Andrew D. Irvine

Frege on Number Properties

Abstract. In the Grundlagen, Frege offers eight main arguments, together with a series of more minor supporting arguments, against Mill’s view that numbers are “properties of external things”. This paper reviews all eight of these arguments, arguing that none are conclusive. Keywords: Frege, Mill, mathematical realism, scientific realism, number properties.

Introduction Frege is someone not often noted for his sympathy to the view that mathematics involves the study of physical properties. Even so, he thought the idea worth considering, at least in the case of arithmetic. In the Grundlagen [10] he introduces his discussion of the position as follows: In language, numbers most commonly appear in adjectival form and attributive construction in the same sort of way as the words hard or heavy or red, which have for their meanings properties of external things. It is natural to ask whether we must think of the individual numbers too as such properties, and whether, accordingly, the concept of Number can be classed along with that, say, of colour. (§21, p. 27)1 Should number be classified along with colour and other physical properties? Mill’s answer to this question had clearly been in the affirmative. According to Mill [18], each numeral connotes a physical property belonging 1 The failure to distinguish between use and mention appears in the original. All passages by Frege quoted in this paper come from this translation. Contemporary discussions of Frege include Geach [11], Resnik [20], Wright [24], Dummett [8], and Burgess [7]. 2

The literature on mereology does not contain terminological uniformity. By “mereological whole” I mean largely the same as Mill [18] and Frege [10] mean by “agglomeration”, as W.V. Quine [19] means by “scattered object”, as David Lewis [16] means by “fusion”, as Joan Weiner [23] means by “conjunctiva”, as Tyler Burge [6] and David Armstrong [2] mean by “aggregate”, and as Peter Van Inwagen [22] and David Armstrong [1] mean by “sum”. Occasionally, when confusion seems unlikely to arise, I use the term “object” as a synonym. Special Issue: Philosophy of Mathematics Edited by Andr´e Fuhrmann, Ivan Kasa and Manfred Kupffer

Studia Logica (2010) 96: 239–260 DOI: 10.1007/s11225-010-9285-z

© Springer 2010

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to a particular kind of agglomeration or mereological whole.2 As he explains it, each mathematical definition is composed of two things, the explanation of a name, and the assertion of a fact: of which the latter alone can form a first principle or premise of a science. The fact asserted in the definition of a number is a physical fact. Each of the numbers two, three, four &c., denotes physical phenomena, and connotes a physical property of those phenomena. Two, for instance, denotes all pairs of things, and twelve all dozens of things, connoting what makes them pairs or dozens; and that which makes them so is something physical; since it cannot be denied that two apples are physically distinguishable from three apples, two horses from one horse, and so forth: that they are a different visible and tangible phenomenon. (bk 3, ch. 24, §5, in vol. 7, p. 610)3 Mill goes on, “What, then, is that which is connoted by a name of number? Of course, some property belonging to the agglomeration of things which we call by the name; and that property is, the characteristic manner in which the agglomeration is made up of, and may be separated into, parts” (bk 3, ch. 24, §5, in vol. 7, p. 611). This “characteristic manner” of separation or division, Mill believes, has to be physical since it is present within the physical phenomena itself. Frege disagreed. As he sees it, “Number is not abstracted from things in the way that colour, weight and hardness are” (§45, p. 58). As a result, “number is neither spatial and physical, like Mill’s piles of pebbles and gingersnaps, nor yet subjective like ideas, but non-sensible and objective” (§27, p. 38).4 In other words, according to Frege, Mill is mistaken to think of number as a natural property of ordinary, physical objects. In defence of his view, Frege gives a protracted discussion in §§21-27, 29-31 and 46 of the Grundlagen, explaining why he believes numbers are not “properties of external things”. His discussion can be divided into eight main arguments, together with a series of more minor supporting arguments. Central to his main arguments are the following eight claims: 1. that there is no single or “characteristic” manner by which an agglomeration or mereological whole may be divided or separated into parts; 3

The failure to distinguish between use and mention again appears in the original. All passages by Mill quoted in this paper come from this edition. Contemporary discussions of Mill include Britton [5], Kessler [13], Kitcher [14], Resnik [20], Kitcher [15], and Bostock [3]. 4 Later Frege makes much the same claim in §45 when he comments that “Number is not anything physical, but nor is it anything subjective (an idea)” (p. 58).

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2. that the actual manner by which an agglomeration or mereological whole is divided or separated into parts is not physical; 3. that, unlike physical properties, number properties do not have division or composition of application; 4. that one, although a number, is not a physical property; 5. that zero, although a number, is not a physical property; 6. that very large numbers are not physical properties; 7. that the word “one” cannot function as a predicate; and 8. that unlike physical properties, number properties also apply to nonphysical entities. Each of these claims is worth considering. It turns out that none is conclusive against the view that mathematics involves the study of physical properties. Frege’s basic observation is that, unlike the identification of physical properties such as hardness or heaviness or redness, counting needs to occur qua some characteristic manner of division, qua some way of dividing a mereological whole into parts. Frege refers to the ground of these divisions as concepts. Like Plato’s forms, Frege’s concepts are intended to be objective but non-physical. But must these “manners of division” be, as Frege suggests, separate from the physical world? Or might they be found in the nature of physical objects themselves? In contrast to Frege’s view, scientific realism is the view that there exist in nature observer-independent physical properties, relations and particulars, and that science is a fallible but reliable guide to discovering their existence. We thus have two distinct theories. The first is that in nature there exists only a single, undifferentiated mass of stuff, a mereological whole that obtains its structure only through the imposition of non-physical concepts. The second is that in nature there exist objects structured by physical properties and relations.5 Frege’s account favours the first of these two views; Mill’s account favours the second. In what follows, each of Frege’s arguments suggesting that nonphysical concepts are preferable to Mill’s “properties of external things” is shown to fail. It is then suggested that in place of Frege’s nonphysical concepts, it is at least as plausible to rely on the existence of observer-independent physical 5

In the remainder of this paper I will use the term “properties” to refer to both properties and relations. See Kessler [13] for a more nuanced account.

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properties, and that it is these physical properties that give the world the mathematical and scientific structure it has.6

1.

Frege’s First Argument

Frege’s first and most often-cited argument is that there is no single or “characteristic” manner by which an agglomeration or mereological whole may be divided or separated into parts. He makes the point in several ways. For example, just as a pile of playing cards may be viewed as a fixed number of cards or as a different number of packs (§22, pp. 28-29), the same will be true of other objects as well: the Iliad may be viewed as one poem, as twentyfour books or as some large number of verses (§22, p. 28); a pile of straw may be thought of as some small number of bundles, as some larger number of individual straws or as some even larger number of cells or molecules (§23, p. 30); a grove of trees may be thought to be a single copse or five individuals trees (§46, p. 59); four companies may be thought of instead as 500 men (§46, p. 59). Frege tells us that in all such cases the number assigned depends on more than just the mereological whole itself. It also depends on the word or concept we choose to apply to it. There are, after all, only three possibilities: 1. the mereological whole alone will contain a unique way of dividing itself into units; 2. the mereological whole alone will contain multiple ways of dividing itself into units; or 3. the mereological whole alone will contain no way of dividing itself into units (§22, p. 28). As the above examples show, the first of these three possibilities is clearly false. If a mereological whole contained only a single, unique way of dividing itself into units, a single poem could not also be twenty-four books. Four companies could not also be 500 men. The second possibility must also be false, says Frege, since, if true, it would entail a contradiction: just as no object can have incompatible physical properties, no object can have incompatible arithmetical properties. No poem can be both one and twentyfour. No pile can be both four and fifty-two. As a result, we’re forced to conclude that the third possibility is correct, that the whole alone contains no way of dividing itself into units, that there must be something separate 6

For example, see Franklin [9].

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from, or external to, each object (namely a word or concept) that does this division for us. As Frege summarizes, “If I can call the same object red and green with equal right, it is a sure sign that the object named is not what really has the green colour; for that we must get a surface which is green only. Similarly, an object to which I can ascribe different numbers with equal right is not what really has a number” (§22, p. 29). But surely this is too quick: contra Frege, there may be any number of purely physical factors that allow a mereological whole to be divided into units. Just as the area of a large square may be identified with the area of four smaller squares (or quadrants), and at the same time also may be identified with the area of two mid-sized rectangles (each of which is identified with the area of two of the smaller quadrants), a mereological whole may turn out to be composed of a variety of different parts, each having different number properties associated with them. In both cases it will be the complexity of physical composition that allows us to focus our attention on one number rather than the other. In other words, it is because the number of straws (or cells or molecules) in a bundle is fixed independently of the words or concepts an observer might use that the observer may be mistaken in claiming that there are six straws when in fact there are five. Simply put, it is the bundle itself that is composed of straws and cells and molecules, all of which exist independently of the observer. Physical objects may thus possess a numerical complexity that Frege finds implausible. Identifying this complexity may not always be easy. Pointing to an object, for example, will never by itself be sufficient to indicate a unique number property, just as pointing to an object will never by itself be sufficient to indicate an object’s shape, as opposed to its colour or volume, or even the object itself.7 The pointing will have to be accompanied by some way of recognizing, or drawing our attention to, the fact that it is the shape or colour or volume, or the object itself, that we want to indicate. But this need not detract from the physical reality of such properties. Nor need it detract from the physical reality of an object’s corresponding number properties. The fact that there may exist more than one division of shape or division of number associated with a single mereological whole becomes irrelevant. The claim that there is no “characteristic manner” by which an agglomeration or mereological whole may be divided or separated into parts is thus at best misleading. Even if literally true (in the sense that there is no 7 For example, recall the famous discussion in Quine [19] of “gavagai” (ch. 2). In contrast, Frege’s comment “I can point to the patch of each individual colour without saying a word, but I cannot in the same way point to the individual numbers” (§22, p. 29) appears to rely on a distinction that doesn’t exist.

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single or unique such property), this is not a reason to doubt that number terms involve reference to the properties of “external things”. Put another way, Frege is mistaken to believe that a mereological whole containing multiple ways of dividing itself into units must necessarily be selfcontradictory. Provided that these ways of dividing an object into units are appropriately independent of one another, they may coexist without contradiction. Of course a card and a pack are not independent in the sense that they are autonomous of one another, i.e., in the sense that the destruction of the card would leave the pack unaffected; but they are independent in the sense that the existence of the card is not incompatible with the existence of the pack, in a way that the existence of a green surface is incompatible with the existence of a red surface. Our challenge is thus not that mereological wholes fail to present us with unique manners of division; rather, it is that until we focus upon only one such manner of division, each mereological whole presents us with many such manners of division and with a degree of numerical complexity unlike that of colour. (At least this appears to be true of the phenomenological experience of colour, if not of that portion of the underlying electromagnetic spectrum that we refer to as “coloured light”.) The reason this complexity is not contradictory lies in the kind of independence the underlying physical properties have from one another. The properties of being both a card and a pack, or of being both a cell and a molecule, are distinct from one another in a way that the properties of being both red and green are not, and it is this distinction that allows for the existence of distinct number properties.8 As a result, Frege is wrong to confuse objective manners of physical division with unique manners of physical division (calling them both characteristic manners of division), and it is this equivocation that leads to his mistaken assumption that number properties must be ontologically separate from their corresponding mereological wholes. Put yet another way, Frege understands counting to be counting qua some attribute. Because of this, he concludes that “Number is not anything physical” (§45, p. 58), and so he requires the postulation of his nonphysical “concepts”. But this is a clear non sequitur. Just because counting turns out to be qua some attribute, some feature of the mereological whole, it need not follow that this feature must be external to the object in question. If the physical world is rich enough to include a variety of properties (the property of being a pack, of being a suit, of being a red card, etc.) then, even if Frege 8 In the terminology of Johnson [12], red and green are incompatible determinates of the determinable colour. But since for “any given determinate, there is only one determinable to which it can belong” (p. xxxv), being a cell and being a molecule are not determinates of a determinable such as being a particular or being a physical object.

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is correct that counting must occur qua some feature of the whole, there still will be no need to postulate some non-physical concept rather than (or in addition to) a purely physical property.9 In other words, scientific realism the acceptance of observer-independent physical particulars and properties is surely no less plausible an explanation than the suggestion that, in nature, there exists only a single, undifferentiated mass of stuff, a mereological whole that obtains its structure solely through the imposition of some external, non-physical concept.

2.

Frege’s Second Argument

Frege’s second argument is that the actual way an agglomeration or mereological whole is divided or separated into parts depends on the way in which we choose to regard it. As a result, says Frege, this division depends on something that is not physical, namely the mind: The Number 1, on the other hand, or 100 or any other Number, cannot be said to belong to the pile of playing cards in its own right, but at most to belong to it in view of the way in which we have chosen to regard it; and even then not in such a way that we can simply assign the Number to it as a predicate. What we choose to call a complete pack is obviously an arbitrary decision, in which the pile of playing cards has no say. But it is when we examine the pile in the light of this decision, that we discover perhaps that we can call it two complete packs. (§22, p. 29)10 It is in this context that Frege approvingly quotes Berkeley: “According as the mind variously combines its ideas, the unit varies; and as the unit, so the number, which is only a collection of unity, doth also vary. We call a window 9

Although Mill would disagree with Frege’s claim that “the content of a statement of number is an assertion about a concept” (§46), it’s important to understand the point of the disagreement. Frege’s underlying claims concerning the formal structure of arithmetic are not what is in dispute. For example, recall Frege’s observation that statements involving number are not of the form “ϕa”, where “ϕ” is to be replaced by an expression standing for a first-level concept, but instead are of the form “(∃n x)ϕx”, where the numerical quantifier “(∃n x)” stands for a second-level concept in whose scope falls the first-level concept designated by the predicate replacing “ϕ”. What would Mill say of such a claim? Only that Frege’s so-called “concepts” (whether first- or second-order) must be replaced by properties that are purely physical (something that Frege repeatedly denies), and that such properties (or connotations) are fully present within the mereological whole itself. The basis of the dispute is thus one of ontology, not logic. 10 But note Frege’s comment regarding a flower’s petals in §26 (p. 34), discussed below.

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one, a chimney one, and yet a house in which there are many windows, and many chimneys, hath an equal right to be called one, and many houses go to the making of one city” (§25, p. 33). There is something right and something wrong about these observations. While it is obviously correct that we are free to regard a pile of playing cards as either fifty-two cards or one pack, and equally correct that we are free to define words however we please, neither of these observations is sufficient to show that numbers are to be assigned to a whole simply as a result of the way in which “we have chosen to regard it”. Simply put, it is an error to conclude that counting is an “arbitrary decision, in which the pile of playing cards has no say”. If this were true, we could arbitrarily regard the same mereological whole as fifty-three cards rather than fifty-two, or as six packs rather than one. Since this is clearly false, Frege is too quick to conclude that number “cannot be said to belong to the pile of playing cards in its own right”. To see this, consider more carefully Frege’s claim that “What we choose to call a complete pack is obviously an arbitrary decision”. How shall we interpret Frege here? He may intend this claim in any of three separate senses: 1. He may mean our choice to be arbitrary in an unrestricted ontological sense. In other words, he may mean that we are arbitrarily able to regard a pile as fifty-three cards, or as any number of cards, rather than as fiftytwo. He may mean that there are no features internal to the mereological whole that restrict our choice of number. 2. He may mean our choice to be arbitrary in a restricted ontological sense. In other words, he may mean that although it is an arbitrary decision whether we regard a pile as one pack or as fifty-two cards, we are not free to regard it as six packs or fifty-three cards. He may mean that there exists a restricted range of possibilities from among which, and only which, we are free to choose. 3. He may mean our choice to be arbitrary in a purely definitional sense. In other words, he may mean that we are free to define our words and phrases (“one pack”, “fifty-two cards”) however we want. He may mean that our use of language is unrestricted in the sense that words and phrases can be used to mean anything we want them to mean, so the phrase “two packs” might mean what we now refer to in standard English as “one pack”. Each of these three possible interpretations can be shown to be insufficient for Frege’s purposes.

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If Frege means our choice to be arbitrary in a completely unrestricted ontological sense then his position is obviously wrong. A pack of fifty-two cards no more contains fifty-three cards than it contains one hundred and fifty-three cards. There is something objective, something in nature, literally in the world, that determines how many cards are in the pile and this is something over which our words and concepts have no influence. No one, simply by changing how a mereological whole is considered, is able to change it from a pile of fifty-two cards to a pile of fifty-three cards. If Frege means our choice to be arbitrary in a restricted ontological sense, then we are entitled to ask what it is that governs the restrictions under which we are required to function. For example, why are we entitled to select between one pack and fifty-two cards, but not between one pack and six packs, or between fifty-two cards and fifty-three cards? The most natural response, once again, is to note that it is the physical structure of the world (and not any arbitrary application of non-physical concepts or of the mind) that plays this role. A wide variety of distinct numbers may be discovered to apply to each mereological whole, but these numbers will be fixed by the physical structure contained within the mind-independent, mereological whole itself, and nothing else. Seen from this point of view, while it is true to say that we as observers may wish to concentrate our attention upon one of these numbers rather than another, and while it is true to say that this will be a function of “the way in which we have chosen to regard it”, this is an entirely separate issue from what it is that gives the object the number (or numbers) it has. Finally, if Frege means our choice to be arbitrary in a purely definitional sense, then he is correct that we are free to define our words however we may choose; but this observation has nothing to do with how many cards or packs appear in front of us. Of course, the language we use has everything to do with how we choose to report such facts; but even though we are free to choose to report the number of cards in front of us using a familiar language such as English or some other more exotic notation in which “five” means six and “six” means five, this has nothing to do with what it is that makes the number of cards what it is. Put simply, just as talk of inches rather than centimeters has nothing to do with whether a three-inch peg will fit into a two-inch hole, talk of 26-card decks rather than 52-card decks has nothing to do with how many cards or decks exist.11 11

As Abraham Lincoln famously asked, if you call a calf’s tail a leg, how many legs does it have? His answer: Four. Calling a tail a leg doesn’t make it one. See Lincoln’s “Five-legged Calf” in McClure [17].

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Thus in all three cases, the division of a mereological whole into units has everything to do with the underlying nature of the mereological whole itself and nothing to do with arbitrary linguistic or conceptual choices made by the observer. Frege also misleads us when he suggests that “what changes here from one judgement to the other is neither any individual object, nor the whole, the agglomeration of them, but rather my terminology” (§46, p. 59). The same happens when he says “it is quite true that, while I am not in a position, simply by thinking of it differently, to alter the colour or hardness of a thing in the slightest, I am able to think of the Iliad either as one poem, or as 24 Books, or as some large Number of verses” (§22, p. 28). No one, simply by thinking of an object differently, is able to alter the number of books from twenty-four to, say, twenty-five. All we can do is change our attention from, say, the number of books to the number of verses, just as we can change our attention from the colour of an object to its shape, or change the language we choose to use to report such facts, for example by switching from decimal notation to binary notation, so that “10” no longer represents ten. But, once again, these remain entirely separate matters. Even if counting needs to occur qua some feature (as seems plausible), it doesn’t follow that this feature must be mental or linguistic or in some other way external to the mereological whole itself. Similarly, when Frege draws our attention to how physical arrangement relates to numbers, we must again be careful not to be misled. Consider again Frege’s example of a given number of straws: [N]eed the straws form any sort of bundle at all in order to be numbered? Must we literally hold a rally of all the blind in Germany before we can attach any sense to the expression “the number of blind in Germany”? Are a thousand grains of wheat, when once they have been scattered by the sower, a thousand grains of wheat no longer? Nor does it make any difference whether the events occur together or thousands of years apart. (§23, p. 30) Here Frege is correct to point out that physical arrangement, whether in time or in space, is something distinct from number, but this shows only that our discovery or observation of number properties may depend on physical or temporal arrangement in a way that the existence of number properties does not. Similarly, it is a mistake to conclude, as Frege does, that differences of number appear where no physical differences exist:

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Mill is, of course, quite right that two apples are physically different from three apples, and two horses from one horse; that they are a different visible and tangible phenomenon. But are we to infer from this that their twoness or threeness is something physical? One pair of boots may be the same visible and tangible phenomenon as two boots. Here we have a difference in number to which no physical difference corresponds; for two and one pair are by no means the same thing, as Mill seems oddly to believe. (§25, pp. 32–33) But this is too quick. For his part, Mill would surely deny being guilty of any such error. As Mill explains, The expression “two pebbles and one pebble”, and the expression, “three pebbles”, stand indeed for the same aggregation of objects, but they by no means stand for the same physical fact. They are names of the same objects, but of those objects in two different states: though they denote the same things, their connotation is different. Three pebbles in two separate parcels, and three pebbles in one parcel, do not make the same impression on our senses; and the assertion that the very same pebbles may by an alteration of place and arrangement be made to produce either the one set of sensations or the other, though a very familiar proposition, is not an identical one. (bk 2, ch. 6, §2, in vol. 7, p. 256) In other words, the key difference between Frege and Mill is that Mill sees the ground of number claims as being internal to a mereological whole in a way that Frege does not. As both Frege and Mill agree, “two” and “one pair” refer to something identical. Both terms pick out exactly the same mereological whole or physical extension. But from this it needn’t follow that it is something external to this extension, to the “aggregation of objects”, that provides the basis for distinguishing between the two terms. Mill’s physical connotations12 appear to work every bit as well as Frege’s nonphysical concepts.13 In other words, just because we are able to distinguish between distinct expressions referring to the same extension, it need not follow that such expressions automatically will require the existence of non-physical properties. So far, at least, Frege has given us no reason to think they will. 12

That Mill intends his connotations to be physical is clear. See not just the above quotation, but also bk 3, ch. 24, §5 (in vol. 7, p. 610). 13 That Frege intends his concepts to be nonphysical is also clear. See not just the above quotation, but also §27 (p. 38) and §45 (p. 58).

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This brings us to Frege’s discussion of objectivity. It is Frege’s view that, although statements involving number are not about properties of external things, they are nevertheless objective. As he explains it, the botanist intends to assert something just as objective when he gives the number of a flower’s petals as when he gives their colour. As a result, “The one depends on our arbitrary choice just as little as the other” (§26, p. 34). It follows that “There does, therefore exist a certain similarity between Number and colour; it consists, however, not in our becoming acquainted with them both in external things through the senses, but in their being both objective” (§26, p. 34). Frege goes on to say that it is important to distinguish things that are “objective” from “what is handleable or spatial or actual” (§26, p. 35). For example, the axis of the earth and the centre of mass of the solar system are both objective, but they are not physical. Similarly, we can talk of the equator as an imaginary line, but as Frege reminds us, “it would be wrong to call it a fictitious line; it is not a creature of thought, the product of a psychological process, but is only recognized or apprehended by thought” (§26, p. 35). As he further explains, “If to be recognized were to be created, then we should be able to say nothing positive about the equator for any period earlier than the date of its alleged creation” (§26, p. 35). As a result, he understands the term “objective” to mean that which “is independent of our sensation, intuition and imagination, and of all construction of mental pictures out of memories of earlier sensations, but not what is independent of the reason” (§26, p. 36). As insightful as these remarks are, they make it difficult to reconcile Frege’s understanding of objectivity with his earlier remarks about the arbitrary application of concepts. On the one hand, we are told that the botanist intends to assert something wholly objective when he gives the number of a flower’s petals, that the counting of petals “depends on our arbitrary choice just as little as” the assignment of colour (§26, p. 34, emphasis added). On the other hand, we are told that number “cannot be said to belong to the pile of playing cards in its own right, but at most to belong to it in view of the way in which we have chosen to regard it” (§22, p. 29, emphasis added) and that “What we choose to call a complete pack is obviously an arbitrary decision, in which the pile of playing cards has no say” (§22, p. 29, emphasis added). One way to reconcile this tension is to conclude that it is in the purely definitional sense that Frege intends his remark that “What we choose to call a complete pack is obviously an arbitrary decision, in which the pile of playing cards has no say”; but if so, his conclusion that “it is when we

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examine the pile in the light of their decision, that we discover perhaps that we can call it two complete packs” (§22, p. 29) becomes even more clearly a point merely about linguistic usage, having nothing to do with the actual division of objects into parts. In contrast, if it is Frege’s intention merely to remind us that being objective means being independent of sensation, intuition and imagination, then this leaves completely open the question of the ground of this objectivity and, so far at least, Frege has given us no reason to look beyond the physical. Simply put, the denial of psychologism by itself need not lead to Platonism. For the anti-Platonist, since the actual way an agglomeration or mereological whole is divided or separated into parts is a function solely of the nature of the mereological whole itself, it will be the mereological whole and nothing else that serves as the ground for this objectivity. Surely this is just as plausible an explanation as the postulation of non-physical concepts, if not more so.

3.

Frege’s Third Argument

Frege’s third argument is that, unlike physical properties, number properties do not have division or composition of application. For example, the division of a coloured object will result in objects sharing that same colour, but the division of a numbered object will not result in objects sharing that same number. (At least this appears plausible in finite cases.) Similarly, the composition of an object from other objects all sharing one colour will result in a new object of that same colour, but the composition of an object from other objects all sharing one number will not result in a new object of that same number. (Again, this is something that appears plausible in finite cases.) As Frege puts it, Is it not in totally different senses that we speak of a tree as having 1000 leaves and again as having green leaves? The green colour we ascribe to each single leaf, but not the number 1000. If we call all the leaves of a tree taken together its foliage, then the foliage too is green, but it is not 1000. To what then does the property 1000 really belong? It almost looks as though it belongs neither to any single one of the leaves nor to the totality of them all [i.e. the agglomeration]; is it possible that it does not really belong to things in the external world at all? (§22, p. 28) Later, he makes much the same point when he says, “This is even clearer if we take the plural. Whereas we can combine ‘Solon was wise’ and ‘Thales

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was wise’ into ‘Solon and Thales were wise’, we cannot say ‘Solon and Thales were one’. But it is hard to see why this should be impossible, if ‘one’ were a property both of Solon and of Thales in the same way that ‘wise’ is” (§29, pp. 40–41). The problem with this argument is that there is no reason why we should accept the assumption that composition and division apply uniformly to all physical properties. Individual water molecules need not (and in fact do not) share the same freezing point as water cooled by the gallon; adding a second, equal source of electrical power results in a change of voltage, not a constant; and even though each member of a pair of shoes is shoe-shaped, the fact that the pair is not does not disqualify shape (or mass or weight or electrical charge or charm or any other physical property) from being physical. In the case of Solon and Thales, although it is possible to say that each member of the pair is one even though the pair itself is two, it is also possible to say of the pair that it is one (as Frege himself has already noted); but accepting either of these observations as a fundamental insight into the nature of number would be as mistaken as Frege’s apparent acceptance of universal composition and division. The preservation of properties through composition and division turns out to be a defining characteristic of neither physical nor nonphysical properties, so Frege’s third argument is misconceived right from the start.

4.

Frege’s Fourth Argument

Frege’s fourth argument is that one, although a number, is not a physical property. In support of this claim, Frege gives two quite separate kinds of argument. The first is that it can hardly “make sense to ascribe the property ‘one’ to any object whatever, when every object, according as to how we look at it, can be either one or not one” (§30, p. 41). Approached from a slightly different direction, he also makes the argument as follows: It must strike us immediately as remarkable that every single thing should possess this property. It would be incomprehensible why we should still ascribe it expressly to a thing at all. It is only in virtue of the possibility of something not being wise that it makes sense to say “Solon is wise”. The content of a concept diminishes as its extension increases; if its extension becomes all-embracing, its content must vanish altogether. (§29, p. 40) In other words, if properties are used to distinguish one thing from another, and if the property of being one literally applies to everything, one can hardly be thought a property. Put in yet other words, if the application of oneness is

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completely arbitrary, i.e. if it applies regardless of the nature of the object in question, or if the predicate’s “extension becomes all-embracing”, then “its content must vanish altogether”. For scientific realists, this is the beginning of an attractive argument. Even so, it is a mistake to claim that the number one applies universally. Twenty-four books are twenty-four, not one. Five trees are five, not one, and it is an error to confuse one pack with four suits, as Frege himself has repeatedly reminded us. In other words, since “is one” does not apply universally, the property of being one may be used to distinguish between objects in just the same way as any other physical property. Put another way, if “every object, according as to how we look at it, can be either one or not one” (§30, p. 41), we are entitled to look for the reason or ground that the object sometimes can be looked at as being one, and sometimes can be looked at as not being one. In each case, Frege has failed to give us a convincing reason for thinking that this ground must be non-physical. Frege’s second argument in this context concerns the purported difference between skill at individuation and knowledge of the number one. For Frege, the two must be separate since, on his view, statements about number are ultimately assertions about concepts, not about physical objects. As he puts it, if Mill’s theory is correct and if number assertions are really about physical objects or properties, then “we should have to expect animals, too, to be capable of having some sort of idea of unity” (§31, p. 41) simply because they are capable of interacting with and distinguishing between individual objects. But this, says Frege, appears completely implausible: Can it be that a dog staring at the moon does have an idea, however ill-defined, of what we signify by the word “one”? This is hardly credible and yet it certainly distinguishes individual objects: another dog, its master, a stone it is playing with, these certainly appear to the dog every bit as isolated, as self-contained, as undivided, as they do to us. It will notice a difference, no doubt, between being set on by several other dogs and being set on by only one, but this is what Mill calls the physical difference. We need to know specifically: is the dog conscious, however dimly, of that common element in the two situations which we express by the word “one”, when, for example, it first is bitten by one larger dog and then chases one cat? This seems to me unlikely. (§31, pp. 41–2) The lesson we should draw from this, says Frege, is that it is only through a distinct act of the mind that human beings are able to link ordinary physical

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I infer, therefore, that the notion of unity is not, as Locke holds, “suggested to the understanding by every object without us, and every idea within”, but becomes known to us through the exercise of those higher intellectual powers which distinguish men from brutes. Consequently, such properties of things as being undivided or being isolated, which animals perceive quite as well as we do, cannot be what is essential in our concept. (§31, p. 42) The problem with this argument is that it again involves a clear non sequitur. Does it really follow from the fact, if it is a fact, that skill at individuation is something distinct from knowledge of the number one, that number claims will not involve assertions about properties of physical objects? Is it not possible—in fact, plausible—that both the skill and the knowledge result from the same underlying physical cause? At the same time we might also ask, why is simple individuation not sufficient for an animal (whether dog or man) to “have an idea, however ill-defined, of what we signify by the word ‘one’ ”? Just as we need not know everything about Frege to conclude that we have knowledge of the person signified by the phrase “the author of the Grundlagen”, is it not also possible that we will be able to understand the meaning of “one” without being able to do more than individuate appropriate instances of the term’s application? If so, the suggestion that skill at individuation and knowledge of the number one are necessarily distinct can hardly serve as evidence in favour of Frege’s claim that number assertions are really assertions involving concepts, rather than physical properties.

5.

Frege’s Fifth Argument

Frege’s fifth argument is that zero, although a number, is not a physical property. If it were, it would have to be a property of some physical thing; but this is impossible. Frege makes the case as follows: This is perhaps clearest with the number 0. If I say “Venus has 0 moons”, there simply does not exist any moon or agglomeration of moons for anything to be asserted of; but what happens is that a property is assigned to the concept “moon of Venus”, namely that of including nothing under it. If I say “the King’s carriage is drawn by four horses”, then I assign the number four to the concept “horse that draws the King’s carriage”. (§46, p. 59)

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This does indeed seem to be a serious ontological challenge. Since the proposition “Venus has zero moons” is true, there does not exist any moon or agglomeration of moons to serve as the subject of such an assertion. Even so, just because the number zero fails to signify a physical property, it need not follow that it depends upon a non-physical concept. Another, better option will be that, rather than indicating a property or concept of any kind, the number zero will signify the absence of a property and, in particular, the absence of a physical number property. If so, explaining the function of the word “zero” will be no different than explaining the function of negation more generally. Claiming that the physical world alone is insufficient to serve as a truth-maker for negative existentials such as “∼ (∃x)M xv” (or “Venus has zero moons”) thus becomes equivalent to claiming that lacks, absences and deficiencies more generally all require non-physical truthmakers, and surely this is too much for even most Platonists to accept.14

6.

Frege’s Sixth Argument

Frege’s sixth argument is that very large numbers are not physical properties since, if they were, they would be unknowable, something that is clearly not so. Frege summarizes the argument as follows: Mill’s theory must necessarily lead to the demand that a fact should be observed specially for each number, for in a general law precisely what is peculiar to the number 1,000,000, which necessarily belongs to its definition, would be lost. On Mill’s view we could actually not put 1,000,000 = 999,999 + 1 unless we had observed a collection of things split up in precisely this peculiar way, different from that characteristic of any and every other number whatsoever. (§7, pp. 11–12) As Frege sees it, Mill’s view requires that arithmetical knowledge be based on empirical observation. From this it appears to follow that each distinct knowledge claim must involve a distinct act of empirical observation, even though this is implausible in the extreme. Given our limited observational abilities, arithmetical knowledge will very soon outrun empirical observation. As Frege put is, “Who is actually prepared to assert 14 Even if not, to move from the claim that zero is not based on a physical property to the claim that no non-zero numbers are based on physical properties would be an induction of the most staggering kind. The refusal in India for many centuries to recognize zero as a number (e.g., see Boyer and Merzbach [4], p. 213) may have been related to these kinds of considerations.

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that the fact which, according to Mill, is contained in the definition of an eighteen-figure number has ever been observed, and who is prepared to deny that the symbol for such a number has, none the less, a sense?” (§8, p. 11) It follows according to this interpretation of Mill’s theory that either we have no arithmetical knowledge of large numbers or large numbers are not to be identified with physical properties. Mill’s response to this argument would be that it is physical facts, rather than observed facts, that are essential for his theory of mathematical truth. Admittedly, this distinction between physical facts and observed facts is one upon which Mill sometimes appears to equivocate. As Frege himself asks, “what in the world can be the observed fact, or the physical fact (to use another of Mill’s expressions), which is asserted in the definition of the number 777864?” (§7, p. 9) Even so, once the distinction is made and it becomes clear that it is physical facts (rather than observed facts) to which Mill is referring, Frege’s argument loses its force. According to Mill, it is physical facts, whether observed or not, that serve as the truthmakers for arithmetical definitions. It is then observations, or (lower-level) observations together with arithmetical definitions and proofs, that serve as the inductive justification for (higher-level) arithmetical knowledge. In either case, arithmetical knowledge is obtainable in at least some cases without direct observation.

7.

Frege’s Seventh Argument

Frege’s seventh argument is that the word “one” cannot function as a predicate. In his words, If it were correct to take “one man” in the same way as “wise man”, we should expect to be able to use “one” also as a grammatical predicate, and to be able to say “Solon was one” just as much as “Solon was wise”. It is true that “Solon was one” can actually occur, but not in a way to make it intelligible on its own in isolation. It may, for example, mean “Solon was a wise man”, if “wise man” can be supplied from the context. In isolation, however, it seems that “one” cannot be a predicate. (§29, p. 40) At one level, not much time needs to be spent on this argument since Frege’s claim here appears to be straightforwardly false. Not only can we successfully assert, “Solon was one” in the relevant sense, we can also assert “They are one” or “They are not one”, for example in response to someone who is uncertain whether Tully and Cicero are one or two. “One” appears to

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function quite normally as a numerical predicate, and just as easily as other numbers. But underlying Frege’s claim may also be the distinction between what we now refer to as logical and non-logical predicates, the distinction between predicates that express identity and difference on the one hand and predicates that stand for genuine physical or scientific properties (what Frege refers to as being “sensible”) on the other. If so, the modern-day Millian will be free to accept either horn of the dilemma. On the one hand, he may accept the distinction but deny that accepting identity as a purely logical predicate opens the door to non-physical concepts generally. On the other hand, he may conclude that identity remains a purely physical relation, one that sometimes holds and sometimes fails to hold, just like every other physical property and relation. In either case, application of the predicate need not require the postulation of non-physical concepts.

8.

Frege’s Eighth Argument

Frege’s final argument is based on the observation that, unlike physical properties, number properties also apply to non-physical entities. Frege reminds us that, in addition to applying to ordinary physical objects, number properties also apply to nonphysical items such as immaterial ideas, events and concepts. As a result, says Frege, number terms cannot be said to involve reference to ordinary physical properties. As he puts it, “This brings us to another reason for refusing to class number along with colour and solidity: it is applicable over a far wider range” (§24, p. 30). In other words, should a property that has been abstracted from external (i.e. physical) things be transferred without any change of sense to events, ideas and concepts, “The effect would be just like speaking of fusible events, or blue ideas, or salty concepts or tough judgements”. Thus, “It does not make sense that what is by nature sensible should occur in what is non-sensible” (§24, p. 31). In justifying this claim, Frege makes use of two separate sub-arguments, one ontological, the other epistemological. He summarizes his ontological argument as follows: In order to suppose that there is in the same way, when we look at a triangle, something sensible corresponding to the word “three”, we should have to commit ourselves to finding that same thing again in three concepts too; so that something non-sensible would have in it something sensible. (§24, p. 31)

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In other words, non-sensible items cannot have sensible properties since, if they did, they would be, at least in part, sensible, which is impossible. The epistemological version is closely related: The three in it [i.e. in a non-sensible object] we do not see directly; rather, we see something upon which we can fasten an intellectual activity of ours leading to a judgement in which the number three occurs. How is it after all that we do become acquainted with, let us say, the Number of figures of the syllogism as drawn up by Aristotle? Is it perhaps with our eyes? What we see is at most certain symbols for the syllogistic figures, not the figures themselves. How are we to be able to see their Number, if they themselves remain invisible? (§24, p. 32) In other words, we discover number properties, not by observing non-sensible objects but by an activity of the mind. Two points are worth noting in response to these arguments. The first is that, whatever their merits, these arguments will have force only if we have independent reason to reject physicalism. If ideas, events, concepts and figures of the syllogism all turn out to be reducible to the physical (as the physicalist will want to maintain), Frege’s so-called “wider range” of application won’t be wider at all. In short, in answer to Frege’s question, “Do such things really exist as agglomerations of proofs of a theorem, or agglomerations of events?” (§23, p. 30), the physicalist’s natural reply will be yes. Until it can be shown that there is in fact a range of application wider than that of the physical world—something Frege merely assumes at this point in his argument—Frege’s final argument remains lacking in force. The second point is that Frege’s final argument contains within it the seeds of a clear reductio against Frege’s own position. If numbers must be non-physical because they apply to non-physical objects, it will follow by exactly parallel reasoning that numbers must also be physical since they apply to physical objects. In other words, since numbers apply to ordinary physical objects—Frege’s fifty-two cards, five trees, four companies and five-hundred men—and since only physical properties can apply to physical objects, it follows that number properties must be physical. To paraphrase Frege’s ontological version of the argument, if we assume that there is something non-sensible (a Fregean concept) corresponding to our applications of the number “three”, we should have to commit ourselves to finding that same thing again in three cards or three trees, just as we find it in three concepts or three events. But then something non-sensible would be found in something sensible, which (according to Frege) should be impossible. To paraphrase

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his epistemological version of the argument, how is it that we might become acquainted with, say, the number of leaves on a tree? Is it by reason, rather than with our eyes or sense of touch? Frege is inclined to say yes, but how could we calculate such a number by reason alone when it is a purely contingent, physical fact about the world? The answer is that if, as Frege proposes, the physical world only provides us with a background upon which we “can fasten an intellectual activity of ours leading to a judgement” (§24, p. 32), a judgment in which the number occurs, then we will also need to accept his suggestion that the number of leaves on a tree “does not really belong to things in the external world at all” (§22, p. 29) and that counting objects like leaves becomes an “arbitrary decision” of the mind in which the object itself “has no say” (§22, p. 29). For most of us, this is far too great a departure from how things in fact are. The number of leaves on a tree is something fixed in nature, and something that will remain fixed regardless of whether an observer is present to observe and count the leaves or not. For anyone having even a modest amount of what Russell [21] calls a “robust sense of reality” (p. 170), this amounts to as clear a reductio as one might hope to find in such abstract matters as those normally studied in philosophy. Acknowledgements. Early drafts of this paper were read at the University of Saskatchewan and Vancouver Island University. I’m grateful to the audiences at both universities, as well as to Roberta Ballarin, Jim Franklin, Brian Hepburn, John Woods and several anonymous referees for their helpful comments. References [1] Armstrong, D. M., Truth and Truthmakers, Cambridge: Cambridge University Press, 2004. [2] Armstrong, D. M., A World of States of Affairs, Cambridge: Cambridge University Press, 1997. [3] Bostock, David, ‘Empiricism in the Philosophy of Mathematics’, in A. D. Irvine (ed.), Philosophy of Mathematics, Amsterdam: Elsevier/North Holland, 2009, pp. 157–229. [4] Boyer, Carl B., and Uta C. Merzbach, A History of Mathematics, 2nd edn, New York: John Wiley & Sons, 1991. [5] Britton, Karl, ‘The Nature of Arithmetic: a Reconsideration of Mill’s Views’, Proceedings of the Aristotelian Society, 48 (1947-48), II, 1–12. [6] Burge, Tyler, ‘A Theory of Aggregates’, Nous, 2 (1977), 97–117. [7] Burgess, John, Fixing Frege, Princeton: Princeton University Press, 2005.

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[8] Dummett, Michael, Frege: Philosophy of Mathematics, Cambridge, Mass.: Harvard University Press, 1991. [9] Franklin, James, ‘Aristotelian Realism’, in A. D. Irvine (ed.), Philosophy of Mathematics, Amsterdam: Elsevier/North Holland, 2009, pp. 103–155. [10] Frege, Gottlob, Die Grundlagen der Arithmatik (1884), trans. by J. L. Austin as The Foundations of Arithmetic, 2nd rev. edn, Oxford: Blackwell, 1980. [11] Geach, Peter, ‘Frege’s Grundlagen’, Philosophical Review, 60 (1951), 535–544. [12] Johnson, W. E., Logic: Part I, Cambridge: Cambridge University Press, 1921. [13] Kessler, Glenn, ‘Frege, Mill, and the Foundations of Arithmetic’, Journal of Philosophy 77 (1980), 65–79. [14] Kitcher, Philip, ‘Arithmetic for the Millian’, Philosophical Studies 37 (1980), 215–36. [15] Kitcher, Philip, ‘Mill, Mathematics and the Naturalist Tradition’, in J. Skorupski (ed.), The Cambridge Companion to Mill, Cambridge: Cambridge University Press, 1998, 57–111. [16] Lewis, David, Parts of Classes, Oxford: Blackwell, 1991. [17] McClure, Alexander K., Lincoln’s Yarns and Stories, Chicago and Philadelphia: The John C. Winston Company, 1900. [18] Mill, John Stuart, A System of Logic: Ratiocinative and Inductive (1843), Collected Works of John Stuart Mill, vols 7 & 8, Toronto: University of Toronto Press, 1973. [19] Quine, W. V., Word and Object, Cambridge, Mass.: MIT Press, 1960. [20] Resnik, Michael D., Frege and the Philosophy of Mathematics, Ithaca, NY: Cornell University Press, 1980, pp. 137–60. [21] Russell, Bertrand, Introduction to Mathematical Philosophy, London: George Allen and Unwin Ltd, 1919. [22] Van Inwagen, Peter, Ontology, Identity, and Modality: Essays in Metaphysics, Cambridge: Cambridge University Press, 2001. [23] Weiner, Joan, Frege in Perspective, Ithaca, NY: Cornell University Press, 1990. [24] Wright, Crispin, Frege’s Conception of Numbers as Objects, Aberdeen: Aberdeen University Press, 1983.

Andrew D. Irvine Department of Philosophy University of British Columbia Vancouver, BC V6T 1Z1 Canada [email protected]