a) The Category of Number

To what category do the natural numbers belong? So far, we have met with three quite different answers. According to Kant, numbers are concepts. Kant, of course, does not have much of a choice: a number can only be either an intuition or a concept. If it were an intuition, then it would be an individual thing, that is, spatio-temporal. If it were an individual thing, then one could ask where it is located and how long it has existed. Since there are no reasonable answers to these questions, it seems clear that numbers cannot be intuitions. But if not intuitions, then they must be concepts.

Kant is not the only philosopher whose choice is severely limited. Within the Platonic framework, numbers must be either temporal individuals or eternal forms. Since they cannot be the former, as I just argued, they must be the latter. Within the Aristotelian ontology, numbers must be either parts of substances or accidents. If the former, then they must be either prime matter or essences. They could not possibly belong to the category of prime matter, for prime matter by itself is not separated into particular entities as the individual numbers are. Thus numbers can only be either essential properties or accidental properties.

Even Bolzano is confined in his categorization by the tradition. We saw that he holds that (abstract) numbers are properties of peculiar sorts of wholes, namely, of what he calls sums. Mill, on the other hand, argues that they are perceptual structures. Thus while Bolzano thinks of them as “abstract things,” Mill conceives of them as being concrete. Very roughly, while Bolzano assigns numbers to Plato’s realm of being, Mill locates them in the realm of becoming. However, in either ontology, numbers somehow arise from wholes; in Bolzano’s ontology, from abstract wholes, in Mill’s from concrete wholes. The reasons for this affinity is clear: Numbers are “multiplicities” of some sort or other, and wholes have a “multiplicity” of parts.

Bolzano and Mill cannot bring themselves to claim that numbers are properties of ordinary individual things. They are forced to break out of the ontological straightjacket of the tradition. Bolzano introduces the category of sum, Mill makes use of the category of structure. Bolzano, as we would expect, is the more radical of the two. His sums are really unique “arithmetic” entities. Mill’s structures are ordinary wholes, formed by ordinary relations; for example, by spatial relations or relations among color shades. I have argued that both of these views are false.

It is important to realize that neither Bolzano’s sums nor Mill’s structures are sets. A number of possibilities opens up as soon as one breaks out of the ontological circle of the tradition and acknowledges the category of set. We shall now look at some of these possibilities.

1) NUMBERS AS MULTITUDES: HUSSERL

At the beginning of Book IV of the Elements, Euclid gives the following explanation of number: A unit is that by virtue of which each of the things that exist is called one. A number is a multitude composed of concepts.

A number is a multitude. But what is a multitude? If multitudes are sets, then Euclid holds that a number is a set of units. I believe that this is very close to what Husserl maintains in his Philosophie der Arithmetik. Consider the number three. According to Husserl, there are concrete multitudes (Vielheiten). The color red, the moon, and Napoleon form such a multitude of three things. Another such multitude consists of a certain pain, an angel, and Italy. By abstracting from the particular things contained in the multitudes, we are supposed to arrive at the notion of a multitude of this sort by reflecting on the characteristic relation that obtains in each case between the parts of the multitude. In other words, we discover what kind of whole a concrete multitude is by paying attention to the peculiar relation which is characteristic of multitudes. In this fashion, we find out that multitudes are formed by the relation of collective connection (kollektive Verbindung). A multitude is a whole whose parts are collectively connected. Here is how Husserl describes the process by means of which we get from concrete multitudes to multitudes in general and, hence, to “sets”:

Somehow, determinate single contents [read: particular things] are given as collectively connected; by proceeding through abstraction to the general concept, we do not pay attention to them as such and such determinate contents; the main interest is concentrated, rather, on their collective connection, while they themselves are merely viewed as some contents or other, each one of them, as something, as some one thing. We shall take advantage of this result by relating it to an earlier remark, according to which the collective connection can be linguistically indicated in a completely clear and intelligible way by means of the conjunction and. Multitude in general, as we can now express it simply and straightforwardly, is nothing else but: something and something and something, etc.; or any one and any one and any one, etc.; or, for short: one and one and one, etc.

(Husserl, 1970b, pp. 79-80)

Before we take a closer look at Husserl’s conception of the number three, let me call your attention to the fact that this conception plays an essential part in mathematical intuitionism. Compare Husserl’s words with the following quotation from Heyting:

INT. We start with the notion of the natural numbers 1, 2, 3, etc. They are so familiar to us, that it is difficult to reduce this notion to simpler ones. Yet I shall try to describe their sense in plain words. In the perception of an object we conceive the notion of an entity by a process of abstracting from the particular qualities of the object. We also recognize the possibility of an indefinite repetition of the conception of entities. In these notions lies the source of the concept of natural numbers.

(A. Heyting, Intuitionism, An Introduction, p. 13)

Mathematical intuitionism, we may say tongue in cheek, is Husserl’s philosophy of arithmetic taken seriously by brilliant mathematicians.

The multitude one and one and one, according to Husserl, is the number three, Similarly, other natural numbers are multitudes with different numbers of ones. We arrive at the notion of number in general, according to his view, by a second act of abstraction: we simply abstract from the particular number of times that the one, the unit, occurs in the multitude.

Husserl’s analysis contains two essential steps. Firstly, he claims that abstraction leads from the notion of the concrete multitude to the notion of something and something and something. Secondly, he identifies this notion with the notion of one plus one plus one. This identification has two parts: the notion of something is identified with the notion of one, and the concept of and is identified with the concept of plus. I think that all three steps are mistaken.

As Frege explains in great detail, abstraction will not yield the notion of a multitude in general (see G. Frege, The Foundations of Arithmetic, pp. 45-51). If we abstract from every characteristic, property, or feature that distinguishes the color red from both the moon and from Napoleon, we do indeed get the notion of a mere something or other. But if we do the same for the moon and for Napoleon, then we get again in either case the mere concept of something or other. By abstraction, we arrive three times at the same notion (of a mere something). We do not get the notion of something and something and something, that is, of some sort of triple. On the other hand, if we do not press the process of abstraction to its limit, so that we still have three different notions of three different things, then we arrive at the concept, not of something and something and something, but of the concept of something and something else and something else again. And from this notion we cannot jump to the notion of a multitude consisting of three units; for one unit is supposed to be the same as any other unit. In short, abstraction does not yield the notion of something and something and something.

Next, we must take a look at the “collective connection” which, according to Husserl, characterizes concrete multitudes. It seems to me that these multitudes are sets all but in name, because the collective connection is a mere wisp of a relation:

A multitude comes about when a uniform interest, and in and with it simultaneously a uniform awareness, sets off and comprehends different contents. . . . The fullest confirmation for our conception comes again from inner experience. If we ask in what the connection consists when we, for example, think a multitude of such dissimilar things as redness, the moon, and Napoleon, we receive the answer that it merely consists in that we think these contents together, think them in one act.

(Husserl, 1970b, p. 74)

Compare this way of “collecting” things with the following quotation from Kitcher:

One way of collecting all the red objects on a table is to segregate them from the rest of the objects, and to assign them a special place. We learn to collect by engaging in this type of activity. However, our collecting does not stop there. Later we can collect the objects in thought without moving them about. We become accustomed to collecting objects by running through a list of their names, or by producing predicates which apply to them.

(P. Kitcher, The Nature of Mathematical Knowledge, pp. 110-11)

It seems to me that Kitcher describes here two quite different “operations,” and that the second one does not at all deserve to be called a “collecting.” To arrange things in spatial patterns is one thing, to think of their names or to think of “predicates which apply to them,” it seems to me, is quite a different thing. In the first instance, we do produce a certain spatial structure with definite spatial relations among its parts. In the second case, no such structure or whole is produced. Notice that we are not even supposed to run through a list of the objects themselves in order to “collect” them, but merely through a list of their names. By thinking of a list of names, we are said to “collect” the objects so named. If Husserl’s “collective connection” is a mere wisp of a relation, then Kitcher’s relation is hardly there at all: it consists of nothing else but that the names of the objects to be collected are thought together.

But it is obvious, I think, that the alleged relation created by thinking of things together is not the sum relation of arithmetic. To think of things in one mental act is not the same as to add two numbers. By thinking of redness, the moon, and Napoleon as a group of things, I do not sum them up. I do not think of a sum at all. At best, I think of a set of things. What else could it mean to think of these things together, to think of them in one mental act?

Perhaps we are supposed to think of the constituents of the multitude as its constituents. Perhaps we are supposed to think of a list of its constituents. Perhaps we are supposed to think that this particular multitude consists of the color red and the moon and Napoleon. But the and of which we are then thinking is not the sum relation either. (Nor is it a relation among the members of the set.) It is the familiar conjunction. We are thinking (asserting, judging) that this multitude consists of the color red and that it consists of the moon and that it also consists of Napoleon. This relation holds between states of affairs and only between states of affairs, while the sum relation holds between numbers and only between numbers. Furthermore, while the conjunction is a two-term relation, the sum relation is a three-term relation. Husserl’s analysis fails, because the collective connection is not the sum relation of arithmetic.

But it also fails, I submit, because the notion something is not the same as the notion one. More succinctly, something is not the same thing as one. One, of course, is the familiar number. Something, on the other hand, is a complex entity, consisting of the quantifier some (to be contrasted with such different quantifiers as all, no, almost all, quite a few, etc.) and of the variable thing (either in the sense of entity, or else in the sense of object of the mind). Something is of the same sort as one thing. (It is merely an accident of language that we do not write ‘onething’.) In the latter expression, too, we can distinguish between the quantifier, one, and the variable thing. I should mention that the very nature of the topic forces me to diverge from customary terminology. For example, I call the entity some, not the word ‘some’, a quantifier. Furthermore, I do not think of the complex phrase ‘something’ (‘some thing’) as a quantifier. And, finally, I think of the entity thing (object, entity) as the variable; the expression ‘thing’, I call “the expression representing the variable.” Thus I sharply distinguish for “quantifiers” and “variables,” as well as for other expressions, between the expressions and what they represent.

Back to Husserl. He seems to have some misgivings about his cavalier identification of something with one. He admits that the two expressions do not mean the same thing (Husserl, 1970b, p. 84). But this difference is due, he claims, to our paying attention to different aspects of one and the same thing. We call something “one,” he says, when we conceive of it as a part of a multitude; and we call it “something” when we do not relate it to a multitude. It seems to me that there may be a smidgen of truth to this claim. The number one is indeed a constituent of the structure formed by the natural numbers, while this is not the case for the thing something. But this is hardly what Husserl had in mind.

To summarize our discussion of Husserl’s notion of number: Husserl comes very close to saying that the number three is a set consisting of units. But such a set, as Frege points out, collapses into a set with just one member, namely, the unit. In other words, a set consisting of “three ones” contracts into a set which consists just of the number one. On the other hand, if we assume that the members of the set are different from each other, that they are not all identical with the number one (or the thing something), then we do not have a set of units or of number ones.

2) NUMBERS AS PROPERTIES OF INDIVIDUAL THINGS

Frege as we know, gives a series of arguments against the view that numbers are properties of individual things. He argues, for example, that the number one cannot be a property (in Frege, 1974, p. 40). If it were a property, then everything would have to exemplify it. Frege then wonders: “It is not easy to imagine how language could have come to invent a word for a property which could not be of the slightest use for adding to the description of any object whatsover.” I do not find this consideration very convincing. Even though a word does not add to “the description of an object,” it may still be extremely useful. The term ‘entity’ (‘thing’, ‘being’), for example, is extremely useful, even though it is true that everything is an entity. [But not every object (of thought, imagination, etc.) is an entity, since some objects do not exist.] We may have to talk about entities of very different kinds. However, the analogy between the case for ‘one’ and my case for ‘entity’ is not perfect. We are supposed to contemplate the possibility that ‘one’ is the name of a property, while I hold that ‘entity’ is not the name of a property, but is the name of a unique and altogether extraordinary entity, namely, of existence (see my The Categorial Structure of the World, pp. 387-416 and also Phenomenology and Existentialism, pp. 178-95). For this reason, the analogy with identity may be more appropriate. Being self-identical is a characteristic of everything (of every entity, though not of every object of the mind). Yet to say of something that it is self-identical does not add to the description of that thing. Would Frege therefore also hold that ‘identity’ is not the name of a relation, since this relation “could not be of the slightest use for adding to the description of any object whatsoever”? (Frege, we should gleefully remember in this context, argues at one point that existence is nothing but self-identity, because to say of something that it exists, does not add to the description of that thing! See his Dialogue with Puenjer on Existence.)

Be that as it may, I find Frege’s second argument against the view that the number one is a property more convincing. If ‘one’ were a property, he says, then we should be able to use ‘one’ as a grammatical predicate. (G. Frege, 1974, pp. 40-41). I would prefer to put it somewhat differently: If ‘one’ were a property, then it would be exemplified; but it is not exemplified. If it were a property, according to Frege, then ‘Solon is one’ should make just as much sense as ‘Solon is wise’, but it does not. Of course, there are contexts in which ‘Solon is one’ makes sense, but then ‘one’ does not occur as a predicate. What ‘one’ means then is that Solon is one person. In such contexts, what we predicate of Solon is, not the number one, but the “property” of being one person, one thing. The number one is then only a part of this property. But here, too, we must be cautious. We cannot assume without further ado that there is such a property as the property of being one person. That is why I put ‘property’ in quotation marks. Rather, we must ask: What kind of entity is one person, one individual, one entity? How, precisely, does Solon is wise differ from Solon is one person? The question after the nature of number is, in part, a question about the structure of such facts.

Frege also points out that properties behave differently from the number one in regard to plural expressions. While it is true that Solon and Thales are wise, it is not true that Solon and Thales are one. They are two persons and not one. Solon is one person, and Thales is one person; and one person plus one person are two persons. This shows that the ‘and’ in ‘Solon and Thales are two persons’, unlike the ‘and’ in ‘Solon and Thales are wise’, represents the arithmetic plus.

Now, if it is true, as I have contended, that the number one is not a property because it is not exemplified, then it stands to reason that no natural number is a property; for we may assume that what holds for ‘one’, holds for the rest of the natural numbers. But let us look at some further possibilities.

Frege considers the possibility that numbers, in general, are properties of individual things. His example is an individual tree, and he assumes that this tree, at a certain moment, has precisely one thousand leaves (G. Frege, 1974, p. 28). He asks: What is the individual thing in this situation that has the property one-thousand? There simply is no plausible answer to this question. It is clear that any particular leaf does not have this propertyy, for it is one leaf and not a thousand. Nor can the foliage of the tree—a certain spatio-temporal structure—be said to have this property, for it, too, is one and not a thousand: the tree has one such foliage. But there is no other individual which could reasonably qualify for the alleged property one-thousand. Hence we are forced to conclude that this number is not a property at all.

Frege’s second example concerns a deck of (German) playing cards. If we hand someone this deck of cards and ask her how many things she holds in her hand, she may be baffled by our question. To make our question clear, we must tell her what kind of thing she is supposed to count: decks, suits, or individual cards. If numbers were properties of individual things, so Frege seems to reason, then one and the same individual thing, the deck of cards, would have to exemplify different numbers. It would have to be one, four, and thirty-two. But it can only have one number.

It has been said in response to Frege’s argument that the deck of cards may well have all three numerical properties (see D. Armstrong, Universals and Scientific Realism, vol. 2, pp. 71-74). The deck is simultaneously one-parted, since it is one deck, four-parted, since it consists of four suits, and thirty-two-parted, since it consists of thirty-two cards. In general, Armstrong claims, a complex entity has as many numerical properties as it has parts. But this reply to Frege overlooks the distinction which we have drawn between the number four, on the one hand, and four suits, on the other. It is granted that the one deck has four suits as parts and also has thirty-two cards as parts. But these two features are not the same as the numbers four and thirty-two. The feature of having four suits as parts is not the number four. The deck also has the feature of having four aces as parts. Now, if the first feature were the number four, then there would be no reason to deny that the second feature also is the number four. Then the first feature would be identical with the second feature. But it is not.

Numbers are not properties of individual things. Are they properties of things from a different category? Although Frege holds that numbers are what he calls “objects” rather than “functions,” there are passages in the Foundations which point to a different view. He speaks repeatedly of assigning numbers to concepts and of numbers as belonging to concepts (G. Frege, 1974, pp. 59, 63, 64, and 66). He even writes in a footnote (p. 80) that instead of saying that the number of F’s is the extension of the concept similar to F, he could have said that it is the concept similar to F. This suggests the view that the number of F’s is the relational property of being similar to F. Numbers, according to this view, are relational properties of properties. The number two, for example, is the relational property of being similar to the property of being a shoe which I am now wearing. This relational property is of course shared by all and only those properties which are exemplified by precisely two things. In other words, all of these properties have something in common, namely, the property of being similar to the property of being a shoe which I am now wearing. A moment’s reflection shows, though, that this view will not do. Consider the property of being similar to the property of being a thumb of mine. This property as well is exemplified by those and only those properties which are exemplified by precisely two things. We should conclude, therefore, that this property, too, is the number two. But how could it be, since the two relational properties are obviously not the same? Or else we shall have to conclude that there is not just one number two, but many two’s. And this conclusion, it seems to me, is quite obviously false.

3) NUMBERS AS PROPERTIES OF PROPERTIES: CANTOR

Another treatment of numbers as properties of properties can be found in Hilbert and Ackermann’s Principles of Mathematical Logic. They agree that numbers are not properties of individual things: “For example, the fact that the number of continents is five cannot be expressed by saying that each continent has the number five as a property; but it is a property of the predicate “to be a continent” that it holds for exactly five individuals” (D. Hilbert and W. Ackermann, Principles of Mathematical Logic, p. 136). And then they conclude: “Number thus appears as the properties of predicates, and in our calculus every number is represented as a predicate constant of second level” (Hilbert and Ackermann, p. 136).

As you can see, I have taken liberties in describing Hilbert’s and Ackermann’s view. When I spoke of properties, they speak of predicates. But their characterization of numbers is at any rate merely a preliminary to the real task at hand which consists of providing “expressions for the numbers 0, 1, and 2, i.e., for the second level predicates 0 (F), 1 (F), 2 (F).” I shall give these expressions here in words rather than in symbols:

0 (F); There is no x for which F is true,

1 (F): There is an x for which F holds, and any y for which Fy holds is identical with x,

2 (F): There are two different x and y for which F is true, and any z for which Fz holds is identical with x or with y.

(Hilbert and Ackermann, pp. 136-37)

Contrary to what Hilbert and Ackermann say, they do not give expressions for 0, 1, and 2. Rather, they give so-called contextual definitions. At this point, two issues arise which we must keep apart. Firstly, there is the question of the ontological efficacy of “contextual definitions.” I have argued elsewhere at some length that such “definitions” have no reductive power (see my Ontological Reduction, pp. 109-16, and The Categorial Structure of the World, pp. 302-304). My argument comes down to this. If ‘o (F)’, for example, is an arbitrary expression which has no prior meaning, then the “contextual definition” turns out to be a mere abbreviation proposal to the effect that ‘o (F)’ shall be nothing but a shortened version of the expression ‘There is no x for which F is true’. In this case, of course, ‘o (F)’ says nothing about the number zero. The occurrence of ‘o’ in this expression is totally gratuitous. On the other hand, if ‘0’ is supposed to represent the number zero, and if ‘F’ is taken to stand for a property, and if concatenation is meant to represent exemplification, then the so-called contextual definition turns out to be an equivalence statement to the effect that a property F has the property zero if and only if there is nothing which has the property. In this case, it is always legitimate to ask whether or not the so-called contextual definition is true. In this case, it is clear, no ontological reduction of something to something else takes place.

The second issue is the more specific question of whether or not a property F can truly be said to exemplify the property zero. Is zero a property of properties? I shall try to answer this question in connection with Cantor’s conception of the category of number.

Cantor states in his review of Frege’s Foundations: “I call ‘Cardinal Number [Maechtigkeit] of a multitude [Inbegriff] or set of elements’ (where the latter may be homogeneous or not, simple or complex) that general concept [Allgemeinbegriff] under which all and only those sets fall which are similar to the given one” (G. Cantor, Gesammelte Abhandlungen, p. 441). If we make allowance for the Kantian terminology (“concept” instead of “property”), then what Cantor maintains is that the number of a set is a property exemplified by all and only those sets which are similar to the given one.

I believe that this is an extremely plausible view. It answers Frege’s question to what we attribute the number one-thousand when we say that the tree has one thousand leaves. It is attributed, not to the tree or the foliage, but to the set of leaves. (Alternatively, it may be said to be attributed to the property of being a leaf of this tree.) It also explains how different numbers can be attributed to Frege’s deck of cards. The number one, for example, is a property of the set whose only member is the deck of cards. The number four, on the other hand, is a property of the set of suits contained in the deck. However, in spite of these and other advantages, Cantor’s view seems to me to be mistaken: Numbers are not properties of sets (or properties of properties). And this for at least two reasons.

Firstly, as Frege points out, we cannot form plurals for number words as we can for property words. To put it less linguistically:

The fact that 1/2 is neither spatial nor real notwithstanding, it is not a concept in the sense that objects could fall under it. One cannot say: “this is a 1/2”, as one can say: “this is a right angle,” nor are expressions like “all 1/2”, “some 1/2” admissible; rather, 1/2 is treated as a determinate single object. . . .

(G. Frege, “On Formal Theories of Arithmetic,”
in Collected Papers on Mathematics,
Logic, and Philosophy, p. 120)

I think that what these facts show is that numbers cannot be properties because they are not exemplified by something. I take it to be one of the fundamental laws of ontology that all and only properties are exemplified, so that an entity cannot possibly be a property unless it is exemplified by something. (Using one of our two important notions of necessity, we could say: It is necessary that entities which are not exemplified are not properties!) When we talk about elephants in the plural, we are talking about things which are elephants; we are talking about things which exemplify the property of being an elephant. An elephant is a thing which is an elephant. If the number three were a property, then there would have to exist things which are three in the very same sense in which there are things which are elephants. But there are no things which are three in this sense.

It may be replied, in defense of Cantor’s view, that our example is unfair. What correspond to elephants are certain sets (or properties), not individual things. But this reply brings us to the second reason why, in my opinion, Cantor’s view is false. Consider a set of three things, say, Husserl’s “concrete multitude” consisting of the color red, Napoleon, and the moon. Is this entity a three? Does this thing have the property three? I do not think so. This set is not a three. What is true is that this set has the “property” of having three members. It is “three-membered.” It is a triple. In this context, I shall concede that being three-membered or being a triple is a property of sets, just as being an elephant is a property of individual things. But the property of being a triple is not the same as the number three. To treat the number three as if it were a property of sets is to confuse the number with the property of being a triple.

As soon as we distinguish, as I think we must, between a number, on the one hand, and the “property” of having so-and-so many members (or the “property” of being exemplified by so-and-so many things), on the other, Cantor’s view is stripped of its initial plausibility. Surely, from the fact that having three members may be conceived of as a property of sets it does not follow that the number three, by itself, is a property. The number three, as I pointed out earlier, is merely a part of that “property.” This becomes obvious if we compare the “property” of having three members with the “property” of having three brothers. These are undoubtedly quite different properties, yet both contain the number three.

Having disposed of the view that numbers are properties of sets or of properties, we must next consider Frege’s view that numbers are sets.

4) NUMBERS AS SETS: FREGE

Frege’s view is contained in the famous “definition”:

(Frege, 1974, pp. 79-80)

There are a number of problems connected with Frege’s use of the term ‘extension’ (‘Umfang’), and I shall briefly allude to them. Firstly, Frege speaks of “value ranges” (“Wertverlaeufe”) rather than sets, and this raises the question of whether the former differ from the latter and, if so, in what way. I shall take for granted, in our general discussion of logicism, that value-ranges are sets. But I must call attention to the fact that much in Frege speaks against this identification (see, for example, N. Cocchiarella, “Frege, Russell, and Logicism: A Logical Reconstruction”). And I shall presently offer an interpretation of Frege’s peculiar brand of logicism which rests on a sharp distinction between value-ranges and sets. Secondly, there is the question of whether in the “definition” (F) Frege meant by ‘extension’ value-range. I shall assume that he did. It follows then that (F) asserts that numbers are sets. But this creates, thirdly, a most important philosophical problem for Frege’s program. If numbers are sets, according to Frege’s view, then arithmetic reduces to set-theory (and logic) and not, as Frege claims, to logic. Logicism, therefore, fails for this reason alone. Frege was aware of this objection. In the article cited earlier, for example, he says:

Therefore if arithmetic is to be independent of all particular properties of things, this must also hold true of its building blocks: they must be of a purely logical nature. From this there follows the requirement that everything arithmetical be reducible to logic by means of definitions. So, for example, I have replaced the expression ‘set’, which is frequently used by mathematicians, with the expression customary in logic: ‘concept’.

(Frege, 1984, p. 114)

There are two ways of looking at this and similar passages in Frege. It is natural to assume that Frege means to say that his “definition” could alternatively be formulated in terms of concepts rather than extensions of concepts, so that numbers turn out to be concepts and, therefore, logical rather than mathematical things. But this interpretation runs counter to Frege’s whole approach. On the other hand, there is an “unnatural,” rather Byzantine, interpretation which agrees well with the rest of his philosophy. I call it “Byzantine” because it rests on Frege’s idiosyncratic view that ‘the concept horse’ represents, not a concept, but a so-called concept-correlate. Assume that concept-correlates are the extensions of concepts. Assume further that concept-correlates are Frege’s value ranges. What Frege then claims, in the passage cited above and in similar passages, is that he could have said that the number of the concept F is the concept “equal to the concept F” rather than that it is the extension of the concept “equal to the concept F,” bearing in mind that, according to his idiosyncratic view, ‘the concept equal to the concept F’ represents a concept correlate (that is, an extension) rather than a concept. Either way, according to this interpretation, numbers turn out to be extensions, that is, value-ranges. And value ranges, we must emphasize, are by Frege conceived of as logical rather than mathematical entities.

This point is so important that we must stay with it a moment longer. Frege, I submit, would have insisted that value-ranges are “constituted in their being” by the respective concepts. It is this fact that characterizes them as “logical” things. Sets, as conceived of by the mathematician, on the other hand, are “constituted in their being,” not by concepts, but by their members (cf. Cocchiarella). Thus Frege’s philosophy of mathematics requires a sharp distinction between value-ranges and sets. And his logicism demands that value-ranges, in distinction from sets, are “logical” rather than “mathematical” things. I do not think there are such things as Frege’s value-ranges. But I do believe that there are sets. Hence, there is no possibility of “reducing” numbers to value-ranges, but there is the possibility of “reducing” them to sets. I shall therefore take this latter possibility seriously, and I shall pretend, contrary to fact, that Frege’s “reduction” was meant to be a reduction to sets. Precisely speaking, and putting our discussion in a nutshell, Frege’s logicism fails, first of all, not because he reduces numbers to sets, but because (as Russell’s paradox proves) there are no value-ranges.

But it also fails for another reason. Frege calls (F) a “definition,” but it is obvious that it is a description. It is an identity statement which may or may not be true. Most certainly, it is not a harmless abbreviation proposal, a mere stipulation for which the question of truth cannot arise. (F) amounts to:

(F¹) For any property f, the number of things which are f is (identical with) the set of all those properties which are similar to f.

Consider the number of planets, namely, the number nine. According to (F¹), this number is the set consisting of all and only those properties which are similar to the property of being a planet. The number nine is described as a set of properties, namely, as the set of all and only those properties which are exemplified by precisely nine things.

Frege’s view appears to be rather far-fetched. The number nine does not seem to be a set at all, to say nothing of a set containing such properties at the property of being a planet. How could one possibly arrive at such an outlandish view? Well, there are two straightforward ways. One could try to derive (F¹) from other perhaps more plausible statements. Or else one could try to argue for it directly by showing that this identity statement is true because the number and the corresponding set share all of their attributes (properties and relations).

Assume that ‘A’ is a mere conventional abbreviation for ‘B’. If so, then ‘A’ represents whatever ‘B’ represents, and conversely. It follows that A is identical with B. Every abbreviation implies in this fashion a true identity statement. Is (F¹) of this sort? Is (F¹) simply a consequence of a harmless abbreviation? Obviously not. It seems to me to be obvious that the expression ‘the number of planets’ is not a conventional abbreviation for ‘the set of properties which are similar to the property of being a planet’. It is preposterous to believe that there exists an implicit or explicit agreement among English-speaking people to use the phrase ‘the number of F’s’ whenever they wish to talk about certain sets.

But perhaps (F¹) follows, not from an abbreviation, but from other true propositions. Consider the following two true equivalences:

(E₁) For all properties f and g: f is similar to g if and only if the set determined by the property of being similar to f is the same as the set determined by the property of being similar to g.

(E₂) For all properties f and g: f is similar to g if and only if the number of f’s is the same as the number of g’s.

Now, if you mistakenly think of these two equivalences as identities, then you may conclude that the right side of (E₁) is identical with the right side of (E₂), for both are then identical with: f is similar to g. And if you identify the right sides, then you get, in effect, Frege’s “definition.” But the right sides are not identical. We can only derive the following equivalence:

(E₃) For all properties f and g: the set determined by the property of being similar to f is the same as the set determined by the property of being similar to g if and only if the number of f’s is the same as the number of g’s.

And from (E₃) we cannot get by uncontroversial means to (F¹).

What, then, about a direct defense of (F¹)? We accept the law of the identity of indiscernibles as well as the law of the indiscernibility of identicals. Can we show that the number nine has the same attributes as the set of properties which are similar to the property of being a planet? I think that these two entities have quite different properties and stand in quite different relations to things. Numbers, for example, stand in arithmetic relations; sets do not. Nine is the sum of six and three, but the set of properties just mentioned is not the sum of anything. On the other hand, sets have members (elements); numbers do not. The set just mentioned contains as a member the property of being a pencil on my desk, since there are nine pencils on my desk at this moment; the number nine, on the other hand, has no members at all. Furthermore, sets form unions, numbers do not. The set of pencils on my desk is the union of the set of yellow pencils on my desk and the set of blue pencils on my desk. The number nine, by contrast, is not the union of anything. While numbers form one kind of structure, sets form an entirely different kind of structure.

There are, however, and this is a most important point, certain similarities between these two kinds of structures. This similarity is succinctly characterized when we point out that a certain portion of set theory and a certain portion of arithmetic are both “Boolean algebras.” But this analogy between, say, the sum relation, on the one hand, and the union relation on the other, must not blind us to the fact that these are two different relations. We can no more add sets than we can eat them; and we can no more form the union of two numbers than we can join them in marriage. What we can add are the numbers of the members of two sets. A set which is the union of two nonoverlapping sets of two members each, for example, has four members because two members plus two members are four members. And two members and two members are four members because two plus two is four.

Our refutation of the view which we have attributed to Frege in the last few pages is straightforward: Numbers cannot be sets because arithmetic relations do not hold among sets. But there is an extremely popular response to this objection. Arithmetic relations, one claims, can be defined in set-theoretic terms. If this were true, then there really would be no arithmetic relations. And if there are no arithmetic relations, then they cannot hold among numbers. And if they do not hold among numbers, then they cannot distinguish between numbers and sets.

That arithmetic relations can be defined in terms of set-theoretic ones is a dogma for most mathematicians and many philosophers. Russell, to mention just one prominent example, maintains in the Principles of Mathematics that the “chief point to be observed is that logical addition of classes is the fundamental notion, while the arithmetic addition of numbers is wholly subsequent” (B. Russell, Principles of Mathematics, p. 119). Russell bases this claim on the following sort of “definition”:

Notice, firstly, that Russell says that m + n “is the number of a class,” but speaks of u as “having m terms.” m + n, to be precise, is not the number of a class, but is the number of terms of a class. Secondly, it is obvious that (R) can be read either as an equivalence or as an identity. According to the first, it states that the number which is the sum of m and n is the same as the number of the class w if and only if w is the union of the two sets u and v. This is true, but implies that we are here concerned with two quite different relations. According to the second reading, (R) says that the number (sum) m + n is identical with the number of the members of the set w. This, too, is true; but it, too, does not show that the sum relation is the same as the “logical sum relation” among sets.

My case against the view that numbers are sets rests, in one sentence, on this contention. The sum relation among numbers is not identical with any relation among sets.

5) NUMBERS AS QUANTIFIERS

How one categorizes numbers depends on what categories one has available, that is, on one’s ontology. I have argued that numbers are neither individual things, not properties, nor structures of certain kinds, nor sets. What categories remain? Well, there are relations and there are facts. I think that numbers are quite obviously not facts. I also believe that they are not relations, but hasten to add that there are plausible views which categorize numbers as relations, such as the one recently outlined by David Armstrong (see D. M. Armstrong, A Combinatorial Theory of Possibility, chap. IX: Mathematics, manuscript). If I am correct, then numbers do not belong to any one of the standard and familiar categories. And this suggests that they form a category of their own. What can we say about this new category? How shall we describe it? The so-called relational property of being exemplified by three things provides a clue, for it bears a striking resemblance to the “properties” of being exemplified by some (at least one) things, being exemplified by all things, being exemplified by nothing, being exemplified by almost all things, etc. This similarity suggests that numbers resemble such things as all, some, almost all, no, etc. These things are sometimes called “quantifiers,” and we shall therefore say that numbers are quantifiers.

Unfortunately, the term ‘quantifier’ is not without ambiguity, and we must clarify our use. Firstly, according to our terminology, a quantifier is not a linguistic entity; it is not a sign or expression. Rather, it is what a certain expression represents. For example, ‘all’ is not a quantifier, but rather a word that represents the quantifier all, just as ‘3’ is not a number but a numeral that represents the number 3. Secondly, we must distinguish between the quantifier (and the quantifier expression), on the one hand, and what such expressions as ‘All things are such that’, ‘Some things are such that’, etc. represent (and these expressions themselves), on the other. Some call the whole expression (or what it represents) a “quantifier.” We do not. Only the word ‘all’, for example, represents a quantifier; the expression ‘all things are such that’ represents an entity that is quite different from this quantifier. In this expression there occurs, in addition to the word for the quantifier, a word that represents what is quantified, namely, ‘things’, and also a phrase that indicates the peculiar nexus which connects the quantified things with the rest of the state of affairs, namely, ‘are such that’. Let us call the whole expression a “quantifier phrase.” According to our analysis, we can then distinguish between three parts of a quantifier phrase: (1) the quantifier expression, (2) the expression for the kind of entity that is quantified, and (3) an expression for the characteristic nexus.

By calling numbers “quantifiers,” we try to accomplish two things. Firstly, we describe what numbers do (in Dedekind’s words: “Was sie sollen”): They quantify. Secondly, we also try to shed some light on their nature (“Was sie sind”) by emphasizing their similarity to the well-known quantifiers of logic. For example, I hold that the sentences:

‘There are four persons in this room’, and

‘There are nine planets’

represent facts of the form:

Four things (entities) are such that: they are persons and they are in this room,

and

Nine entities are such that: they are planets.

One can easily represent these facts in a Principia-style symbolism.

What about statements like “Two persons plus two persons are four persons? We have already noted that this is an instance of the arithmetic truth that two plus two is four. It is also, but in a different sense, an instance of the law that two entities plus two entities are four entities. This law we can represent by:

‘All entities are such that: two entities plus two entities are four entities’. Our assertion about persons becomes:

All persons are such that: two persons plus two persons are four persons. How shall we analyze the fact that Solon is one thing (rather than two things or three things)? I think that this fact amounts to:

One thing is such that: it is identical with Solon.

The assertion that Solon and Thales are two things becomes:

One thing is identical with Solon, and one thing is identical with Thales, and the former plus the latter are two things.

These examples show that numbers behave like the familiar quantifiers. We are in the fortunate situation that the behavior of the ordinary quantifiers is so well known that much light is shed on the category of number by pointing out that it encompasses the ordinary quantifiers. By the same token, though, whatever obscurity surrounds the ordinary quantifiers is bound to rub off on the numbers. Russell, at one point, offers an argument that there is no such thing as the quantifier all. It may further illuminate our view if we take a look at his argument. Russell states:

If u be a class-concept, is the concept “all u’s” analyzable into two constituents, all and u, or is it a new concept, defined by a certain relation to u, and no more complex than u itself? We may observe, to begin with, that “all u’s” is synonymous with “u’s”, at least according to a very common use of the plural. Our question is, then, as to the meaning of the plural. The word all has certainly some definite meaning, but it seems highly doubtful whether it means more than the indication of a relation. “All men” and “all numbers” have in common the fact that they both have a certain relation to a class concept, namely, to man and number respectively. But it is very difficult to isolate any further element of all-ness which both share, unless we take as this element the mere fact that both are concepts of classes. It would seem, then, that “all u’s” is not validly analyzable into all and u, and that language, in this case as in some others, is a misleading guide. The same remark will apply to every, any, some, a, and the.

(Russell, 1964, pp. 72-73)

Russell argues here directly against our contention that ‘all u’s’ is short for ‘all entities which are u (are such that)’, and that the latter represents a complex entity, involving the quantifier all, the variable entity, and the property u. According to him, ‘all u’s’ is long for ‘u’s’. But this seems to be mistaken, for the plural can occur with many different quantifier expressions. In addition to ‘all men’, we have ‘some men’, ‘several men’, ‘seven men’, ‘quite a few men’, etc. The singular, on the other hand, occurs only with ‘a’, ‘the’, ‘one’, and ‘no’. If the entity all men were the same as the thing men, then the entity some men would have to be the thing some all men. But there is no such thing. Of course, this is not to deny that we may speak of men being mortal rather than of all men being mortal. But this is possible, not because ‘all men’ simply means the same as ‘men’, but because ‘men’ means on these occasions what ‘all men’ usually means. As to how Russell’s argument is to be applied to the quantifiers every, any, some, etc., I am at a loss.

It may be objected, in the spirit of well-established logical practice, that my example involving some in addition to all is gratuitous, for ‘some’ can be defined in terms of ‘all’. There exists, therefore, only the quantifier all, and that quantifier reduces, according to Russell, to the plural of the respective property. Once again, we have run into the mistaken habit of reading an ontological reduction into a mere equivalence, for that is what the so-called “definition” consists in. As everyone knows, the following equivalence is true:

All properties f are such that: some things are f if and only if it is not the case that all things are not f.

But this equivalence does not show at all that the quantifier some (or, alternatively, the quantifier all) does not exist. Quite to the contrary, if this equivalence has any ontological significance, then it consists in showing that both the quantifier all and the quantifier some exist. It shows this by showing that certain facts involving the former are equivalent to certain facts involving the latter. According to our point of view, no quantifier is “ontologically reducible” to any other quantifier. All of the quantifiers so far mentioned, including the numbers, exist, but one can do logic with just a few of them.

6) THE EXISTENCE OF THE REAL NUMBERS

I have argued that natural numbers belong to the category of quantifier. What about the (positive) rational and irrational numbers? I have no doubt that the (positive) real numbers exist. Or, at least, I see no reason why one should believe otherwise. It is an axiom of my ontology, if you like to talk that way, that the real numbers exist. And if they exist, then obviously they also belong to the category of quantifier. I do not think that too many philosophers would disagree with me about the existence of the real numbers. However, many probably would be offended by my firm conviction that the rational and irrational numbers are not “constructed” out of the natural numbers. In my view, all of the real numbers are there, all of them “at once,” the irrational numbers as well as the rational numbers, the rationals as well as the natural numbers. We must resist the fashionable talk about a piecemeal construction of the rational numbers out of the natural numbers, and of the irrational numbers out of the rational numbers. Such talk rests, once again, on a misconception of the ontological power of certain “definitions.”

One customarily “defines” the rational numbers in terms of equivalence sets of ordered pairs of natural numbers. A relation R is “introduced by definition,” as one so glibly says:

(D)[a, b] R [c, d] if and only if ad = bc.

(D) states that two ordered pairs of natural numbers stand in the relation R to each other if and only if the product of a and d is the same as the product of b and c. Of course, one cannot “introduce” a relation R, if there exists no such relation, and there is no reason to believe, in my view, that R exists. But even if it existed, (D) is a straightforward equivalence and as such does not “reduce” anything to anything. The true state of affairs is, rather, that the fraction a/b is that number which stands in the division relation to a and b. The fraction can be described as that number which stands in a certain arithmetic relation to two numbers. This number may be identical with the number which stands in the same arithmetic relation to two other numbers c and d. For example, the number which stands in the division relation to I and 2, namely, the fraction 1/2, is identical with the number which stands in that same relation to the numbers 2 and 4, namely, the fraction 2/4. One and the same number can be described in two different ways. Since division and multiplication are two sides of a coin, we get the following version of (D):

The fact that we can form the description expression ‘the number which stands in the division relation to 1 and 2’ does not guarantee, of course, that there exists such a number. If we believe that the fraction 1/2 exists, and of course we do, then our belief must be based on something other than our ability to form the expression. As usual, there are two obvious ways in which we can justify our belief in the existence of the fraction. We may claim that we are acquainted with the fraction; for example, that we perceive it. Or else we may try to present an argument which proves the existence of the fraction. I think we know that (some) fractions exist because we are indeed acquainted with them. We see the number 1/2, for example, when we see that only 1/2 of the pizza we bought for dinner is left, just as we see the color midnight blue when we see that our son’s sweater is midnight blue.

Euclid’s tenth Book contains a proof that the square root of two is not a rational number. From our pedantic but ontologically pure point of view, this means that a certain number, namely, the number which when multiplied with itself is two, is not identical with any number that stands in the division relation to two natural numbers. And this amounts to the surprising discovery that there are numbers which are neither natural numbers nor fractions. Dedekind’s famous “definition” of the irrationals must be assessed in the light of our view that the irrational numbers were discovered and not created by definition. From this point of view, Dedekind, too, discovered something. What he discovered is that irrational numbers can be described as determining certain sets of rational numbers. For example, the square root of two divides all real numbers into two sets, namely, into those numbers which are smaller than it, on the one hand, and the rest of the real numbers, on the other. It can therefore be described as the smallest number of this latter set. Alternatively, we can think of it as dividing all the real numbers into the set of numbers which are larger than it is and the rest of the real numbers. In this case, we can describe it as the largest number of the latter set. In either case, though, we think of the square root of two as dividing up all of the real numbers. There is no “filling in the gaps between the rational numbers by postulating the existence of irrational numbers.” We can no more “create” the irrationals than we can “create” the rationals, than we can “create” the natural numbers. Nor can we bring them into being by “postulating them.” If we “postulate” that there are irrational numbers, then, if there are such numbers, our postulate is true. Otherwise, our postulate is simply false. Nothing could be further from the truth than Kronecker’s aphorism that while the natural numbers are made by God, the rest of the numbers are our invention.

Much of what I have just said is in the spirit of Frege’s philosophy of arithmetic. But not all of it. Frege, for example, points out that even if a concept does not contain a contradiction, we cannot conclude that something falls under it. Thus even if we could prove that the concept square root of two is without contradiction, we could not infer that there must be such a number. This observation corresponds to our insistence that the fact that we can form certain noncontradictory description expressions does not imply that these expressions describe something (Frege, 1974, pp. 105-107). Frege correctly goes on to argue that the mathematician can no more create numbers at will than the geographer can create continents at will. The mathematician, too, can only discover what is there and give it a name (Frege, 1974, pp. 107-108).

How then does Frege show that there are fractions or irrational numbers? How does he show, for that matter, that there are natural numbers? Frege faces a dilemma:

How are complex numbers to be given to us then, and fractions and irrational numbers? If we turn for assistance to intuition [Anschauung], we import something foreign into arithmetic; but if we only define the concept of such a number by giving its characteristics, if we simply require the number to have certain properties; then there is still no guarantee that anything falls under the concept and answers to our requirements, and yet it is precisely on this that proofs must be based.

(Frege, 1974, p. 114)

I do not think that Frege can escape from the horns of this dilemma, contrary to his own assessment of the situation. For our view, one of the two horns does not exist: We maintain that we know that certain numbers exist because we perceive them, and we do not consider this recourse to perception (Anschauung) an import of something foreign to arithmetic. Quite to the contrary, it is of the essence of our empiricism that knowledge of all entities of whatever kind, of their existence as well as their nature, must ultimately rest on perception or introspection.

In the end, Frege cannot but embrace rationalism in order to escape from the dilemma: numbers are presented, he maintains, not to the senses, but to reason:

On this view of numbers the charm of work on arithmetic and analysis is, it seems to me, easily accounted for. We might say, indeed, slightly changing the well-known words: reason’s true object is reason itself. In arithmetic, we are concerned with objects which are known to us, not through the medium of the senses as something foreign from the outside, but which are given immediately to reason to which, as its very own, they are utterly transparent.

(Frege, 1974, p. 115, my translation)

b) Acquaintance with Numbers

Our view that numbers are quantifiers may appear to cast a pall over our professed empiricism. If numbers are quantifiers, then surely they cannot be “sensible things.” Quantifiers, if there are such things at all, traditional wisdom pronounces, are part of the most abstract furniture of the world, and abstract things, as everyone knows, are not sensible. It is a long-standing dogma of philosophy that all sensible things are spatio-temporal (concrete). Since quantifiers are not spatio-temporal, they cannot be sensible. Thus we seem to be forced into the arms of rationalism. As I have pointed out before, it is this dialectic that accounts for Mill’s insistence that numbers are somehow perceptual structures. Nor can there be any doubt that it is responsible for Husserl’s theory of eidetic intuition. Even Frege, as we shall see in a moment, was unable to resist its power. But it is a measure of his greatness that he added an important twist to that dialectic.

Powerful as the Platonic tradition is, its hold on the minds of philosophers can be assailed by a very simple and straightforward argument. Colors, I maintain, are abstract entities; they are not located in space and/or time. The color shade midnight blue, for example, has no spatial attributes: it has no shape, no size, and does not stand in spatial relations to other things. Nor does it have temporal attributes: it has no duration and does not stand in temporal relations to other things. If you do not agree with this categorization, then we have a disagreement that lies much deeper than the issue of rationalism versus empiricism. So, grant me for the moment that colors are indeed abstract things. But colors, I maintain, can be perceived. It is obvious, is it not, that we see with our very eyes that the sweater before us is midnight blue. It follows, therefore, contrary to the Platonic dogma and its Kantian refinements, that we perceive abstract entities, that is, things which are not spatiotemporal. I see only one way of avoiding this conclusion: You must deny that the colors which we see are abstract things. This is precisely what Husserl does. But he has to pay a dreadful ontological price: He has to invent a second sort of color, abstract colors, in addition to concrete colors. The colors we see, he holds, are concrete, individual things. But in addition, there also exist abstract, universal, colors, which we do not see, but which are grasped in eidetic intuition.

My strategy is to draw a parallel between the case I just made for colors and the case that can be made for certain (small) numbers. Numbers, like colors, are abstract things. Yet, like colors, they can be perceived. I appeal to the case of colors because it seems to me well suited to cast doubt upon the Platonic dogma that abstract things are not sensible. And as soon as this dogma is thrown into doubt, my view that numbers can be perceived will appear not only plausible, but inevitable.

When I claim that numbers can be perceived in precisely the same way in which we perceive colors, I do not mean to say, of course, that numbers belong to the same category as colors. Colors are properties (of individual things), while numbers are quantifiers. Rather, what I mean is that colors and numbers are presented to us in the same kind of mental act, for example, in an act of seeing. Nor do I wish to claim that a color or a number is ever presented to us in isolation, separate from all other things. We see colors when we see colored things, and we see numbers when we see numbered things. I see the color midnight blue when I see that my son wears a midnight blue sweater, and I see the number two when I see that there are two blue pencils on my desk.

As I just pointed out, colors and numbers are categorially different. But there is also the following important difference: While colors can only be seen, numbers can be seen, they can be heard, they can be felt, etc. Just as you can see that there are two pencils on the table, so can you hear that there are two dogs barking and you can feel two pains, one in each leg. In short, while colors are presented to us through just one sense, numbers are given to us through all of the senses.

1) THE SENSIBLE NATURE OF NUMBERS

Frege argues that number is not a sensible property like color (Frege, 1974, pp. 27-32). We agree that numbers are not properties and, therefore, could not be sensible properties. But Frege claims more, namely, that number, no matter to what category they belong, are not sensible. What convinces him is an argument that may be called “the argument from the lack of sense-impressions.” In outline, the argument runs as follows. (1) Perception depends on sensations. (2) While there are color sensations, there are no number sensations. (3) Therefore, numbers, unlike colors, cannot be perceived. In support of (1), Frege says, for example: “When we see a blue surface, we have an impression of a unique sort which corresponds to the word “blue”; this impression we recognize again, when we catch sight of another blue surface” (Frege, 1974, p. 31). He then goes on to argue that there is no corresponding impression for the word “three” when we look, for example, at a triangle and see its three sides.

Frege gives two reasons for his contention that no sense-impression corresponds to the word ‘three’, but they come down to the same thing. Firstly, if we assume that something sensible corresponds to the word ‘three’, then that same sensible thing must be found also in three concepts. Hence, we should find something sensible, whatever corresponds to ‘three’, in something that is not sensible, namely, in concepts. “The effect would be,” Frege says, “just like speaking of a fusible event, a blue idea, a salty concept, or a stringy judgment” (Frege, 1974, p. 31). Secondly, Frege asks how it is that we become acquainted with the number of figures of Aristotelian syllogisms. He states that it cannot be by means of our eyes, for what we literally see is at most certain symbols for the syllogistic figures, not the figures themselves. And how can we see their numbers, he asks rhetorically, if the figures themselves remain invisible?

Both of these considerations, I think, make the same point: Nothing sensible corresponds to the word ‘three’ because even nonsensible things can be three. Frege seems to take for granted that something sensible cannot be presented together with something nonsensible. Why does he take this for granted? Surely, the analogy to the blue idea misses its mark. A blue idea is impossible because ideas do not have colors. The problem is not that something sensible, the color, is supposed to be presented together with something nonsensible (the idea). A blue tone or a square with a pitch are just as impossible as a blue idea, even though we have here sensible things, namely, tones and pitches. We may agree with Frege that ideas (concepts) can be numbered, and that concepts are not sensible. But we fail to see how it follows that sensible numbers cannot be presented together with nonsensible concepts. The conclusion seems to be inevitable that Frege has fallen prey to the Platonic dogma, according to which the understanding contemplates the nonsensible forms and the senses acquaint us with the sensible individuals. The world is divided into two realms and the mind is split up into two separate faculties.

How deeply Frege is steeped in that unfortunate tradition can be seen from his conception of colors. Frege, we know, distinguishes between objective things and subjective (mental) things. This is his anti-Kantianism. But he does not escape completely from the Kantian spell. Objective things, he holds, divide into objects and concepts (more precisely: into objects and functions). The choice of the term ‘concepts’ betrays the Kantian influence, for concepts are nothing but properties; and while the word ‘concept’ has associations of subjectivity, no such connection with the mind adheres to the term ‘property’. Is the color shade midnight blue, to go to the heart of the matter, an object, a concept, or else a subjective idea? Our answer is straightforward: It is a property and as such neither mental nor nonmental, neither subjective nor objective. Furthermore, it is a property both of mental things, namely, of sensations, and of nonmental things, namely, of perceptual objects. Frege’s answer is not straightforward. It betrays all of the confusions of the Kantian tradition.

Frege distinguishes between blue as a sense-impression and as a concept. The color concept, what we would call the “property,” he says, belongs to a surface quite independently of us. It consists of the power to reflect certain light rays (Frege, 1974, p. 31). The color concept, we are surprised to read, is not at all the property which we see with our eyes, but is a certain physical property of surfaces. Frege thus accepts the argument from physics and must suffer all of its catastrophic consequences. The color concept, therefore, is not sensible. What is sensible is the so-called color sense-impression.

But what, then, does the word ‘blue’ represent? What am I saying when I say that your sweater is midnight blue? In the Foundations, Frege seems to waver between two different answers. At times he seems to be saying that ‘midnight blue’ represents not the objective concept, but the sense-impression (“which we cannot know to agree with anyone else’s,” Frege, 1974, p. 36). But there are also many passages which seem to imply that all expressions purporting to represent subjective ideas in reality represent objective concepts. (This is also an essential premise of Frege’s refutation of idealism. Cf. my Reflections on Frege’s Philosophy, pp. 33-43.)

Color appears in Frege’s philosophy either as a physical property of surfaces (which even a color-blind person could distinguish, Frege, 1974, p. 36), or else as a subjective sensation in the mind. The visible color of perceptual objects has disappeared from the world. Frege, like most philosophers of the last four hundred years, accepts unquestioningly the argument from physics. This comes as no surprise to us. But something else does. When Frege distinguishes between objects and concepts (functions), he applies this distinction only to the objective realm, not to the mind. But it is clear, as I have emphasized before, that a sensation, ontologically speaking, is just as much an object as a chair or the number two. And it is also perfectly obvious that this object has properties, that is, that it falls under certain concepts. Frege, we see, also makes the common mistake of thinking of the “color sensation” as being neither an individual nor a property. Or perhaps it would be more accurate to say that he simply never faces the question of whether it is an object or a concept.

To return to numbers, the number three, according to Frege, is not a sensible thing. There is no sense-impression as there is for the color midnight blue. But as soon as we distinguish, as we must, between a sensation and its properties, this claim is stripped of its plausibility. In what sense is there a sense-impression of the color? When we see a blue sweater, we experience under normal circumstances a blue sensation. We experience a sensation which has the same color as the sweater before us. The perceptual object shares the color with the sensation. The question, therefore, cannot be whether or not the number three is a sensation. The number three cannot be a sensation, just as the color cannot be a sensation. The question is, rather, whether or not sensations can be numbered, just as they can be blue. The answer to this question, I submit, is clear. Just as one may experience a blue sensation, so may one experience two sensations. When we look at the triangle of Frege’s example and see that it has three sides, we experience three sensations (corresponding to the three sides), just as we experience a blue sensation if the triangle happens to be blue. (I do not wish to claim, of course, that these are all of the sensations which we experience, nor that we do not experience patterns of sensations rather than isolated sensations.) We conclude that, contrary to Frege’s view, numbers are just as “sensible” as colors.

2) FREGE’S CONTEXT THESIS

In the Foundations, Frege’s world divides into the realm of subjective ideas and the realm of objective objects and concepts. Numbers belong to the objective realm. How are we acquainted with them? Since numbers are not sensible things, according to Frege, it cannot be the senses that acquaint us with numbers. Thus it must be the understanding. Frege, as I said earlier, is forced to embrace rationalism. But I also said then that he adds a brilliant twist to the traditional dialectic.

When we look at a triangle, Frege says, we have a sense-impression that corresponds to the word ‘triangular’, but we do not have a corresponding sense impression for the word ‘three’; we do not see the three. Rather, we see “something upon which can fasten an intellectual activity of ours leading to a judgment in which the number three occurs” (Frege, 1974, p. 32). Frege is rather vague at this point. What precisely is it that we see? Is it the impression which we have? Or is it something else? If something else, what is it, and what is the difference, if any, between sensing, seeing, and judging? These important questions are not answered. But one thing is relatively clear: Frege claims that the number three appears as part of a judgment. From this remark and others, I have the impression that he distinguishes between the experience of subjective things, on the one hand, and judgments involving objective things, on the other. He holds that sensibility acquaints us with subjective things, while judgment is the eye that sees objective things. A color, for example, has both a subjective side and an objective side. It is a sensation in the mind and a concept in the objective world. The sensation is known through experience, that is, through sensibility. The objective concept, on the other hand, is known in a judgment. A number, in contrast to a color, is an objective thing (an object); there is no corresponding sensation. Hence it can only be known through judgment.

If judgment is the eye of the understanding, then it follows that all objective things are given to us within a context that is represented by a sentence; for judgments are expressed by sentences, not by words or phrases. This means that we shall not find the (objective) meaning (or referent) of a word, if we look for it outside of such a context. I believe that this is the point of Frege’s famous principle, stated early in the Foundations (Frege, 1974, p. x.), that words have meaning only in the context of a sentence. Frege does not mean to subscribe to some kind of “meaning is use” doctrine as Dummett, for example, alleges (see M. Dummett, “Nominalism”). Rather, he claims that even though one can distinguish between different parts of the context represented by a sentence, these parts are never given separately to a mind. The only place where one finds the objective referent of a word is within the context of a judgment. Let us, for a moment, anticipate Frege’s later distinction between the sense and the reference of a declarative sentence. In these terms, Frege claims that we are acquainted with objective things only by being presented with a whole thought, a thought which contains objective things—objects and concepts (functions)—as parts. (As I mentioned in connection with a discussion of Bolzano’s “propositions,” I think that the most fundamental flaw of Frege’s ontology is his insistence that so-called thoughts are made up of senses rather than referents. This, of course, is a consequence of his view that a sentence refers to a truth-value. Cf. my Reflections on Frege’s Philosophy, pp. 181-223.)

Objective things are presented to us in mental acts of judgment. Why does Frege hold this view? There are two clues in the Foundations. Firstly, he remarks that if one does not accept this view, then “one is almost forced to take as the meanings of words mental pictures or acts of the individual mind” (Frege, 1974, p. xxii). Secondly, he also says that only by adhering to his view can one “avoid a physical view of number without slipping into a psychological view of it” (Frege, 1974, p. 116). Let us try to reconstruct his train of thought.

Recall Kant’s distinction between presentations, on the one hand, and judgments, on the other. Judgments, it is said, are based on presentations. Frege, I have just argued, changes this pattern. According to him, all presentations are mental. Acts of presentation acquaint us with nothing but subjective things. This leaves judgment as the only source of knowledge of objective entities. Thus Frege comes to hold that if we do not accept the view that objective things are presented in judgments, then they must be given through presentation. But if they are given in presentation, then they cannot really be objective; they must turn out to be “mental pictures or acts of the individual mind.”

In order to understand Frege’s thought behind the second quotation, we must again appeal to the Kantian foil. Presentations, according to Kant, divide into intuitions and concepts. Intuitions are particular (singular, individual), while concepts are universal (general). In this respect intutions and concepts resemble Frege’s objects and concepts (functions). Now, Frege insists that numbers are objects rather than concepts. But this means, in Kantian terms, that they are intuitions rather than concepts. A Kantian would therefore conclude that if numbers are intuitions rather than concepts, they must be either “outer” or “inner” intuitions, that is, they must be either perceptual objects or else mental pictures. Frege argues that numbers are not perceptual objects (for example, against Mill). If he is correct, then they must be, according to the Kantian way of thinking, mental pictures. Frege tries to avoid this conclusion by arguing that even though numbers are “particular,” they are not intuitions. They are neither “independent” outside objects nor “independent” inner pictures. Rather, they always occur dependently in certain contexts which are presented to judgment. In this fashion, I submit, he avoids “a physical view of number without slipping into a psychological one.”

Frege’s way out of the dilemma posed by the two alternatives that numbers are either individuals in the perceptual world or else individuals in the mind, becomes quite transparent in his later philosophy, which is centered around the category of thought. So-called thoughts (what a horrible misnomer!) are parts neither of the mind nor of the perceptual world. They are neither fish nor fowl. But they form the realm in which numbers dwell. They cannot be perceived; they can only be grasped by the understanding. This is the core of Frege’s rationalism. But Frege, as we have just seen, adds what I earlier called a “twist” to the traditional rationalistic position: The eye of the understanding is propositional; it sees, not abstract things in isolation, but propositions (thoughts).

c) The Reduction of Arithmetic to Logic

1) THE IMPORTANCE OF DEFINITIONS

In the Foundations, Frege’s aim is to make it “probable that the laws of arithmetic are analytic judgments and consequently a priori. Arithmetic thus becomes simply a development of logic, and every proposition of arithmetic a law of logic, albeit a derivative one” (Frege, 1974, p. 99). Frege contradicts Kant: Arithmetic is not synthetic, as Kant had proclaimed. How does Frege propose to show that arithmetic is analytic? A proposition is analytic, Frege explains, if and only if, in finding a proof for it, “we come only on general logic laws and on definitions, . . . bearing in mind that we must take account also of all propositions upon which the admissibility of any of the definitions depend” (Frege, 1974, p. 4). His task is perfectly clear: Frege must show that arithmetic propositions are really propositions of logic, and this can be achieved if it can be shown that the crucial arithmetic terms can be defined in logical terms.

Logicism stands or falls with the proper conception of definitions. In the Foundations, alas, there is no such conception. Instead, we find hints, allusions, and metaphors. In the last quotation, Frege speaks of propositions on which the admissibility of definitions depends. Assume for a moment that Frege were correct, that the number of F’s is really the same entity as a certain extension. If this identity statement were true, then we could build a definition upon it. We could then conclude that a given numeral is just another sign for a certain extension, namely for the respective number. Schematically and in principle, assume that A = BC. Then we can of course use ‘A’ as just another expression for BC, and ‘BC’ as just another expression for A. Notice, however, that this is no longer true if ‘A’ or ‘BC’ are description expressions. From the fact that ‘the number of F’s describes the same thing as ‘the extension of G’, we must not conclude that these two expressions represent the same thing, that is, the same description. Rather, they represent different descriptions, but descriptions which describe the same thing (if Frege were correct). This is the vexing complication forced upon us by the existence of descriptions (and hence, of description expressions). If Frege thinks at this point of definitions as flowing from true identity statements in the manner just described, then his “reduction” of arithmetic to logic stands or falls with the truth or falsehood of the relevant identity statements. I have argued earlier that this particular and crucial identity statement is false.

But the Foundations also contains remarks to the effect that the crucial definitions are nothing but harmless abbreviation proposals. For example, on p. 78 he says: “The definition of an object does not, as such, really assert anything about the object, but only lays down the meaning of a symbol.” In the same vein, we read in the Grundgesetze: “We introduce a new name by means of a definition by stipulating that it is to have the same sense and the same reference as some name composed of signs that are familiar” (G. Frege, The Basic Laws of Arithmetic, p. 82). On the whole, Frege is not too clear about the nature of definitions in the Foundations. He insists that definitions do not just specify concepts in terms of a list of characteristics, and then tries to illustrate their function by means of a geometrical example (Frege, 1974, pp. 100-101). Let concepts be represented by certain well-demarkated areas in a plane. A concept, defined by a list of characteristics (the “marks” of the concept), corresponds to an area which is common to all of those areas that represent the defining characteristics. No new line is drawn. We merely trace already existing lines in such a way that the new area, enclosed by the lines we trace, contains several of the original areas. For example, consider the two concepts (characteristics) green and round. We can introduce a new word ‘ground’ by stipulating that it shall be short for ‘green and round’, so that we can say ‘A is ground’ instead of ‘A is green and round’. What we have done, in terms of the geometric picture, is to trace the lines which surround the two areas green and round, thus creating a larger area consisting of these two areas. That nothing new emerges in this way is obvious. In particular, it is clear that Frege’s definition of the number of F’s is not of this sort. Frege, therefore, wishes to contrast the “fruitful” definitions of the Foundations with these trivial conventions. The type of definition employed in the Foundations, he asserts, consists of drawing boundaries which did not previously exist. “What we shall be able to infer from it, cannot be inspected in advance; here, we are not simply taking out of the box again what we have just put into it” (Frege, 1974, pp. 100-101). But does this drawing of new boundaries not amount to the “creation” of concepts? And, surely, according to Frege’s view, we can no more create concepts than we can create numbers. It took a while, as we shall see, before Frege fully realized that his reduction of arithmetic to logic requires a detailed analysis of the nature of definitions.

2) DETAILS OF THE REDUCTION

We know that Frege’s attempt to reduce arithmetic to logic fails. It fails, for one, because extensions in Frege’s sense do not exist. Rather, what there is that closely resembles such extensions are sets. But even if Fregean extensions existed, numbers are not identical with such extensions. If we try to improve on Frege’s view by substituting sets for extensions, logicism does not fare any better. Firstly, numbers are not sets either. Secondly, even if they were sets, arithmetic would have been reduced to set theory rather than to logic.

But even though we know that Frege’s philosophical goal cannot be achieved, we cannot but admire his ingenuity in regard to the details of the reduction. To exhibit this ingenuity, let us reconstruct Frege’s steps from our point of view.

We start out with seven abbreviations:

Next, we have a number of true statements in which these abbreviations occur:

The following truths allow us to “connect” numbers with the abbreviations listed earlier:

With these abbreviations and theorems at the ready, let us take another look at Frege’s so-called definition. We shall assume that Frege’s extensions are sets. And we shall also pretend for the time being that it makes sense to speak of the property of there being so-and-so many things which exemplify a certain property. In other words, we shall assume that numbers are properties of properties. If so, then it is true that a given “numerical property” U is coextensive with another property, namely, with the property of being similar to a certain property. Frege uses this fact, one may say, to get his definition: He identifies the numerical property with the set determined by the corresponding number. His definition, after a fashion, corresponds to our (N₅). But Frege reverses the order of introduction. He does not base (N₅), as I did, on certain statements about numbers. However, that does not really matter for my criticism. The fact remains that (N₅) is not a mere convention.

To round out the picture, let us briefly look at some steps of Frege’s “reduction.”

Frege here treats his definition as a mere abbreviation proposal. But this appearance is deceptive. The word ‘number’ must always be taken in its ordinary and unproblematic meaning. With this ordinary meaning, it enters into the description expression ‘the number which belongs to the concept F’. Since we understand this expression, we are in a position to decide whether or not Frege’s so-called definition of the number of a concept is true. We are also in a position to decide whether or not Frege’s “definition” of the concept number is true; for that “definition” is really an equivalence statement: All entities e are such that: e is a number if and only if there is a concept of which e is the number.

This corresponds to our (N₁), but differs from it importantly in that Frege mentions a definite “property” under which, as we know, nothing falls. This “property,” in the spirit of the reduction, is conceived of as a “logical property.” I would raise a minor objection: Identity, I think, is an ontological relation rather than a logical one. The law of self-identity, in my view, is an ontological rather than a logical law. It holds for all entities whatsoever.

Here again we have a true equivalence statement rather than a linguistic convention.

Frege mentions in passing that only by means of this “definition” is it possible to reduce the argument from n to (n + 1) to the general laws of logic. The “reduction” proceeds along the following line. Applying the general “definition” to the special case of the series of natural numbers, we discover that Y follows in the series after X if and only if Y has all the properties which (a) belong to X, and (b) are such that if they belong to any given number, they belong also to the number which follows in the series of natural numbers directly after the given number. Furthermore, it is also true that N is a natural number if and only if N is either the number o, or else N has all the properties which (a) belong to 0, and (b) are such that if they belong to any number M, they also belong to whatever follows in the series of natural numbers directly after M. If one reformulates the principle of mathematical induction according to these “definitions,” then the principle turns into the following truth:

If F is a property which (a) belongs to o, and (b) is such that if it belongs to any number M, it also belongs to whatever follows in the series of natural numbers directly after M, then F belongs to everything which is either identical with o or has all the properties which (a) belong to o, and (b) are such that if they belong to any object M, they belong also to whatever follows in the series of natural numbers directly after M.

After this brief survey of Frege’s brilliant chain of “reductive steps,” let us cast one last look at the principle of his reduction. As our example, we shall take his “definition” of o:

This is an informative identity statement. The number o is described as the number of a certain concept (property). Add to (1) the following proposition:

(3) describes the number o in purely logical terms, assuming that the notion of the extension of a concept is a purely logical notion. But in order to arrive at this description, we started with (1) which is not a logical truth, and we had to add (2) which, according to our view, is not even true! What Frege does in this case, from our point of view, is to start with a true description of the number o, add to this description the false assertion that the number is an extension, and conclude, mistakenly, that o is an extension. None of these three statements is a pure logical truth.

3) FREGE’S LATER THOUGHTS ABOUT DEFINITIONS

The so-called definitions of the Foundations are informative identity statements (or informative equivalences) and, as such, are not derived from harmless abbreviations. Like all such statements, they must either be shown to be true or else admitted as axioms. In the Foundations, Frege had as yet not developed his view about informative identity statements in terms of the distinction between sense and reference. It is clear that this view does not change the general picture which I have drawn of Frege’s attempt to reduce arithmetic notions to logical ones. The dialectic remains the same. Definitions will serve Frege’s purpose only if they are either straightforward abbreviation proposals or else identity statements derived from such proposals. Any other kind of statement, in particular, informative identity and equivalence statements, will not do.

It is not surprising, given Frege’s high stakes, that he returned time and again to the question of what, precisely, the powers and limits of definitions are. For example, in a letter to Hilbert of 1900, Frege remarks: “In thinking about definitions, I have been tightening my requirements more and more, to the point where I have moved so far from the opinions of most mathematicians that communication has become very difficult” (G. Frege, Philosophical and Mathematical Correspondence. p. 45). But his most detailed attempt to describe the nature of definitions can be found in the unpublished manuscript Logic in Mathematics (probably from 1914) (In G. Frege, Posthumous Writings).

A genuine definition (eigentliche Definition), Frege says there, has the following characteristics: (1) It introduces a simple expression for a complex expression; (2) the simple expression receives the same sense as the complex expression; (3) such definitions are not really necessary for a system, since the new signs do not really add anything new, but merely allow for more manageable expressions; (4) after the simple expression has received its meaning (sense), the definition is transformed into an identity statement; (5) but this identity statement is tautological and does not extend our knowledge (Frege, 1979, pp. 207-208). Frege here describes what I have called “abbreviation proposals.” (There is a difference between us, though, because I do not make his sense-reference distinction.) It is obvious, as I have emphasized time and again, that these definitions are ontologically harmless. However, Frege’s so-called definitions in the Foundations and elsewhere are obviously not of this sort. And Frege comes close to admitting this fact when he says:

Indeed, no truth must become provable by means of a definition which otherwise would be unprovable. Whenever something that purports to be a definition makes the proof of a truth possible, we do not have a genuine definition. Rather, this definition must contain something which must be either proven as a theorem or else acknowledged to be an axiom.

(Frege, 1979b, p. 208; my translation)

Genuine definitions, according to Frege, must be distinguished from logical analysis:

As little as it can be indifferent whether or not I analyze a body chemically in order to see what its elements are, as little can it be unimportant whether I undertake a logical analysis of a logical structure in order to learn what its components are, or leave it unanalyzed, as if it were simple, when in fact it is complex.

(Frege, 1979b, pp. 208-209; my translation from the German)

Definitions, conceived of as instances of logical analysis, are far from harmless, for they make proofs possible which otherwise would have been impossible. The importance of such analyses can be seen if we recall Frege’s treatment of the quantified propositions All F are G, Some F are G, etc. His analysis, far from being harmless or trivial, had the most far-reaching consequences for logic and ontology.

Frege concludes his investigation into the nature of definitions by distinguishing between two cases. Firstly, we may build up a sense from other senses and introduce a completely new and simple sign for the complex sense: “One could call this constructive definition (aufbauende Definition); but we shall simply call it definition” (Frege, 1979b, p. 210). Secondly, an expression may have been in use for some time, but we believe that its sense can be analyzed in such a way that we can more perspicuously represent this sense by a certain complex expression (with the same sense as the original expression). Frege thinks that it would be best not to speak in this second case of definition, because we are here dealing with an axiom: “In this second case, there remains no room for an arbitrary decision, because the simple sign already has a sense. One can only arbitrarily assign a sense to an expression which does not already have a sense” (Frege, 1979b, p. 210).

Let us take another look at Frege’s “definition” of zero:

(Z) o = the number which belongs to the concept not identical with itself. It is obvious that this is not a harmless abbreviation proposal or the result of one. Rather, the right side of (Z) is supposed to elucidate the ordinary meaning of ‘o’. Is it then an instance of logical analysis as understood by the later Frege? It appears so. An entirely new and different conception of the reduction of arithmetic to logic emerges in this case; for Frege gives us the following method of deciding whether or not a commonly used expression—in our case, ‘o’—has the same sense as a complex expression arrived at after analysis—in our case, ‘the number which belongs to the concept not identical with itself. We are to introduce a new sign, say ‘#’, as an abbreviation for the longer expression, and ask whether or not this sign has the same sense as ‘o’. Frege claims that we can avoid giving a direct answer to this question by rebuilding our system (of arithmetic, in our example), not using ‘o’ but using ‘#’ instead:

If we now succeed in this fashion to build the system of mathematics without needing the sign ‘A’ [‘o’], then we can leave it at that and we do not need to answer the question: With what sense has ‘A’ [‘o’] been used previously? This is the unobjectionable way of doing it. However, it may be convenient to use the sign ‘A’ [‘o’] instead of the sign ‘C’ [‘#’]. But then we have to treat it as a newly introduced sign which had no sense before it was defined.

(Frege, 1979b, p. 211; my translation)

A straightforward application of Frege’s method is complicated by the fact that in (Z) we have, on the one hand, a name of the number zero and, on the other, a description expression for this number. According to Frege’s sense-reference distinction, the name may be associated with any one of a great number of senses. Let us assume, however, that it does have the same sense as the description expression and, hence, also the same reference. But we shall pretend that we do not know that it has the same sense. Everything now depends on whether or not we succeed in building the system of arithmetic by using ‘#’ instead of ‘0’. But what is this system? Is it the system which, among other things, contains such truths as: 1 is the immediate successor of 0? If so, how can we decide whether our new system contains this proposition? The new system will presumably contain the sentence: ‘1 is the immediate successor of the number which belongs to the concept not identical with itself, but does this sentence represent the proposition that 1 is the immediate successor of 0? The only way to answer this question, it seems to me, is to decide whether or not ‘o’ has the same sense as the description expression. Thus we are back to our original question. Following Frege’s instruction, we tried to avoid a direct answer to the question of whether or not ‘o’ has the same sense as the description expression by constructing arithmetic in terms of the description expression rather than the name. But then we are faced with the question of whether or not we have indeed constructed arithmetic rather than something else. And this question, it seems to me, cannot be answered unless we can first decide whether or not the name has the same sense as the description expression.

Within our view, naturally, the situation is quite different. Since ‘o’ does not have a sense, but only a reference, namely, the number zero, the question cannot be whether or not the name and the description expression have the same sense. However, for a description expression we must distinguish between the description which it represents and the thing which it describes. Obviously, the number is not identical with the description represented by the expression. Thus the question can only be: is the number zero the same as the thing described by the description? In this case, the answer is affirmative. (Z) is a true (informative) identity statement. Since (Z) is true, it is also true that since I is the successor of o, I is the successor of the number of the concept not being identical with itself. But notice that on our view these propositions, though both true, are not the same. The state of affairs that I is the successor of o is not the same as the state of affairs that I is the successor of the number of the concept not identical with itself. Do we have a system of arithmetic if we replace the name by the description expression (and other names by other expressions)? The answer is obvious: As long as we talk in our system about the number zero, by means of whatever description expression, we are doing arithmetic. It does not matter how we describe the number, as long as we talk about it and not about something else. There are weird descriptions of zero and, hence, weird ways of saying that I is the successor of o, but what is said, as distinguished from how it is said, is all that matters to the subject matter called “arithmetic.”

In summary Frege’s reduction of arithmetic to logic fails because numbers are not extensions. They are not extensions because, as Russell’s paradox shows, there are no extensions. (But there are sets!) Even if there were extensions, numbers would not be extensions, because numbers are quantifiers. All of this assumes that if there were extensions, they would be “logical things.” If we think of extensions as sets, constituted in their being, not by properties (concepts), but by their members, then the reduction fails for these two reasons: Firstly, once again, it is false that numbers are sets; secondly, even if they were sets, the reduction would at best reduce arithmetic to set theory, not to logic.