REFERENTIAL USES OF ARABIC NUMERALS

Abstract

Is the debate over the existence of numbers unsolvable? Mario Gómez-Torrente presents a novel proposal to unclog the old discussion between the realist and the anti-realist about numbers. In this paper, the strategy is outlined, highlighting its results and showing how they determine the desiderata for a satisfactory theory of the reference of Arabic numerals, which should lead to a satisfactory explanation about numbers. It is argued here that the theory almost achieves its goals, yet it does not capture the relevant association between how a number can be split up and the morphological property of Arabic numerals to be positional. This property seems to play a substantial role in providing a complete theory of Arabic numerals and numbers.

Full text

Manuscrito – Rev. Int. Fil. Campinas, v. 43, n. 4, pp. 142-164, Oct.-Dec. 2020.
REFERENTIAL USES OF ARABIC NUMERALS
_________
MELISSA VIVANCO
https://orcid.org/0000-0001-9923-7208
University of Miami
Department of Philosophy Miami, FL
U.S.A.
melisa.viva@ciencias.unam.mx
Article info
CDD: 401
Received: 25.08.2020; Revised: 09.09.2020; Accepted: 14.09.2020
https://doi.org/10.1590/0100-6045.2020.V43N4.ME
Keywords
Arabic numerals
Reference
Arithmetical epistemology
Numbers
Abstract: Is the debate over the existence of numbers unsolvable?
Mario Gómez-Torrente presents a novel proposal to unclog the
old discussion between the realist and the anti-realist about
numbers. In this paper, the strategy is outlined, highlighting its
results and showing how they determine the desiderata for a
satisfactory theory of the reference of Arabic numerals, which
should lead to a satisfactory explanation about numbers. It is
argued here that the theory almost achieves its goals, yet it does not
capture the relevant association between how a number can be split
up and the morphological property of Arabic numerals to be
positional. This property seems to play a substantial role in
providing a complete theory of Arabic numerals and numbers.

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1. THE GAME: HOW NUMERALS COULD REFER INSTEAD
OF
WHETHER NUMERALS REFER
There is a well–known debate about the metaphysics
of
natural numbers. Typically, the discussion takes place in a
match whose players belong to one of two predefined teams:
the Realist and the Antirealist. If you choose the Realist team,
as Frege (1884), Burgess and Rosen (1997), Hale and Wright
(2009), and
others have done, prepare yourself to commit to the
existence of natural numbers as abstract, objective, and (not
necessarily but most likely) mind–independent entities. The realist
player holds that arithmetical sentences are true in virtue of facts
about the denotations of their singular terms and predicates. Her
challenge
in this game is to explain by virtue of what do we gain
knowledge of arithmetic sentences (since we don’t have the same
type of contact with abstract entities as we do with whatever
entities that are supposed to make
empirical sentences true).
Naturally, you might like the Antirealist team better. The spirit
of this popular
team is to deny the existence of entities such as
numbers (see Field (1989), Yablo (2010), Bueno (2016) Once
you choose to become an antirealist, your challenge is to
explain in virtue of what are arithmetical sentences true. This game
has spawned a diverse variety of accounts in which each
team
shows off their most sophisticated tactics, even reaching
extreme positions with consequences such as
that the only
possible result is that both teams ‘win’
(for example, defending
that only radical realism and
radical antirealism are tenable, as
Balaguer (1998) does) or that both teams ‘lose’ (as in a case of
unsolvable epistemic disagreement (see Rosen (2001)).
In the fourth chapter of Roads to reference, Gómez-Torrente
(2019) presents an attractive and novel account where the
starting point is to put aside the traditional game— which has
come to seem bogged down—and starts a new one. The
opening move of this game consists of taking at face value our
linguistic intuitions bearing on the question of how the

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referents of the numerals could get fixed.
The interesting question in this context is not, I
think, whether skeptical or nominalist doubts can
be assuaged to the full satisfaction of the doubters,
but rather whether
there is an intuitive ontology
and epistemology of the numbers and the
numerals that can underwrite a sensible account
of
reference fixing for the numerals. (p.108)
The winner of the new game will be the one who can
provide an explanation of the reference fixing of
the numerals
that can be adequately accompanied by a plausible ontology and
a satisfactory epistemology
about numbers. This is the
objective that Gómez-Torrente tries to achieve in this
chapter.
2.
T
HE TEAMS
:
D
ESCRIPTIVISTS VS
.
R
EFERENTIALISTS
As Gómez-Torrente points out, there are two kinds
of
accounts of Arabic numerals that are seriously con
sidered in the
literature on reference. These are the Descriptivist team, which
holds that Arabic numer
als have semantic structure, being
semantically equiva
lent with certain mathematical descriptions or
related
phrases, and the Referentialist team, which holds that
Arabic numerals are not equivalent with descriptions and are
instead semantically unstructured singular terms such as proper
names, pure indexicals and
simple demonstratives are likely to
be.
Along with other non–descriptiveness semantic views
developed in the book, Gómez-Torrente argues that
numbers
are not descriptions. From the referential side
on numerals, the
first move consists of ruling out the
most prominent

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descriptivist semantics. The strategy
is to analyze paradigmatic
examples of different de
scriptivisms and raise objections that
would presum
ably span classes of the most plausible theories of
this kind.
As said, the procedure to fix the reference of Arabic
numerals
is not completely description-free. The
author is pushed to
adopt a kind of descriptivist ‘fixes- the-reference’ view.
Nonetheless, since the reference
of an Arabic numeral is not a
description itself, this procedure must be carefully carried
out.
2.1
Fixing the reference by verbal numerals
The defender of this view holds that Arabic numerals
either
abbreviate or have their reference conventionally
fixed by their
corresponding verbal numerals. If so,
the advocate of semantic
non–descriptivism for Arabic
numerals is about to lose because
once she accepts a method of fixing the reference using
complex verbal numerals, there seem to be no better
candidates for
the semantic values of Arabic numerals than
those or very similar descriptions. In such a view, ‘765’
abbreviates or has its reference conventionally fixed by a
description such as ‘(seven (times) (one) hundred) (plus)
(sixty
(plus) five)’. What Gómez-Torrente argues is
that there are
many more intelligible Arabic numerals than the conventional
verbal numerals in any typical speaker’s idiolect: eventually, we
would run out of verbal numerals to fix the references of Arabic
numerals.
Another problem with this crude version of the view is that it
might lead to certain classic epistemological
puzzles. Imagine
that a friend of mine from France, Margot, and I are competent
users of Arabic numerals but are not very well versed in
arithmetic. Margot knows that ‘99’ stands for quatre-vingt-dix-
neuf
whose corresponding description in English would be

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something like (four (times) twenty (plus) ten (plus)
nine). On the
other hand, I know that ‘99’ stands for (ninety (plus) nine).
The statement ‘99 is (ninety (plus) nine)’ is necessarily true and
I know it a priori; Margot knows a priori that ‘99 is (four (times)
twenty (plus) ten (plus) nine)’ is true, this identity is also
necessary; we both know that ‘99 = 99’ is true, but we
may
not know that ‘(ninety (plus) nine) is (four (times)
twenty (plus)
ten (plus) nine)’ is true. Would we say
in this case that Margot
and I
know
the semantic value
of ‘99’?
The advocate of fixing the reference by verbal numerals has
to explain how the finite resources from
verbal numerals can
be sufficient to fix the references of the infinite Arabic numerals,
as well as the relation between the descriptions used to fix the
references and the semantic values of the Arabic numerals, in
such a way that epistemic problems such as those mentioned
above are avoided. The ball is in their court.
2.2
Two more moves by the Descriptivist
What is relevant for current purposes is that Gómez-
Torrente builds his own referential strategy based on rules
deployed from descriptivist moves. Due to space
limitations, I
present broadly the two most promi
nent descriptivisms
1
that
Gómez Torrente addresses
and highlight the rules that are
derived from them.
DP
2 Arabic numerals are semantically
equivalent with cer
tain
mathematical descriptions,
1
Gómez-Torrente mentions other theories, as well as other
variations of these ones, and he provides arguments to show that
all of them are challenged by the same sort of objections.

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generally polynomial
expansions
such as ‘  
      ’
in the case of ‘765’.
DP 3 The description giving the
meaning or fixing the reference
of a numeral N later than ‘0’ in
the natural order would be of the
form of ‘The number of numerals
between ‘0’ and ‘M.’ (Where for
‘M ’one would substitute the
Arabic numeral coming just
before N in the natural order.)
The first move above exhibits the Kripkean anti–
descriptivism argument about the semantic detachment for
names applied to numerals. This is analogous to the case
where the name ‘Gödel’ still would refer to Gödel even if the
description ‘The discoverer
of the incompleteness of
arithmetic’ were not satisfied
by Gödel.
Suppose that Arabic numerals are semantically equiv
alent to
polynomials such as the one used in DP2. In the case of natural
numbers, exponentiation does not constitute a fundamental
arithmetic operatio
2
, but rather an extension of multiplication
2
Some mathematicians struggle with the fact that any number
raised to the zero power is 1. Some justifications can be provided:
we can say that the zero power is just the product of no numbers
at all, which is the multiplicative identity, or that pn is the number
of functions from a set of cardinality n to a set of cardinality p, if
n = 0 the number of functions at play is 1 (the empty set); Gómez-
Torrente offers another justification in the footnote 17. All these
reasons carry a sense of artificiality to be mathematical grounds of
the fact that p0 = 1 for all integers p. Basically, what is said is that

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(even multiplication could be questioned since it is an
extension of addition that is dispensable in the construction
of the inductive structure of natural numbers). Given this
consideration, we can imagine a case where someone
discovered that the exponentiation of natural numbers is not
well defined after all. Would this imply
that a typical user of
the Arabic numerals would have
never referred to 765 by ‘765’,
given that the description ‘5(9 + 1)0 + 6(9 + 1)0+1 + 7(9 +
1)0+1+1’ would
have turned out to determine no number? The
answer
seems to be that this would not be implied and ‘765’
would still refer to 765.
Two more observations from DP
2 are noteworthy. Typical
English speakers seem to have de re attitudes
involving the natural
numbers when they entertain cer
tain attitudes via Arabic numeral
representations (at
least when the numerals are not too long),
but not
when they entertain them via other representations
which are more clearly descriptive (see Ackerman (1978) and
Kripke (1992)). This is the case of the polynomial in DP 2.
One way to make our de re attitudes more apparent is by
analyzing How-many questions. Standard
theories of the
semantics of interrogatives character
ize questions as sets of
answers, where an answer to
a question is the semantic value of
an expression that
would count as a (correct or felicitous)
response to the corresponding interrogative, (cfr.
Groenendijk and Stokhof (1997); Krifka (2011)).
Imagine that I text Margot with the question ‘How many
customers went to your restaurant today?’. Con-
sider the
answers: a) ‘795’, and b) ‘5(9 + 1)0 + 6(9 + 1)0+1 + 7(9 + 1)0+1+1’.
They both are correct in some
sense. Nevertheless, the only
one that is clearly felic
itous is a), since this is the only one that
p0 = 1 because otherwise un-desirable practical consequences
would hold. So we must accept the fact as a necessary stipulation.

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rules out
an invitation to another How-many question (which
would reflect that the expectations of the asker have not been
satisfied). If Margot texts me back b), I may very well have to
ask her ‘And how many are those?’
A felicitous answer to a How-many question (note
from the
example that the How-many question can be
replaced by a
What-is-the number-of question) seems to be the one that
concerns our de re attitude toward the number that turns out to
be the semantic value of the correct answer. The best candidate
is the answer
with Arabic numerals. Now let’s move on to the
second
observation.
According to Gómez-Torrente, what makes the poly
nomial in
DP 2 attractive with respect to others (for example, the
polynomial whose existence is guaranteed by the Fundamental
Theorem of Arithmetic) is that once the reference of the digits
is fixed (and the
meaning of the operations is established) the
reference
of complex Arabic numerals is determined. That
seems to be a good reason to choose that kind of polynomial,
but the way I see it, there is a deeper (related) reason to make
that choice. The form of such a polynomial is reflected in the
morphological nature of the
corresponding numeral: the term
whose power has
one
numeral determines the digit that
occupies the first place from right to left in the numeral, the term
whose power has two numerals occupies the second place from
right to left in the numeral, and so on. For reasons explained
in a subsequent section, I would say that this is a potential
point in favor of DP 2 and against
the Gómez-Torrente’s
account. Indeed, the author ac
knowledges that complex Arabic
numerals have a formal complexity that must somehow be relevant to
their content (p.112).
DP3 move raises another concern. Some descriptivisms
seem to be too conceptually demanding to ex
plain everyday uses
of Arabic numerals (think of Margot and I having trouble with
number communication). For example, people can be
competent with numerals for specific numbers without having

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much arithmeti
cal background or the general concept of a
numeral,
which is necessary to understand descriptions such as
DP
3. As Gómez-Torrente points out, these descriptions do not
seem to be part of the semantic structure behind
the numerals
or even of the conceptual wherewithal of
the users of the
numerals as reference fixers.
To conclude, it is convenient to draw a final moral from
descriptivism. According to the Kripkean anti-
descriptivism
arguments for standard proper names, the existence of a
cognitive contact between the speaker and the referent makes the
question of whether a cer
tain object is that referent in the
speaker’s idiolect detachable from the question of whether
that object
satisfies those descriptions (this explains, for example,
why in cases of faulty uses of descriptions the speaker
still
manages to refer). However, this can’t generally be applied to
cases in which a speaker-baptizer fixes the referent with the
help of a description. Such a speaker will associate descriptive
material with the name in an undetachable way. Unlike the
paradigmatic Kripkean story about proper names, the
procedure by which Arabic numerals obtain their reference
does not appear to be completely free of descrip
tive content. If
the Referentialist accepts a descriptivist ‘fixes-the-reference’
view, in order to guarantee a successful reference to numbers
through the use of Arabic numerals, she must account for a
cognitive contact bet
ween the speaker and the numbers that
turns out to
be at least in some degree detachable from
descriptive
content.
3
T
HE RULES
:
N
EW CHALLENGES FOR THE
R
EFERENTIALIST
As a response to descriptivist strategies, Gómez-Torrente
presents a ‘fixes-the-reference’ descriptivist view of con
tent fixing
for the Arabic numerals that will appeal to less sophisticated

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descriptions, a view that will be accompanied by the
postulation of a significant role for certain elements that can
plausibly be said to provide
a non-descriptive cognitive contact
between speakers
and numbers (p.115).
RP
1 The correct view of the Arabic numerals is a
de
scriptivist ‘fixes-the-reference’ view, but
the numerals are not semantically equivalent
with de
scriptions
The Referentialist move, RP
1, must comply with the rules
that have been set by the lessons learned from the descriptivist
moves:
1.
The detachment argument (RP1 is exempt
from
this because the procedure of fixing the
reference
involves descriptive content)
2.
The descriptions involved in the fixing-the-
reference
procedure should not be
sophisticated to a degree that requires
extensive conceptual (linguistic or
mathematical) resources on the part of the
speaker
3.
Complex Arabic numerals must get their
interpretations by means of a general
procedure which
exploits in some way their
morphological constituents and the meanings
4.
The speaker must be successful in referring to
the
numbers in a way that is independent of the
fact that the numbers satisfy the descriptions
involved
in the procedure for fixing the
reference of Arabic
numerals
5.
The fixing-the-reference procedure must
lead to an ontology and epistemology that
accounts for
our de re attitudes toward

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numbers through the use of Arabic numerals
3.1
Fixing the reference
As mentioned in §1, the Realist used to be required to
explain
our epistemic access to what the semantic values of the numerals
could be, which would ultimately explain our knowledge of
arithmetic truths. Such a requirement cannot be overlooked by
the Referentialist. In the new game, the procedure for fixing the
reference
of the Arabic numerals should at least partially help to
provide a satisfactory answer to the question.
Gómez-Torrente’s proposal is based on the existence of
the ability to generate the series of Arabic
numerals without
recourse to sophisticated arithmeti
cal knowledge:
A. Small Arabic numerals, like ‘1’, get their
refer
ents either via translation to their
corresponding verbal numerals, or
directly via descriptions similar to those
that presumably fix the referents of
small verbal numerals, such as ‘the
number of these fingers’, or ‘this
number’
B.
Larger Arabic numerals get their
referents fixed
in a typical speaker’s
idiolect when she masters
systematic
ways to identify those referents in terms
of the referents of the smaller
numerals. For example, by means of
the speaker’s disposition to associate
Arabic numerals (coming after ‘1’ and
following ‘M ’), with descriptions of the
form ‘The number greater by one
than M
’

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Number Rule.
The Arabic numeral
‘1’
refers to the number one; and whenever
an Arabic numeral refers to a certain
number, the Arabic numeral that follows it
in the generating order refers to the number
greater by one than that number.
A competent user could in principle associate with each
Arabic numeral with which she is acquainted with one or more
particular utterances of reference-fixing descriptions for that
numeral which intuitively follows
from the Number rule. The
ability to generate and interpret bigger numbers via the
mechanism is, not coincidentally, related to the ability to count;
it is ul
timately this ability that provides our most basic
con
ceptions of number (pp.127, 128).
This move to fix the reference of Arabic numerals
suggests
an answer to the epistemological question: access to numbers will
be partially explained by certain abilities typically identified as
linguistic, which can be combined with abilities of a different
cognitive nature
(for example, the ability to identify small
multiplicities), and playing the ability to count–which plays a
privileged role.
The mastery of these abilities eventually induces the
appearance of particular conceptions of the numbers in a
progressively larger set, and also of a minimal
conception of the
general notion of number that, ac
cording to Gómez-Torrente,
includes the idea that the
number of a multiplicity of things is
an aspect of it that is common to other multiplicities of things
that can be counted by means of the same verbal numeral
(among others related ideas). Additionally, I would include
the idea that a multiplicity of things can be
arranged into
smaller multiplicities of things, each
of those corresponding to
a verbal numeral; the relation between the smaller
multiplicities and the to
tal is associated with the relation
between the verbal numerals that correspond to the smaller

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multiplicities and the verbal numeral that corresponds to the
total. For instance, a multiplicity of 125 things whose
corresponding numeral is ‘one hundred twenty-five’ can be
arranged (in particular) into three multiplicities of things: one of
100 things whose corresponding numeral is ‘one hundred’, one of
20 things whose corresponding
numeral is ‘twenty’, and one of 5
things whose corresponding numeral is ‘five’.
3.2
What the numbers could be
For the ontological move, Gómez-Torrente offers a wide
variety of reasons to support the claim that numbers are
cardinality properties of pluralities. Contrary to
other potential
candidates of what numbers could be,
these properties stand in
the same relation as numbers do to the corresponding
pluralities. I consider that this is a compelling reason to hold
that numbers are cardinality properties of pluralities.
Provided the numbers are any things at all,
a natural
number must by its nature be susceptible of
being had by pluralities of things, susceptible of
being the number of
pluralities. Thus, 17 is so
susceptible, and
in fact it is the number of the major
moons of Jupiter. (Gómez-Torrente 2015, p.
317)
Whatever numbers are, there is an intrinsic rela
tion between
them and the pluralities that have them
as their numbers. This
relation is the same as that between
cardinality properties (and
only them) and any
plurality that has them as properties.

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3.3
Following the rules of the game?
It is not difficult to concede that Gómez-Torrente’s
account does not face a major problem regarding rules
1, 2, and
4. In my view, rules 3 and 5 deserve more attention.
The question regarding rule 3 is, to what extent does
the
proposed fixing-the-reference procedure exploit the
morphological components and meanings of
the Arabic
numerals?
To answer this question, it is worth recalling the basics of
the procedure. Let’s call this set B:
•
The speaker’s general conception of number and
of
adding one
•
The ability to generate and interpret bigger num
bers via
the Number Rule (related to the ability to count)
-
Having the reference of ‘1’
-
Once the reference of
‘
M
’
has been fixed,
the
reference of the following numeral in the
natural
order
‘
N
’
is fixed in virtue of the fact that N is
greater by one than M
The procedure certainly reflects
some
relevant association
between the morphological nature of numerals within a
positional system of Arabic numerals and the numbers that
constitute the structure of natural num
bers. Namely, that the
transition from
‘
M
’
to
‘
N
’
is attached to the transition from M
to N given by adding 1 to M. The association captured by B
is reflected in the similarity between the procedure and what it
takes and the characterization of natural numbers as
the

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inductive set
3
such that 1 is the first element,
and the others
are produced by adding 1 successively.
But what does this have to
do with the morphological components and meanings of
Arabic numerals? There seems to be no reason to think that
the procedure
would not work for other numbering systems
that are essentially different from Arabic numerals. In particular,
for systems in which numerals have a substantially different
morphology. If so, it is very unclear that rule
3 is being followed.
Furthermore, since the last rule requires that the theory at play
accounts for our de re attitudes toward numbers via Arabic
numerals, this also jeopardizes the move with respect to rule 5.
The procedure may
seem adequate for consecutive
numerals whose morphology does not change significantly
after the transition by adding 1 of the corresponding
numbers. Consider for example, ‘764’ and ‘765’ (since ‘5’ is
the numeral that follows ‘4’ in the
natural order, whose referent
is obtained from the fact
that 5 is the number greater by one than
4), we have
a straightforward account for the reference of ‘765’
in terms of their constituents. Nonetheless, the account
seems
to lack an explanatory element for cases like ‘999’ and ‘1000’
whose morphologies (determined by
the constituents) differ
more substantially. It seems
that Gómez-Torrente’s account
underestimates the fact
that Arabic numeral systems are
positional, which is
a salient aspect regarding their morphology.
I think that unsurprisingly, this property is not only associated
with a relevant characteristic of numbers but also plays an
important role in how Arabic numerals manage to refer to
numbers.
3
A nonempty partially ordered set in which every element has a
successor.

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To illustrate the aforementioned point, consider a
non-
positional system (one in which the position a symbol
occupies in the numeral bears no relation to its value
4
). For
instance, the system where ‘1’ stands for the number one, ‘11’
stands for the number two, ‘111’ stands for the number three,
and so on. The
procedure offered by Gómez-Torrente
succeeds in accounting for how the numerals in this system get
their
references fixed. The speaker picks out the referent of
‘1111111111’ in virtue of the fact that the numeral ‘1’ refers to
the number one and the ability to recognize that ‘1111111111’
follows to ‘111111111’ because its referent (the number ten)
is the number greater by one than the referent of
‘111111111’, which has been
already fixed by the same
procedure. That which con
stitutes B is apparently all that is
needed to explain
how almost any system whose numerals refer
to num
bers manages to do so.
Certainly, such numerals do not seem to entertain
de re
attitudes toward the number 102 as the numeral
‘102’ does in the
positional decimal Arabic system. (Or
as the numeral ‘1212’
allegedly would do if we had been trained in the Arabic
numeral system in base 4). If I ask Margot how many
costumers went to her restaurant today and she texts me a
sequence of 765 ones, this does not seem to be a felicitous
answer to my question: either I have to count the quantity of
ones or text her again ‘And how many are those?’.
4
A positional numbering system is a numeral system in which the
contribution of a digit to the value of a number is the product of
the value of the digit by a factor determined by the position of the
digit. In early numeral systems, such as Roman numerals, a digit
has only one value: “I” means one, “X” means ten and “C” a
hundred. In modern positional systems, such as the decimal
system, the position of the digit means that its value must be
multiplied by some value: in “555”, the three identical symbols
represent five hundreds, five tens, and five units, respectively, due
to their different positions in the digit string.

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The offered ‘fixing-the-reference’ theory is not
sensi
tive to the difference between non-positional
numbering systems and positional Arabic numeral systems.
This seems to be a potential problem for the current strategy,
which seems to support a realist position since its success
depends on a satisfactory reference theory for Arabic numerals.
It would be suspicious
that a general theory of reference for
almost any numbering system can provide a solution to the
epistemological problem. (Think of the arithmetic limitations
we have when using Roman numerals.)
3.4
Adjusting the Referentialist’s move
According to Gómez-Torrente, numbers, as the plural
cardinality properties they are likely to be, need
not stand in
any quasi-graphical structural relation
ship with the numerals
(p.13). They may not need to, however, they do stand in a
relation like that with
Arabic numerals, which plays an important
role in this particular way of answering the epistemological
ques
tion. It is partially in virtue of that relation that we
entertain de re thoughts toward numbers via Arabic numerals.
The potential move that I outline here is
intended to ‘complete’
a successful Referentialist move. The aim is that the procedure for
fixing the reference of the Arabic numerals also captures their
morphological
property of being positional.
Firstly, the Number rule needs the referents of at least two
symbols to make sense of the fact that a symbol’s position in
a complex numeral is related to its value. The natural way to do
so is adding the condition that ‘0’ refers to the number zero
5
.
The second
part of the Number rule could be more difficult to
complement, but some observations derived from the fact that
5
As Gómez-Torrente mentions in the chapter, there are many
alternative ways to fix the reference of ‘0’.

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numbers are cardinality properties of pluralities
may help (This
would further support the view that
numbers are plural
properties, then we have an intuitive ontology that can
underwrite a sensible account
of reference fixing for Arabic
numerals.):
A plurality α with the property of having n as its
number
also has the property of admitting partitions
as follows,
•
For each number m, (n m > 1) there is a partition
of α such that at least one of its ‘sub- pluralities’
has m as its number. The symbol placed in the
first position from left to right is the numeral that
corresponds to the number of pluralities whose
cardinality is m. If there are
m
sub-pluralities whose
cardinality is
m
, a new
position on the right is
generated in the corre
sponding Arabic numeral in
base
m
. The symbol
placed in the new position is
the numeral that
corresponds to the number of the
pluralities left
whose cardinality is 1, which is less
than m.

If there are
m times m
sub-pluralities whose car
dinality
is m, a new position is generated on the right in
the corresponding Arabic numeral in base m. The
symbol placed in the first position is the numeral
corresponding to the number of sub-pluralities
whose cardinality is m times m.
If there are (
m times
m
)+
k
sub-pluralities whose
cardinality is m (k < m
times m), the symbol
placed in the second position is
the numeral that corresponds to the number of the
pluralities left
whose cardinality is m, (i.e., (m times
m + k) - (m times m) = k). The symbol placed in
the third position is the numeral that corresponds
to the number of pluralities whose cardinality is
1,
which is less than m.
•
If there are m times (m times m) · · ·

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Certainly, this process has been described by ‘brute
force’.
The ideal would be to present the general rule by which this
process occurs in terms of pluralities; I
am quite confident that
in the right space this task can be accomplished, especially if the
rule is centered on a
fixed base (preferably base 10). For now,
let’s use an example to illustrate the broad idea,
Let α be a plurality whose cardinality is n = 5 and m the
number that determines the base of the Arabic numeral system
such that 1 < m
≤
n
6
:
•
For m = n, the partition in play has 1 plurality
whose
cardinality is five and 0
pluralities whose cardinalities
are less than five. The correspond
ing Arabic
numeral in base five is ‘10’.
•
For m = 4, the partition in play has 1 plurality
whose cardinality is four and 1
remaining plurality
whose cardinality is one. The corresponding
Arabic
numeral in base four is ‘11’.
•
For m = 3, the partition in play has 1 plurality
whose cardinality is three and 2 remaining
pluralities whose cardinality is one. The
corresponding Arabic numeral in base three is ‘12’.
•
For m = 2, the partition in play has 1 two-times-
two
plurality whose cardinality is two, 0
remaining
pluralities whose cardinality is greater than
two times
two, and 1 remaining plurality whose
cardinality is
one. The corresponding Arabic nu
meral in base two
6
If m is a base greater than n = 5, there are no sub-pluralities whose
cardinality is m, therefore the corresponding Arabic nu- meral is
not complex.

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is ‘101’
7
.
The resulting principle may not be as simple as the
Number
rule, but this does not imply that it is too conceptually
demanding for a competent user of Arabic
numerals. All she
needs, in addition to the other abilities mentioned, is the ability to
associate the idea that
a not-too-large plurality can be
partitioned in a certain way with the idea that the position of a
symbol in a complex Arabic numeral is related to its value. The
speaker masters the procedure because of the same reason as in the
original account: she develops the ability (related to counting) to
systematically associate Arabic numerals, within a given Arabic
positional numeral
system, to the corresponding numbers.
4
G
AME OVER
...?!:
C
ONCLUDING REMARKS
Gómez-Torrente’s theory hypothesizes an association
between the Arabic numerals and a system of unso
phisticated
reference-fixing phrases that single out the
corresponding
numbers. The ability to generate the
series of Arabic numerals
underlies this association and is responsible for giving to the
speakers the non- descriptive cognitive contact with the
referents of the
Arabic numerals that would seem required in
view of
the existence of de re attitudes toward numbers via
those numerals.
The reference-fixing theory, along with the epis
temic and
ontological remarks, does justice to our use
of Arabic numerals
as well as to a variety of intuitions
about numbers. Adding the
observation that the prop
erty of being part of a positional
system of numerals plays a crucial role in the contact we have
7
Note that a plurality of cardinality 5 contains two sub- pluralities
whose cardinality is two. As a result, a new position is generated
on the right in the corresponding Arabic numeral in base two.

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with num
bers through Arabic numerals, I maintain that the
proposal successfully accounts for the following intuitions:
1.
The cardinality property of a plurality (i.e., its
number) is an aspect of it that is common to
other pluralities that can be counted by means of
the same Arabic numeral
2.
The cardinality properties (numbers) are
homogeneous in nature and things that are generable
from an initial item by recursive application of
an
homogeneous operation that can plausibly be
seen
to correspond to the intuitive operation of adding
one. It is no coincidence that this is the case with
the Arabic numerals system
3.
The cardinality property (number) of a plurality
can
be splitup according to the following fact:
the
plurality can be partitioned into smaller plu
ralities,
each of which corresponds to an Arabic
numeral.
The relation between the cardinality
properties of
the sub-pluralities and the cardinality property of
the total plurality is associated with the relation
between each of the con
stituents and the complex
Arabic numeral that
corresponds to the total
plurality. The way these partitions can be selected is reflected
in the morphology of the corresponding Arabic numeral. In
particular, in the value that each constituent has, according
to its position.
The epistemological concern of the traditional An
tirealist
has been addressed from a completely new perspective. As it
is explained by Gómez-Torrente: by the time a typical speaker
is exposed in earnest to
the Arabic numerals, she has learned to
recite at least
a moderately large initial segment of the sequence
of
existing verbal numerals and has learned to count plu
ralities of
objects with it. The development of these abilities precedes

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the grasp of the concept of quantity or number as such. The
mastery of these abilities
eventually induces the appearance of
particular conceptions of the numbers in a progressively larger
set,
and also of a minimal conception of the general notion of
number. A system of numerals with suitable properties, such
as the Arabic decimal system, plays
a crucial role in the evolution
towards mastering such
abilities.
I have no doubt that the game is not over yet and
the
Antirealist (or the Descriptivist) will have more to
say. Now it is
her move.
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