What s in a Numeral? Frege s Answer
Joan Weiner
Frege wanted to define the number 1 and the concept of number. What is required of
a satisfactory definition? A truly arbitrary definition will not do: to stipulate that the
number one is Julius Caesar is to change the subject. One might expect Frege to
define the number 1 by giving a description that picks out the object that the nu-
meral  1 already names; to define the concept of number by giving a description that
picks out precisely those objects that are numbers. Yet Frege appears to do no such
thing. Indeed, when he defends his definitions, he does not argue that they pick out
objects that we have been talking about all along the issue never comes up. The
aim of this paper is to explain why. I argue that, on Frege s view, our numerals do
not, antecedent to his work, name particular objects. This raises an obvious ques-
tion: If (like  Odysseus ) the numerals do not name particular objects, how can Frege
write (as he does) as if sentences in which numerals appear state truths? One central
concern of this paper is exegetical to answer these questions. But my aim is not
solely exegetical. For these questions direct us to something that, I believe, creates
only an apparent problem for Frege but an actual problem for many contemporary
philosophers: the assumption that singular terms appearing in statements about the
world must actually have referents. Another aim of this paper is to suggest that the
problem as well as a solution that can be found in Frege s writings should be of
import to contemporary philosophers.
Frege begins The Foundations of Arithmetic, the work in which he intro-
duces the project which was to occupy him for most of his professional
career, with the question  What is the number 1? It is a question to
which even mathematicians, he says, have no satisfactory answer. And
given this scandalous situation, he adds, there is small hope that we will
be able to say what number is. Frege intends to rectify the situation by
providing definitions of the number one and the concept number. But
what, exactly, is required of a definition? Surely a truly arbitrary defini-
tion will not do. Surely to stipulate that the number one is Julius Caesar
is to change the subject. It seems reasonable to suppose that an accepta-
ble definition must be a true statement that contains a description that
picks out the object to which the numeral  1 already refers. Similarly, it
seems reasonable to suppose that an acceptable definition of the con-
cept of number must contain a description that picks out precisely
those objects that are numbers those objects to which our numerals
already refer. But, while this may seem reasonable to us, there is a
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doi:10.1093/mind/fzm677
678 Joan Weiner
respect in which it may not seem reasonable to expect this of Frege dur-
ing the period in which he was writing Foundations.
 Refer and its cognates are understood by most contemporary phi-
losophers as technical philosophical terms whose origin as technical
philosophical terms lies in Frege s renowned 1892 paper,  Über Sinn
und Bedeutung .1 In that paper, Frege introduces a distinction between
two sorts of content an object expression has: Sinn (typically translated
 sense ) and Bedeutung (typically translated either as  reference or as
 meaning 2). Although there are currently debates about how  Bedeu-
tung should be translated and about whether or not it is meant to
play a role in semantic theory there is no question that the Bedeutung
of an object expression is the object (if any) to which it refers.3 But
 Über Sinn und Bedeutung was written well after Frege wrote Founda-
tions, and the Sinn/Bedeutung distinction he introduced there is one
that, Frege himself says (1891b, 1893), he had not yet formulated when
he wrote Foundations. Thus it may seem anachronistic to say that, in
the project as envisioned in Foundations, Frege required his definition
of the number one to pick out the object to which the numeral  1
already refers. Indeed one might suspect that this requirement is not
even formulable in the language of Foundations.
But this is a mistake. In the first sentence of Foundations, Frege intro-
duces two statements of a question,  what the number one is, or what
the symbol 1 means (die Frage, was die Zahl Eins sei, oder was das
1
Because the correct translation of this title, particularly of the word  Bedeutung , is a matter of
controversy, I have chosen to leave the title untranslated. In the interests of clarity, I have also used
 Bedeutung and  Sinn untranslated in many of the discussions that follow.
2
In the standard translations of Frege s texts,  Bedeutung and its cognates are translated by
 meaning and its cognates. However, many of the philosophers who write about Frege use  refer-
ent and its cognates. In the quotations that follow, I have used the standard translations but, when
 meaning is used as a translation of  Bedeutung , I have included the German as well.
3
This far, I go with the traditional interpretation. I part company with the traditional interpre-
tation over the issue of whether Frege has a semantic theory on which there is a reference relation
that holds between linguistic expressions and pieces of reality. The reason is that this relation
would have to hold, not just between object-expressions and objects, but also between function-
expressions and either functions or some sort of sets of ordered tuples. The latter, however, is
something that, Frege says explicitly, cannot be stated. And, indeed, all primitive Begriffsschrift ex-
pressions, that is, all the Begriffsschrift expressions for which one would expect a clause in a truth
definition, are function-expressions. Thus Frege would be opposed to any attempt to give a theory
of reference for his Begriffsschrift. Here, and in what follows, I have used the term  Begriffsschrift
without italics to refer to Frege s logical language. I use  Begriffsschrift (italicized) to talk about
Frege s 1879 monograph. For a discussion of the relevant passages from Frege s writings and their
significance, see Weiner 2002 and 1990 chapter 6.
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Mind, Vol. 116 . 463 . July 2007
What s in a Numeral? Frege s Answer 679
Zeichen 1 bedeute) (Frege 1884, p. i).4 The first statement of the
question what the number one is appears to have nothing in par-
ticular to do with language or meaning. In order to answer this ques-
tion we need a description that picks out a particular object and
distinguishes it from all others. Since Frege immediately restates the
question as a question about a linguistic expression, there can be no
doubt that, in order to answer the question  what does the symbol 1
mean (bedeute)? , we need to pick out an object and distinguish it from
all others. The object in question is what we are using the symbol  1 to
talk about. Thus, although  bedeuten is not a technical term in Founda-
tions, it is clear from the context that what he means by  was das Zeichen
1 bedeute is: which object the symbol  1 names or refers to.
How are we to begin to answer this question? Frege s second state-
ment of the question is not simply a clarification of the original state-
ment, it is the introduction of a strategy. He introduces, as a
fundamental principle that is to constrain his project, the renowned
 context principle :  never to ask for the meaning (Bedeutung) of a word
in isolation, but only in the context of a proposition (Frege 1884, p. x).
We need not look far for propositions containing the symbol  1 and the
other numerals. Such propositions are omnipresent in our everyday
discussions, both informal and scientific. When we say, for instance,
that 0 g 1 (that 1 is not an even number; that 1 is the successor of 0, that
the Earth has 1 moon, etc.) we are, presumably, talking about some-
thing. Frege s context principle suggests, then, that the definition must
say what we are talking about when we use the symbol in these familiar
sentences. Now it may seem that the locution  what we are talking
about is needlessly vague. For it may seem obvious that, if we are talk-
ing about something when we use the symbol  1 , then there is some
thing that we are talking about. That is, the symbol  1 must refer to a
unique object and the definition is supposed to pick out the object we
are talking about when we use the symbol in these familiar sentences.
This is also suggested by Frege s claim that the number one  looks like a
definite particular object, with properties that can be specified (Frege
1884, p. ii). It seems reasonable to suppose that Frege is assuming that
the symbol  1 that appears in our everyday propositions refers to a
unique object. And, if this is so, surely an acceptable definition must be
one that picks out that particular object.
4
Although, in the English translation, the fact that these are two characterizations of the same
question is only implicit, it is explicit in Frege s actual words. Frege s sentence begins,  Auf die
Frage, was die Zahl eins sei, oder was die Zeichen 1 bedeute &  . Both characterizations are charac-
terizations of die Frage.
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680 Joan Weiner
Given all this it may seem only reasonable to attribute to Frege the
following apparently obvious faithfulness requirement:
The apparently obvious faithfulness requirement: A definition of an
object expression (concept expression) must pick out the object to
which the expression already refers (objects of which the expression
already is true).
If this is a requirement, an acceptable definition of  1 must be a true
statement that contains a description that picks out the particular
object that we use the numeral  1 to name; that we are already using  1
to talk about. Similarly, an acceptable definition of the concept of
number must contain a description that picks out precisely those
objects that are numbers  those objects that we are talking about
when we use the word  number ; those objects that we use our numer-
als to name. Another way of characterizing this requirement is that the
definition must preserve reference. For all the apparent obviousness of
this requirement, however, Frege never explicitly says that his defini-
tions must satisfy it; nor does he ever argue that his definitions do sat-
isfy it. One might suspect that this is an oversight on Frege s part. But I
think that there is a different explanation. For the assumption that
there is a definite particular object that we are already talking about
when we use the numeral  1 is not as obvious as it seems. There is, I will
argue, a great deal of evidence that Frege would have denied that there
was a unique object that antecedent to his work was named by the
numeral  1 . But if this is so, how can these definitions teach us anything
about our science of arithmetic? And what criteria must these defini-
tions satisfy? To answer these questions, we need to understand what it
is that Frege thinks we need to learn about the science of arithmetic.
1. Why define the number one and concept number?
Definitions of the number one and the concept of number are neces-
sary, on Frege s view, if we are to prove the truths of arithmetic from
primitive truths. What are primitive truths? And why does Frege think
we should prove the truths of arithmetic from primitive truths? In the
early sections of Foundations, Frege offers two motivations for attempt-
ing to provide such proofs. The first, which he characterizes as mathe-
matical, is a desire for increased rigour for proof, wherever proof is
possible. The second, which he characterizes as philosophical (and in
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What s in a Numeral? Frege s Answer 681
Begriffsschrift as epistemological5) is a desire to classify the truths of
arithmetic as a priori or a posteriori, synthetic or analytic.
On Frege s view what is at issue in this scheme of classification is  the
ultimate ground upon which rests the justification for holding [a prop-
osition] to be true (Frege 1884, Sect. 3, p. 3). If a truth is provable, its
classification as analytic or synthetic, a priori or a posteriori, is deter-
mined by its most economical (most general) proof by the proof
requiring the fewest specific assumptions. The least economical (least
general) sort of proof requires an appeal to facts: to truths about partic-
ular objects that cannot and need not be proved; unprovable truths that
are not general. A posteriori truths, for example, truths of the empirical
sciences, require appeals to facts (Frege 1884, Sect. 3, p. 4). Truths that
are not a posteriori are a priori. The grounds that justify a priori truths
are more general than those that justify a posteriori truths. But not all a
priori truths have maximally general grounds. There are some, for
example the truths of geometry, that cannot be proved  without making
use of truths which are not of a general logical nature, but belong to the
sphere of some special science (Frege 1884, Sect. 3, p. 4). These truths
are synthetic a priori. But there are also a priori truths that can be
proved without any appeals to facts or to truths of a special science.
These truths truths that can be proved using only general logical laws
and definitions are analytic.
What about the primitive truths the truths at which our proofs
stop? How are we to recognize which truths are primitive? And how are
we to identify which place a primitive truth occupies in Frege s classifi-
catory scheme? One of the striking features of Frege s discussion of this
system for classifying truths is that he never answers these questions.
We can, however, make some inferences from what Frege does write
about how these questions should be answered. If the point of proving
truths of arithmetic from primitive truths is to enable us to determine
the correct classification of the arithmetical truths, there will be eligibil-
ity conditions that determine what can be taken as a primitive truth.
5
See, for example, the opening remarks of the introduction to Frege 1879. Although Frege had
not yet formulated the project in quite this way when he wrote Begriffsschrift, the project of finding
gapless proofs of the truths of arithmetic from logical laws is described and it is described as an
epistemological project his aim is to show that the ground of arithmetic is the ground upon
which all knowledge rests. It is also important to note that there is no evidence that Frege changed
his mind about this epistemological motivation of the project. The characterization of his motiva-
tion as epistemological also appears in the opening remarks of the introduction to Grundgesetze.
For more on this, see n. 35 below.
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682 Joan Weiner
One obvious eligibility condition is that it must not require proof its
truth must be evident without proof.6 Another is that the expression of
the primitive truth should include only simple, undefinable expres-
sions. For these simples are the ultimate building blocks of the disci-
pline. Frege writes that  the properties belonging to these ultimate
building blocks of a discipline contain, as it were in a nutshell, its whole
contents (Frege 1885b, p. 96).
Finally, there must be some means, other than examining a proof, of
determining whether the truth in question is a fact about particular
objects, a primitive general truth of some special science, or a general
logical law. For, if there is to be a definite answer to the question about
the correct classification of the truths of arithmetic, then there must be
some way to classify correctly the primitive laws on which the truths of
arithmetic depend. One strategy is simply to take each primitive truth
to constitute its own trivial one-line proof. Primitive truths about par-
ticular objects are a posteriori. Primitive general laws of special sciences
are synthetic a priori. And primitive logical laws are analytic. But, even
so, we will need a way to recognize whether the truths in question are
about particular objects, laws of special sciences or primitive logical
laws. This gives us a third eligibility condition on primitive truths: it
must be evident, without proof, which is the correct classification of
that truth. That is, it must be evident, without proof, whether the truth
in question is analytic, synthetic a priori or a posteriori.
What, then, of the simplest truths of arithmetic? Are they primitive
truths or not? One thing we need, in order to answer this question, is a
6
The issue of the role self-evidence plays in Frege s project has been much discussed in the liter-
ature. See, for instance, Burge 1998; Jeshion, 2001 and 2004; Schirn 2006; Weiner 2004. Although it
is beyond the scope of this paper to rehearse these debates here, I think that this much is evident: It
is obvious that, if the proofs based on an unproved primitive law are to establish the truth of their
conclusion, the truth of the primitive law must be evident without proof. And although Frege does
not explicitly state this, it is suggested by a number of his comments. He suggests (1884 Sect. 5) that
first principles must be immediately self-evident. In 1903a Sect. 60, he says that in arithmetic  every
assertion that is not completely self-evident should have a real proof  . He also says, at least once,
that the truth of an unproved axiom must be certain. See, 1903b, p. 319. He is more explicit when
the unproved laws are laws of logic. See, for example, 1903a, pp. xvi xvii and 1923/6, p. 50.
Moreover, I agree with Burge and Jeshion on this much: in his talk of self-evidence, Frege did not
mean to be talking about something purely psychological. Frege did, however, believe that a self-
evident truth would be recognized by anyone who understood it. He writes, for instance,
I can only say: so long as I understand the words  straight line ,  parallel , and  intersect as I do,
I cannot but accept the parallels axiom. If someone else does not accept it, I can only assume
that he understands these words differently. Their sense is indissolubly bound up with the
axioms of parallels (Frege 1914, p. 266/247).
This is not to say that the parallels axioms need be immediately obvious to anyone who understands
the words, elucidation could be required, as I argue in Weiner 2005b. And we can go wrong here.
Most notably, Frege mistakenly takes Basic Law V to be a primitive truth of logic. And Basic Law V,
far from being self-evident, is the law that introduces an inconsistency into the system.
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What s in a Numeral? Frege s Answer 683
method for recognizing gapless proofs. Otherwise, a proof that appar-
ently has only general logical laws and definitions among its premisses
might actually contain an implicit appeal to something that is neither a
logical law nor a definition. The task is  that of finding the proof of the
proposition, and of following it up right back to the primitive truths
(Frege 1884 Sect. 3, p. 4). In the process, Frege says,
[W]e very soon come to propositions which cannot be proved so long as we
do not succeed in analysing concepts which occur in them into simpler con-
cepts or in reducing them to something of greater generality. Now here it is
above all Number which has to be either defined or recognized as indefina-
ble. (Frege 1884, Sect. 4, p. 5)
Thus a central aim of Frege s project is to answer an epistemological
question. That is, among the requirements his definitions of the num-
bers must satisfy is:
The epistemological requirement: the definitions allow us to prove
truths of arithmetic from primitive truths.
But one might suppose that, in this process, the concept of number will
be recognized as indefinable. For one might suppose that the axioms
that underlie all truths of arithmetic are basic truths about the num-
bers. Moreover, a casual look at Foundations suggests what these basic
truths might be. Frege suggests that all positive integers can be defined
from one and increase by one.7 Hence, one might think, some axioms
will be about the number one. Other likely candidates are general laws
about numbers (e.g. the commutative law for addition). Why, then,
should we need definitions of the number one and the concept
number?8
In order to address this question, it may help to begin with some-
thing that clearly is eligible to be taken as a primitive truth: the claim
7
This is somewhat misleading since, when Frege actually gets to the point of defining the num-
bers, he begins with 0 rather than 1. I suspect that Frege s reason for directing our attention to the
number 1 rather than the number 0 is that he thinks that some of his audience, particularly those
with an empiricist bent, will be suspicious of the number 0. See, for instance, Frege 1885b, p. 97;
Frege 1884, section 8.
8
I mention these candidates for primitive truths because they appear in Foundations discus-
sions. But one might also think that there are more obvious candidates: the Dedekind-Peano
axioms. Using these axioms, we seem to be able to get arithmetic from two primitive undefined
terms ( 0 and a sign for the successor function). Although Frege does not actually discuss these
axioms, it is pretty clear that the considerations I raise below apply in exactly the same way. In fact,
of course, Frege begins his definitions with  0 , not  1 , and the Dedekind-Peano axioms are among
the truths he wants to prove from logical laws and definitions.
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684 Joan Weiner
that every object is identical to itself.9 Since its truth is self-evident, it
satisfies the first of the eligibility requirements for primitive laws (Frege
1893, pp. xvi xvii). Supposing this to be a primitive law, is it analytic
that is, is it a general logical law? The hallmark of a logical law is its gen-
erality. Logic is independent of the special sciences. The laws of logic,
Frege writes in Begriffsschrift, are those  upon which all knowledge rests
(Frege 1879, p. 5). Analytic truths unlike other truths cannot be
denied in conceptual thought. He suggests, in Foundations, that the
truths of arithmetic must be analytic because  Here, we have only to try
denying any one of them, and complete confusion ensues (Frege 1884,
p. 21).10
The law that every object is identical to itself exemplifies these fea-
tures. First, this law surely tells us, not just about every actual (spatio-
temporal) object or every intuitable object, but about every object. Sec-
ond, it seems that we cannot deny it in conceptual thought that is, we
cannot deny it  without involving ourselves in any contradictions when
we proceed to our deductions (Frege 1884, p. 20). Given these criteria,
the law in question is analytic; it belongs to logic. The axioms of geom-
etry, in contrast, are synthetic on Frege s view because we can assume
the contrary of an axiom of geometry without involving ourselves in
contradictions (Frege 1884, pp. 20 1).
What of the basic truths about numbers and the number one? Frege
seems to think that we can see, without proof, that the fundamental
truths of arithmetic are true. Indeed, he suggests without argument
that the fundamental propositions of the science of number have the
same status as logical laws that denying them will involve us in con-
tradictions (Frege 1884, p. 21). But do they have the requisite maximal
generality? Frege says that the truths of arithmetic govern the widest
9
Although this law is easily derivable from two of Frege s basic laws, it is not itself one of his ba-
sic laws. Since one of Frege s aims is to prove everything from the smallest possible number of
primitive laws (see, for example, Frege 1893, p. vi), many laws that are eligible to be taken as prim-
itive laws of logic are derived, rather than primitive, laws of Frege s system.
10
Although it should be noted that Frege seems to have retreated from this criterion (probably
because it sounds too psychological) in Frege 1903. He writes there that we may say we must ac-
knowledge the law of identity  unless we wish to reduce our thought to confusion and finally re-
nounce all judgement (1903, p. xvii), but then adds  I shall neither dispute nor support this view .
However, he writes, in one of his last papers,
The assertion of a thought which contradicts a logical law can indeed appear, if not nonsensical,
then at least absurd; for the truth of a logical law is immediately evident of itself, from the sense
of its expression. (Frege 1923/6, p. 50)
It is, however, also worth noting that there is a non-psychologistic interpretation of the 1884
remark that the confusion involved in denying such a truth is simply that it forces us to involve
ourselves in contradictions when we proceed to our deductions.
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What s in a Numeral? Frege s Answer 685
domain of all [das umfassendste] (Frege 1884, Sect. 14, p. 21).11 For we
can, and do, use numbers in discussions of virtually every subject mat-
ter. Everything that can be an object of thought, including mental and
physical phenomena, spatial and non-spatial phenomena, can be
counted.12 It seems to follow that these truths belong to logic. But, for
all that, Frege never makes the simple decision to add, for example, the
Dedekind-Peano axioms to his list of primitive logical laws. Why not?
Frege never explicitly tells us why truths of arithmetic cannot be
added to the list of primitive logical laws. However, there is a difference
between number statements and the logical law that tells us that every
object is identical to itself. For the latter statement is obviously general.
Were someone to ask why this statement applies to everything, it is
difficult to see how to give a substantive answer. But, for all Frege s
remarks about the universal applicability of arithmetic, there is clearly
something of substance to be said about why arithmetic is general and,
indeed, something that Frege goes on to say.
Why is arithmetic applicable everywhere? Frege s answer requires us
to see, first, that statements of number are assertions about concepts.
To say that Venus has 0 moons, for instance, is to say nothing more nor
less than what is captured by the (partially regimented) statement
Ź( x)(x is a moon of Venus). As is clear from the regimentation, this
tells us something about the concept moon of Venus (i.e. that nothing
falls under it).13 There is no mention of a particular object or of any-
thing else that does not belong to logic. Moreover, concepts will be
involved in any statement about any subject matter it is not insignifi-
cant that Frege calls his logical notation  Begriffsschrift , or  concepts-
script . Thus once we recognize that statements of number are asser-
tions about concepts, it makes sense that arithmetic should be applica-
ble in every domain. One difference, then, between primitive logical
laws and truths of arithmetic, is that the generality of primitive logical
laws but not of simple truths of arithmetic is evident.
11
This is not to contrast the variables of arithmetic with variables of other sciences. For Frege,
all variables range over an unrestricted domain. Rather, to say that the truths of arithmetic govern
the widest domain of all is to say that they do not express, for example,  the peculiarities of what is
spatial as Frege says in 1885b, pp. 94 5. In his later writings, Frege describes this maximal general-
ity somewhat differently. Laws of logic, he says,  prescribe universally the way in which one ought
to think if one is to think at all (1893, p. xv), as opposed to laws of geometry or physics, which pro-
vide a guide to thought only in restricted fields. See also, Frege 1897, pp. 145 6/157 8.
12
See, for example, Frege 1884, Sections 24, 48. 1885b, p. 94.
13
As is well known today, a similar sort of regimentation can be used to express what is stated
by assertions of other numbers. For example, to say that Venus has 1 moon is to say: ( x)(x is a
moon of Venus) & ( y)(y is a moon of Venus d y = x).
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686 Joan Weiner
Indeed, there is a sense in which the discovery that a statement of
number is an assertion about a concept introduces a new mystery. For,
when Frege redescribes the statement that Venus has 0 moons so that it
will be apparent that the statement is an assertion about a concept, the
numeral  0 disappears. This looks like a recipe for eliminating
numerals the numeral  1 would disappear from the statement that
Venus has 1 moon, etc. But the recipe works only for a certain sort of
statement in which numerals appear. There is no similar recipe for
eliminating numerals from statements of pure arithmetic (e.g.
 1 + 1 = 2 ). And this gives rise to a new question: what is the relation
between the  1 that appears in  Venus has 1 moon and the same symbol
in  1 + 1 = 2 ? Given Frege s interest in the applications of arithmetic,
this cannot be ignored. For we think we are entitled to make inferences
that involve both statements of number and statements of pure arith-
metic. Given that Venus has 0 moons, that the Earth has 1 moon, and
that 0 < 1, for example, we think that we are entitled to infer that the
Earth has more moons than Venus. But unless there is some strategy for
redescribing the content of  0 < 1 , there is no obvious explanation of
why this is a good inference. And it is far from obvious that Frege s dis-
covery about statements of number will help us out here.
Indeed, at first glance, the truths of pure arithmetic (among them the
Dedekind-Peano axioms) seem clearly not to have the requisite maxi-
mal generality. For, as Frege himself argues, numbers are objects (Frege
1884, Sects. 55 61). And if the number 0 is, as it seems to be, a particular
object, the claim that 0 is not the successor of any number, for instance,
would seem to be not a general truth, not a truth that tells us some-
thing that holds for every object but a particular truth that tells us
something about a particular object. This feature, given Frege s earlier
stated criteria, would seem to mark it as an a posteriori truth, for he
writes,
For a truth to be a posteriori, it must be impossible to construct a proof of it
without including an appeal to facts, that is, to truths which cannot be
proved and are not general, since they contain assertions about particular
objects. (Frege 1884, Sect. 3)
Nor do laws about numbers seem maximally general. These laws seem
to govern, not the widest domain of all, but the peculiar domain of
numbers. Inferences by mathematical induction, Frege says, appear to
be  peculiar to mathematics and one of his tasks is to show that such an
inference is  based on the general laws of logic (Frege 1884, p. iv).
Frege s discovery that statements of number can be understood as
statements about concepts shows us why we  have no choice but to
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What s in a Numeral? Frege s Answer 687
acknowledge the purely logical nature of arithmetical modes of infer-
ence (Frege 1885b, p. 96). But, he continues,  Together with this admis-
sion, there arises the task of bringing this nature to light wherever it
cannot be recognized immediately, which is quite frequently the case in
the writings of mathematicians . And, as we have just seen, the discov-
ery that statements of numbers are assertions about concepts does not
yet show us how it can be that the claim that 0 is not the successor of
any number is a maximally general statement. Nor does it show us that
the laws of arithmetic are maximally general they still look like state-
ments that apply to a peculiar domain, the domain constituted by the
natural numbers. In fact, Frege specifically mentions one of the Dede-
kind-Peano axioms, mathematical induction, as an example of a case of
a statement that appears to be, and is ordinarily taken to be, peculiar to
arithmetic but can be shown to be derivable from general laws of logic
(Frege 1884, p. iv, Sect. 80, 108). One of the important achievements of
Begriffsschrift was to have shown that mathematical induction can be
derived from laws that are recognizably laws of logic. The same must be
done for all statements of arithmetic that appear to be about particular
objects or to be laws governing a limited domain. In order to do this, he
needs to show how to define numerals and the concept of number from
recognizably logical notions and to prove the truths of arithmetic using
only these definitions and logical laws.
We have now seen one criterion that must be satisfied by Frege s defi-
nitions. They must enable him to provide gapless proofs of the truths of
arithmetic from primitive truths from primitive logical laws, if he is
to substantiate his conviction that they are analytic. But what would
count as gapless proofs of truths of arithmetic from primitive truths? If
the proofs are to show us the epistemological nature of the truths of our
arithmetic, then the definitions must satisfy some sort of faithfulness
requirements. Otherwise, the introduction of Frege s definitions might
simply transform arithmetic into some new and foreign science.
2. What are Frege s explicit faithfulness requirements?
Frege s new logical language, which he introduced several years earlier
in Begriffsschrift, is designed to express conceptual content [begriff-
lichen Inhalt], the content associated with an expression that has  signif-
icance for the inferential sequence (Frege 1879, p. 6). Since this lan-
guage was developed for the purpose of carrying out the proofs
required by Frege s project, one might suppose that what must be pre-
served by definitions, if we are legitimately to regard them as faithful to
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688 Joan Weiner
arithmetic is not reference, but whatever conceptual content is inherent
in our pre-systematic views about arithmetic. What evidence is there
that Frege held this view?
Although Frege does not use the expression  conceptual content in
Foundations, he does repeatedly write of the  content [Inhalt] of a
proposition and of the  content [Inhalt] of a symbol or word. 14 More-
over, the link between the content he wants to preserve and significance
for inference is evident. He wants to convince us that intellectual effort
is needed if we are to understand the content of the expression
 number and the numerals. And one sort of evidence he offers is that,
while we constantly make everyday inferences from numerical formu-
lae, these inferences do not seem immediately licensed by logical laws
alone. Nor is it evident what else is required. That is, it is not evident
how these inferences can be made gapless. The link between content
and inference is also apparent in other discussions of Foundations. For
example, in his discussion of the content of the proposition  All whales
are mammals , he argues that the proposition is not about animals
because,
We cannot infer from it that the animal before us is a mammal without the
additional premiss that it is a whale, as to which our proposition says noth-
ing. (Frege 1884, Sect. 47, p. 60)
Several sections later, he argues that the ideas we associate with an
expression cannot constitute its content, because the associated ideas
do not support our inferences. Thus when Frege begins Foundations
with the question  What is the number one? he seems to be asking
what conceptual content is associated with the word  one . Moreover,
the idea that Frege wants definitions that preserve conceptual content
fits with some explicit faithfulness requirements that figure in the Foun-
dations discussions.
Frege defends his definitions, after sketching definitions of the
numerals and the concept number, in a group of sections in Founda-
tions labelled  the completion and testing [Ergänzung und Bewährung]
of our definition . One of the things Frege tests is whether his defini-
tions meet what we can call
Faithfulness requirement 1: the definitions must allow us to derive
 the well known properties of numbers (Frege 1884, p. 81).
14
See, for example, Frege 1884, sections 3, 8, 47, 62, 64. In later sections (see, for example,
Sects. 70, 74, 104), he also uses the locution  judgeable content [beurtheilbarer Inhalt]. I owe
thanks to the reviewer for suggesting I talk about the use of  Inhalt in Foundations.
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What s in a Numeral? Frege s Answer 689
Frege wants to prove that the numbers, as defined, have the properties
that seem to underlie the uses we make of arithmetic, both in science
and in everyday life. For example, it must be possible to prove, from the
definition of 0, that 0 is the number that belongs to a concept if and
only if no object falls under it (the number that belongs to a particular
concept is the number of objects that fall under the concept) (Frege
1884, Sect. 75, p. 88). We must be able to prove such claims as: if 1 is the
number which belongs to a concept, then there exists an object which
falls under that concept (Frege 1884, Sect. 78, p. 91). Or, as he also says,
which I will call
Faithfulness requirement 2: the definitions must provide a basis for
an arithmetic that meets the demand  that its numbers should be
adapted for use in every application made of number (Frege 1884,
Sect. 19, p. 26).
Another way of describing these requirements is to say that what we
take to be simple truths and applications of our arithmetic must be
reproducible in an arithmetic based on Frege s definitions. No accepta-
ble definitions of  0 and  1 will make it true that 0 = 1, or false (failing
new astronomical events) that the Earth has 1 moon. It may seem that
what Frege s actual defence shows is that his aim is to formulate defini-
tions that preserve what we antecedently think of as the simple truths of
arithmetic. But it would be misleading to characterize his faithfulness
requirements as requirements that his definitions preserve truth. For
Frege does not simply try to convince us that it is true, given his defini-
tion of  1 , that if 1 is the number which belongs to a concept, then some
object falls under the concept. He tries to show us that it is provable.15
The definitions must not only preserve what we regard as the truths of
pre-systematic arithmetic, they must also provide support for its infer-
ences. If, before Frege s definitions are offered, we regard ourselves as
entitled to make inferences about the numbers of moons of Venus and
the Earth, the introduction of these definitions should enable us to
replace our original, enthymematic arguments with gapless argu-
ments.16
15
In Sect. 78 1884, Frege lists a number of simple facts about numbers that  are to be proved by
means of his definitions. He writes,  Ich lasse hier einige Sätze folgen, die mittels unserer Definitio-
nen zu beweisen sind .
16
For these reasons, one might be inclined to think that the criterion of correctness for Frege s
definitions is that the claims of ordinary pre-systematic arithmetic be logically equivalent to state-
ments in Frege s systematic science of arithmetic. This is the position that Patricia Blanchette
(1994) defends. The problem with this sort of interpretation, as I will argue shortly, is that it does
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690 Joan Weiner
What all this suggests is that we have the answer to our question
about Frege s faithfulness requirements. Faithful definitions must be
definitions on which those sentences that we take to express truths of
arithmetic come out true and on which those series of sentences that
we take to express correct inferences turn out to be enthymematic ver-
sions of gapless proofs in the logical system. It does not seem unreason-
able to describe the content this captures I will call it  Foundations-
content  as (at least a part of) conceptual content (or, content that
has significance for the inferential sequence). Foundations-content is
not, of course, exactly the notion of conceptual content introduced in
Begriffsschrift. This is not a bad thing. For the notion of conceptual
content will not do the work that it is introduced to do in Begriffsschrift.
One problem, which seems evident from the above discussion, is that
Frege really needs distinct terms for the object that we use a name to
talk about and the content that affects the use of the name in gapless
inference.17
The sort of content that Frege seems to be using in Foundations
appears to be one step towards the later separation of Sinn and Bedeu-
tung. For merely having Foundations-content will not suffice to guaran-
tee that an object name has a referent. After all, it is not impossible to
go wrong about whether an object expression has a referent. Should we
find out, for instance, that Waverley was a collaborative work, we would
surely want to re-evaluate our beliefs and inferences, but this would not
deprive the expression  the author of Waverley of Foundations-content.
not fit well with Frege s actual remarks about the different roles played by Begriffsschrift and nat-
ural language. For neither the conceptual content nor the sense originally carried by the expression
need be preserved by the definition.
17
This need comes out even in the discussions of Begriffsschrift. For, when he introduces the
identity sign in Begriffsschrift, which he calls the  identity of content [Inhaltsgleichheit], he gives an
example of two distinct expressions that both name the same point. He then explains that the fact
that the two expressions pick out the same point (which he also calls the content [Inhalt] of the
two expressions) is to be expressed by putting the identity sign between them. The section ends
with the statement that what  AhB means is that  the sign A and the sign B have the same con-
ceptual content [begrifflichen Inhalt], so that we can everywhere put B for A and conversely (1879 sec-
tion 8). However, the object named by a term cannot be the conceptual content of that term;
sameness of conceptual content is much too strong to be a requirement for the truth of identity
statements. To see why, consider the statement that the morning star is identical to the evening
star. Since  the morning star = the morning star is an instance of the logical law a=a, while  the
evening star = the morning star is not, the two expressions do not have the same significance for
inference (or conceptual content). Hence, using Frege s  official definition of identity statements,
the identity statement should be false. But the discussion that precedes this official statement
makes it clear that, since  the morning star and  the evening star are different names of the same
object, the identity statement should be true. For an explanation of the problems with the Begriff-
sschrift account that, I argue, led Frege to adopt his Sinn/Bedeutung distinction, see Weiner 1997b.
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What s in a Numeral? Frege s Answer 691
Moreover, it is not difficult to find people who think that certain state-
ments about Hamlet express truths and who make inferences about
Hamlet. There is, then, some content associated with the name  Ham-
let that fits the description of Foundations-content.
This is not to say that Frege would have agreed that there are truths
or correct inferences about Hamlet he would not have done but
only that the notion of content that Frege exploits in Foundations is
something that fictional names can have. Indeed, this Foundations-con-
tent is very close to Frege s later notion of Sinn (or sense) and Frege
claims explicitly that fictional proper names have sense.18 Frege s argu-
ment that his definitions must preserve Foundations-content that is,
that what we take to be the truths of presystematic arithmetic must
turn out to be true given the definitions and that what we take to be
correct inferences of presystematic arithmetic must be replaceable by
gapless proofs of systematic arithmetic does not require us to assume
that the numerals of presystematic arithmetic referred to particular
objects, antecedent to Frege s work. Nor is there any argument in
Frege s explicit defence of his definitions that, antecedent to his defini-
tions, each numeral already referred to a unique object and that his def-
inition of each numeral, in fact, picks out the object to which the
numeral already referred.
3. Is sense-preservation the real faithfulness requirement?
There is another interpretation that appears to fit all these features of
Foundations and also offers an explanation of why Frege never articu-
lates the apparently obvious faithfulness requirement. The interpreta-
tion is that, when he wrote Foundations, Frege was groping for a
constraint on definitions that he could not yet express: that the defini-
tions must express the sense these expressions already have. The exist-
ence of the notion of Foundations-content suggests that Frege was
already trying to formulate the notion of sense in Foundations. For
Foundations-content seems to be a transitional notion  one that
unofficially supplants the notion of conceptual content only to be sup-
planted itself when Frege introduces the notion of sense in Function
and Concept. Moreover, in his later writings, Frege makes it clear that
18
See, for example, Frege 1892a, pp. 32 3. It may well be that Frege meant the notion of Sinn to
capture this aspect of conceptual content. Frege writes, both in 1893, p. x and in a letter to Husserl
(Frege 1891b, p. 63/96), that what he used to call judgeable content has been now been divided into
thought and truth-value. It is plausible to suppose that he regards conceptual content of all object
expressions as having been divided into Sinn and Bedeutung. However, the evidence is
inconclusive.
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692 Joan Weiner
the definition of an object-expression determines, not only its referent,
but also its sense. He writes, for instance, that the sense of the definien-
dum  is built up out of the senses of the parts of the definiens (Frege
1914, p. 208/224)19. On this interpretation, Frege s later introduction of
the notion of sense is needed in order to express his intended faithful-
ness requirement. Since the sense of an expression picks out the object
(if any) to which the expression refers, definitions of the numerals that
preserve sense and pick out objects will thereby pick out objects to
which we have been referring all along. That is, on this interpretation,
the apparently obvious faithfulness requirement must be satisfied by
any acceptable definition. And we can explain why Frege does not
bother to articulate the apparently obvious faithfulness requirement:
the satisfaction of this requirement will be a consequence of the preser-
vation of sense.
Of course, in order to make this interpretation out, some work is
required. Thus far I have made only two claims about sense: that the
sense of an object expression picks out the referent (if any) and that
sense is some sort of content connected with inferences. Although this
is not much of an account of a technical term, it is also not much less
than Frege actually says about sense. Thus it is difficult to be clear on
exactly how he understands this notion.20 However, there is one variety
of remark that I have not yet mentioned: the remarks that suggest that
the sense of an expression is what a speaker who understands the
expression grasps.21 These remarks make the notion of sense more
19
See also, Frege 1903b, p. 323, 1895, p. 75, 1918/19b, p. 150.
20
Perhaps the most common contemporary characterization is that sense is  mode of presenta-
tion (Art des Gegebenseins). For example, Tyler Burge writes ,  A sense is a mode of presentation
that is  grasped by those  sufficiently familiar with the language to which an expression belongs
(Burge 1990, p. 243). This is not entirely correct: Frege says not that sense is mode of presentation
but that sense contains mode of presentation. It is also worth noting that in Frege s first introduc-
tion of the notion of sense (1891a), there is no talk of mode of presentation. Nor in fact, is this a lo-
cution that appears in many of Frege s other writings. Frege mentions mode of presentation only
four times in 1892a, after which he seems to have dropped the locution although there is men-
tion, in  Thoughts , of how an object gegeben ist (1918/19a, pp. 65 6) and, Frege continues, such a
Weise corresponds to a particular Sinn.
21
Perhaps the most famous remarks connecting sense with what we grasp when we understand
an expression are those in which Frege suggests that, because  the evening star and  the morning
star have different senses, someone might hold one of  the morning star is a body illuminated by
the Sun and  the evening star is a body illuminated by the Sun true and the other false. (1892a,
p. 32) He also writes, in the same paper,  The sense of a proper name is grasped by everybody who
is sufficiently familiar with the language or totality of designations (1892a, p. 27). See also, Frege
1892a, pp. 28 9. Talk of grasping thoughts also appears throughout Frege 1918/19; see, for example,
pp. 64, 66, 70, 74, and 76. Other remarks include the remarks about mode of presentation in the
beginning of Frege 1892a, pp. 26 7 and, Frege s remark that propositions expressing different
thoughts tell us different things [Verschiedenes besagen] in Frege 1891a pp. 13.
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What s in a Numeral? Frege s Answer 693
understandable and familiar. But how are we to square them with the
fact that (as Frege realizes) his definitions will not immediately be rec-
ognized as correct by his readers?
One possibility is that we have an imperfect grasp of the sense of our
statements.22 That is, it may well be that if we substitute Frege s defini-
tions for the defined terms in our statements, the thoughts expressed
are exactly the same as the thoughts expressed in our original state-
ments. Our not recognizing this could simply be an artefact of our lack
of understanding; of our imperfect grasp of the sense of our statements.
It could be Frege s view that, once we grasp the thoughts, we will see
that numbers really are extensions. In this way, Frege might view his
definitions as giving an answer to the question  what is a number?
which is both uniquely correct and also something that most of his
audience will not immediately recognize as correct.
Frege never explicitly says that his definitions of the numerals are
meant to express a sense these terms already have. However, in his 1914
unpublished notes for a course on the foundations of arithmetic there
are passages that provide some support for this interpretation. Frege
characterizes mathematicians (including Weierstrass) as having only an
imperfect understanding of the basic building blocks of mathematics.23
And in his criticism of Weierstrass s account of the concept of number,
Frege tries to explain how various mathematicians all of whom are,
presumably, engaged in the same science can have offered different,
and inconsistent, accounts of the concept of number (Frege 1914,
pp. 215 22/232 40). Frege writes,
Perhaps the sense appears to both through such a haze that when they make to
get hold of it, they miss it. One of them makes a grasp to the right perhaps and
the other to the left, and so although they mean to get hold of the same thing,
they fail to do so. (Frege 1914, p. 217/234)
22
The view I discuss here is very close to Tyler Burge s view of Frege s understanding of sense.
See, in particular, Burge 1984, 1990 as well as pp. 54 8 of the introduction of Burge 2005. Burge at-
tributes to Frege a kind of anti-individualism on which we can be expressing senses that are not
grasped by any members of our linguistic community. But Burge never addresses the obvious
question. Supposing that there are two distinct senses (say, of  number ), each of which is entirely
consistent with accepted mathematical thought. What is it that determines which of these two
senses is the one we are currently expressing? It is explicable that Burge never addresses this ques-
tion, given that he thinks that there is overwhelming evidence that Frege thought that most every-
day expressions particularly the expressions of the sentences of arithmetic have a definite
sense. But Burge does not argue for this and never acknowledges the existence of the extended
passages including Frege 1896 and Frege 1903a, Sects 56 67 in which Frege appears to contra-
dict it. I address this issue below.
23
Although the most notable example is the criticism of Weierstrass in Frege s 1914 course
notes, the ignorance of mathematicians about arithmetic is actually a recurring theme in his writ-
ings. See, for example, the introduction of Frege 1884 and Frege 1899, pp. 24 5.
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694 Joan Weiner
This suggests that the term  number has a unique correct sense at
which the mathematicians are aiming: that, when we use sentences of
arithmetic, we are expressing thoughts, but not fully grasping the
thoughts we express. If this is so, we can think of Frege s task as that of
enabling his readers to grasp these thoughts.
Moreover, if Frege does think that each numeral had a unique, deter-
minate sense antecedent to his definitions, he seems to be committed to
the view that each numeral also had a unique, determinate referent. To
see this, suppose each numeral had, antecedent to Frege s definitions, a
unique determinate sense that was both imperfectly understood by
mathematicians and did not pick out a unique, determinate referent.
On Frege s view, a definition is required to pick out a referent. Given
that the original sense does not pick out a referent, no acceptable defi-
nition can preserve the original sense. That is, Frege s definition will
express a new sense. But, in this case, what possible reason can there be
to say that there was a unique (different) sense originally expressed by
the numeral? It cannot be that the original sense is determined by what
mathematicians antecedently understood since, by hypothesis, the
original sense was imperfectly understood. It cannot be that the origi-
nal sense picked out the right referent because, by hypothesis, the origi-
nal sense did not pick out a referent. Thus, if Frege is committed to the
view that is suggested in the passage quoted above, then he seems to be
committed to the view that each numeral already has, not just a sense,
but a referent-fixing sense. I call this a  suggestion , however, not just
because he does not explicitly state it, but also because it is undermined
by many of his statements throughout his published and unpublished
work, including the 1914 course notes in which the passage quoted
above appears.
4. Why Frege would reject the apparently obvious faithfulness
requirement
One such remark appears in Frege s critique, in Foundations, of a pro-
posed definition of the concept of number. He writes  it must be noted
that for us the concept of number has not yet been fixed, but is only due
to be determined in the light of our definition of numerical identity
(Frege 1884, Sect. 63, p. 74). This is an odd choice of words if we suppose
that each numeral already has a determinate sense hence already
refers to a particular object and that to be a natural number is simply
to be one of these objects. For Frege writes that all that can be
demanded of a concept is that it should be determined, for each object,
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What s in a Numeral? Frege s Answer 695
whether or not it falls under the concept (Frege 1884, Sect. 74, p. 87). If
each numeral already refers to a particular object, and if the numbers
are simply the objects to which the numerals refer, then the concept of
number is already fixed. We may be lacking a definition that identifies
this fixed concept of number.24 We may imperfectly grasp the concep-
tual content (or, the sense) that is already associated with the term
 number (and already determines the objects to which it applies). But
it certainly does not follow that the concept of number is due to be
determined in the light of our definition. Indeed, on the interpretation
under discussion, it is not the concept of number that is to be deter-
mined by the definition; it is our understanding of the senses that our
sentences already express. The role of the definition is to get us to
understand a sense that is already associated with the word  number .
Were this Frege s only remark of the sort, one might dismiss it as
merely an odd choice of words. But it is not.
The numbers on which Frege concentrates in most of the discussions
of Foundations are the natural numbers.25 However, definitions of the
natural numbers will not suffice to accomplish Frege s ultimate goal
to show that analysis is analytic. In the last part of Foundations, Frege
turns to the issue of defining the complex numbers. He considers the
possibility of stipulating that the time-interval of one second is the
square root of  1, and he adds, in a footnote, that we are entitled to
choose any one of a number of objects to be the square root of  1. The
reason is that
the meaning [Bedeutung] of the square root of  1 is not something which was
already unalterably fixed before we made these choices, but is decided for the
first time by and along with them. (Frege 1884, Sect. 100, p. 110)26
Here there is no ambiguity. If the choice decides for the first time what
the square root of  1 is, then it cannot simply be that we have imper-
fectly grasped the (determinate) sense of our statements about the
complex numbers. It is not consistent with the above passage to take
 the square root of  1 as already having a determinate (referent-fixing)
24
It is also odd that Frege says that  for us the concept is not fixed rather than, for example,  by
this procedure .
25
He actually says that he is writing about  positive whole numbers (see, for example, Frege
1884 p. 119) but 0 is among the numbers defined.
26
It is not entirely clear how  Bedeutung should be translated in this passage. However, it is
clear that one of the things that, Frege is saying, was not fixed before the choices in question, is
what it is to which  the square root of  1 refers. It is also clear that, if the expression in question
does have a sense or conceptual content antecedent to the choice being made, once the choice has
been made, the sense or conceptual content that is chosen will not be the same as whatever sense
or conceptual content was originally associated with the expression.
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696 Joan Weiner
sense. Rather, it has to be that our symbols for complex numbers
antecedent to Fregean definitions do not have determinate referents.
Indeed, he goes on to suggest as much in the next section. Moreover,
since the choice in question is of a referent-fixing sense, in so far as the
symbols for complex numbers have fixed and determinate senses or
conceptual contents, these senses or conceptual contents will not be
entirely preserved by the definitions. One might suspect that this marks
a difference between the complex and natural numbers. But Frege gives
us no indication that there is such a difference. The language he uses is
virtually the same in both discussions.
This is not to say that anything will do as a definition of the square
root of  1. For he goes on to suggest that there is a problem with defin-
ing the square root of  1 as a time-interval. To do so is to import into
arithmetic  something quite foreign to it, namely time and to make
arithmetic synthetic (Frege 1884, Sect. 103, p. 112). In order to show that
arithmetic is analytic, Frege suggests using the same solution for com-
plex numbers that he used for natural numbers: to define them as
extensions of concepts. The notion of extension of concept is a logical
notion, on Frege s view, and definitions of the numbers as extensions of
concepts should make it possible to prove truths about the numbers
from logical laws. He ends Foundations with the following remark
about offering such definitions:
Once suppose this everywhere accomplished, and numbers of every kind,
whether negative, fractional, irrational or complex, are revealed as no more
mysterious than the positive whole numbers, which in turn are no more real
or more actual or more palpable than they. (Frege 1884, Sect. 109, p. 119)
This would be an odd remark if, for example,  1 had all along referred
to a particular extension of a concept while the symbol  i refers to an
extension of a concept only because of an arbitrary stipulation. But,
again, this may be simply an odd choice of words. What other evidence
is there?
As we have seen, Frege views the definiens of a definition as express-
ing a sense. When Frege defines the numbers as extensions of concepts,
it becomes (if it is not already) a part of the sense of each numeral that
the number in question is an extension of a concept. If we assume, as
we must on the interpretation under discussion, that Frege s definitions
are meant to capture a determinate sense that our numerals already
express (whether or not we grasp this sense), then it is already part of
the sense of our numerals and of the term  number that numbers are
extensions. Frege s definition is, in part, a discovery about the nature of
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What s in a Numeral? Frege s Answer 697
numbers. But a closer look at Frege s discussions of definitions in later
writings reveals a number of conflicts with this interpretation.
First, Frege does not think that, in general, the fact that a term is used
in everyday science (science that precedes the sort of systematic treat-
ment that he advocates) is any indication that the term has determinate
sense. For in a discussion of the enterprise of defining a scientific terms
in  On the Foundations of Geometry II , he writes,
Now it may happen that this sign (word) is not altogether new, but has al-
ready been used in ordinary discourse or in a scientific treatment that pre-
cedes the truly systematic one. As a rule, this usage is too vacillating for pure
science. (1906a, pp. 302 3)
Thus it is certainly possible, on his view, that the numerals have a usage
that is too vacillating for pure science. If this is so, then not only does
that usage fail to express a determinate referent-fixing sense, a proper
definition will introduce a sense that requires a reform of our use of it
in a science. That is, our pre-systematic usage will not be consistent
with a proper definition. It may be that, none the less, there is a unique
correct sense for each of the symbols of arithmetic a unique sense
that is appropriate for pure science and that fits with what we are doing
in our pre-systematic science. But Frege s comments here make it clear
that he does not regard this as in any way an automatic consequence of
the fact that our pre-systematic science works. If he thinks that our use
of the symbols of arithmetic does have this character, it would not be
something that goes without saying.
And supposing the symbols of arithmetic do have this character
(whether or not it goes without saying), then there is another problem
with attributing to Frege the view that his definitions are meant to
express the unique correct sense of each symbol of arithmetic that is
consistent with our use of it. Frege is adamant that definitions (as
opposed to axioms) are stipulative.27 But he does acknowledge that
sometimes we may want to define a simple term that is already in use,
in which case we cannot give an arbitrary stipulation. In the 1914 course
notes, he identifies two different situations in which we might be said to
be giving a definition. In the first, the definition is stipulative. We con-
struct a complex sense and use a new sign to express the sense. Such a
definition, he says, may be called constructive [aufbauende] but, he
adds,  we prefer to call it a  definition tout court  (Frege 1914, p. 210/
227). He also describes a second sort of situation one in which we
27
Frege is most explicit and adamant in his series of papers on the foundations of geometry.
See, 1903b, pp. 320 1; 1906a pp. 294 9, 383. But he also makes such comments elsewhere. See, for
example, 1885c, p. 52.
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698 Joan Weiner
want to give a logical analysis of the sense of a simple sign with a long
established use. In such a situation, he says, we might speak of an  ana-
lytic [zerlegende] definition (1914, p. 210/227)28 But, he continues,
[I]t is better to eschew the word  definition altogether in this case, because
what we should here like to call a definition is really to be regarded as an ax-
iom & So we shall stick to our original way of speaking and call only a con-
structive definition a definition.29
Thus if our pre-systematic science does determine a unique correct
sense for our numerals, we should have a systematic science that begins,
not with definitions, but with axioms expressing the senses of these
terms. Moreover, the truth of such axioms, Frege claims, is recognizable
only by an  immediate insight [unmittelbares Einleuchten] (1914, p. 210/
227). These remarks are difficult to square with the idea that it might be
a substantive discovery that numbers actually are extensions of
concepts a discovery that we can be made to understand but will not
see immediately.
Moreover, these remarks are completely in accord with Frege s treat-
ment of his definitions in Foundations. In the discussions that follow
his introduction of definitions of the natural numbers, Frege acknowl-
edges that the correctness of his definitions is not evident for, he says,
we  think of the extensions of concepts as something quite different
from numbers (Frege 1884, Sect. 69, p. 80). Were it a substantive dis-
covery that numbers are extensions of concepts, we would expect his
defence of his definitions to include an argument that shows that the
statements we make about the numbers commit us to the view that
numbers really are extensions of concepts. But no such argument
appears among Frege s defences of his definitions.
Frege s observation that we think of extensions of concepts as differ-
ent from numbers appears in Sect. 69 of Foundations and the rest of the
section contains a brief response a response that seems to amount to
the employment of a version of the context principle. Instead of
28
It is important to note that the word  analytic here is a translation of  zerlegende and not
 analytische . The issue here is not about the epistemological status of definitions, since there could
be no such thing, on Frege s view, as a synthetic or a posteriori definition.
29
It is interesting to note that Frege says something very similar in his review of Husserl s Phi-
losophy of Arithmetic. He writes,
A definition is also incapable of analysing the sense, for the analysed sense just is not the original
one. In using the word, to be explained, I either think clearly, everything I think when I use the
.
defining expression: we then have the  obvious circle ; or the defining expression has a more
richly articulated sense, in which case I do not think the same thing in using it as I do in using
the word to be explained: the definition is then wrong. (1894, p. 319)
The words used here are almost exactly the same as the words used in his later course notes. In this
case, however, it is not entirely clear that Frege is speaking in his own voice, rather than Husserl s.
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What s in a Numeral? Frege s Answer 699
addressing the question  Are numbers extensions? he considers the
question,  Are the assertions we make about extensions assertions we
can make about numbers? There are, he says, two sorts of assertions we
can make about extensions. The first sort is an identity statement. And
just as we can assert the identity of extensions, he says, we can assert the
identity of numbers. The second sort of assertion is that one extension
is wider than another. This, he admits, is not something we say of num-
bers. But it is also, he notes, not a relation that can hold between the
extensions that, according to his definitions, are numbers. It follows
that the assertions we make about the extensions that (given Frege s
definitions) are numbers are also assertions that we can make about
numbers. Frege turns next, in sections 70 83 to the  testing and comple-
tion of his definitions, where he tries to show that the definitions of
numbers as extensions allows us to derive the appropriate number
statements.
There is, then, no argument that it is part of the nature of numbers to
be extensions. And there is no indication, here or elsewhere, that the
definition of each numeral picks out the unique object to which the
numeral already refers. Indeed, there is a remark that looks to be a
denial of this. Frege claims later in Foundations that he attaches no deci-
sive importance to bringing in the extensions of concepts.30 This claim
is completely mysterious if we assume that, when we use the numerals
in our current pre-systematic language, we are talking about particular
objects, and if we assume that Frege s task is to provide definitions that
pick these objects out. For if so, either we are already talking about (our
numerals already refer to) extensions of concepts (in which case it
would be essential to bring in extensions) or we are already talking
about (our numerals already refer to) objects other than extensions of
concepts (in which case it would be wrong to bring in extensions).
Frege s comments are simply not consistent with the assumption that
his definitions are meant to pick out objects that we have been talking
about all along.
Unless we are prepared to engage in interpretive contortions, the
appropriate conclusion is that, when Frege asks for a definition of the
concept number, he is not asking for an explicit description of the
objects to which our numerals already refer. And, given this, it is
implausible to attribute to Frege the view that there is a unique concept
to which  number refers, and unique objects to which the numerals
30
Frege 1884, Sect. 107, p. 117. Frege does come, later in his career, to attach more importance to
bringing in extensions of concepts. But his reason is that  we just cannot get on without them
(1893, p. x), not that numbers really are extensions.
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700 Joan Weiner
refer, antecedent to his introduction of his definitions. That is, anteced-
ent to Frege s introduction of his definitions, the concept number is not
fixed.
Of course, if the view we get from Frege s explicit remarks is absurd,
there may be a compelling reason to engage in interpretive contortions.
But is this view absurd? A number of contemporary philosophers sub-
scribe to least one part of this view that our numerals do not refer to
particular objects.31 One reason is that there are many distinct set theo-
retic definitions of the numbers that fit our understanding both eve-
ryday and scientific of the numbers. Nothing in our understanding
of the truths of arithmetic seems to offer grounds for deciding between
alternative systems of set theoretic definitions or, for that matter,
grounds for saying that numbers are (or are not) sets. Given this,
Frege s explicit remarks do not seem absurd at all. There is every reason
to believe that the numerals do not refer to particular objects and, con-
sequently, that the content associated with the numerals can be cap-
tured by offering definitions that are at least partly stipulative.
5. But what about the relation between truth and reference?
There is another problem, however. Frege writes, about the sentence
 Odysseus was set ashore at Ithaca while sound asleep ,
[A]nyone who seriously took the sentence to be true or false would ascribe to
the name  Odysseus a meaning [eine Bedeutung zuerkennt], not merely a
sense; for it is of what the name means [der Bedeutung dieses Namens] that the
predicate is affirmed or denied. (Frege 1892a, pp. 32 3)
To doubt that  Odysseus has a referent, he suggests, is to doubt that the
sentence has a truth value. For if  Odysseus does not have a referent,
there is nothing of which having been set ashore at Ithaca is being pred-
icated. And, if this is Frege s view about truth and reference, there is a
problem for his writings about arithmetic. As we have just seen, there is
reason to think that Frege does doubt that  0 and  1 already have deter-
minate referents. But, by the reasoning in the passage from  Über Sinn
und Bedeutung , this is also a reason to doubt that  0 is not equal to 1
has a truth value. For, if there are no objects to which  0 and  1 refer,
there are no objects which are said, in the sentence  0 is not equal to 1 ,
31
Most such views have their roots in Paul Benacerraf  s 1965, where Benacerraf concludes, from
the fact that there are distinct set theoretic definitions of the natural numbers, that numbers are
not objects. For some other examples of philosophical view on which arithmetic is not about ob-
jects to which numerals refer, see Field 1980; Kitcher 1983, chapter 6; Resnik 1997; Shapiro 1997.
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What s in a Numeral? Frege s Answer 701
to be distinct. Yet Frege seems to assume that our everyday sentences of
arithmetic express truths. And he thinks that numbers are objects.
Similar reasoning has bothered contemporary philosophers who
worry about arithmetic, many of whom have inferred that the mistake
is to take the truths of arithmetic to be about numbers. On this view,
the grammar of sentences of arithmetic misleads us. If our aim is to
understand arithmetic, defining the numerals will not help. The task,
instead, is to restate the content of statements of arithmetic without
using numerals. The upshot should allow us to find an account of the
contribution made by numerals to the truth-value of sentences in which
they appear an account that does not require them to refer to objects.
Frege, in contrast, never addresses this problem. The explanation,
one might suspect, is that he simply did not notice that one conse-
quence of his views is that our everyday sentences (including simple
truths of arithmetic) do not have truth values.32 But this explanation is
not entirely convincing. For in his later work, he comes very close to
explicitly acknowledging this consequence. In the second volume of
Grundegesetze, he writes  would the sentence  any square root of 9 is
odd have a comprehensible sense at all if square root of 9 were not a
concept with a sharp boundary (Frege 1903a, Sect. 56)? A sentence that
does not have comprehensible sense cannot have a truth-value. One
might suspect that Frege takes it to be obvious that  any square root of 9
is odd does have a comprehensible sense and, hence, that the concept
square root of 9 does have a sharp boundary. However, a look at the con-
text in which the question appears shows that this interpretation is
incorrect. A concept, on Frege s view, has a sharp boundary just in case
it determinately holds or not of each object. For example, in order for
greater than zero (or positive) to be a proper concept, Frege says,  it
would have to be determinate whether, for example, the Moon is
greater than zero (Frege 1903a, Sect. 62). He continues,
We may indeed specify that only numbers can stand in our relation, and
infer from this that the Moon, not being a number, is also not greater than
zero. But with that there would have to go a complete definition of the word
 number , and that is just what is most lacking.
32
Gary Kemp (1996) makes this argument. Many of the following arguments are responses to his
objections. In particular, I think that Kemp s argument that Frege just did not notice this conse-
quence is plausible only if we ignore the passages from volume ii of Grundgesetze I discuss below.
Kemp also argues that the sort of interpretation I am advancing here does not fit Frege s view that to
judge is to accept as true. One consequence, Kemp claims, is that whenever a Begriffsschrift is una-
vailable,  we should, preposterously, have to refrain from judging and asserting altogether (Kemp
1996, p. 178). I address this issue in the comparison of arithmetic with research on obesity below.
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702 Joan Weiner
Frege argues, not just that such concepts must be defined, but that the
sort of definitions offered by mathematicians are unacceptable. In the dis-
cussions of Grundgesetze that immediately follow, he suggests that such
expressions as  greater than and  + are used by mathematicians in such a
way that they have neither determinate sense nor determinate reference.33
It is difficult to imagine that Frege said all this without noticing the con-
sequence that sentences in which  greater than 0 ,  greater than and  +
appear have no truth-value.34 But, whether he noticed this or not, there is a
problem here that must be addressed. For given his requirement that each
predicate pick out a concept with a sharp boundary, few, if any, of our eve-
ryday sentences have comprehensible sense or truth-values. And even if
Frege did not mind making such apparently implausible claims about nat-
ural language, there is a puzzle about Frege s conception of his project.
Frege s avowed project, which he describes in detail in Foundations, is
to show that the truths of arithmetic are analytic. And while, in his later
writings, he abandons the expression  analytic , both the details of the
project showing that the truths of arithmetic can be derived from
definitions and logical laws  and its epistemological pretensions
remain. For example, he begins Grundgesetze with the claim that he will
show that arithmetic is a branch of logic  and need not borrow any
ground of proof whatever from either experience or intuition (Frege
1893, p. 1)35 That is, he wants to exhibit the grounds on which the proofs
of the truths of arithmetic depend. Yet if the sentences that we take as
33
Most of the discussions in these sections are about Bedeutung. For example, Frege claims that
the expression  something the half of which is less than one has no Bedeutung because the func-
tion 1/2x is not defined for the Moon as argument. But there are some mentions of sense. In Sect.
56, he says that the sentence  all square roots of 9 are odd does not have sense if  square root of 9
does determinately hold or not of each object. The implication of later remarks is that it does
not if we do not know, for example, whether the moon is less than ½ of 1, then we presumably
also do not know whether it is a square root of 9. In Sect. 65 he also mentions sense, claiming that,
because the plus sign is not defined for all objects, the antecedent of  if a + b is not equal to b + a
and a is a number, then b is not a number is senseless.
34
See also the discussion of the universal generalization of  (x > 2) d (x2 > 2) in Frege 1896.
There Frege argues that, if this is to have a truth-value, then whenever a name that designates an
object is substituted for  x in  x2  , the resulting expression must designate an object; that is, the
function in question must be defined for all objects.
35
It is worth noting that, in Foundations, he characterizes analytic truths as those whose proof
requires only general logical laws and definitions (1884, Sect. 3, p. 4). That is, to show that truths of
arithmetic are reducible to logical truths is to show that they are analytic, in the sense described in
Foundations. Thus it does not seem to be a misrepresentation to continue to characterize the
project of the Grundgesetze as an attempt to show that the truths of arithmetic are analytic. One
might suspect that there is a hidden change: that Frege abandoned the word  analytic because his
later work had no epistemological motivations. But, as the remark quoted above indicates, this is
not so. Moreover, this is not the only such remark that appears in Frege s later writings. He also
says that the proofs must be gapless because this will allow him to insure that there are no unno-
ticed presuppositions and it is precisely these, he says,  that obstruct our insight into the epistemo-
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What s in a Numeral? Frege s Answer 703
expressing truths of arithmetic do not actually have truth-values, it is
difficult to see what the point of this project is. Indeed, it is not even
obvious that we know any truths of arithmetic. Although Frege does
not address this problem explicitly, there are solutions to it be found in
his writings. The key to understanding these solutions can be found in
his discussions of natural language, Begriffsschrift, and science.
6. How does natural language differ from Begriffsschrift?
Frege characterizes his logical language, Begriffsschrift, as a tool that
will enable us to avoid some of the difficulties inherent in natural lan-
guage. When we use natural language, he says, even careful use of logi-
cal laws will not prevent us from making errors. Mistakes, he writes,
 easily escape the eye of the examiner, especially those which arise from
subtle differences in the meanings of a word (Frege 1882, p. 51). He
continues,
That we nevertheless find our way about reasonably well in life as well as in
science we owe to the manifold ways of checking that we have at our
disposal. Experience and space perception protect us from many errors.
Frege does not suggest that there is anything wrong with relying on the
manifold ways of checking. Nor does he suggest that the existence of
subtle differences in the meanings of a word is a defect that ought to be
eliminated from natural language. Instead, he suggests that, for some
purposes, these features of natural language are virtues. He says they
are rooted  in a certain softness and instability of language which never-
theless is necessary for its versatility and potential for development . He
continues,
In this respect, language can be compared to the hand, which despite its
adaptability to the most diverse tasks is still inadequate. We build for our-
selves artificial hands, tools for particular purposes, which work with more
accuracy than the hand can provide. And how is this accuracy possible?
logical nature of a law (Frege 1893, p. 1). And he characterizes the gaplessness of the inferences as
what enables us  to gain a basis upon which to judge the epistemological nature of the law that is
proved [der erkenntnistheoretischen Natur des bewiesenen Gesetzes] (1893, p. vii). Indeed, in one of
the last written works that survive, Frege 1924/5b, he discusses the same issues that begin Founda-
tions. Do proofs of the truths of arithmetic require appeals to sense perception or to intuition, or
to logical laws alone? Although, in the other work of this last year of his life, Frege 1924/5a, he does
not use the term  intuition (Anschauung), he does refer to the  geometrical source of knowledge
(die geometrische Erkenntnisquelle). In virtually all Frege s other writings, he identifies intuition as
the geometrical source of knowledge. Thus, in spite of the absence of the term  analytic in his later
work, it is difficult to see that there is any change in his view or his understanding of the epistemo-
logical questions he wanted to answer.
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704 Joan Weiner
Through the very stiffness and inflexibility of parts the lack of which makes
the hands so dextrous. Word-language is inadequate in a similar way. We
need a system of symbols from which every ambiguity is banned, which has
a strict logical form from which the content cannot escape. (Frege 1882,
p. 52)
Frege s attitude is clear. Neither natural language nor a logically perfect
symbolic language is suitable for every purpose. Whether certain fea-
tures of a language count as virtues or defects will depend on the pur-
pose for which we want to use the language. The suitability of
Begriffsschrift for Frege s specialized purposes does not make it an ideal
language. For while some features of natural language are defects, given
Frege s specialized purposes, these same features are virtues for most
purposes. He makes the same point in Begriffsschrift, where he com-
pares his logically perfect language to a microscope, which is useless for
making normal observations. He does not celebrate Begriffsschrift as
an ideal language. Rather, he says that Begriffsschrift  is a device
invented for certain scientific purposes , and adds that  one must not
condemn it because it is not suited to others (Frege 1879, p. v).
Begriffsschrift is to be a language suitable for the expression and eval-
uation of inferences. It must be capable of expressing all content of any
statement that has significance for the inferences in which it can figure.
Once our inferences are expressed in Begriffsschrift, the employment of
Frege s logical laws and rules are to make it a mechanical task to deter-
mine whether an inference is correct and gapless, or whether it requires
an unstated premiss. Because this is a mechanical task, no presupposi-
tion can sneak in unnoticed. Only if we have such a language and logi-
cal system, will we be in position to produce identifiably gapless proofs
of the truths of arithmetic. And only identifiably gapless proofs from
primitive truths will enable us to carry out Frege s project: to determine
the epistemological nature of the laws of arithmetic.
In order to classify the truths of arithmetic according to their episte-
mological nature, we must define all terms of arithmetic from primi-
tive, undefinable terms and we must construct a list of axioms or
primitive truths from which all truths of arithmetic can be proved by
gapless logical inferences. To provide such definitions, axioms and
proofs for a science is, to use Frege s later expression, to provide a sys-
tematic science. And science, Frege claims, comes to fruition only in a
system (Frege 1914, p. 242/261). Arithmetic is to be regarded as a science
in its early stages a science whose sentences have not yet been associ-
ated with precise thought content. But this is not to say that Frege
thinks that arithmetic is less developed than other sciences. Arithmetic,
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What s in a Numeral? Frege s Answer 705
on Frege s view, is as highly developed as any science.36 But arithmetic
does not satisfy the standards for systematic science. In fact, there are
no systematic sciences Euclidean geometry comes closest, but its
proofs are not gapless.37 Frege s systematic science of logic, of which
arithmetic is a part, will be the first.
We can now see why it need not be absurd, from Frege s point of
view, to say that the everyday sentences of natural language do not have
truth-values. Frege s view seems to be that truth is what we get, not in
everyday circumstances, but rather at some ideal end of inquiry.38 And
the language for this ideal end, the language for systematic science, is
not natural language but Begriffsschrift. But while natural language
may not be a good vehicle for expressing truth, it is an essential tool in
the early stages of our attempts to express truths. In a diary entry, Frege
wrote, of his attempt to say what the numbers are,
[O]ne might think that language would first have to be freed from all logical
imperfections before it was employed in such investigations. But of course
the work necessary to do this can itself only be done by using this tool, for all
its imperfections. Fortunately as a result of our logical work we have ac-
quired a yardstick by which we are apprised of these defects. Such a yardstick
is at work even in language, obstructed though it may be by the many illog-
ical features that are also at work in language. (Frege 1924, p. 266/285)
To systematize a science, we begin with the everyday sentences that are
regarded as its basic truths such sentences as  0 is not equal to 1 . Our
everyday view that this sentence expresses a truth is not quite right. The
content associated with it is not yet precise enough; the science is not
yet sufficiently well worked out. But much of the science of arithmetic
is worked out. Many of the standards by which we judge sciences have
been met. Given this, and given the role played by our sentence in the
everyday science of arithmetic, this sentence provides a guide for sys-
tematizing arithmetic. For it places constraints on our definitions of the
36
At least this seems to be the view in most of his writings. For example, he writes, in 1885a, that
 since mathematical ideas have been developed into a richer and more subtle structure than else-
where, this science is especially suited to serve as a basis for epistemological and logical investiga-
tions . And,  No science is in such command of its subject-matter as mathematics and can work it
up into such a perspicuous form (Frege 1914, p. 242/261). On the other hand, in some of his later
writings (see, especially, 1898/9) he suggests that mathematics is in a particularly bad state. How-
ever, most of these statements seem to be expressions of Frege s frustration at the lack of impact his
work has had on mathematics.
37
See for example, Frege s discussion, in 1882, pp. 50 1, of the tacit presuppositions made by Eu-
clid.
38
This is very close to Burge s view. See, Burge 1990. Although I have more sympathy with this
view than with the other views of Burge that have been mentioned, I think for the reasons dis-
cussed below that this view is not quite right.
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706 Joan Weiner
terms  0 and  1 . The only acceptable definitions are those on which the
sentence  0 is not equal to 1 expresses a truth. Since it will help to have a
label for this attitude in the discussion that follows, I will say that Frege
regards these sentences as true. It is important to see, here, the difference
between regarding these sentences as true and believing things we do
not fully grasp. Someone can be in the latter state only if there is some
particular definite thought that s/he believes but does not fully grasp.
And, in such a situation, the appropriate stance to take towards the sen-
tence is to discover what the content of that thought is. In contrast, a
sentence that is regarded as true in the sense I have described here may
or may not express a definite sense. There is, as we have already seen,
good reason to believe that Frege did not think that sentences contain-
ing such expressions as  greater than have definite sense. Thus, while it
is a consequence of Frege s view that few, if any, of our everyday sen-
tences actually express truths, none the less it is consistent with his view
that we can regard some of these sentences particularly those that are
the results of pre-systematic research as expressing truths.
One might suspect that the view outlined here must conflict with
Frege s statements in  Über Sinn und Bedeutung , which includes exten-
sive discussion of natural language. Frege introduces his renowned
Sinn/Bedeutung distinction by talking about words and sentences of
everyday language. And the Bedeutung of an object expression is what-
ever object that expression designates. Yet it is difficult to find any
actual inconsistency. Although Frege writes as if the terms of everyday
language have Bedeutung and the sentences of everyday language have
truth-values, he never actually says that they do. It is not because the
subject never comes up. For example, although he raises the question of
whether  the Moon has a Bedeutung, he does not go on to say that it
does. He says only that we  presuppose a meaning [Bedeutung]
(Frege 1892a, p. 31). Nor does he say that such presuppositions are
always or generally, or even sometimes correct. Rather, all he says
is,
[T]he question whether the presupposition is perhaps always mistaken need
not be answered here; in order to justify mention of that which a sign means
[von der Bedeutung eines Zeichens] it is enough, at first, to point to our inten-
tion in speaking or thinking. (We must then add the reservation: provided
such a meaning exists.) (Frege 1892a, pp. 31 2)
We do, of course, presuppose that our terms have Bedeutung, that there
is something we really are talking about and that our sentences really
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What s in a Numeral? Frege s Answer 707
have truth-values.39 As we have seen, Frege s comments about the
numerals and  number indicate that he thinks there are scientific con-
texts in which this presupposition is incorrect.40 The incorrectness of
this particular presupposition has not, however, impeded our everyday
use of arithmetic. It appears not even to have impeded even such
sophisticated mathematical uses as Weierstrass s. It is the project of sys-
tematization that requires both that all presuppositions be eliminated
and that the necessary work be done to guarantee that each term has
Bedeutung.
This is not to suggest that the issue of having Bedeutung is unimpor-
tant in our use of natural language. Rather, in our use of natural
language even in scientific contexts it is no prerequisite that our
terms have Bedeutung. If this view is Frege s view, it may seem to be one
of his least plausible views. For surely, one might think, it is essential
that terms used in scientific contexts have Bedeutung. But a closer look
suggests that this is not right at all. Indeed, it is this apparently absurd
view that, at least in some cases, best fits our conception of good scien-
tific practice. To see this, it will help to look at an example.
7. Is Frege s apparently absurd view really absurd?
The observation that, among people with heart disease, those who are
younger tend to be obese has motivated research into the relation of
obesity and increased risk of heart disease. Today, as a result of a good
deal of research, it is widely regarded as a well-established truth that
obesity increases one s risk of heart disease. Yet  obese no more desig-
39
There is another passage, later in 1892a, in which Frege issues a warning against apparent
proper names without any meaning. An examination of this discussion, however, reveals no sup-
port for interpreting this warning as applying to the terms of natural language. The imperfection
that concerns him is not that language has proper names with no Bedeutung, but rather that it is
possible to form proper names with no Bedeutung. For, Frege claims, whether a proper name has
Bedeutung must not depend on the truth of some sentence. Given the importance to science of a
term s having Bedeutung, if we use a language some of whose terms do, and some of whose terms
do not, have Bedeutung, then it will be important to show that the terms we use in scientific con-
texts have Bedeutung. He writes,
A logically perfect language (Begriffsschrift) should satisfy the conditions, that every expression
grammatically well constructed as a proper name out of signs already introduced shall in fact
designate an object, and that no new sign shall be introduced as a proper name without being
secured a meaning. (1892a, p. 41.)
That is, Frege s discussion of the requirement that all terms have Bedeutung is in a discussion of the
requirements his Begriffsschrift must satisfy. For further discussion of the significance of Frege s
general requirement that every Begriffsschrift term must have a Bedeutung, see Weiner 2002.
40
See also the discussion of definitions in 1903a sections 56 67, where Frege argues that the
addition sign does not have a Bedeutung because there is no determinate value for the sum of the
Moon and the Moon.
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708 Joan Weiner
nates a fixed concept than  number . Although the medical researchers
studying obesity agree that obesity is some weight-related characteristic
that is associated with increased morbidity and mortality, several dis-
tinct sorts of definitions are used in medical research. In some work,
obesity is defined in terms of body mass index, an index calculated
using measurements of height and weight.41 Insurance tables, and some
scientific studies, consider frame size as well as height and weight.
However, since muscle weighs more than fat, the recent popularity of
fitness programs has led some researchers to feel that defining obesity
in terms of body mass index is a bad idea. Some think obesity should be
defined in terms of percentage of body fat. But if obesity is defined in
terms of body fat, it is not obvious that it should be defined in terms of
percentage of body fat since the location of body fat affects its signifi-
cance for morbidity and mortality.42
As this indicates, if  obesity is to be defined, several decisions will
have to be made among them, which are the terms to be used in the
definition. This decision will be made, in part, on the basis of research.
But not all decisions are determined by research. Some will be some-
what arbitrary. For example, the criterion of obesity might be body
mass index 20% (or 21% or 25%) greater than the ideal. Moreover,
research must be conducted in order to determine ideal body mass
index. And, here as well, there will be considerable latitude in deter-
mining the point at which to draw the line.
In any case, the search for a good definition of obesity continues,
along with the investigation of various hypotheses about obesity. Yet it
would be unreasonable to halt all investigation of the effects of obesity
on morbidity and mortality on the grounds that, since the concept has
yet to be fixed, the hypotheses have no truth-values. It would be unrea-
sonable to give up our view that obesity increases risk of heart disease
(i.e. that it is true that obesity increases risk of heart disease). That is, an
apparently absurd view that Frege seems to hold that we are entitled
to regard certain sentences as expressing truths, in spite of the fact that
some of their terms do not have determinate reference (or sense) is
not absurd at all. It is, in fact, a perfectly apt way of describing how
some scientific research actually is (and should be) carried out. But this
is not to say that the issue of a term s having determinate reference is of
41
Body mass index (or Quetelet index) is defined as: [weight in kg]/[height in meters]2. The
most commonly used definitions of obesity are in terms of BMI.
42
The indexes typically used to measure percentage of body fat are calculated from measures of
skinfold thickness. Estimates of obesity using these indexes have generally been found less useful
than those using BMI or girth measurements in combination with Waist/hip ratio. For an example
of studies of this issue, see Mueller, 1991.
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Mind, Vol. 116 . 463 . July 2007
What s in a Numeral? Frege s Answer 709
no concern to this sort of science. In fact, the problem with requiring
that all terms used in scientific investigation already have fixed refer-
ence is precisely that it can be part of the scientific enterprise to fix the
reference associated with a term that is already in use.43 The procedure,
as we have noted already, involves a combination of research and stipu-
lation. At the ideal end of this research all terms appearing in the sen-
tences that express the established truths would have fixed reference.44
What, then, is Frege s view of truth? It may seem that the appropriate
moral to draw is not that Frege thinks that the sentences of everyday
language have no truth values, but rather that he has two notions of
truth: a strict sort of truth that is the aim of science (expressible in
Begriffsschrift) and a different sort of truth that applies to sentences of
natural language. For, as we have seen, on Frege s view there are restric-
tions on acceptable definitions. Among the restrictions is that, on an
acceptable definition, the natural language statements we regard as true
need to come out true. This other sort of natural-language-truth is very
like the supervaluationist notion of truth: a sentence containing a term
that does not have fixed reference is true if and only if it is true on every
acceptable way of defining that term.45 After all, the significance of
Frege s regarding it as true that each number has unique successor is
that on every acceptable definition (precisification) of the term
43
For some discussion of how this works in epidemiology, see Weiner 2007.
44
It is worth noting that there are many differences between the scientific treatment of the term
 obese and Frege s treatment of the term  number . One obvious difference is that the attempt to
define obesity is important for the pursuit of research on obesity whereas, as Frege s example of
Weierstrass shows, there is no such need for someone who wants to carry out research in number
theory. These differences, however, do not affect the point I wish to make here: that we can see by
looking at obesity research that it is not a general requirement for scientific research that the ex-
pressions we use have reference. Moreover, if this is not a general requirement for scientific re-
search, how could it be a general requirement for our everyday use of the terms of natural
language? Of course in everyday life we do assume that our terms have reference. But what we can
see if we look at scientific research is that this assumption is generally mistaken. And it is the as-
sumption that this is a general requirement of our use of language that makes Frege s view seem
absurd. I am indebted to Luca Ferrero for suggesting that this point should be clarified.
45
In recent years, supervaluationist views have played a prominent role in the enterprise of ad-
dressing the problem of truth in a vague language. See, for example, Kit Fine 1975. Central to this
approach is the notion of precisification or a sharpening of the bounds of a predicate. The super-
valuationist strategy is to say that a sentence containing a vague predicate is true just in case it is
true given any admissible precisification. Given a particular precisification of, for example, the
term  bald , each person is either bald or not bald. If someone is bald on some precisifications but
not all, it is neither true nor false that s/he is bald. One of the key achievements of this approach is
to deal with the idea that in such penumbral cases there can be, in addition to statements without
truth values, statements that do have truth values. Suppose, for example that both Al and Bob are
in the penumbra of baldness it is neither true nor false that Al is bald; neither true nor false that
Bob is bald. Suppose, also, that Al has fewer hairs that Bob. Although neither the antecedent nor
consequent of  if Bob is bald, then Al is bald has a truth value, the conditional can, none the less,
© Weiner 2007
Mind, Vol. 116 . 463 . July 2007
710 Joan Weiner
 number , it will be provable, hence true, that each number has a unique
successor.46
There is, however, an important difference between the supervalua-
tionist view of natural language and Frege s view. Although the super-
valuationist and Frege share the view that there is something right
about many of our everyday sentences, the supervaluationist wants to
preserve both the presupposition that these sentences are true and the
presupposition that their constituents have fixed meaning in the fol-
lowing sense: the contributions these constituents make to the truth-
values of sentences in which they appear are fixed. The supervaluation-
ist s task is to offer an account of truth on which these presuppositions
are correct. On Frege s view, in contrast, such an enterprise is mis-
guided. Although there is something right about the sentences that we
regard as setting out fundamental truths of pre-systematic arithmetic,
this is not something to be explained in an account of truth. For the
demands of truth, as Frege understands them, show us that there is also
something wrong with these sentences of arithmetic. His aim is to sat-
isfy these demands, using what is right about pre-systematic arithmetic
as a starting point. This requires the construction of an actual precisifi-
cation. Frege s task is to replace imprecise pre-systematic sentences with
precise systematic sentences for example, to introduce definitions of
 number and  successor , from which it can be proved that each
number has unique successor. There are, Frege recognizes, many
notions of truth, but his interest is in a particular sort of truth,  the sort
of truth which it is the aim of science to discern (Frege 1918/1919, p. 59).
And, for this, we cannot rest content with the standards of pre-system-
atic arithmetic.
Thus to say that our statements do not now satisfy Frege s demand
that all constituents have fixed reference is merely to say that we are not
finished. Our sciences have not yet reached fruition problems remain
to be solved. The demands that Frege identifies as the demands of truth
be true. For, supposing that baldness supervenes on numbers of hairs, every admissible precisifi-
cation on which Bob is bald must also be one on which Al is bald. On the view to which, as we have
seen, Frege is committed, there is no special penumbral category every sentence of ordinary lan-
guage arithmetic falls into the penumbral category. And, at the same time, these penumbral state-
ments must have truth values. Thus the supervaluationist strategy can be seen as one on which the
(natural language) statements of truths of arithmetic come out true for the same reason that  if Bob
is bald then Al is bald comes out true.
46
I do not mean to suggest, by using the term  precisification that the term  number is vague in
the way  obese is vague. However, both terms fail one of Frege s tests they fail to be determi-
nately true or false of each object. And the supervaluationist s notion of precisifying a term
amounts to modifying it in such a way that it passes this test.
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Mind, Vol. 116 . 463 . July 2007
What s in a Numeral? Frege s Answer 711
can be seen, in this way, as part of a regulative ideal for science. But
there is no reason (or need) to assume that any sentences of natural lan-
guage actually satisfy the demand. One upshot is that we can reconcile
Frege s conception of his project with his statements about truth. The
project of showing that the truths of arithmetic are analytic is not a
project that is external to the development of the science of arithmetic.
It is a further the final step in bringing this science to fruition.
Because Frege s project failed in such a dramatic way, he is much
more lauded today for his insights into language than he is for his
insights into arithmetic.47 For many years his most widely read works
were  Über Sinn und Bedeutung and  The Thought , works that were
typically read, not as contributions to his overall project, but as contri-
butions to the philosophy of language if not of natural language as it
is, then of natural language as it should be. And these contributions
have had an important influence on contemporary writings about lan-
guage. I have tried to show that, in order to understand Frege s actual
insights, we need to focus on the role Frege s views on natural language
play in his overall project. This is, of course, of historical interest. But
not only of historical interest. For I have also argued that Frege s actual
views, for all their initial appearance of absurdity, are extremely accu-
rate to our actual use of natural language more accurate than the
views about reference traditionally attributed to him. If so, then there is
every reason to believe that we have only begun to learn what Frege has
to teach us.48
Department of Philosophy joan weiner
Sycamore Hall 026
1033 Third Street
Indiana University
Bloomington, IN 47405-7005
USA
[e-mail]
47
It is worth noting, however, that although Frege regarded his logicist project as having been
decisively refuted, a version of this project has been resurrected and currently forms an important
strand of contemporary philosophical thought about arithmetic. See, Hale and Wright 2001.
48
I am indebted to Luca Ferrero, Mark Kaplan, Eva Picardi and Thomas Ricketts for helpful
comments on earlier drafts. I am especially indebted to the referee for Mind, who gave me unusu-
ally detailed and very helpful comments. Versions of this paper were read at the History and Phi-
losophy of Science department of Indiana University, the Ludwig Maximilian Universität
München, the University of Bologna, the University of Sheffield, and the University of Trieste. I
would like to thank the many participants in those discussions.
© Weiner 2007
Mind, Vol. 116 . 463 . July 2007
712 Joan Weiner
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Mind, Vol. 116 . 463 . July 2007
What s in a Numeral? Frege s Answer 713
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Mind, Vol. 116 . 463 . July 2007
714 Joan Weiner
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