interests me more than eleven.
b. Twelve is very interesting; five is not.
There are two possible explanations. First, the numerals in (1) are actually focused, which
may mean that the sentences for the subject position presuppose a domain of pre-individuated
number objects and the numeral just serves to pick out one of those objects. Second, there
may be an implicit sortal number in subject position, which for some reason may not be
present in object position. Given either explanation, it is also possible that the numeral in (1a,
b) is in fact in topic position, as in (2):
.
(2) Twelve, that number is very interesting.
16’
For the intensionality of count, see Moltmann (1995).
17
Simple numerals are also unacceptable with certain expressions of identity that are fine with
explicit number-referring terms:
(1) a. ?? Two plus two is the same number as four.
b. The number four is the same number as the number four.
18
The common formalization of (71a) within approaches using the adjectival strategy is by
making use of quantification over concepts, as below ‘(Hodes 1984, 1990):
(1) F G (2x Fx & 2x Gx &  x(Fx & Gx))  4x (Gx v F x)
This presupposes the view that counting means counting objects that fall under a concept. See
Bigelow (1988) for a critique of that view.
19
One might think of a linguistic explanation why numerals, predicative complements, and
that-clauses require w-pronouns in German. Predicative complements, that-clauses, and
numerals perhaps are not gender-marked, and w-pronouns are selected whenever the
expression modified lacks gender-marking. However, that-clauses and numerals show neutral
gender in their agreement with neutral pronouns; and thus they must be gender-marked:
(1) a. Dass p und dass nicht p impliziert seine eigene Widerlegung.
‘That p and not p implies its own refutation.’
b. Zehn ist nicht groesser als die Summe seines Vorgaengers mit eins.
‘Ten is not greater than the sum of its predecessor with one.’
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