Synthese (2013) 190:1099–1112 DOI 10.1007/s11229-011-9883-y

What are numbers? Joongol Kim

Received: 23 May 2009 / Accepted: 20 January 2011 / Published online: 8 February 2011 © Springer Science+Business Media B.V. 2011

Abstract This paper argues that (cardinal) numbers are originally given to us in the context ‘Fs exist n-wise’, and accordingly, numbers are certain manners or modes of existence, by addressing two objections both of which are due to Frege. First, the so-called Caesar objection will be answered by explaining exactly what kind of manner or mode numbers are. And then what we shall call the Functionality of Cardinality objection will be answered by establishing the fact that for any numbers m and n, if there are exactly m Fs and also there are exactly n Fs, then m = n. Keywords Modes of existence · The Caesar objection · The Functionality of Cardinality · Gottlob Frege 1 Introduction Imagine a mother teaching her small child how to count. She shows her child a picture of cars, and then asks how many cars there are, not as a genuine question the child is expected to answer, but as an indication of what they are doing. Then in no time she starts uttering ‘1, 2, 3’, and so on till she counts off all the cars in the picture. Notice that although she utters each number word ‘n’ in isolation, it implicitly occurs in the context ‘(so far) there are n cars’. That is, the utterance of ‘1’ means ‘(so far) there is one car’; the utterance of ‘2’ means ‘(so far) there are two cars’; and so on. So it would seem that the original context in which numbers are first given to us is (1) there are n Fs,

J. Kim (B) Department of Philosophy and Religious Studies, Western Illinois University, Macomb, IL 61455, USA e-mail: [email protected]

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which is here understood in the sense of ‘so far there are n Fs’, or equivalently, ‘there are n Fs, and possibly more’. Indeed, each individual number n can be introduced in the context of (1) in purely logical terms. That is, the condition in which we can utter the expression ‘there are n Fs’, for each n, can be specified in purely logical terms. First, we can utter ‘there is 1 F’ when there is an F. Or in symbols: (2) ∃1 x(F x) (‘there is 1 F’) ↔ ∃x(F x). And we can utter ‘there are n  Fs’ for any n ≥ 1 when there are n Fs and there is one more F. Or in symbols: (3) ∃n  x(F x) (‘there are n  Fs’) ↔ ∃1 y(F y ∧ ∃n x(F x ∧ y = x)). And, lastly, we can utter ‘there are 0 Fs’ when there is not even 1 F: (4) ∃0 x(F x) (‘there are 0 Fs’) ↔ ¬∃1 x(F x). (2)–(4) provide for a definition of every individual number word ‘n’ in purely logical terms. However, Frege raised some objections to the view that definitions like (2)–(4) could be adopted as a basis for arithmetic. In Sect. 55 of his Grundlagen Frege considered the following definitions: (5) the number 0 belongs to a concept F ↔ for any x, x is not an F; (6) the number 1 belongs to a concept F ↔ the number 0 does not belong to F, and for any x and y, if x is an F and y is an F, then x = y; (7) the number (n + 1) belongs to a concept F ↔ there is an x such that x is an F, and the number n belongs to the concept ‘is an F but is not identical with x’. Then he put forward two objections. First, “we can never—to take a crude example—decide by means of our definitions whether any concept has the number Julius Caesar belonging to it or whether that same familiar conqueror of Gaul is a number or is not” (Frege 1884, Sect. 56). The definitions (5)–(7) do not explain what the word ‘number’ means in the expression ‘the number n belongs to a concept F’. Especially, they do not say anything about what kind of thing numbers are, and so one cannot determine whether, say, Julius Caesar is a number or not merely by knowing that numbers are those things which can be introduced through (5)–(7). And if one cannot determine whether such a thing as Julius Caesar is a number, then one does not really understand what it is to be a number. That is, one cannot attain an understanding of the concept of number through definitions like (5)–(7). It would seem that Frege’s Caesar objection to (5)–(7) applies with equal force to our definitions (2)–(4), though not exactly in its original form. If numbers are those things which can be introduced in the context ‘there are n Fs’, then since, unlike the sentence ‘Julius Caesar belongs to a concept F’, the sentence ‘there are Julius Caesar Fs’ is not even well-formed, it follows that Julius Caesar cannot be a number. So we can indeed determine, by (2)–(4), whether Julius Caesar is a number. However, the basic point of Frege’s Caesar objection remains unaffected. We cannot determine by the definitions (2)–(4) whether, say, red is a number, or to take an even better example, whether many is a number. To be sure, we know that such things as red and many are

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not cardinal numbers and will never show up on the list of ‘there are 2 Fs’, ‘there are 3 Fs’, and so on; but that is not thanks to those definitions. The second objection, which I consider to be a more crucial one, is this: Moreover we cannot by the aid of our suggested definitions prove that, if the number a belongs to the concept F and the number b belongs to the same concept, then necessarily a = b. Thus we should be unable to justify the expression “the number which belongs to the concept F”, and therefore should find it impossible in general to prove a numerical identity, since we should be quite unable to achieve a determinate number. (Frege 1884, Sect. 56) Again, it would seem that this objection applies not just to (5)–(7) but also to our definitions (2)–(4). Notice that in order to establish, for instance, that the number of primes between 1 and 10 is 4, it would not suffice to show that there are exactly 4 primes between 1 and 10. Since the definite article in ‘the number of primes between 1 and 10’ indicates that 4 is the only number n such that there are exactly n primes between 1 and 10, we would also need to show that for any n, if there are exactly n primes between 1 and 10, then n = 4. Of course, it is the case that for any property F, there can be no two distinct numbers m and n such that there are exactly m Fs and also there are exactly n Fs. This is one of the most fundamental facts about cardinal numbers, and we shall henceforth refer to it as the Functionality of Cardinality, or FC for short: FC ∀F∀m∀n ((there are exactly m Fs ∧ there are exactly n Fs) → m = n). Given FC, all that is needed to establish the fact that the number of primes between 1 and 10 is 4 would be to show that there are exactly 4 primes between 1 and 10. But how could we go about proving FC by means of the definitions (2)–(4)? The consequent of the conditional in FC is a numerical identity. Could there be a criterion of numerical identity formulated in the spirit of the definitions (2)–(4) that allows us to derive the consequent, namely that m = n, from the antecedent, namely that there are exactly m Fs and there are exactly n Fs? The answer is not at all obvious. Indeed, from the Fregean viewpoint, the answer would be a clear ‘no’. For Frege, identity is a relation that only holds between objects, and objects can only be denoted by singular terms. Thus, if numbers are originally given to us in the context ‘there are n Fs’, then since they are not denoted by singular terms, they cannot be objects, which means that no genuine identity statements could be made about them. We shall refer to this second objection as the FC objection. The Caesar objection and the FC objection, considered together, pose a formidable challenge to the view that numbers originally appear in the context of (1), namely ‘there are n Fs’. The purpose of this paper is to meet this challenge squarely. First, as an answer to the Caesar objection, I will present and defend a conception of numbers as ‘adverbial entities’. Then I will show how FC can be proved in terms of an intuitive criterion of numerical identity that naturally follows from the adverbial conception of numbers.

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2 The Caesar objection What sort of thing can and should numbers be if they are originally given to us in the context ‘there are n Fs’? We have already eliminated the possibility that numbers may be (Fregean) objects, for objects can only be denoted by singular terms and the ‘n’ in ‘there are n Fs’ does not function as a singular term. If it did, then we should be able to replace it with a singular term like ‘Julius Caesar’ without affecting the grammaticality of sentences of that form. However, as noted before, sentences like ‘there are Julius Caesar moons of Mars’ are not well-formed. Also, numbers cannot be properties of individual objects like red. Because red is a property of individual objects, the sentence ‘there are red planets’ can be rephrased as ‘there are planets that are red (individually)’. However, the sentence ‘there are eight planets’ cannot be similarly rephrased as ‘there are planets that are eight (individually)’. Some have argued that numbers are collective properties of objects, that is, those properties which certain objects possess together but not individually.1 On this view, the sentence ‘there are eight planets’ can indeed be rephrased as ‘there are planets that are eight (collectively)’, or equivalently, ‘there are some things such that they are planets and they are eight’. However, as Frege (1884, Sects. 46 and 52) emphasized, the case of the number 0 shows that a number cannot be a collective property of objects. For instance, the sentence ‘there are zero moons of Venus’ cannot be rephrased as ‘there are moons of Venus that are zero’, for there are no moons of Venus in the first place that can possess the collective property of being zero. The advocate of the view of numbers as collective properties of objects normally tries to avoid the difficulty by not counting zero among numbers.2 However, the expression ‘number’ means the same in ‘the number of planets is eight’ and ‘the number of moons of Venus is zero’. Hence, any satisfactory view of numbers should be able to provide a uniform account of ‘there are eight planets’ and ‘there are zero moons of Venus’.3 The view of numbers as (individual or collective) properties of objects commits the error of severing the tie between the expression ‘there are’ and the number word ‘n’ in ‘there are n Fs’. The phrase ‘there are n’ functions as a logical unit, namely as a higher-level predicate attached to a first-level predicate ‘F’. This fact might lead one to think that numbers are higher-level properties denoted by higher-level predicates.4 However, it is the entire phrase ‘there are n’ that functions as a higher-level predicate. The ‘n’ in ‘there are n Fs’ is merely an element of the higher-level predicate ‘there are n’, and does not function in and by itself as a higher-level predicate. Hence, numbers cannot be higher-level properties.5

1 See, among others, Yi (1998, 1999) for an exposition and defense of the view. 2 See, for instance, Yi (1999, p. 189). 3 For a detailed critical discussion of the conception of numbers as collective properties, see Kim (2010). 4 An exposition of this view can be found in Carnap (1931) or Dummett (1991, Chaps. 9 and 11). 5 For details of this argument against the conception of numbers as higher-level properties, see Kim (2004,

Chap. 6).

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At this point one might conclude that there could be no such things as numbers over and above higher-level properties of the form ‘there being n’. Frege raised this possibility in his late period when it became evident to him that his life’s work ended in failure: Since a statement of number based on counting contains an assertion about a concept, in a logically perfect language a sentence used to make such a statement must contain two parts, first a sign for the concept about which the statement is made, and secondly a sign for a second level concept. … How can we get from these [second level] concepts to the numbers of arithmetic in a way that cannot be faulted? Or are there simply no numbers in arithmetic? Could the numerals help to form signs for these second level concepts, and yet not be signs in their own right? (Frege 1979, p. 257) The thought is that the ‘n’ in ‘there are n Fs’ combines with the expression ‘there are’ to form an expression that denotes a higher-level property as a whole, and yet does not denote anything in and by itself.6 However, consider what FC says: any numbers m and n must be the same if there are exactly m Fs and also there are exactly n Fs. FC can be true only if the numerical variables m and n range over the same quantifiable items in ‘there are exactly m Fs’ and ‘there are exactly n Fs’ as in ‘m = n’. Hence, since FC is a fundamental fact to be reckoned with in any philosophical foundation of arithmetic, we should find a way to make it plausible to hold that the ‘n’ in ‘there are n’ has its own denotation. If a number word ‘n’ in a higher-level predicate of the form ‘there are n’ has denotation in its own right, it must play a certain logical role of its own. What exactly is the role of the ‘n’ in ‘there are n Fs’? A clue can be obtained by considering the logical behavior of frequency adjectives. Consider the following statement: (8) There are occasional hurricanes in the southeast. The word ‘occasional’ here seems to function as an adjectival modifier of the noun phrase ‘hurricanes’. However, that is only appearance. If it were an adjectival modifier, it would mean an individual or collective property of some hurricanes. But being occasional, or occasionality, can be neither. It cannot be a property of an individual hurricane, for it makes no sense to say such things as that Katrina is occasional. Katrina happened once, and will never happen again. Indeed, something like Katrina can happen again, but it would not be Katrina. Also, occasionality cannot be a collective property of some hurricanes, for it makes no sense to say that some hurricanes such as Isabel, Ivan, Emily and Katrina are occasional (collectively). They may be said to have happened collectively in a time span of many years in the past; yet, they happened only once and will never happen again in the future. Some hurricanes will happen collectively in a time span of many years in the future, and they could indeed look like Isabel, Ivan, Emily and Katrina; yet they would not be the same hurricanes as Isabel, Ivan, Emily and Katrina. So the frequency word ‘occasional’ in (8) is not an adjectival modifier of the noun phrase ‘hurricanes’. It rather belongs to the verb phrase ‘there are occasional’, which 6 This thought has been defended in one way or another in Bostock (1974), Hodes (1984), and Rayo (2002).

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is a colloquial equivalent of the phrase ‘exist occasionally’. That is, what (8) says is the following: (9) Hurricanes exist (or happen) occasionally in the southeast. Thus, despite appearances, the adjective ‘occasional’ in (8) functions in fact as a verb-modifying adverb. Now, compare (8) with this statement: (10) There are eight planets in the solar system. Like the ‘occasional’ in (8), here the word ‘eight’ appears as an adjectival modifier of the noun phrase ‘planets’; but really it is an element in the higher-level predicate ‘there are eight’. And just as the ‘occasional’ in (8) really functions as an adverbial modifier of the verb ‘exist’, so does the ‘eight’ in (10). That is, if we use the expression ‘n-wise’ as a numerical adverb corresponding to a number word ‘n’, what (10) really says can be put as follows: (11) Planets exist eight-wise in the solar system. Thus, just like frequency ‘adjectives’, number ‘adjectives’ in fact function as verbmodifying adverbs. Verb-modifying adverbs typically denote ways or manners in which things are done, happen or obtain, and such ways or manners might be broadly called modes. Thus, the ‘n’ in ‘there are n Fs’ denotes a certain mode in which Fs exist, namely n-wise. For instance, to say that there are eight planets in the solar system is to say that planets exist in a certain manner or mode in the solar system, namely eight-wise. The number eight is that mode of existence: eight-wise. Numbers are modes of existence. There are other modes of existence than numbers. The frequency adverb ‘occasionally’ in (9) functions as an adverbial modifier of the verb ‘exist’, and so denotes a mode of existence. Indeed, any frequency word that can serve as an answer to the question ‘How often are there Fs?’ would denote a mode of existence. But number words serve to answer the question, ‘How many Fs are there?’. If we use the term ‘quantity’ broadly in the sense of any aspect of a thing that allows of comparison in size, we could say that the ‘How many?’ question concerns a certain quantity of Fs. So numbers are quantitative modes of existence. Still, not all quantitative modes of existence are numbers. Consider the sentence (12) There is little water on Mars. Here the word ‘little’ does not function as an adjectival modifier of the noun ‘water’. Compare the word ‘clean’ in the sentence ‘There is clean water in the ocean’. It functions as an adjectival modifier of ‘water’, and so we can rephrase the sentence as ‘There is water in the ocean that is clean’. (12), however, cannot be rephrased as ‘There is water on Mars that is little’. What it says is rather the following: (13) Water exists scarcely—or to coin a new word, little-wise—on Mars. So the ‘little’ in (12) functions as an adverbial modifier, and denotes a certain way or mode in which water exists on Mars. Hence, little is a mode of existence. Also, little can serve to answer the question ‘How much F is there?’. Since the ‘How much?’

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question is as much about quantity as the ‘How many?’ question in our broad sense of the term ‘quantity’, little is a quantitative mode of existence, too, and yet is not a number. What distinguishes numbers from other quantitative modes of existence like little is the fact that they can answer a ‘How many?’ question. A ‘How many?’ question concerns those things that fall under a countable concept, namely a concept F such that it is possible, in principle, to determine whether any given Fs are identical. That is, a countable concept is a concept for which there is a criterion of identity. If we call any things falling under a countable concept individuals, then we can say that numbers are quantitative modes of existence of individuals. Such things as little and much serve to answer a ‘How much?’ question, which concerns a certain stuff, namely something like water that lacks a criterion of identity and so is uncountable. Thus, little, much, and the like are quantitative modes of existence of stuffs. Even all quantitative modes of existence of individuals are not numbers. Consider the following sentence: (14) There are many moons of Jupiter. Here the word ‘many’ functions like the expression ‘sixty three’ in ‘there are sixty three moons of Jupiter’, namely as an adverbial modifier of the phrase ‘there are’, that is, ‘exist’. Thus, (14) can be rephrased as follows: (15) Moons of Jupiter exist in the mode of many, or to use a neologism, many-wise. Also, many applies to countable concepts like moon of Jupiter. Hence, many, few and the like are quantitative modes of existence of individuals. Yet, they are not cardinal numbers. Cardinal numbers are definite in that they satisfy the following condition: (16) ∀F∀G ((there are exactly n Fs ∧ there are exactly n Gs) → there is a one-one correlation between the Fs and the Gs).7 Such things as many and few fail to satisfy the definiteness condition, and as such, are indefinite quantitative modes of existence of individuals. Numbers are definite quantitative modes of existence of individuals. One might worry that on our conception of numbers as modes of existence, the number 0 should not be a number, since 0 indicates non-existence. However, as Frege correctly pointed out, “In answering ‘Zero’ [to ‘How many?’], we are not denying that there is such a number: we are naming it” (Frege 1894, p. 206). When we answer “0” to the question, ‘How many moons of Venus are there?’, we are not just denying that there are any moons of Venus, but affirming that there are 0 moons of Venus, or equivalently, that moons of Venus exist zero-wise. That is, we are naming the particular manner in which moons of Venus exist. To put it another way, the word ‘zero’ behaves differently from the word ‘no’. The word ‘no’ merely abbreviates ‘not any’, and so the statement that there are no moons of Venus is a simple negation. In contrast, the statement that there are 0 moons of Venus positively identifies the manner in which moons of Venus exist. Hence, 0 is indeed a mode of existence—though a rather peculiar one in that if any things exist zero-wise, then they do not exist. 7 In the following section it will be shown that cardinal numbers indeed satisfy this definiteness condition.

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This completes our response to Frege’s Caesar objection. The challenge brought by the objection is to provide a principle by which we can determine whether any given thing is a cardinal number or not, namely a criterion of application for the concept of a number. Our adverbial conception of numbers suggests the following criterion of application for the concept of a number: any given thing is a cardinal number if and only if it is a definite quantitative mode of existence of individuals. 3 The FC objection If numbers are originally given to us in the context ‘there are n Fs’, and so are certain ‘adverbial entities’ or modes, how can we obtain determinate numbers like the number of primes between 1 and 10 and make statements about them such as that the number of primes between 1 and 10 is 4? We can, as noted in Sect. 1, by establishing the Functionality of Cardinality, namely that FC ∀F∀m∀n ((there are exactly m Fs ∧ there are exactly n Fs) → m = n). In this section it will be shown that FC can be derived from some intuitive principles that naturally arise from our adverbial conception of numbers. A proof of FC requires a criterion of identity for numbers. Since numbers are a certain kind of manner or mode of existence, numbers should be recognized as the same if and only if they are the same manner or mode of existence. More specifically, any numbers m and n are to be recognized as the same if and only if, for any property F whatever, if Fs exist m-wise, then they must also exist n-wise, and vice versa. Here the modal force of the verb ‘must’ should be logical, for each individual number can be defined strictly in logical terms through (2)–(4). Thus, if we use ‘⇒’ for logical implication (and ‘⇔’ for logical equivalence), the criterion of identity for numbers could be formulated as follows: (17) m = n ↔ ∀F(∃m x(F x) ⇔ ∃n x(F x)). That is, m = n if and only if, for any property F, that there are m Fs logically implies that there are n Fs, and vice versa. Since individual numbers can be defined in purely logical terms, identity between numbers must be a strictly logical relation, and should not be affected by how many things there are in the given domain of discourse. This is indeed the case on our definition of numerical identity. For instance, even if there is only one object in the domain, still 2 = 3, given (17). For suppose, for reductio, that ∀F(∃2 x(F x) ⇒ ∃3 x(F x)). Let Gx be x = a ∨ x = b. Then ¬(∃3 x(Gx)), that is, it is logically false that there are (at least) 3 Gs. By the assumption for reductio, ¬(∃2 x(Gx)), that is, it is logically false that there are (at least) 2 Gs, which is absurd. The conception of numbers as modes of existence also suggests that m can be recognized as greater than n if and only if, for any property F, if Fs exist m-wise, then they must also exist n-wise, and yet it is not the case that for any property F, if Fs exist n-wise, then they must also exist m-wise. Or in symbols: (18) m > n ↔ (∀F(∃m x(F x) ⇒ ∃n x(F x)) ∧ ∃F(∃n x(F x)  ∃m x(F x))). And ‘m ≥ n’ can be defined as usual:

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(19) m ≥ n ↔ (m > n ∨ m = n). The less than relation and the less than or equal to relation are inverses of the greater than relation and the greater than or equal to relation, respectively: m < n ↔ n > m, and m ≤ n ↔ n ≥ m. In order to prove FC, it suffices to show that if there are exactly m Fs and also there are exactly n Fs, then for any property G, ∃m x(Gx) ⇔ ∃n x(Gx). And to show this, we need to analyze the expression ‘there are exactly n Fs’. Since the word ‘exactly’ in this context means the same as ‘at least and at most’, we need to analyze the expressions ‘there are at least n Fs’ and ‘there are at most n Fs’. If there are at least n Fs, then for any m such that 1 ≤ m ≤ n, there must be an F that can be associated with m and no other number. In other words, ‘there are at least n Fs’ means that there is a left-total, left-unique relation between the numbers from 1 to n and the Fs: (20) there are at least n Fs ↔ ∃R(∀m(1 ≤ m ≤ n → ∃x((F x ∧ Rmx) ∧ ∀l((1 ≤ l ≤ n ∧ Rlx) → l = m)))). And if there are at most n Fs, that means that for every F, there is a number m such that 1 ≤ m ≤ n and m is associated with the F and no other F. In symbols: (21) there are at most n Fs ↔ ∃R(∀x(F x → ∃m((1 ≤ m ≤ n ∧ Rmx) ∧ ∀y((F y ∧ Rmy) → y = x)))). And ‘there are exactly n Fs’ can be defined as ‘there are at least n Fs and there are at most n Fs’, which amounts to saying that there is a one-to-one correlation between the numbers from 1 to n and the Fs: (22) there are exactly n Fs ↔ ∃R(∀m(1 ≤ m ≤ n → ∃x((F x ∧ Rmx) ∧ ∀l((l ≤ n ∧ Rlx) → l = m))) ∧ ∀x(F x → ∃m((1 ≤ m ≤ n ∧ Rmx) ∧ ∀y((F y ∧ Rmy) → y = x)))). Notice that the definiteness condition for cardinal numbers—(16)—is a trivial consequence of (22). With these definitions at hand, we can prove FC along the following lines. Suppose there are exactly m Fs and there are also exactly n Fs. Then there is a one-one correlation between the numbers from 1 to m and the Fs, and there is also a one-one correlation between the numbers from 1 to n and the Fs. By symmetry and transitivity of one-one correlation, it follows that there is a one-one correlation between the numbers from 1 to m and the numbers from 1 to n. If so, then m must be equal to n, that is, for any property G, there are m Gs ⇔ there are n Gs. For, suppose, first, that there are (at least) m Gs. Then there is a left-total, left-unique relation between the numbers from 1 to m and the Fs, and so, since there is a one-one correlation between the numbers from 1 to m and the numbers from 1 to n, there should be a left-total, left-unique relation between the numbers from 1 to n and the Fs, which is to say that there are (at least) n Gs. Similarly, if there are n Gs, then there must be m Gs. This completes our response to the FC objection, namely that if numbers are introduced in the context ‘there are n Fs’ as in (2)–(4), then it would be impossible to obtain determinate numbers of the form ‘the number of Fs’. If there are exactly n Fs, then by FC, it follows that for any m, if there are also exactly m Fs, then m = n. But

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to say that the number of Fs = n is to say that there are exactly n Fs and for any m, if there are also exactly m Fs, then m = n. Hence, if there are exactly n Fs, then the number of Fs must be n. Since the converse is true by definition, we have established the following equivalence: (23) there are exactly n Fs ↔ the number of Fs = n. So whenever there are exactly n Fs, we can safely talk about the number of Fs. 4 Ontology and syntax The mainstream, Fregean approach to ontology and syntax is to assign different kinds of entities to different categories of expressions. According to Frege, objects are what singular terms denote; properties of objects are what monadic first-level predicates denote; second-level properties, namely properties of properties of objects, are what second-level predicates denote; and so on. The driving motivation of our own approach to ontology and syntax regarding arithmetic was to explain the (possibility of the) following fact: FC ∀F∀m∀n ((there are exactly m Fs ∧ there are exactly n Fs) → m = n). In FC the numerical variables m and n occur in adverbial position and in singular term position simultaneously. Thus, in order to explain the very possibility of the Functionality of Cardinality, it was imperative to reject the Fregean approach to ontology. In this section I will provide a more principled justification of the view that numbers can be denoted by expressions of two different logical types, namely singular numerical terms and numerical adverbs. According to Frege (1892), singular terms denote objects, namely what he called complete entities, as opposed to concepts, or more broadly, functions, namely what he called incomplete entities. It would seem, however, that anything can be denoted by a singular term if it is capable of being singled out among many. But anything can be singled out among many if it can be distinguished from all others, that is, if it is not identical with any of the others. To put it in terms of the notion of a countable concept introduced in Sect. 2, anything can be denoted by a singular term if it falls under a countable concept, namely a concept for which there is a criterion of identity. Now, there are some things that fall under a countable concept, although they are primarily denoted by certain expressions that are not singular terms. Colors are a prime example: color words primarily occur as predicative adjectives as in ‘Blood is red’, but it is possible to determine, barring borderlines cases, whether any given colors are identical. That is why we can single out certain colors using singular terms, and talk about them as in, say, ‘Red is darker than pink’. Those who follow Frege in believing that genuine singular terms cannot denote what predicates denote would attempt to paraphrase away the substantival use of color words. For instance, a Wittgensteinian might argue that ‘Red is darker than pink’ is not a genuine statement that talks about red and pink themselves, but is in fact a rule of inference that says that if one thing is red and the other pink, then it can be inferred that the former is darker than the latter.8 8 For an exposition of this view of color words, see Baker and Hacker (2005, Chap. V, esp. pp. 95–97).

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However, it would seem that such an attempt is ill-motivated. For, insofar as some things fall under a countable concept, they can be singled out using singular terms as their names. Consider the expression ‘the average American household’. This is not a genuine singular term since the average American household is not something that falls under a countable concept. But colors do fall under a countable concept, as there are quite often determinate answers to questions of the form ‘How many colors do you see in …?’, despite the fact that colors are essentially predicative or ‘incomplete’ entities. Likewise with numbers: number words ultimately function as adverbial modifiers in sentences of the form ‘Fs exist n-wise’, and yet it is possible to determine whether any given numbers are identical, for as argued in Sect. 2, identity between m and n is simply a matter of logical equivalence between ‘there are m Fs’ and ‘there are n Fs’. And the fact that there is a criterion of identity for numbers means that we should be able to single them out using numerical singular terms. For instance, since for any property F, if there are 4 Fs, then there must be also 2 + 2 Fs,9 and vice versa, we can single out 4 as equal to 2 + 2 and make the identity statement that 4 = 2 + 2. The expression ‘4’ functions as a singular term in ‘4 = 2 + 2’, and as an adverbial modifier in ‘there are 4 Fs’. However, despite that, it denotes one and the same thing, namely the number 4, in both. So the difference between its two grammatical uses does not lie in what it denotes. It rather lies in what purpose it serves: in ‘there are 4 Fs’ it serves to indicate a certain manner or mode in which Fs exist, namely four-wise, whereas in ‘4 = 2 + 2’ it serves to talk about the mode of existence itself. To sum up, singular terms do not necessarily denote (Fregean) objects. Rather, they denote what we have previously called individuals, namely those things that fall under a countable concept. Numbers, as adverbial entities, would fall on the side of (Fregean) functions or ‘incomplete’ entities. However, insofar as they are individuals, they can be denoted by numerical singular terms. To put it differently, objects, that is, ‘complete’ individuals, are not the only things that can be denoted by singular terms; ‘incomplete’ individuals such as numbers can also be denoted by them. Generally, adverbs are not known as denoting quantifiable items like numbers. But that is only because most adverbs do not denote items that can be distinguished from one another and so can be counted. Take, for instance, the expression ‘a lot’ in the sentence ‘The nose tackle weighs a lot’. It does not denote any item that can be recognized as the same; if it did, then it would follow that a car that weighs the same as the nose tackle weighs a lot, which is absurd. And most adverbs are like ‘a lot’ in that respect. However, there are adverbials—words or phrases that occur in adverb position—that denote items that fall under a countable concept, that is, individuals. Consider the following sentences: (24) The nose tackle weighs a lot—he weighs 375 pounds. (25) Mt. Everest is very high—it is 8848 meters high. (26) The car is running fast—it is running 150 miles per hour.

9 The addition operation can be defined in the context ‘there are n Fs’ thus: ∃ m+n x(F x) (‘there are

m + n Fs’) ↔ ∃G(∃m x(F x ∧ Gx) ∧ ∃n x(F x ∧ ¬(Gx))).

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The two italicized expressions in each of (24)–(26) function in the exact same manner. For instance, both ‘a lot’ and ‘375 pounds’ in (24) function as adverbial modifiers of the verb ‘weighs’. Yet, unlike ‘a lot’, ‘375 pounds’ denotes an item that can always be recognized as the same. That is why if a car weighs the same as the nose tackle, then we can say rightly that it weighs 375 pounds. Similarly, the adverbial ‘8848 meters’ in (25) and the adverbial ‘150 miles per hour’ in (26) denote certain items that fall under countable concepts—a height and a speed, respectively. These examples suggest that expressions for magnitudes including—but not limited to—weights, heights and speeds primarily function as adverbial modifiers, and yet can denote individuals. And the fact that magnitudes like weights fall under a countable concept and so are individuals means that we should be able to single them out using singular terms and talk about them, even though they are primarily denoted by adverbials. For instance, we should be able to predicate something of a weight as in ‘375 pounds is a heavy weight even for a nose tackle’. Also, we should be able to make an identity statement about weights as in ‘375 pounds = 170 kilograms’. Moreover, in order to capture the logical structure of certain valid inferences involving magnitude expressions like ‘375 pounds’, it is necessary to allow those expressions to occur both in adverbial position and in singular term position simultaneously. Consider the inference from (24) and the sentence ‘No one who weighs more than 250 pounds can play quarterback’ to the sentence ‘The nose tackle cannot play quarterback’. Using w as a variable for weights, we can symbolize the second premise as follows: (27) ∀x∀w ((x weighs w ∧ w > 250 pounds) → x cannot play quarterback). The conclusion follows from (24), (27) and the fact that 375 pounds > 250 pounds. Also, consider the sentence (28) ∀x∀v∀w ((x weighs v∧ and x weighs w) → v = w), where v and w are variables for weights. (28) states an important fact, namely that weight is functional in the sense that nothing can have no two distinct weights (at the same time). It is because of this fact that we can infer from (24) that the weight of the nose tackle is 375 pounds. Note that the variable w in (27) and the variables v and w in (28) occur both in adverbial position and in singular term position. A lesson to be drawn from these considerations is that the standard practice of interpreting singular terms as names of objects (as opposed to properties, relations, and functions in general) should be revised so as to allow singular terms to denote different sorts of individuals including adverbial entities like numbers and magnitudes. Such a revision would in practice mean adopting a many-sorted logic with many—that is, two or more—sorts of variables. For instance, consider our criterion of numerical identity: (17) m = n ↔ ∀F(∃m x(F x) ⇔ ∃n x(F x)). Here m and n occur as parameters, but we can bind them by universal quantifiers and thereby turn (17) into a proposition, namely: (29) ∀m∀n(m = n ↔ ∀F(∃m x(F x) ⇔ ∃n x(F x))). Two sorts of individual variables occur in (29): the standard individual variable x, which occurs in singular term position, and the numerical variables m and n, which

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occur both in adverbial position and in singular term position. The latter might be called specific individual variables in the sense that they range over individuals of a specific sort, namely numbers, and the standard x and its ilk might be called generic individual variables in the sense that individuals of any sort whatever can be their values. Thus, an interpretation of (29) would require two separate domains of discourse: one for what are used to count, namely numbers, and the other for what are counted, namely individuals of any given sort. Since all and only countable things are individuals, the fact that individuals of any sort can be values of generic variables means that any countable things, including numbers, can be counted in this two-sorted logic. 5 Conclusion In Grundlagen Sect. 56 Frege rejected the definitions (5)–(7) because he could not see how the Caesar objection and especially the FC objection to them could be overcome. Then he went on to argue that in arithmetic number words most typically appear in identities like ‘1 + 1 = 2’ (Sect. 57), and so numbers should be introduced in the context of an identity—more specifically, in the context ‘the number of Fs = the number of Gs’ (Sect. 62). Of course, in ordinary discourse number words commonly appear in adjectival constructions as in ‘There are four moons of Jupiter’. But, Frege insisted, “to arrive at a concept of number usable for the purposes of science” those ordinary statements involving an adjectival use of a number word should be converted into identity statements like ‘The number of moons of Jupiter is four’ (Sect. 57). However, the conversion strategy allowed Frege to prove FC only by trivializing it and so evading the real challenge it posed. To see this, note that given the conversion strategy, FC should be rephrased as follows: (30) ∀F∀m∀n ((the number of Fs = m ∧ the number of Fs = n) → m = n). This is a truism based on the laws of identity. However, the advantage Frege thus gained from the conversion strategy is comparable to the advantage of theft over honest toil. (30) employs the expression ‘the number of’, and thus assumes the Functionality of Cardinality. It explains nothing as to why cardinality is functional. The FC objection poses a genuine challenge to any philosophy of arithmetic, namely to explain the fact that FC ∀F∀m∀n ((there are exactly m Fs ∧ there are exactly n Fs) → m = n). And to meet this challenge, one cannot use expressions of the form ‘the number of Fs’, and should allow the numerical variables m and n to occur in the context ‘there are n Fs’. To conclude, the fact that there are determinate numbers of the form ‘the number of Fs’ does not support Frege’s conception of numbers as objects. On the contrary, it forces us to reckon with the Functionality of Cardinality, which in turn forces us to conceive numbers as adverbial entities that are originally given to us in the context ‘there are n Fs’, or more precisely, ‘Fs exist n-wise’. Acknowledgments An early version of this paper was presented under the title of “An Adverbial Theory of Numbers” at the 2005 American Philosophical Association Eastern Division Meeting in New York.

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Thanks to the audience at the session, and especially to Agustín Rayo, who provided commentary. I am also grateful to an anonymous reviewer of this journal for helpful comments.

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