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1. Wittgenstein vs. Frege on the Notions of Number, Concept and Extension

What are numbers? This question is addressed by Frege in his Foundations of Arithmetic. Based on syntactical considerations about numerical expressions, such as "the number 1," "1+1" or "the number of the Jupiter's moons," Frege argues that these expressions are used referentially (cf. Frege, 1950: [section]57). Numbers, then, are the referents of these expressions, and a definition of "number" must tell us what kind of objects numbers are. He then famously suggests that such a definition should be derived from an answer to the question as to when two concepts can be ascribed the same number (cf. Frege, 1950: [section]62). Frege observes that this is the case if and only if (henceforth: iff) the concepts are equinumerous in the sense that the objects falling under one concept can be one-one correlated to the objects falling under the other. Numbers are then defined as numerical equivalence classes, i.e. as classes that contain all and only those concepts equinumerous to some specific concept. And it is by reference to such specific concepts that the particular classes to which the number signs correlate are specified. So, the class of all non-instantiated concepts that "0" is supposed to designate can be specified by an incoherent concept such as being different from itself. A possible choice (though not exactly the one that Frege actually chooses) for the canonical representatives of other classes is provided by the concepts for the initial segments of the series of numerals. One could accordingly stipulate that a numeral "n" (other than "0") designates the class which contains all and only those concepts that are equinumerous to the concept numeral between "1" and "n." A number ascription of the form "The number of the Fs is n" can then be interpreted as claiming that the concepts F and numeral between "1" and "n" are representatives of the same numerical equivalence class.

Nowadays few would wish to defend Frege's proposed identification of numbers with numerical equivalence classes. But his referentialism concerning numerical expressions - and, with it, the resulting construal of numbers as objects - is still a highly popular view among contemporary philosophers of mathematics, if not the predominant one. (1)

One early critic of Frege's conception of numbers as numerical equivalence classes, who is usually overlooked in contemporary debates, was Wittgenstein. In the Tractatus, Wittgenstein's discussion of mathematics was primarily focused on the intra-mathematical use of number signs in arithmetical equations (cf. 1922: 6.02-6.031, 6.2-6.241). His proposed reduction of arithmetic to a theory of logical operations was conceived as an alternative to the set-theoretic reductions of Frege and Russell, which are based on an analysis of the extra-mathematical use of number signs in number ascriptions.

In his middle period, Wittgenstein finally addresses the extra-mathematical use of number signs. He agrees with Frege that numbers are ascribed to concepts (1975: [section]99; 1974: 332, 345; 1978: V, [section]49). But he criticizes Frege's analysis of number ascriptions. And he maintains his rejection of Frege's definitions of the general notion of number and of Frege's definition of the individual numbers. Wittgenstein now rejects the very idea of defining the notion of a cardinal number:
What we are looking for is not a definition of the concept of number,
but an exposition of the grammar of the word 'number' and of the
numerals. (1974: 321)

And it is such an exposition that Wittgenstein seeks to provide with the aid of the following remarks:
We might say, parenthetically, that a number is an external property of
a concept and an internal property of its extension (the list of
objects that fall under it). A number is a schema for the extension of
a concept. (1974: 332) Numbers are pictures of the extensions of
concepts. (1975: [section]100)

From these remarks we can derive the following three claims, which relate the notions concept, extension and number:

([T.sub.1]) Numbers are external properties of concepts,

([T.sub.2]) Numbers are internal properties of extensions, and

([T.sub.3]) Numbers are pictures - or: schemes - of extensions.

Wittgenstein's somewhat cryptic claims about the use of number signs in ascriptions of numbers to concepts have received rather little attention from his commentators, partly, perhaps, because Wittgenstein's main concern in his middle period is to clarify the nature of mathematical propositions and mathematical proofs. The idea that numbers are pictures of extensions, for instance, is not discussed in Marion (1998). And Frascolla's short discussion of it does not elaborate the underlying attack on Frege's conception of numbers (1994: 47-49). In this paper we shall offer an interpretation and critical evaluation of the three claims ([T.sub.1]), ([T.sub.2), and ([T.sub.3), along with an assessment of Wittgenstein's arguments against Frege. And we shall argue that a slightly modified version of Wittgenstein's account offers a coherent and fruitful analysis of the use of number signs. As it will turn out, this analysis is not only at odds with Frege's proposed identification of numbers and numerical equivalence classes. It also challenges his referentialism, which is still widely accepted. For this reason, there will very likely not be many to whom this analysis will immediately appeal. However, if the arguments we shall present are correct, Wittgenstein's unorthodox account of number signs is of more than merely historic interest.

2. Number Ascriptions

Number signs must be distinguished from the corresponding numerical quantifiers or, if you prefer, numerical predicates. One and the same number sign "n" may determine number ascriptions of different types, namely propositions of the form "There are exactly n Fs," "There are more than n Fs" and "There are fewer than n Fs", which shall be abbreviated by[there exists]=n(F),[there exists]>n(F) and[there exists]<n(F) respectively. In what follows, we shall focus our attention on the so-called definite numerical quantifiers "[there exists]=n". So, unless otherwise stated, we shall reserve the labels "number ascription" and "numerical predicate" for corresponding closed and open sentences.

In Philosophical Grammar, Wittgenstein proposes analytic definitions of numerical predicates, which show how to transform number ascriptions into complex existential propositions (1974:333). Using orthodox notation instead of Wittgenstein's own symbolism, these definitions can be formulated as follows:

(NA) [mathematical expression not reproducible]


In proposing these definitions, Wittgenstein still presupposes a Tractatus convention: identity is not expressed by a corresponding sign but by using the same variables or names for the same objects. The orthodox treatment of identity, which we will subsequently adopt, requires the following modification of (NA):

(NA1) [mathematical expression not reproducible]


As Frascolla observes, these are in fact the explanations of the definite numerical quantifiers that Frege envisaged (Frascolla, 1994: 47; Frege, 1950: [section]55).

What makes these definitions appealing is that they correctly represent the inferential relations between number ascriptions and the corresponding complex existential propositions. In his middle period, however, Wittgenstein claims that a definition has to satisfy a stronger requirement than this kind of inferential adequacy: a definition must show the way to the verification of corresponding propositions:
Definitions are signposts. They indicate the path to a verification.
[...] A definition reduces one concept to another or to several others,
which again are reduced to others and so on. The direction of this
reduction process is fixed by a certain method of verification.

Definitions which do not fulfil that purpose are without significance.
(1979: 221)

Now, it is true that the later Wittgenstein no longer equates meaning with verification but rather with use. Nevertheless, he still maintains that the grounds of judgement are grammatically related to the proposition and tell us - partially, at least - what proposition it is (cf. 1981: [section]437). Moreover, knowing how to verify a proposition remains one criterion for understanding its sense (cf. 2010: [section]353). It therefore seems that Wittgenstein, even after abandoning the strong verificationism of his middle period, would still require an explanation of the numerical predicates to be epistemically adequate in the following sense: the explanation must indicate the rule which is followed in verifying number ascriptions and whose mastery constitutes understanding the numerical predicates. We will now try to show that the explanations that Wittgenstein himself proposed do not satisfy this requirement.

(NA1) shows how a number ascription can be derived from a corresponding complex existential statement, the correspondence being such that the number ascribed to the concept must match the number of positively existentially quantified variables in the existential proposition. In the case of larger numbers, this derivation would require counting the variables. Now, although it is indeed possible to proceed this way, the usual and most basic method for verifying "[THERE EXISTS]=n(F)" certainly consists in directly counting the Fs. And the understanding of numerical predicates does not consist in knowing how to derive number ascriptions from complex existential propositions. In fact, these latter propositions do not play any significant role in our practices of generating or deciding number ascriptions. What has to be mastered is the technique of transitive counting. And what has to be known is that counting the Fs establishes the truth of "[THERE EXISTS]=n(F)" if this process terminates with "n," and establishes the falsehood of this proposition if it terminates with a different numeral than "n."

The next task, then, is to provide an explanation of the numerical predicates which makes it explicit that number ascriptions are verified by the method of (transitive) counting. Any formulation of such an explanation faces two interconnected difficulties. First, the use of numerical predicates is usually not verbally explained but practically taught. The pupil is shown how to verify number ascriptions with a range of examples, and his own attempts are corrected or approved. Second, verifying number ascriptions by the method of counting does not involve any equivalence transformation of number ascriptions into sentences of a more basic type (whether complex existential propositions or otherwise).

The first point implies that we must conceive of the verbal explanation being sought, not as a formula to be given to the pupil, but as a codification of the verification procedure which the pupil is supposed to practise. What is required is an explanation that could serve as a reminder for the initiated, not necessarily as a recipe for the uninitiated. The second point implies that the explanation of numerical predicates must remind us, not how to derive number ascriptions from sentences of other types, but how to verify number ascriptions by the method of counting. So, if the explanation is to connect number ascriptions to equivalent sentences of a certain type, the latter must not be conceived as the premises from which the pupil is supposed to derive number ascriptions. Rather, they have to be interpreted as expressions of the conditions whose fulfilment is verified when one establishes the truth value of corresponding number ascriptions by counting the relevant objects.

Arguably, this second requirement is not fulfilled by (NA1). Even if the right-hand sides are not conceived as premises for corresponding number ascriptions, they can hardly be said to express the condition whose fulfilment is verified by counting the relevant objects. Frege once observed that counting objects consists in correlating them one-by-one with the elements of an initial segment of the series of numerals (1972: 326). Accordingly, the corresponding verification procedure for number ascriptions can be codified by the following explanation:

(NA2) [??] The Fs are one-one correlatable to the numerals between "1" and " n" inclusive.

The method of one-one correlation is indeed the method of numerical comparison. For one verifies how two concepts F and G compare in respect of number - that is, whether there are more Fs than Gs, fewer Fs than Gs, or just as many Fs as Gs - by correlating the Fs either directly or indirectly to the Gs. So, despite Wittgenstein's qualms, it must not only be conceded to Frege that equinumerosity - the notion expressed by the phrase "just as many" - is one-one correlatability (2) It must also be granted that numerical predicates can be explained by reference to the notion of equinumerosity (cf. Wittgenstein 1974: 355-356). For (NA2) can be rephrased as follows:

(NA3) [??] is equinumerous to the concept numeral between "1" and "n" inclusive.

Recall, however, that Frege interprets number ascriptions as statements about the identity of numerical equivalence classes. Thus, by ascribing a number n to a concept F, one would not merely claim that the two concepts F and numeral between "1" and "n" inclusive are equinumerous. According to Frege, one would also refer to all concepts, claiming that any concept is equinumerous to F just in case it is also equinumerous to numeral between "1" and "n" inclusive. To this account Wittgenstein rightly objects that we do not verify such a number ascription by checking case by case whether any concept is equinumerous to F iff it is equinumerous to numeral between "1" and "n." We only check whether or not these two concepts are equinumerous (1979: 221).

This objection is even more serious if it can be assumed that there are indefinitely many concepts with which F and numeral between "1" and "n" could, in principle, be numerically compared. For in that case there is no such thing as checking case by case how any concept numerically compares to the two concepts in question. Such an infinite generalisation must either be postulated as a rule, or it must be derived from such postulations, perhaps in conjunction with other propositions. And this is what can be done in the present case. For in conjunction with the general rule that equinumerosity is symmetric and transitive, the equinumerosity of the concepts F and numeral up to "n" indeed implies that any concept must be equinumerous either to both or to neither of them.

At any rate, what "[[there exists].sub.=]n(F)" claims is simply that the two concepts F and numeral up to "n" are equinumerous. This is the condition whose fulfilment is verified when one establishes the truth value of the proposition. And the identity of the corresponding numerical equivalence classes can only be conceived as a consequence of, not as a criterion for, the truth of the number ascription.

The argument against construing number ascriptions as class identities can be transferred to the case of ascriptions of infinite numbers such as "The number of the rational numbers is[??]" or "The number of the real numbers is [??]." Wittgenstein's contention that expressions for numerical relations - "just as many," "more than" and "fewer than" - undergo a change in meaning when applied to infinite concepts is defensible (Wittgenstein, 1974: 464; 1976: 160/1). What "There are as many tables as chairs" claims is that the tables and the chairs can actually be one-one correlated (possibly indirectly, through the direct correlation of their names). An equinumerosity statement about infinite concepts such as "There are just as many natural numbers as rational ones" cannot be interpreted in (exactly) the same way. For there is no such thing as actually correlating all the natural numbers to corresponding rational ones. The equinumerosity of these two concepts can only be established by specifying an appropriate rule for correlating natural numbers to rational ones.

This complication, however, does not affect our argument. The crucial fact is that it is still only the two concepts in question that are numerically compared in order to establish the truth of "There are just as many natural numbers as rational ones." The identity of the corresponding numerical equivalence classes can again only be construed as a consequence of this statement and the fact that equinumerosity remains an equivalence relation even after it is extended to infinite concepts.

Thus, as far as number ascriptions such as "The number of the rational numbers is [??]" or "The number of the real numbers is [??]" are concerned, the same analysis as in (N[A.sub.3]suggests itself: these sentences claim that the specified concepts are equinumerous to concepts determined by the number signs "[??]d "[??]s, for example, the concepts natural number and subset of the set of natural numbers).

3. Numbers and Concepts

In order to evaluate Wittgenstein's claims ([T.sub.1]), ([T.sub.2]) and ([T.sub.3]) properly, we have to understand them correctly. We shall therefore begin by giving an interpretation of the key terms employed. According to Wittgenstein, the three notions number, concept and extension stand for grammatical rather than ontological categories. Wittgenstein is explicit about this with regards to the formal counterparts of the notions of a concept and of an extension: the sign for a concept is a predicate, and the sign for an extension is a list (1974: 332). Wittgenstein's use of the word "number" appears to be slightly ambiguous. Wittgenstein's first two claims are best understood as claims about numerical predicates: ([T.sub.1]) and ([T.sub.2]) could have been expressed by saying that numerical properties are external properties of concepts and internal properties of extensions. However, as will become clear, his claim that numbers are pictures of extensions is best interpreted as a claim about number signs.

The distinction between external and internal properties can be understood in terms of the bipolarity of the corresponding propositions. A proposition is bipolar iff it is capable of being true and capable of being false. Accordingly, F is an external property of a iff "a is F" is true and bipolar. And F is an internal property of a iff "a is F" is true but not bipolar. The following definitions extend this conception to cover false propositions as well: F is externally related to a iff "a is F" is bipolar. And F is internally related to a iff "a is F" is not bipolar (i.e. is either tautological or contradictory).

According to Wittgenstein's thesis ([T.sub.1]), numerical properties are externally related to concepts (as opposed to extensions). In order to account for the intended contrast between concepts and extensions, we must introduce one further definition. To a list of pairwise different singular terms [f.sub.1],... [f.sub.n] there corresponds the predicate "x is identical with [f.sub.1],..., or [f.sub.n]," which shall be abbreviated as ([f.sub.1],... [f.sub.n])(x). We shall call such expressions "extensive predicates." And ([T.sub.1]), it seems, can then be interpreted as follows: number ascriptions to genuine - i.e. non-extensive - predicates are bipolar.

It is not clear whether Wittgenstein thought that ([T.sub.1]) and ([T.sub.2]) hold without exception (cf. 1974: 332). With respect to ([T.sub.1]), at any rate, such a general claim would be difficult to maintain. It is true that some - or perhaps even most - number ascriptions to genuine predicates are bipolar. But there are exceptions to this rule. For there are predicates which exclude or imply certain numerical determinations.

One class of examples is provided by those predicates that actually contain numerical determinations, such as "x is a top-ten tennis player." The numerical phrase included in this predicate renders the number ascription "There are exactly eleven top-ten tennis players," for instance, self-contradictory and hence incapable of being true. A second class of examples is provided by predicates whose analysis shows that they exclude or imply certain numerical determinations. Incoherent predicates such as "x is a married bachelor" belong to this class. For ascribing the number 0 to such a predicate is tautological. And the ascription of any other number is contradictory. Another, more complicated example can be derived from Wittgenstein's conception of how mathematical proofs form concepts (cf. 1978: I, [section][section]25-28). Since pentacles are memorable figures that can be recognized without counting their corners, the concept of a pentacle can be explained ostensively. Such an explanation does not include a specification of how many corners a pentacle has. The visual criterion, which it introduces, only determines that all instances of the concept pentacle have the same number of corners. However, by counting the corners of one particular instance, one can prove that every pentacle necessarily has exactly five corners (cf. 1978: I, [section][section]25-28). This proof may not establish that "This pentacle has exactly 5 corners" is tautological. The sentence may still count as bipolar, since the presupposed existence of the pentacle to which it refers is a factual matter. But the proof certainly establishes that a sentence of the form "This pentacle has exactly n corners" is self-contradictory if n[not equal to]5.

In reply to this objection, Wittgenstein could argue that predicates that imply or exclude numerical determinations should actually count as extensive predicates, in so far as they allow for appropriate analyses. The predicate "x is a top-ten tennis player," for instance, can be analysed thus: x is identical with the number one tennis player, or with the number two tennis player or...or with the number ten tennis player. Such analyses, however, are certainly less plausible in other cases. And they are not available at all in the case of incoherent predicates. Clearly, then,

([T.sub.1]) must be replaced by a weaker formulation, such as the following: typically -though not universally - numerical properties are externally related to concepts.

In the preceding section it was argued that Frege's account of number ascriptions has to be freed from reference to numerical equivalence classes. This suggests that reference to classes must also be omitted from his definitions of the particular numbers signs and from his definition of the general notion of a number. The fact that numerical properties are externally related to most concepts provides a further argument to that effect.

Crucial for this argument is the observation that if F is externally related to a, then it is logically possible that a exemplifies F at one time but not at another. Numerical properties, in particular, are exemplified by different concepts at different times. For many concepts are such that different numbers of objects fall under them at different times. The concept astronaut, for example, exemplifies the property of being equinumerous to the concept not self-identical in the year 1900, but not in the year 2000. As a result, talk of being a numerical equivalence class and of being the numerical equivalence class of a certain concept is well defined only when relativized to times. The class of the concepts that in 1900 are equinumerous to the concept not self-identical can be said to be a numerical equivalence class at this time but not in 2000, since the elements of this class are pairwise equinumerous only at the former but not at the latter time.

Now, when Frege proposed to identify the notion of a number with that of a numerical equivalence class, he certainly intended his definition to comply with the principle that if n is a number at one time, then it is also a number at any other time. This, however, is not the case. For one and the same class of concepts can be a numerical equivalence class--and hence a number, according to Frege's definition--at one time and not at another. And it seems reasonable to assume that when Frege claimed that numerals designate the extensions of corresponding numerical properties, he intended any particular numeral to designate one and the same class at any time. The reference of "0", for instance, is supposed to be the same in any context of utterance. But, due to the fact that numerical properties have different extensions at different times, Frege's proposed designation rules map number signs onto different classes as a function of time. "0," in particular, would not designate the same class in 1900 as in 2000.

This difficulty for Frege's definitions could be circumvented if, instead of relativizing talk of numerical equivalences classes to times, one relativized concepts to times. For suppose that astronaut is treated as a relation between persons and times, as Frege in another context actually seems to suggest (Frege, 1950: [section]46). Then, there being a different number of astronauts in 1900 than in 2000 is not a matter of two different numerical properties applying to the same concept astronaut. Rather, it is a matter of two different numerical properties applying to two different concepts, viz. astronaut in 1900 and astronaut in 2000.

However, this strategy has little appeal. It goes against our principles for the individuation of concepts in that it implies that by saying "a is F" at time [t.sub.1] and again at time [t.sub.2] one applies different concepts each time. And it excludes the possibility of speaking of changing substances, i.e. of one and the same object falling under contrary concepts at different times. Accordingly, we have to omit the step from numerical properties to their extensions in Frege's explanations in order to avoid the consequence that being a number is a temporal matter. For, atemporally--in the required sense--a numeral relates only to the corresponding numerical property.

In this case, Frege's definitions of the particular number signs reduce to (N[A.sub.3]), the explanations of the corresponding numerical predicates proposed at the end of section 2. For the formulation that "n" designates the class that contains all and only those concepts equinumerous to the concept numeral between "1" and "n" inclusive has to give way either to (N[A.sub.3]) or to some material reformulation of this definition, such as: "n" determines the property which is exemplified by all and only those concepts that are equinumerous to the concept numeral between "1" and "n" inclusive.

And the characterization of the numerical equivalence classes, which was supposed to serve as the definition of the general notion of a number, reduces to a characterization of the (definite) numerical properties. Thus, we do not say of a class of concepts that it is a number just in case one of its elements, F, is such that any concept G belongs to the class iff G is equinumerous to F. But one might say of a property of concepts [phi] that it is a (definite) numerical property if it satisfies the following condition: if a concept F exemplifies [phi] at time [t.sub.1], then a concept G exemplifies [phi] at time [t.sub.2] iff there are as many Fs at [t.sub.1] as Gs at [t.sub.2]. For this kind of invariance under equinumerosity sets the definite numerical properties expressed by "[[there exists].sub.=]n" apart, for instance, from the properties that are expressed by "[[there exists].sub.<]n" or "[[there exists].sub.>]n".

Wittgenstein uses the corresponding equivalence

[??] [AND] OR [CONJUNCTION] F is equinumerous to G in an attempt to explain the notion of equinumerosity by reference to number ascriptions (cf. 1974: 357-358). However, his idea that number ascriptions are explanatorily prior to equinumerosity statements is motivated by his ill-founded refusal to explain the notion of equinumerosity in terms of one-one correlations. And, contrary to what one may be tempted to think, this idea is not supported by the (correct) observation that we often - or even mostly - verify equinumerosity statements by deriving their truth or falsity from corresponding number ascriptions. For the validity of this procedure derives precisely from the fact that [[there exists].sub.=]n(F) and [[there exists].sub.=]n(G) claim that F and G are each equinumerous to the concept numeral between "1" and "n" inclusive.

We thus conclude that Frege is right as far as explanatory order is concerned, and that it is only the appeal to classes that must be resisted. Number ascriptions are to be explained by reference to equinumerosity statements, and not the other way round, since there being exactly n Fs consists in there being just as many Fs as numerals between "1" and "n" inclusive. And these explanations, then, imply the invariance rules characteristic of numerical properties, the rules which allow us to conclude that there are exactly n Gs if there are exactly n Fs and just as many Fs as Gs. But pace Frege, one must not construe these inferential rules as a characterization of the objects to which number signs putatively refer. And one should not interpret propositions of either kind--equinumerosity statements and number ascriptions--as claims about the identity of such putative objects.

4. Numbers and Extensions

According to ([T.sub.2]), numerical properties are internally related to extensions. This thesis allows for two different interpretations. First, it could be interpreted as claiming that number ascriptions to extensive predicates are not bipolar. This claim, in turn, could be made more precise in the following way. Let [f.sub.1], [f.sub.2], ..., [f.sub.n] be (pairwise different) singular terms and ([f.sub.1],..., [f.sub.n])(x) the corresponding extensive predicate. Then, under this interpretation, ([T.sub.2]) could be taken to be equivalent to the conjunction of the following two principles:

If n=m, then [[there exists].sub.=]m(([f.sub.1],..., [f.sub.n] )) is tautological.

If n[not equal to]m, then [[there exists].sub.=]m(([f.sub.1],...[f.sub.n])) is self-contradictory.

It is not difficult to see, however, that ([T.sub.2]) is problematic under this interpretation. For ease of exposition, let us introduce the following abbreviations for complex logical sums and products:

[mathematical expression not reproducible]

Now, a claim to the effect that there are exactly m objects to which the extensive predicate ([f.sub.1],..., [f.sub.n])(x) applies can be formulated in our notation as: [[there exists].sub.=]m(([f.sub.1],...[f.sub.n])). Such a proposition is true iff the list [f.sub.1], ..., [f.sub.n] contains exactly m singular terms such that (i) none of them is empty, and (ii) no pair of them is co-referential. Accordingly, a proposition of the form

(3) [mathematical expression not reproducible]

which claims that all the terms [f.sub.1], ..., [f.sub.n] fulfil these two conditions, is equivalent to a proposition of the form

(4) [[there exists].sub.=]n(([f.sub.1],..., [f.sub.n])).

It is, of course, unconditionally true that [[there exists].sub.=]m(([f.sub.1],..., [f.sub.n] )) is self-contradictory if m>n. [[there exists].sub.=]3 (Peter, Paul), for instance, would make the incoherent claim that there are exactly three objects to which the predicate "identical with Peter or with Paul" applies. (3) However, the validity of Wittgenstein's principles for the cases n=m and m<n presupposes a kind of ideal notation in which there are neither empty nor co-referential singular terms. And this Tractatus presupposition does not hold where natural languages are concerned. Here it is often an empirical matter whether a singular term is empty or whether two such terms are co-referential. And if this is the case for all the terms of the list, then [[there exists].sub.=]m(([f.sub.1],...[f.sub.n])) is bipolar if m[less than or equal to]n. And it is not (4) itself that is tautological, but only the conditional statement with (3) as the antecedent and (4) as the consequent. Accordingly, the number ascriptions [[there exists].sub.=]2(Peter, Paul), [[there exists].sub.=]1(Peter, Paul) and [[there exists].sub.=]0(Peter, Paul) are bipolar, since there being two, one or no object to which "identical with Peter or with Paul" applies are all genuine possibilities. And the following conditional is a tautology: If Peter exists, Paul exists, and Peter is not identical with Paul, then there are exactly two objects to which the predicate "identical with Peter or with Paul" applies. Thus, we have to conclude that, where the possibility of empty or co-referential singular terms cannot be (logically) excluded, number ascriptions to extensive predicates are bipolar. And, in this sense, numerical properties can be externally related to extensions.

We will now turn to a second, more promising interpretation of ([T.sub.2]). Note that a number ascription [[there exists].sub.=]n(F) is implied by a proposition of the form

[mathematical expression not reproducible]

Propositions of this kind, which shall subsequently be abbreviated by ([f.sub.1][f.sub.2],..., [f.sub.n])(F), can be called extension ascriptions. For what ([f.sub.1], [f.sub.2], ..., [f.sub.n])(F) claims is that the list [f.sub.1],..., [f.sub.n] comprehensively and non-redundantly specifies the extension of F. The inference rules, according to which ([f.sub.1], [f.sub.2],..., [f.sub.n])(F) implies [[there exists].sub.=]n(F), could, for obvious reasons, be called numerical generalizations. These rules imply the following two principles:

If n=m, then [logical not] ([f.sub.1],... [f.sub.n])(F) [disjunction] [[there exists].sub.=]m(F) is tautological.

If n[not equal to]m, then ([f.sub.1],... [f.sub.n])(F) [AND] OR [CONJUNCTION] [[there exists].sub.=]m(F) is contradictory.

As an example, consider the following extension ascription: Peter is an astronaut, Paul is an astronaut, and nothing else is an astronaut, and Peter is not identical with Paul. This sentence implies the number ascription [[there exists].sub.=]2(astronaut). And it is incompatible with [[there exists].sub.=]3(astronaut), for instance. Accordingly, ([T.sub.2]) is true if interpreted as the conjunction of these two principles and hence as a description of the rules of numerical generalization.

It is difficult to establish which of these two interpretations of ([T.sub.2]), if any, Wittgenstein has in mind. We shall adopt the second interpretation, since it is systematically more fruitful and is at least suggested by some passages in Wittgenstein (cf. 1974: 332). Accordingly, we can express the truth which is approximated by Wittgenstein's claims ([T.sub.1]) and ([T.sub.2]) as follows: the number that belongs to a certain concept cannot simply be derived from an analysis of the concept in the normal case, but it can be derived from a comprehensive and non-redundant specification of the concept's extension. In other words, the question of how many objects fall under a concept F cannot usually be answered a priori. But the answer can always be derived from an appropriate answer to the more specific question of which objects fall under F.

5. Numbers as Pictures

It will be recalled that according to ([T.sub.3]) numbers are schemes - or: pictures - of extensions. Here, Wittgenstein is most plausibly referring to number signs rather than to numerical predicates, the claim being that number signs schematize extension signs. This claim is indeed defensible, if restricted to finite numbers, a restriction that Wittgenstein himself acknowledges (1979: 226, 228). Consider first the usual Arabic numerals. With the exception of "0," any such numeral "n" determines a corresponding numeral sequence (1, 2,..., n). And such a sequence can be conceived as a number sign in its own right. For one could systematically employ such formulations as "There are exactly 1, 2,..., n Fs" (abbreviated as: [[there exists].sub.=](1, 2,..., n)(F)) instead of the usual "There are exactly n Fs." The corresponding rules for numerical generalization have the following form:

([f.sub.1]... [f.sub.n])(F) implies [[there exists].sub.=](1, 2,..., n)(F).

These rules show that one can derive a number ascription from a corresponding extension ascription by successively substituting numerals for singular terms.

Accordingly, a numeral sequence does indeed deserve to be called an extension scheme - as opposed to an extension sign. On the one hand, it has the same form and the same multiplicity as a corresponding extension sign, in so far as both expressions are lists with equally many entries. On the other hand, the numeral sequence acquires schematic generality from the fact that the rules of numerical generalization map all and only those extension signs onto the same numeral sequence that have equally many entries. As a result, [[there exists].sub.=](1, 2,..., n)(F) no longer answers the question of which objects are F.

The individual numerals (apart from the exceptional case of "1") no longer have the same multiplicity as the corresponding extension signs. But, in an extended sense, these numerals can likewise be conceived as extension schemes, in so far as they translate into numeral sequences in virtue of their respective positions in the series of numerals. Once the numerals have undergone this translation "they assume the very multiplicity they mean" (Wittgenstein, 1979: 225). Thus, the numeral "3," for instance, schematizes the extension sign (Peter, Paul, Mary) in so far as the multiplicity of the corresponding numeral sequence (1, 2, 3) matches the multiplicity of the extension sign.

This conception can be generalized. As Wittgenstein suggests, one can distinguish quite generally between picture-like number signs (henceforth: pictorial number signs), which straightforwardly schematize extension signs, and non-pictorial number signs, which are transformable into such schematic signs:
Our number signs contain the possibility of being transformed into
other signs that are pictures in an immediate way. That is, our number
signs, together with the rules of syntax, are instructions for the
construction of picture-like symbols.

There must always remain a clear way back to a picture-like
representation of numbers leading through all arithmetical symbols,
abbreviations, signs for operations, etc. The symbolism of the
representation of numbers is a system of rules for translation into
something picture-like. (1979: 225)

In order to render this account more systematic, we propose the following definitions:

(PNS) "n" is used as a pictorial number sign in "[[there exists].sub.=]n(F)" with respect to a certain way of parsing "n" iff the condition--necessary and sufficient--for the truth of "[[there exists].sub.=]n(F)" consists in there being just as many Fs as sub-expressions of "n" corresponding to the parsing of "n."

(SNS) "n" is used as a non-pictorial number sign in "[[there exists].sub.=]n(F)" iff there is another number sign "n*" and a way of parsing it such that the condition - necessary and sufficient--for the truth of "[[there exists].sub.=]n(F)" consists in their being just as many Fs as subexpressions of "n*" corresponding to the parsing of "n*."

So conceived, the distinction between pictorial and non-pictorial number signs classifies signs not according to their form, but according to their use. One and the same sign can be used either as a pictorial or as a non-pictorial number sign. The stroke numeral in "[[there exists].sub.=]IIIIII(F)," for example, is used as a pictorial number sign if there being as many Fs as strokes in the number sign is taken to be the sentence's truth condition. This is the case if the number ascription is supposed to be verified by correlating the Fs to individual strokes. On the other hand, the stroke numeral is used as a non-pictorial number sign if there being as many Fs as stroke numerals between I and IIIII inclusive is taken to be the truth condition of "[[there exists].sub.=]IIIII(F)." That is, if this sentence's truth value is to be determined by counting in the stroke system, i.e. by correlating the Fs not with individual strokes but with stroke numerals I, II, III, etc. In this case, the pictorial number sign associated with "IIIII" would be the corresponding sequence (I, II, III, IIII, IIIII). Thus, whether or not a number sign is used as a pictorial or as a non-pictorial number sign depends on how corresponding number ascriptions are verified.

These remarks indicate how the conception of pictorial and non-pictorial number signs has to be applied to numerals and arithmetic terms more generally. Suppose we are given a certain system of numerals such as the Arabic numerals, the Roman numerals, or the stroke numerals. Then, in order to represent the number of objects falling under a certain concept, the usual way to proceed is to count the objects with the numerals and to incorporate only the last numeral in the number ascription. In this case, the numeral is used as a non-pictorial number sign, whereas the role of the pictorial number signs falls to the corresponding numeral sequence.

Arithmetic terms can be transformed into numerals by applying the relevant arithmetic operations. And usually we take the truth of the number ascription containing the resulting numeral as the condition for the truth of the number ascription containing the corresponding arithmetic term. A number ascription of the form [[there exists].sub.=]((3+1)+1))(F), for instance, is usually verified by counting the Fs and calculating ((3+1)+1)): it counts as true iff both processes lead to the same numeral. If such arithmetic number ascriptions are verified in this way, arithmetic terms are used as non-pictorial numbers signs. And the pictorial number sign corresponding to an arithmetic term is the numeral sequence determined by the numeral to which the arithmetic term reduces. Using an arithmetic term as a pictorial number sign is uncommon, but not impossible. The condition for the truth of a sentence of the form [[there exists].sub.=](((((1)+1)+1)+1)+1)(F), for instance, could be taken to consist in there being just as many Fs as occurrences of "1" in the arithmetic term.

So, in one way or another, number signs of both kinds (numerals and arithmetic terms) provide expressions that serve as standards for numerical comparisons. And, to repeat, such a numerical standard schematizes a corresponding extension sign in so far as it abstracts from the latter's specificity while retaining its multiplicity.

Wittgenstein recognizes that his conception of number signs as directly or indirectly schematizing extension signs cannot be transferred to the infinite case. For there is no such thing as an infinite list. Consequently, there are no extension signs (in Wittgenstein's sense of the term) for infinite concepts, let alone schemes of such signs. So, if "[??](F)" is taken to claim, say, that there is a rule for correlating the Fs one-one to the Arabic numerals, then "[??]" can be described neither as a pictorial nor as a non-pictorial sign in the sense given above. Wittgenstein actually takes this as a reason for not calling "[??]", "[??]", etc., number signs at all (1979: 226, 228). For reasons of space, we cannot elaborate this point here and so restrict our considerations to finite numbers. Given this restriction, it seems correct to say (with Wittgenstein) that the generation of a number ascription to a concept F proceeds by constructing a pictorial number sign, that is, a sign whose subexpressions are one-one correlatable to the Fs and which, therefore, schematizes an eventual sign for the extension of F. And the final number ascription then contains either this sign itself or some conventional equivalent (1979: 225).

Understood along the lines suggested above, Wittgenstein's metaphor of numbers as pictures of extensions can be seen to be defensible, at least as far as pictorial number signs are concerned. For the distinction between pictorial and non-pictorial number signs corresponds to a distinction between pictorial and symbolic representation of the numbers ascribed by the use of these signs. A number ascription using a pictorial number sign can be said to represent the ascribed number ostensively (or: pictorially). A proposition such as "There are exactly IIIII Fs" literally contains the collection of objects which is claimed to match the multiplicity of the Fs. This feature of the use of pictorial number signs becomes particularly striking in such formulations as "There are just as many Fs as there are of these strokes: IIIII" where the demonstrative element involved is explicit. By contrast, a number ascription using a non-pictorial number sign represents the ascribed multiplicity symbolically. The collection of objects which "There are exactly 5 Fs" claims matches the multiplicity of the Fs is given by the pictorial sign (1, 2, 3, 4, 5). And this sign is not contained in the number ascription: it is only related to the non-pictorial sign in the proposition by a symbolic convention.

It is worthwhile comparing this account of the distinction between pictorial and non-pictorial number signs with the treatment of colour signs Wittgenstein discusses in Philosophical Grammar (1974: [section][section]45-49) and in the Big Typescript (2005: 40-47). There, he investigates the difference between using a coloured label as the sign for a colour and using a word. Consider a language in which colour words are explained ostensively using samples:

"black" is the colour of this patch

This definition determines that a proposition of the form "a is black" ascribes the same colour to a symbolically as is ostensively ascribed to a by a is the same colour as this patch

In such a language, colour patches would function as pictures of the ascribed colours and would therefore be analogous to pictorial number signs. For it is these signs themselves - and not some correlated objects - that constitute the samples (or paradigms) with respect to which the subjects of the corresponding number or colour ascriptions are compared.

Wittgenstein was keen to make it clear that samples used as signs are no more immune to misunderstanding than words and that signs which are not directly compared to objects should not be regarded as secondary unless their use involves the use of another sign which is directly compared with the object (2005: 47). These points are both well taken: we do not generally compare objects with colour samples in order to make or verify colour ascriptions. Nevertheless, the consideration of codified languages with similarities to our own can illuminate aspects of actual usage. For once we have seen how number ascription or colour ascription could function without any need to regard numerals and colour words as names of objects, we no longer feel that their function must be explained by appeal to the name-bearer model. More importantly, the use of standard numerals does indeed involve the use of a pictorial number sign. For the verification of number ascriptions involves counting, and the initial number sequence up to the number ascribed is itself a pictorial number sign.

6. The General Notion of a Number

Frege invoked numerical equivalence classes in order to provide number signs with referents. While it is no longer fashionable to identify the supposed referents of number signs with such classes, the assumption that number signs are used referentially is still highly influential. It is in opposition to this assumption that Wittgenstein's conception of number signs as extension schemes has to be understood.

To avoid misunderstanding, it should be pointed out, first, that Wittgenstein acknowledges that it is legitimate to speak of reference in connection with number signs (2010: [section]10). However, he suggests that a proposition such as "'5' refers to a number" employs the trivial notion of reference, according to which any meaningful word can be said to refer to something (2010: [section]13). In this sense, even the use of the connective "and," for instance, could be described by saying that this word refers to a truth-function. For the present discussion, however, the more demanding notion of reference is relevant; and this notion applies only to expressions with a particular function, namely to those expressions whose use accords with the name-bearer model. And Wittgenstein is right that number signs do not refer in this sense.

To ascribe a number to a concept is to represent the multiplicity of the concept's instances. And the function of the corresponding number sign consists in providing a measure for this multiplicity. But, pace Frege, a number sign does not achieve this in virtue of being correlated with some extra-linguistic entity. Rather, it does so by displaying that multiplicity through the complexity of a certain expression, either the number sign itself or a corresponding pictorial number sign. A number ascription of the form "There are exactly 5 Fs," for instance, claims that the multiplicity of Fs is displayed by the elements of the sequence (1, 2, 3, 4, 5) - the sequence which the number sign "5" determines in virtue of its position in the system of Arabic numerals.

At this point, we must consider a possible objection. One might concede that the number sign in a sentence of the form "There are exactly 5 Fs" serves to determine a multiplicity sample but contend that this sample need not itself be a sign or expression. For it seems that the multiplicity sample could be extralinguistic; for instance, when the condition for the sentence's truth is taken to consist in there being as many Fs as fingers on my right hand. However, Wittgenstein's response would certainly be to insist that if the fingers are used as a multiplicity sample, then they are part of the language (1974: 346; 2010: [section]50). This is not the place to discuss this claim, but there is a lot to be said for it. For, if "There are exactly 5 Fs" is used as described above, then the fingers are not represented by the sentence: they belong to the means of representation. And if we can count elements of the means of representation as signs, then the fingers would indeed qualify as the pictorial number signs determined by "5." Nevertheless, even if one were to reject Wittgenstein's broad notion of a sign this would not suffice to rehabilitate the idea that "5" is used as a name for an extralinguistic object. For neither the hand nor any of its fingers could adequately be described as the referent of "5."

Wittgenstein's claim that number signs schematize extension signs is (partly) intended to highlight the fact that the use of number signs does not conform to the name-bearer model. As explained above, one correct idea behind his claim is that number ascriptions can be conceived as generalizations from extension ascriptions. And the transition from an extension ascription to a number ascription is indeed a purely formal operation, a mere transformation of signs which does not introduce any new extralinguistic object.

Frege arrives at his referentialist account of number signs by considering the syntactic form of number ascriptions in his favoured substantival notation: the number of the Fs is n (cf. Frege, 1950: [section]57). Neo-Fregeans argue in a similar way for the same conclusion, although they resist the identification of the supposed extra-linguistic referents of number signs with numerical equivalence classes (cf. Hale and Wright, 2005: 170-171). This line of reasoning, we suggest, has to be rejected. Whether or not a sign purports to refer to an extra-linguistic object cannot be established merely by appeal to the syntactic category to which the sign belongs. Rather, one has to analyse the use of the sign more broadly, including the way we verify sentences in which the sign features. And if the analyses proposed in the preceding sections are correct, then number signs do not purport to refer to extra-linguistic objects.

It is true that these analyses were mostly concerned with the adjectival formulation of number ascriptions: there are exactly n Fs. And, as Hofweber suggests, these formulations may have a slightly different role in communication than the substantival ones (Hofweber, 2005: 210; 2007: 22). But as Frege and his current followers would certainly concede, adjectival and substantival formulations of number ascriptions do not differ in point of verification or truth conditions. And no matter how a number ascription to a concept F is formulated, its existential presuppositions only concern the objects falling under F. Neither the truth of the adjectival formulation nor the truth of the substantival formulation presupposes the existence of an extra-linguistic object to which the number sign, common to both, allegedly purports to refer. In either case, it is sufficient that the Fs match the multiplicity of a certain expression, either the number sign itself or some associated pictorial sign. So, even in a substantially formulated number ascription, the function of the number sign is to provide a linguistic multiplicity sample and not to name an extra-linguistic object.

It is worth noting in this connection that the alleged referential function of number signs cannot be established by insisting on the distinction between numbers and numerical properties. We have already granted Frege the point that the number sign is only part of the (definite) numerical predicate (Frege, 1950: [section]57). A definite numerical predicate such as "There are exactly 5 Fs" contrasts with "There are fewer than 5 Fs" and "There are more than 5 Fs." These latter predicates, however, allow for similar analyses to the ones that (NA3) provides for the definite ones:

There are fewer than 5 Fs: [??] There are fewer Fs than numerals between "1" and "5" inclusive.

There are more than 5 Fs: [??] There are more Fs than numerals between "1" and "5" inclusive.

Accordingly, the number sign "5" does indeed have the same function in all these predicates. But, again, this function does not consist in referring to some extra-linguistic object. Rather, the number sign determines a numeral sequence which serves in each case as a linguistic multiplicity sample. And the difference between the corresponding predicates concerns the numerical relations which hold between the elements of the numeral sequence and the instances of those concepts to which the different predicates apply.

The inadequacy of the name-bearer model as an account for the use of individual number signs implies that the general notion of number does not determine an ontological category. A sentence such as

(5) 5 is a number

cannot be construed as specifying the kind of object to which "5" refers. And a sentence such as

(6) 5 is the same number as V

cannot be construed as claiming that "5" and "V" refer to the same numerical object. Properly interpreted, such sentences express rules for the use of the signs involved: (5) indicates that "5" can be used in the context of number ascriptions, and (6) that "5" and "V" are interchangeable in these contexts. And, to be sure, these rules do not derive from the nature of the alleged referents of these signs. The suitability of "5" for numerical contexts is due to the fact that this expression determines a multiplicity sample, namely the numeral sequence (1, 2, 3, 4, 5). And the interchangeability of "5" and "V" in numerical context derives from the fact that the sequence (I, II, III, IV, V), which is determined by "V," has the same multiplicity as the sequence determined by "5."

In short, number is a grammatical category. The predicate "x is a number" classifies expressions according to their use rather than objects according to their sort. The usual objection against such a meta-linguistic interpretation of the notion of a number is that it cannot account for the fact that there are different number signs for the same number. The proposed interpretation of (6), however, shows that this objection fails. Accordingly, the notion of being the same number must not be confused with the notion of being the same number sign. For (6) is not claiming that "5" and "V" are identical number signs; it only claims that they are interchangeable number signs. And this interpretation of the notion of being the same number is consistent with the proposed meta-linguistic interpretation of the notion of a number.

What, then, are we to make of the initial question as to what numbers are? Arguably, the question itself already suggests referentialism about number signs. For it is most naturally construed as asking to which kind of objects number signs refer. And indeed Frege interprets the question in this way when he answers it with his definition of numbers as numerical equivalence classes. Wittgenstein's rejection of the quest for a definition of the general notion of number is probably best interpreted as directed against this ontological reading of the question as to what numbers are. A proper analysis of the use of number ascriptions shows that number signs are not used to name extra-linguistic entities and thereby exposes the mistaken presupposition of the ontological interpretation of the question. The only remaining alternative, then, is a grammatical interpretation: rather than asking what conditions an object must satisfy in order to be a number, one should ask what conditions a sign must satisfy in order to be a number sign. As was argued above, the correct answer to this latter question is that a number sign must provide a multiplicity sample, i.e. a standard for numerical comparisons which pictorially represents the multiplicity of the instances of a concept. And if we bear in mind the way in which the rules of numerical generalization connect numbers signs to extension signs, then we may answer the initial question as Wittgenstein suggests: numbers are pictures of extensions.


The writing of this paper was not financed by any funding body. The authors are not aware of any financial interests or benefits arising from the direct applications of this research.


(1.) This is not, of course, to deny that there are several important proponents of a property view of numbers (cf. Rundle, 1979; Maddy, 1981; Simons, 1997; Roeper, 2016).

(2.) For a more detailed discussion of Wittgenstein's argument against the explanation of equinumerosity in terms of 1-1 correlations, see Buttner (2016).

(3.) This principle, one may note, is even compatible with the predicate view of proper names, according to which a proper name basically functions like a predicate, applying to any object bearing the name (cf. Burge, 1973). For, in the context of an extensive predicate, a proper name flanks the identity sign and is hence intended to refer to one specific object.


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Universidad del Norte, Barranquilla, Colombia

(corresponding author)


University of Bern, Switzerland

How to cite: Buttner, Kai Michael, and David Dolby (2018). "Numbers as Pictures of Extensions," Analysis and Metaphysics 17: 95-115.

Received 10 February 2018 * Received in revised form 13 March 2018

Accepted 15 March 2018 * Available online 5 April 2018
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Author:Buttner, Kai Michael; Dolby, David
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Date:Jan 1, 2018

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