a) Another Kind of Necessity: Unimaginability

Let us return to Kant, and let me remind you of our explication of his claim that arithmetic truths are a priori. I have assumed that Kant thinks of the particular arithmetic truth under discussion as being general, that is, as being of the sort: Seven things plus five things are twelve things. Kant observes that this generality is not established by induction. It is therefore not merely a general truth, but is universal. Since it is a dogma of Kant’s philosophy that concepts and judgments which contain necessity and universality do not come from experience, he concludes that this arithmetic truth cannot be derived from experience. And then we are invited to a forced march through the jungle of the transcendental aesthetic. According to our view, the universality of the arithmetic generality is due to the fact that it follows from the truth that the sum relation holds between seven, five, and twelve. Kant simply did not realize that a generality may be justified in this way. But this realization proves Kant’s dogma wrong. A judgment with universality can be based on experience. In our example, it is based on experience, because it follows logically from a proposition that is based on experience, namely, on the relational truth about seven, five, and twelve. Just as we see that midnight blue is darker than lemon yellow, so do we see, I claim, that two plus one is three. And just as we could not see that midnight blue is darker than lemon yellow unless we saw the two colors, so we could not see that two plus one is three unless we saw the three numbers. My view thus implies that we are acquainted with (small) numbers. What stands in its way is the common prejudice that one cannot perceive numbers.

So much about the universality of the arithmetic statement. What about its alleged necessity? As I mentioned before, I trace this kind of necessity to the fact that we cannot imagine the statement to be false. While we can imagine that a horse has wings, we cannot imagine that midnight blue is lighter than lemon yellow. We know that horses do not have wings, but we believe that it is not necessary that horses have no wings. And this same sort of necessity attaches to arithmetic propositions of the kind under consideration. We simply cannot imagine that two things plus one thing is anything but three things, and therefore believe, not only that it is true that two things plus one thing is three things, but also that it is necessary.

This sort of necessity plays an important role in Mill’s philosophy. He says:

It is strange that almost all the opponents of the Association psychology should found their main or sole argument in refutation of it upon the feeling of necessity; for if there be any one feeling in our nature which the laws of association are obviously equal to producing, one would say it is that. Necessary, according to Kant’s definition, and there is none better, is that of which the negation is impossible. If we find it impossible, by any trial, to separate two ideas, we shall have all the feeling of necessity which the mind is capable of.

(J. S. Mill, An Examination of Sir William Hamilton’s Philosophy, pp. 260-61)

Mill defends here the “Law of Inseparable Association.” This law explains, for example, why “it is not in our power to think of colour, without thinking of extension” (James Mill, as quoted by John Stuart, 1979a, p. 253). It explains why we cannot imagine an unextended color, or a surface which is both red and green all over at the same time, or “two and two not making four” (Mill, 1979a, p. 263). Mill thinks that he has located the psychological reason for our inability to imagine certain situations. Since he believes, as we do, that this inability is the source of Kant’s necessity of the a priori, he concludes that he has found the psychological basis for the kind of necessity which is supposed to prove “an a priori mental connexion between ideas” (Mill, 1979a, p. 261).

At first glance, Mill’s defense of this piece of association psychology seems to be an instance of love’s labor lost. If the uniform and intimate association between the idea of extension and the idea of color is to account for a feeling of mental necessity to the effect that color must be together with extension, then we should expect the same kind of necessity in many other instances in which, as a matter of fact, it does not occur. For example, we have never seen a stone float on water, yet we have no difficulty imagining such a situation. Here is Mill’s response to this example:

But, in the first place, we have not seen stones sinking in water from the first dawns of consciousness, and in nearly every subsequent moment of our lives, as we have been seeing two and two making four, intersecting straight lines diverging instead of enclosing a space, causes followed by effects and effect preceded by causes. But there is still a more radical distinction than this. No frequency of conjunction between two phenomena will create an inseparable association, if counter-associations are being created all the while. If we sometimes saw stones floating as well as sinking, however often we might have seen them sink, nobody supposes that we should have formed an inseparable association between them and sinking. We have not seen a stone float, but we are in the constant habit of seeing either stones or other things which have the same tendency to sink, remaining in a position which they would otherwise quit, being maintained in it by an unseen force.

(Mill, 1979a, pp. 263-64)

Given these and other refinements of the law of inseparable association, I think that, in the end, it may merely restate the important fact that we cannot imagine certain situations. In connection with the arithmetic example, for example, Mill says at another place that “The experience must not only be constant and uniform, but the juxtaposition of the facts in experience must be immediate and close, as well as early, familiar, and so free from even the semblance of an exception that no counter association can possibly arise” (Mill, 1979a, p. 270).

The fact remains that we cannot imagine certain circumstances, and I maintain that this fact explains the kind of necessity attached to so-called synthetic a priori truths. In order to strengthen our thesis, we must draw a sharp distinction between imagination, on the one hand, and conception, on the other.

Mill does not make this distinction, but talks about different kinds of inconceivability. He gives examples of things the mind “cannot put together in a single image”:

We cannot present anything to ourselves as at once being something, and not being it; as at once having, and not having, a given attribute . . . We cannot represent to ourselves time or space as having an end. We cannot represent to ourselves two and two as making five; not two straight lines as enclosing a space. We cannot represent to ourselves a round square; nor a body all black, and at the same time all white.

(Mill, 1979a, p. 70)

He speaks here of the mind’s putting together images, but he also claims that these things are inconceivable, “our minds and our experience being what they are” (Mill, 1979a, p. 70). In my view, his examples are unimaginable, but not at all inconceivable. I agree that we cannot imagine that something has the property F and also, at the same time, does not have the property F; and we cannot imagine that two plus two are five; and we cannot imagine that something is both round and square at the same time. But I also insist that we can conceive of these three circumstances.

b) Imagination Distinguished from Conception

Mill holds that all cases of inconceivability reduce to two kinds: There is the inconceivability of contradictions, and there is the inconceivability of inseparable association. Let us consider the round square. Mill points out that, strictly speaking, a round square is not a contradictory thing. But he claims that in our experience “at the instant when a thing begins to be round it ceases to be square, so that the beginning of the one impression is inseparably associated with the departure or cessation of the other. Thus our inability to form a conception always arises from our being compelled to form another contradictory to it” (Mill, 1979a, pp. 70-71). The law of inseparable association leads us to associate the idea of not-square with the idea of round; and the inconceivability of contradictions does the rest. The inconceivability of contradictions, therefore, is not a matter of inseparable association. This bears directly on my contention that we can conceive of many situations which we cannot imagine, and, in particular, that we can conceive of contradictory circumstances.

What prevents a clear understanding of the important difference between imagination and conception is a philosophical muddle that reaches from Mill to the present. Mill, like many recent and contemporary philosophers, thinks that contradictions are “unmeaning”: “That the same thing is and is not—that it did and did not rain at the same time and place, that a man is both alive and not alive, are forms of words which carry no significance to my mind” (Mill, 1979a, p. 78). And then he continues: “Whatever may be meant by a man, and whatever may be meant by alive, the statement that a man can be alive and not alive is equally without meaning to me” (Mill, 1979a, p. 78). Contradictions, in short, are said to be meaningless. Since they are meaningless, they are inconceivable. Not only can they not be imagined, as I have granted, but they cannot even be thought; for there is no thought connected with them.

I wish to defend the view that contradictions are perfectly meaningful, intelligible, and conceivable. But I also insist, on the other hand, that they are not imaginable. There are mental acts, other than acts of imagination, which can intend contradictory states of affairs. We can and do deny that it is the case that it is both raining and also not raining at a certain moment at a certain place. We can and do assume that if it were the case, then it would also be the case that: If P, then not-P, and if not-P, then P, where P is the circumstance that it is raining at a certain place at a certain moment. That a contradiction is not meaningless and hence that it can be conceived, is proven by the fact that we draw legitimate inferences from it. There is no difficulty in conceiving what the state of affairs P and not-P “would be like”: P would be the case, and not-P would also be the case. Of course, I cannot imagine what that would “look like,” but I can conceive of it. I know that a contradictory state of affairs cannot obtain. How can I know this, I ask Mill, if I cannot conceive of such a state of affairs?

The case of contradictions illuminates the distinction between what is unimaginable and what is inconceivable. What we cannot imagine is a matter of our perceptual constitution; what we cannot conceive is a matter of our conceptual constitution. Generally speaking, if we can imagine P, then we may or may not be able to imagine not-P, but if we can conceive of P, then we can also conceive of not-P. And this means that we can conceive the negation of any law. For example, we can conceive that bodies move with a speed greater than the speed of light. Much of science fiction would otherwise be unintelligible. We can even conceive of the negations of the laws of arithmetic, set theory, and logic. I can conceive of two plus two being five rather than four. Of course, I cannot imagine a situation in which this would be the case. But I do know what is meant by ‘two plus two is five.’ What is meant is that two plus two is five or, put more pedantically, that the sum relation holds between five, two, and two.

Conceivability extends even beyond the boundaries of arithmetical, set theoretical, and logical falsehoods: It reaches the laws of ontology! For example, it is a law of ontology that the relation of conjunction holds between states of affairs and only between states of affairs. (I assume that there are no “conjunctive properties.”) Yet, I understand perfectly well what the expression ‘P and 7’ purports to represent: It purports to represent a conjunction of the state of affairs P and the number 7. Of course, I know that there is no such fact; for I know the ontological law just mentioned. And I cannot imagine what such a conjunction would be like. To give another example, it is a law of ontology that individual things, though they exemplify properties, are not exemplified by anything. Yet, I can easily conceive of, say, a certain shade of blue exemplifying the pen before me on my desk. According to my assumption, the exemplification relation holds, not between the pen and the color—as it does—but conversely between the color and the pen. Needless to say, I cannot imagine what this situation “looks like.” I cannot paint it, for example, as I can paint a blue pen.

It has been quite the fashion in our century to claim that this or that sentence is meaningless or that this or that expression is unintelligible. Russell’s theory of types is a case in point. That the color midnight blue is not midnight blue seems to me to be, not only a perfectly intelligible proposition, but a true proposition to boot. Nor do I find the negation of this assertion to be nonsense. Not only specific philosophical theories, but conceptions of the very nature of philosophy have been based on exaggerated claims about what is and what is not meaningful. Surely, it takes more than just ordinary philosophical gullibility to agree with Wittgenstein that his pronouncements about the nature of logic and the structure of the world are nonsense. False they may be, but nonsense they are not. Nor would it occur to anyone but the most fanatic philosopher that the philosophical statements of Aristotle, Descartes, and Frege are one and all meaningless.

One cannot but be amazed that so many philosophers have tried to do what, according to their own view, cannot be done, namely, give genuine examples of nonsense. That procrastination eats Wheaties for breakfast may be an outrageous or hilarious statement, but it is hardly nonsensical. Or consider Mill’s example: Humpty Dumpty is an Abracadabra. Mill says that he does not know what this proposition states since he does not know what is meant by ‘Humpty Dumpty’ and by ‘Abracadabra’ (Mill, 1979a, p. 78). Well, the same could possibly be said by Mill about any sentence in Albanian. The question is not whether expressions in an unknown language mean something to us, but whether an expression means something in a language. It is surprising that Mill claims not to know what those two expressions mean. ‘Humpty Dumpty’ is a certain creature in a nursery rhyme who fell off a wall, broke into many pieces, and could not be put together afterwards. An abracadabra is a spell; something that wards off sickness and, perhaps, evil spirits. Humpty Dumpty is not such a spell, for spells do not fall off walls and break into pieces. Thus the sentence ‘Humpty Dumpty is an abracadabra’ is not only quite meaningful, but obviously false. Of course, there are other interpretations of the sentence. ‘Abracadabra’ is often used to signify a meaningless jumble of words, an incantation without meaning. We have here the curious situation that a meaningless string of letters becomes meaningful by being used to represent gibberish. But this makes the sentence meaningful: It states under this interpretation that Humpty Dumpty is a meaningless jumble of letters, and that, again, is simply false.

I have claimed that we can conceive of contradictions. The expression ‘P and not P’, as one says in certain contexts, is well-formed. The expression ‘P and not’, by contrast, is not well-formed. We can also put it this way: While the first sentence represents a state of affairs, the second does not. The so-called formation rules of an artificial language (the rules for well-formedness) prescribe (among other things) what expressions represent propositions (states of affairs). Logic then tells us which of these propositions are tautologies, which are contradictions, and which are neither. This conception, too, contradicts the claim that contradictions are meaningless. But I wish to make another point. The formation rules of a language, it may be noted, reflect the ontological laws of the world. They prescribe what is ontologically possible. For example, the proposition P and not-P is ontologically possible, but logically impossible. The combination P and not, on the other hand, is not even ontologically possible, for it is a law of ontology that the conjunction relation always holds between states of affairs (propositions). The combination P and not is not a proposition. But notice that I speak nevertheless of the “combination” P and not. I do understand this combination: a proposition P is related by conjunction to negation. Thus we must distinguish between a narrower and a wider sense of ‘proposition’. According to the rules of well-formedness, which delineate the narrower notion, the expression ‘P and not’ does not represent a proposition. We may say, with this notion in mind, that it is gibberish. But according to the wider notion of a proposition, the sentence represents a proposition (state of affairs), namely, the proposition that P stands in the conjunction relation to not. Anything of the form X R Y is then a proposition, where R is a relation, and X and Y are any two entities whatsoever.

I shall again look ahead. The distinction we have just made has an important bearing on one of Wittgenstein’s views to the effect that an expression like ‘P and not’ has no meaning, since it gets its meaning from use, and it is not used. In a lecture at Cambridge, Wittgenstein reportedly talked about applying negation to an individual thing, an apple, rather than to a proposition. And then he claimed that we are wavering between two different views. On the one hand, we think that to ‘apple’ and ‘not’ there correspond definite things or ideas, and that these things or ideas may or may not fit (as a certain shape may or may not fit another shape). On the other hand, we think that these words are characterized by their use, and that negation is not completed until its use with ‘apple’ is completed. We cannot ask whether the uses of these two words fit, for their use is given only when the use of the whole expression ‘not apple’ is given (Wittgenstein, as cited by Coffa, in J. A. Coffa, To the Vienna Station, chap. XIV).

As usual, Wittgenstein does not spell out the details of his argument for the last sentence. But it is not hard, I think, to guess how it would proceed. The question before us is: Does the meaning of ‘not’ fit together with the meaning of ‘apple’? We assume that the meanings of these two words are their uses. So the question becomes: Does the use of ‘not’ fit the use of ‘apple’? But the use of ‘not’ is determined, among other things, by whether or not there is a use for ‘not apple’. And this shows that we must already know whether or not the meaning of ‘not’ fits together with the meaning of ‘apple’ before we can answer the question of whether or not they fit. In other words, if we assume that the meanings of words are their uses in connection with other words, then the question of whether or not a given word fits another word cannot be answered. The question “makes no sense.” I suppose that Wittgenstein’s later philosophy rests on the conviction that considerations of this sort show that the meanings of words are their uses. But there is obviously a different possibility. What the argument proves, one may hold, is that the meaning of a word, in the sense relevant for the argument, cannot be its use, as understood for the argument. And this is, indeed, the conclusion which we draw. The expression ‘not apple’ does not represent an ontologically possible entity, because negation, according to a law of ontology, does not attach to properties. This is the reason why ‘not’ and ‘apple’ do not “fit.”

I have argued that our ability to conceive of things and situations is much greater than most philosophers assume. We can conceive not only of contradictory states of affairs, but even of ontologically impossible situations. But we cannot imagine such states of affairs. We cannot imagine logical laws other than ours; we cannot imagine arithmetic laws other than ours; we cannot imagine set theoretic laws other than ours; we cannot even imagine geometric laws other than ours. We cannot imagine non-Euclidian space. But this fact brings another important point to our attention. What we can and cannot imagine is one thing, what is and is not the case, is quite another. From the fact that we cannot imagine non-Euclidian space, it does not follow that the universe has to be Euclidian. The world does not have to conform to our quite limited ability to imagine things. And this means that what we cannot imagine to be otherwise, may nevertheless not be the case. And this implies that what we think to be necessary (in this sense of ‘necessary’), may nevertheless be false. It would never have occurred to Kant, but: Necessity does not imply truth!

c) The Synthetic A Priori and Our Sense Dimensions

Using the word ‘part’ loosely, one can distinguish between separable and inseparable parts of things. The color of a horse is an inseparable part of it, while its head is a separable part. From Berkeley on, some philosophers have stressed this distinction and built ontologies around it (see, for example, B. Smith, ed., Parts and Moments, Studies in Logic and Formal Ontology). One of the most thorough discussions of it occurs in Husserl’s Logical Investigations. In this context, Husserl makes a profound remark that bears directly on our investigation: “It is now immediately plain that all the laws or necessities governing different sorts of non-independent items fall into the spheres of the synthetic a priori: one grasps completely what divides them from merely formal contentless items” (E. Husserl, Logical Investigations, vol. 2, p. 456). He then goes on to say that laws governing the non-independence (read: inseparability) of qualities, intensities, extensions, boundaries, relational forms, etc., must be distinguished from purely “analytic” generalizations. Taking a cue from Husserl, I shall maintain that the most noteworthy area of synthetic a priori truths can be found within the boundaries of our sense dimensions. Synthetic a priori truths rule the structures which are formed by sensible properties and relations.

Bergmann lists six kinds of such truths and then describes the principles of his classification:

(A) Round is a shape. Green is a color. e is a pitch. Of two pitches, one is higher than the other; only a pitch is higher than anything else.

(B) e is higher than c. (This shade of) brown is darker than (that shade of) yellow.

(C) If the first of three pitches is higher than the second and the second is higher than the third, then the first is higher than the third.

(D₁) What has pitch has loudness and conversely. What has shape has color and conversely.

(D₂) Nothing (no tone) has two pitches. Nothing (no area) has two shapes.

(E) If the first of three things (areas) is a part of the second, and the second is a part of the third, then the first is a part of the third.

The truths of (A) are those and only those constituting the several dimensions. Some of them are atomic; some are general. A simple relation obtaining between two (or more) members of a dimension is an atomic fact. (B) is the class of all such facts. The truths of (C) are all general. They are all those and only those which connect the properties of a single dimension with the simple relations between them. The truths of (D₁) are the general truths by which the members of two dimensions depend₂ on each other. The truths of (D₂) are those and only those by which the members of one or two dimensions exclude each other. The last two classes are labeled with the same letter because they have long provided the most popular examples of a priori truths. The class (E) corresponds to (C). The reason for setting it apart is that the spatial relations “in” the facts of E are the only relations of the first type that are mentioned in the list.

(G. Bergmann, “Synthetic A Priori”, pp. 295-96)

It is clear that to (E) also belong certain truths about temporal relations.

Here then we have a list of all kinds of a priori truth about such perceptual properties as colors, shapes, pitches, etc., and such perceptual relations as being higher (in pitch), being darker (between colors), etc., and spatial and temporal relations. All of these truths either are “universal” and “necessary,” or else logically imply “universal” (and “necessary”) truths in the way described earlier. The peculiar necessity Kant discovered and which he describes in the Transcendental Aesthetic reduces to our inability to imagine certain states of affairs. This inability, certainly, is a function of our sensibility. Thus Kant is correct when he claims that the necessity of certain laws is grounded in our constitution. If we had different sense organs, so that we could perceive properties and relations different from the ones we do perceive, then our imagination would also be quite different and, hence, quite different states of affairs would appear to us to be necessary. But Kant is mistaken when he draws from this fact the idealistic conclusion that the world must conform to our imagination. What we can and cannot imagine to be the case, as I said before, does not determine what is the case. The most important lesson to be learned from Kant’s “Irrweg” is that what we cannot imagine to be otherwise may yet prove to be false. We can prove it to be false, because conception, in distinction to imagination, reaches to the limits of the world. We may not be able to imagine the physical structure of space and time, but we can discover it, because we can conceive of it. We cannot imagine the properties of and relations among elementary particles, but we can discover them, because we can conceive of them. We cannot imagine that a property has no extension (set), but we can prove that there is such a property. The Kantian idealist is not only mistaken when he claims that the structure of the world must with necessity conform to our sensibility, he makes an even more fundamental mistake since he believes that what is necessary must be true!

One more word about necessity before we return to Mill’s philosophy of arithmetic. The kind of necessity we have studied attaches to propositions (states of affairs, circumstances). A state of affairs is necessary if and only if we cannot imagine what it would be like for the state of affairs not to obtain; in other words, if and only if we cannot imagine its negation. A possible state of affairs, on the other hand, is a state of affairs which we can imagine to obtain. We may therefore think of being imaginable as a property of certain states of affairs. If we do, then we can distinguish between four kinds of state of affairs:

(1) Imaginable states of affairs,

(2) states of affairs that cannot be imagined,

(3) states of affairs for which we can imagine their negation,

(4) states of affairs for which we cannot imagine their negation.

We may then speak of:

(1’) Possible states of affairs,

(2’) impossible states of affairs,

(3’) contingent states of affairs, and

(4’) necessary states of affairs.

It follows that a state of affairs is possible if and only if it is not the case that its negation is necessary:

(I) P is possible if and only if it is not the case that not-P is necessary. We may also say that a state of affairs entails another state of affairs if and only if we cannot imagine that it is not the case that, if the one state of affairs obtains, the other also obtains:

(II) P entails Q if and only if it is necessary that, if P obtains, then Q obtains. Furthermore, whether or not a state of affairs is imaginable does not depend on whether or not it obtains. This means that the truth of ‘P is possible’ is not a truth function of ‘P’ And this means that none of the following equivalences holds:

P is possible if and only if P,

P is possible if and only if not-P,

P is possible if and only if (P or not-P), and

P is possible if and only if (P and not-P).

Proceeding in the Kantian vein, we may add:

(III) If P obtains, then P is possible.

Next, if we are interested in formulating an axiomatic system that catches these and other “necessary” truths about the possible and the necessary, we shall also have to assume that any theorem of our system is itself necessary:

(IV) If P is a theorem, so is necessarily-P.

Finally, we may believe that whatever follows from a necessary state of affairs is itself necessary and adopt the following equivalent principle:

(V) If it is necessary that if P obtains, Q obtains, then if it is necessary that P obtains, it is necessary that Q obtains.

If we accept the five statements (I) to (V), and add the obvious truths of the so-called propositional calculus, we get a well-known system of “modal logic,” namely, the system T (see G. E. Hughes and M. J. Cresswell, An Introduction to Modal Logic). From my point of view, this is an axiomatization of the Kantian notion of the necessity (and possibility) that attaches to the synthetic a priori. Hence we are dealing, not with a system of logic, but rather with a theory of what is imaginable.

d) Arithmetic and “Physical Facts”

There are two basic kinds of arithmetic propositions: (1) propositions that state arithmetic relations among numbers, and (2) laws about numbers. Frege, in his scathing criticism of Mill’s conception of arithmetic, starts with Leibniz’s proof that 2 + 2 = 4 (G. Frege, The Foundations of Arithmetic, pp. 7-14). This proof starts with the following “definitions”:

This is the way Leibniz puts it and Frege cites it. But I shall take for granted that the ‘and’ here can be replaced by ‘plus’. Frege points out that Leibniz’s proof rests on the association law:

(L) All natural numbers are such that: n₁ + (n₂ + n₃) = (n₁ + n₂) + n₃. We are asking: What kinds of proposition are (D) and (L)? In particular, are they analytic or synthetic? And also: How do we discover these propositions (facts)?

Take for example: (a) 2 = 1 + 1. Leibniz calls this statement a definition, and Frege agrees with him. Mill, on the other hand, observes that this is not a definition in the logical sense. It does not merely fix the meaning of an expression, but asserts an observed matter of fact. He holds that (a) is not just a (consequence of a) stipulation of the sort: Let us agree to let the arbitrary expression ‘2’ stand for whatever ‘1 + 1’ represents. It does not introduce an arbitrary sign, say, ‘#’, as an abbreviation for ‘1 + 1’ (J. S. Mill, A System of Logic, Collected Works, vol. 7, pp. 253-54). At this point, we must be very picky. I take it that ‘1 + 1’ is merely short for (just another expression for) something like ‘the sum of 1 and 1’, that is, for a description expression. Things are rather complicated at this point because description expressions, as Frege was the first philosopher to notice, are “connected” with descriptions as well as with what these descriptions describe. In our case, we must distinguish between what the expression represents, namely, the description the sum of 1 and 1, and what it describes, namely, the number 2. Because of this peculiarity of description expressions, the numeral ‘2’ may be introduced in two quite different ways. It may be a mere abbreviation for the description expression. In this case, it represents, not the number 2, but the description. Given that we mean by ‘2’ the number and not the description, this would be a perverse way of using the numeral. Or else it may be introduced as a name for whatever it is that the description represented by the expression describes: Let us call what is described by the expression: ‘2’. We must note that in this second case, ‘2’ is not really an abbreviation of anything. What happens is, rather, that we give a name to something that has so far only been described.

With this essential refinement in mind, the dispute between Mill and Frege (and Leibniz) comes down to this: Frege claims that in this context ‘2’ is an arbitrary sign, with no prior referent, which is introduced as a name for whatever the description describes, while Mill holds that it is the name of a number, so that we have an assertion to the effect that the number 2 is what the description describes. I think that Mill is correct and Frege mistaken. But if Mill is correct, then the question arises whether the equation ‘2 = 1 + 1’ is analytic or synthetic; the question arises whether the assertion that the number 2 is the same as the sum of 1 and 1 is analytic or synthetic. And according to our explication of analyticity, this question is the same as whether or not it is a logical truth. By calling (a) a definition, Frege and his followers simply try to avoid this unavoidable question.

Why do I think that Mill is correct and Frege mistaken? Compare (a) with the following assertion: (b) 2359 = 2358 + 1.1 think that (b) is indeed a “definition” in the relevant sense. I am not acquainted with the number 2359, I only “know it” as the sum of 2358 and 1 (or, better, as the immediate successor of 2358). Most numbers are indeed presented to us merely through descriptions. We only know them, as Russell might say, by description. That we are not acquainted with them is due to the special constitution of our mind. Very small numbers, on the other hand, are known “by acquaintance.” The number 2, for example, is presented to us in perception; we see it, when we see, for example, two apples. We use the numeral ‘2’ to represent this number with which we are acquainted. The assertion (a), therefore, states that this number is the same number as the number which is the sum of 1 and 1 (or as the number which is the successor of 1). (a) can be used to assert that this number is the same number as the one that stands in a certain relation to 1 and 1 (or to 1).

For numbers the familiar distinction holds between two quite different assertions:

I believe that we are acquainted with certain (small) numbers, just as we are acquainted with certain people. Larger numbers, on the other hand, we only know through the relations which they have to other numbers. Roughly, most numbers are known to us as the successors of the successors of the successors, and so on, of the small numbers with which we are acquainted. What distinguishes our view from the view of Leibniz and Frege is that we hold that some, though not all, of the statements of kind (a) are not the consequences of mere stipulations.

If (a) is not the result of a convention, then we can ask whether or not it is analytic. I think that the answer is obvious: It is not analytic, since it is not a truth of logic. But if it is not a truth of logic, what kind of truth is it? It is in answer to this question that Mill errs and Frege scores his most telling points against him. According to Mill, the proposition 2 + 1 = 3 states an “observed matter of fact.” He even call it a “physical fact” (Mill, 1979b, vol. 7, p. 257). What is this “observed fact”? Mill answers:

And thus we may call “Three is two and one” a definition of three; but the calculations which depend on this proposition do not follow from the definition itself, but from an arithmetical theorem presupposed in it, namely, that collections of objects exist, which while they impress the senses thus, °o°, may be separated into two parts, thus, 00 o. This proposition being granted, we term all such parcels Threes, after which the enunciation of the above mentioned physical fact will serve also for a definition of the word Three.

(Mill, 1979b, p. 257)

To this, Frege replies: “What a mercy, then, that not everything in the world is nailed down; for if it were, we should not be able to bring off this separation, and 2 + 1 would not be 3!” (Frege, 1974, p. 9). Later on, Frege zeroes in on Mill’s expression “physical fact” and points out that if the arithmetic statement were a statement of physical fact, then it would be incorrect to speak of three sensations or of three solutions of an equation. Frege is surely right when he objects that the statement does not assert a “physical” fact about the spatial arrangement of three perceptual objects. But this is not exactly what Mill wishes to say, as it turns out. It would be premature at this point to dismiss Mill’s view out of hand.

Mill maintains that 2 + 1 = 3 is a generalization from experience. What this expression really represents is a general fact:

In one of the most interesting and revealing passages of his book, Mill says:

All numbers must be numbers of something: there are no such things as numbers in the abstract. Ten must mean ten bodies, or ten sounds, or ten beatings of the pulse. But though numbers must be numbers of something, they may be numbers of anything. Propositions, therefore, concerning numbers, have the remarkable peculiarity that they are propositions concerning all things whatever; all objects; all existences of every kind known to our experience . . . That half of four is two, must be true whatever the word four represents, whether four hours, four miles, or four pounds weight.

(Mill, 1979b, pp. 254-255)

We see that Frege’s criticism concerning the three sensations or three solutions to an equation is not justified. Or, at least, it is not justified if we interpret the phrase ‘known to our experience’ sufficiently widely.

Let us take a look at the first two sentences of this quotation from Mill. I agree, of course, that numbers are always numbers of something, just as colors are always colors of something. Numbers, like colors, do not float around by themselves. They are firmly anchored in reality. They quantify kinds of things. But Mill says next: “Ten must mean ten bodies, or ten sounds, . . .”; and this seems to me to be false. Ten’ means the number ten; it represents this number and nothing else. In particular, it does not represent what ‘ten bodies’ represents. Contrary to Mill, I hold that there are “numbers in the abstract” in the sense that they form a kind of entity distinguishable from the properties which they quantify. The expression ’ten bodies’ clearly mentions two things: It mentions the number ten and it mentions bodies. This is obvious from the fact that we can vary the two words involved and also speak of ten sounds and four bodies. This emphasis on the independence of numbers from the things they number is necessary, because we wish to hold that the sum relation, for example, is a unique relation among numbers, and not some kind of “physical” relation among heaps of things or aggregates of things.

Mill seems to think of the numeral ‘3’ as some kind of common name for such aggregates (complexes, wholes, or what have you) as are presumably formed by three bodies, three sounds, three beatings of the heart, etc. The common name ‘number’, then, is thought to name (commonly, indifferently) all such kinds of aggregates (Mill, 1979b, vol. 7, p. 55). From Mill’s point of view, what (G) says is that all 3-aggregates consist of 2-aggregates and 1-aggregates or, for short: All 3’s consist of 2’s and I’S. But what is here the meaning of “consists of’? Since many aggregates are not spatial “heaps,” this relation cannot be the spatial whole-part relation. What then is it? In what reasonable sense can three solutions of an equation be said to consist of two solutions and one solution? I do not think that there is a plausible answer to my rhetorical question.

Mill points out that our familiar way of depicting that 2 plus 1 is 3 as an equation makes it appear as if we are dealing with an ‘‘identical proposition” (Mill, 1979b, vol. 7, p. 256). It makes it look as if we were dealing with the result of a stipulation. Mill wants to dispel this impression. On this point, as I said earlier, we fully agree with him. ‘3’ is not a purely conventional sign for the number which is the sum of 2 and 1. It would be much less misleading, as we see things, to write instead: ‘+(3, 2, 1)’. How does Mill back up his view that this is not a definition? He says that the expression ‘two pebbles and one pebble’ and the expression ‘three pebbles’ “stand indeed for the same aggregation of objects, but they do by no means stand for the same physical fact” (Mill, 1979b, vol. 7, p. 256). He continues: “They are names of the same objects, but of those objects in two different states: though they denote the same things, their connotation is different” (Mill, 1979b, vol. 7, p. 256). What are these different “states”? Three pebbles are three pebbles, as Frege points out, no matter how they are arranged or where they are located. Mill retreats at this point to the different impressions which different arrangements of the pebbles make on our senses. We are reminded of the difference between °o° and 00 o. The arithmetic statement is not a mere definition, according to Mill, because it asserts that the same three things can be arranged in different ways. It states that any aggregate of three things can appear as arranged in one way or in another. And this statement, Mill would insist, is clearly based on experience. But what could possibly be meant here by “arranged”? Of course, pebbles allow for different spatial arrangements. But can we conclude that the statement that two pebbles and one pebble are three pebbles says that three pebbles can be arranged in certain ways? Certainly not; for, as Frege points out, two pebbles and one pebble would be three pebbles if pebbles could not be arranged in any other way at all. Moreover, what kind of arrangement do we have to envisage for three colors, three sensations, or three solutions of an equation? Mill’s view, in a nutshell, clashes with the true nature of the sum relation.

But do we not use pebbles or fingers when we teach children that two plus one is three? Of course, we do. Mill appeals to this undeniable fact in order to defend his view:

The fundamental truths of that science [the science of Number] all rest on the evidence of sense; they are proved by showing to our eyes and fingers that any given number of objects, ten balls for example, may by separation and re-arrangement exhibit to our senses all the different sets of numbers the sum of which is equal to ten. All the improved methods of teaching arithmetic to children proceed on a knowledge of this fact. All who wish to carry the child’s mind along with them in learning arithmetic; all who wish to teach numbers, and not mere ciphers—now teach it through the evidence of the senses in the manner we have described.

(Mill, 1979b, vol. 7, pp. 256-257)

In order to teach a child that two plus one is three, we may indeed arrange three pebbles first like this: o 00, and then like this: 000. But does learning that two plus one is three consist in learning that three pebbles can form these two different spatial patterns? Has the child learned the arithmetic truth when he has grasped the geometric truth? Obviously not! Knowledge that three pebbles can be spatially arranged in countless ways is not knowledge that two plus one is three. We must distinguish between how we teach the child and what we teach it. What the child of our example is supposed to grasp when we rearrange the pebbles is some “abstract” relationship among numbers, not something about what happens when pebbles are arranged in different spatial patterns. The arrangement of the pebbles is merely the “occasion” for seeing that “abstract” relationship. It is a visual aid. We use pebbles, or peas, or oranges, because we can see and feel these things. But the truths we teach on these occasions are not about pebbles, or about peas, or about oranges. They are truths about numbers. We know that the child has understood the relationship between three, two, and one, when he no longer cares either about the particular things numbered or about the particular arrangements of these things.

The situation is quite similar when we teach a child that midnight blue is darker than lemon yellow. We may show her a blue sweater and a yellow ball, and we may show these two items first separately and then together side by side. But we are not trying to teach anything about spatial arrangements. Nor are we interested in the fact that the sweater is made from cotton, manufactured in Hong Kong, and bought as a birthday present, while the ball is made from rubber, manufactured in Indianapolis, and was found in the street. These things are true but unimportant. And in this situation, too, we know that the child has grasped the relationship between the two colors when she no longer cares what particular objects we show her, how these objects are spatially arranged, where they have been made, etc. All that matters is the relationship between the colors of the objects, just as all that matters in the previous case is the relationship among the numbers.

The temptation to think of arithmetic as somehow concerned with operations on perceptual objects lingers on. Hilbert, for example, gives the following account of the equation 2 + 3 = 3 + 2:

2 + 3 = 3 + 2 is intended to communicate the fact that 2 + 3 and 3 + 2, when abbreviations are taken into account, are the self-same numerical symbol 1 1 1 1 1. Similarly, 3 > 2 serves to communicate the fact that the symbol 3, i.e., 1 1 1, is longer than the symbol 2, i.e., 1 1; or, in other words, that the latter symbol is a proper part of the former.

(D. Hilbert, “On the Infinite,” p. 143)

In Hilbert’s philosophy of arithmetic, Mill’s pebbles have been replaced by numerals; the arrangements of pebbles, by arrangements of numerals.

More recently, Kitcher has proposed a conception of arithmetic that harks back to Mill’s emphasis on how we can arrange and rearrange pebbles:

Children come to learn the meaning of ‘set’, ‘number’, ‘addition’ and to accept basic truths of arithmetic by engaging in activities of collecting and segregating. Rather than interpreting these activities as an avenue to knowledge of abstract objects, we can think of the rudimentary arithmetical truths as true in virtue of the operations themselves. By having experiences like that described in the last paragraph, we learn that particular types of collective operations have particular properties: we recognize, for example, that if one performs the collective operation called ‘making two’, then performs on different objects the collective operation called ‘making three’, then performs the collective operation of combining, the total operation is an operation of making five.

(Philip Kitcher, The Nature of Mathematical Knowledge, pp. 107-108)

Kitcher, if I understand him correctly, goes Mill one better: Arithmetic, according to him, is not about the different arrangements of pebbles, but is about the operations of arranging the pebbles. Here are some further quotations that seem to confirm this interpretation:

Knowledge of such properties of such operations is relevant to arithmetic because arithmetic is concerned with collective operations.

(Kitcher, p. 108)

One central idea of my proposal is to replace the notions of abstract mathematical objects, notions like that of collection, with the notion of a kind of mathematical activity, collecting.

(Kitcher, p. 110)

I propose that the view that mathematics describes the structure of reality should be articulated as the claim that mathematics describes the operational activity of an ideal subject.

(Kitcher, p. 111)

In the same vein, I suppose, one may argue that the statement that midnight blue is darker than lemon yellow is not about colors, but about the operation of comparing certain blue and yellow objects. It seems to me, however, that the statement is not about this operation, but about what we recognize (see) as a consequence of it.

Would two plus three be five if nobody performed any “collective operations”? Since I believe that it would, just as midnight blue would be darker than lemon yellow if nobody compared the two colors, I cannot agree with Kitcher’s conception of the nature of arithmetic. If arithmetic were about operations, then there would be no arithmetic, if nobody performed any operations. However, I am convinced that the truths of arithmetic do not depend on there being human beings or other “operators,” just as they do not depend on there being different arrangements of the same pebbles.

On a less global scale, I find fault with Kitcher’s view because I can make little sense, for example, of an addition of operations. Kitcher states that “When we combine the objects collected in two segregative operations on distinct objects we perform an addition on those operations” (Kitcher, p. 112). So you take two peas away from a group of peas, and you take three lentils away from a little heap of lentils, and you put them together in one small pile. It seems to me that you have performed three operations: (1) You have taken the two peas away and put them down, (2) You have taken three lentils away and put them down, and (3) you have combined the peas and lentils into one pile. So far, so good. But is this third operation, the putting together of the peas and the lentils, an “addition on operations,” as Kitcher claims? Surely not. It is clearly not an operation on operations, but an operation with peas and lentils, on a par with the operations of taking away two peas and taking away three lentils. In my view, addition is just as little an operation on the activities of separating peas from peas and lentils from lentils, as it is an operation on the peas and lentils themselves.

e) Arithmetic and Induction

We turn now to the second kind of arithmetic statement, namely, to arithmetic laws. Our example is the so-called law of association (L). I believe that Mill thinks of this law as an expression of the general principle that whatever is made up of parts, is made up of the parts of those parts (Mill, 1979b, vol. 7, p. 613). This reminds us of Bolzano’s thesis that the associative law follows from the explanation of the notion of a sum. But there is also a big difference between Mill and Bolzano. While Mill thinks of that principle as a law of nature, Bolzano introduces it as part of a definition, so that he can claim that the associative law is analytic. Mill, I think, would correctly argue that Bolzano’s explanation does not imply the associative law, for the law presupposes that there are things of the explained sort, that is, that there are so-called sums. Assume that we “define” ‘mermaid’ as something that has a fishtail, a female torso, long blond hair, etc. Does it follow from this explanation that there is a law to the effect that mermaids have fishtails? Obviously not. What follows is merely that if there were mermaids, then they would have fishtails. Bolzano’s “definition,” therefore, merely implies that if there are sums, they obey the associative law. That there are sums, however, is not an analytic truth.

Mill thinks of the associative law as a law about wholes and the parts of their parts. This law, he says, “is obvious to the senses in all cases which can be fairly referred to their decision, and so general as to be coextensive with nature itself, being true of all sorts of phenomena (for all admit of being numbered,) must be considered an inductive truth, or law of nature, of the highest order” (Mill, 1979b, vol. 7, p. 613, my italics). I think that Mill is correct: The associative law is a general truth that can only be justified by induction. Unlike Mill, I would not call it a “law of nature.” Nor would I tie it to the observation of “physical phenomena.” The associative law is a law of arithmetic. It is not a law of nature, if we mean by this expression a law about physical or mental phenomena. Numbers are neither “physical” nor “mental” things. Mill once again builds his empiricism on too narrow a notion of “sense experience.” When I notice that the squares of a chess board, which are parts of the rows of the board, are parts of the chess board, I have not observed an instance of the associative law of arithmetic, as Mill seems to think. What I have noticed is an instance of the law that spatial parts of spatial parts of wholes are spatial parts of those wholes. This law holds for the spatial part-whole relation. It does not hold for every part-whole relation. For example, it does not hold for the membership relation (for sets). It happens to hold also for the sum relation. But, and this is the important point, the sum relation is not the spatial part-whole relation, and to observe an instance of the latter is not to notice an instance of the former.

But this objection does not affect what I consider to be Mill’s main point, namely, that (G) is a general truth which, if it can be justified at all, can only be justified through its instances, that is, as the saying goes, by induction. We observed earlier that this is also Bolzano’s view in regard to the basic laws of logic, mathematics, geometry, and physics (see, for example, B. Bolzano, Wissenschaftslehre, par. 315, 4). In regard to the specific law (G), though, they differ: While Mill treats it as a generality, Bolzano thinks of it as the consequence of a stipulation (definition).

f) A Word about So-called “Recursive Definitions”

I imagine that some readers may have been chafing at the bit for some time now, because they learned a long time ago and as a matter of course that one can prove the associative law from (1) the recursive definition of addition, and (2) the axiom of mathematical induction. Hence (G), they believe, is not a primitive law of arithmetic, and it can be justified by deduction rather than induction. This is perfectly true but does not change the philosophical thrust of our discussion. We merely have to switch from the associative law to the principle of mathematical induction:

What I claimed earlier for the associative law, I now claim for (MI): It is not analytic (but synthetic), it is a true generality, and it can only be justified, if it can be justified at all, by induction. (MI) is not analytic, for it is not a logical truth. It is neither a mere reformulation of a logical truth nor is it an instance of a logical law. It is a true generality. This distinguishes it, in my view, from any particular truth about addition. Finally, since it is a generality, it can only be justified in one of two ways, either by being deduced from other laws, or else by induction. (I take for granted that it cannot be deduced from a relational statement like the truth can that seven things and five things are twelve things.) Since we have assumed that (MI) is an axiom, it cannot be deduced from other laws. Hence it can only be justified, if it can be justified at all, by means of induction from individual cases. I see no other possibility. It is obvious why a logicist must argue that (MI) is analytic, even though it is apparently not analytic. For, only if he can make a case that it is analytic, can he hope to convince us that the associative law (and other such laws) are also analytic.

But even if the impossible were possible, even if a law of arithmetic were a law of logic, this would not suffice to show that the associative law is analytic. This law follows from the principle of mathematical induction plus the so-called recursive definition of addition. This so-called definition, however, turns out to be not an innocent stipulation, not a mere linguistic convention, but another law of arithmetic. We shall have to look into this matter a little more deeply, leaving Mill for the time being, because it touches on many other things and it will pay to clear up, once and for all, some of the technical issues.

The great Dedekind proves in his famous essay Was sind und was sollen die Zahlen? that there is one and only one function R such that the following holds for all natural numbers m and n:

(1) R (m, 1) = the successor of m,

(2) R (m, the successor of n) = the successor of R (m, n).

(The theorem of section 126 of Dedekind’s work is actually more general, but I shall omit what is not essential to our discussion.) The function R is of course our intuitively given sum relation. Looked at from our epistemological point of view, Dedekind merely confirms what we already know from observation, namely, that there exists a certain relation among natural numbers. It is as if someone proved that there exists a certain relation among color shades, namely, the relation of being-darker-than. Dedekind, however, pursues mathematical interests. He introduces the expression ‘sum of the numbers m and n’ for the expression ‘function which fulfills conditions (1) and (2)’. It is clear that he does not pick the term ‘sum’ out of thin air. This word has a well-established meaning. But Dedekind pretends that he does. He acts as if calling his function ‘sum’ is as arbitrary as if he had called it ‘humpty.’ There is no argument; we could have chosen ‘humpty’ instead. Nor is there an argument against pretending—for the sake of making a point—that the word ‘sum’ has no previous meaning. But it is also true that this pretense will not “reduce” the sum relation to anything else, and that it will not transform the assertion that there is such a relation into an analytic truth.

I have just spoken both of functions and of relations. Mathematicians usually speak of the former. I shall from now on use the term ‘relation(s)’, because in my ontology so-called functions are relations. What Dedekind shows, in these terms, is that there is precisely one relation which fulfills the two conditions (1) and (2); and we know, of course, what this relation is, namely, the sum relation.

So-called recursive definitions of functions, as you can now see, are really recursive descriptions of relations. Therefore, the relations so described are as little “reduced” to something else as Scott is when we describe him as the author of Waverley. But recursive descriptions are peculiar; they are different from “ordinary descriptions.” And it is this peculiarity, one may assume, that has led mathematicians and philosophers to attribute magical ontological powers to them.

Consider the first condition of the recursive description:

(1) + (m, 1) = the successor of m.

(1) states that the number which is the sum of m and I is the same number as the number which is the successor of m; (that the number which is the sum of m and I is identical with the number which is the successor of m). One and the same number is described in two different ways, once as a sum, once as a successor. Furthermore, this is a general identity statement which holds for all natural numbers m. Any description expression of the form ‘the sum of M and I’ describes the same number as any description expression of the form ‘the successor of M’, where M is a specific natural number. But this means that any description expression of the first kind “can be replaced” by a description expression of the second kind. Any description expression that mentions the sum relation “can be replaced” by an expression that no longer mentions this relation, but mentions the successor relation instead. Condition (2), similarly, makes it possible “to replace” any description expression of the form ‘the sum of M and the successor of N’ by an expression of the form ‘the successor of the sum of M and N’. And this possibility, combined with the first condition, allows us “to replace” any description expression of a natural number as the sum of two numbers by an expression which no longer mentions the sum relation, but mentions the successor relation instead. For every description of a natural number as a sum there is a description of it as a successor.

Notice that I always put ‘can be replaced’ in quotation marks. We have two different descriptions of the same thing. We do not have two description expressions for the same description. Only if we confuse the descriptions with what they describe can we conclude that the “recursive definition” does away with the sum relation in favor of the successor relation. Only then can we claim that we have here two expressions for the very same thing. As soon as we distinguish, as we must, between a description and what it describes, we realize that although the two descriptions describe the same number, the descriptions themselves are clearly different. How could they possibly be the same, we may ask in wonder, if the first describes the number in terms of the sum relation and the second describes it in terms of the successor relation? Assume that Mary is the mother of John and the spouse of Tom. We can describe Mary either as the mother of John or as the spouse of Tom. We have two descriptions of the same person; but the descriptions are not the same. It is as mistaken to claim that there is no sum relation but only the successor relation, as it is to maintain that there is no mother relation but only a spouse relation. (We assume, for the sake of the analogy, that every woman is a spouse of one man and the mother of one child.)

The mistaken dogma that “recursive definitions” have ontological significance is probably reinforced by the pernicious practice of calling description expressions “singular terms.” According to this mind set, a prejudice so firm that nothing can apparently shake it, our arithmetic example involves two singular terms for the same number, just as Tully and Cicero are two singular terms for the same person. The distinction between descriptions (not description expressions) and what they describe thus gets lost, and with it the insight that one and the same thing may be described in terms of two equally real and important relations.

Condition (1) is a general truth. Properly stated, it says that all natural numbers are such that: the sum of a natural number and one is the same number as the successor of the natural number, This general truth is neither a law of logic nor an instance of a law of logic. Therefore, it is not analytic. Hence it is synthetic. Furthermore, it is not deduced from other laws of arithmetic. Therefore, if it can be justified at all, it can only be justified inductively. We believe that the sum of 1 and any natural number is the same as the next number after the natural number, as Mill probably would say, because we have verified millions of times that the sum of 1 and i is the number after 1, that the sum of 2 and 1 is the number after 2, etc., etc., and we have no reason to believe that the same would not be true for big numbers as well. Mill would probably also claim, and this is where I part company with him, that we have arrived at these individual instances by induction from observing certain “physical phenomena.” However, I agree with him that the general truth under discussion is synthetic, and that it can only be justified by induction.

g) Mill’s Conception of Numbers

We have already touched upon Mill’s conception of numbers, but shall now take a closer look at it. Mill distinguishes between the denotation and the connotation of numerals and says:

Each of the numbers two, three, four, etc., denotes physical phenomena, and connotes a physical property of those phenomena. Two, for instance, denotes all pairs of things, and twelve all dozens of things, connoting what makes them pairs, or dozens; and that which makes them so is something physical; since it cannot be denied that two apples are physically distinguishable from three apples, two horses from one horse, and so forth: that they are a different visible and tangible phenomenon.

(Mill, 1979b, vol. 7, p. 610)

I shall assume that Mill means to say that the numeral ‘two’, not the number two, denotes all pairs of things. It may look at first as if Mill holds that ‘two’ is a name of the set of all pairs, and that he thus anticipates Russell’s conception of numbers. But this impression must be mistaken. Firstly, it is fairly clear that Mill believes that ‘two’ denotes, not the set of pairs, but each pair individually. Thus ‘two’ denotes the shoes I am wearing at this moment. ‘Two’ is therefore a “general name” (or “common name”) of all couples rather than a proper name of a certain set. Secondly, there is Mill’s insistence on the physical nature of the couples, triples, etc. A set, it is generally agreed, is not a “physical thing.”

Perhaps Mill’s use of ‘physical’ should not be taken too narrowly. Two apples, he says, are physically distinguishable from three apples; they are different visible and tangible phenomena. Two tones, we may add, are different from three tones as auditory phenomena. Finally, two pains are distinguishable from three pains, not by means of the senses, but by means of “inner experience.” Perhaps Mill’s claim is merely that we can distinguish, either by outer or by inner sense, between two things and three things. Perhaps Mill is merely saying that, in the case of apples, we know how many apples there are before us by means of perception. His view, in short, may amount to nothing more than a rejection of the Platonic dogma.

We must distinguish between the claim that numerical difference can be known through perception and the question in what this observed difference consists. It is in regard to this latter question that Mill, in my opinion, clearly goes wrong. What is connoted by the name of a number, he tells us, is “some property belonging to the agglomeration of things which we call by the name; and that property is, the characteristic manner in which the agglomeration is made up of, and may be separated into, parts” (Mill, 1979b, vol. 7, p. 611). And a few lines later he says:

What the name of number connotes is, the manner in which single objects of the given kind must be put together, in order to produce that particular aggregate. If the aggregate be of pebbles, and we call it two, the name implies that, to compose the aggregate, one pebble must be joined to one pebble.

(Mill, 1979b, vol. 7, p. 611)

An aggregate of pebbles, so it seems from this passage, is formed by “bringing together” one pebble and another pebble, by putting them side by side. Only in this fashion is formed the whole which is presumably denoted by the word ‘two’. An aggregate of two tones, we may surmise, is formed by playing them together, one after the other. Thus it seems to be an essential part of Mill’s view that certain wholes are the denotations of numerals. What kind of whole is formed depends in a particular case on the nature of the parts. Pebbles, for example, form spatial wholes, while tones form temporal wholes. In the former case, the essential relation among the parts of the whole is spatial; in the latter, it is temporal. Mill’s aggregates clearly are not sets, but are structures of several sorts.

By contrasting sets with structures of these sorts, we can illustrate Mill’s mistake. Frege, we remember, asks Mill rhetorically: “Besides, need the straws form any sort of bundle at all in order to be numbered: Must we literally hold a rally of all the blind in Germany before we can attach any sense to the expression ‘the number of blind in Germany’?” (Frege, 1974, p. 30). We can put it this way: Does not a set of straws have a number (of members), even if the straws do not form a bundle? And does not a set of blind people have a number (of elements), even if these people do not hold a rally?

What happens in Mill’s view to the sum relation? As I understand Mill, the word ‘two’ denotes the spatial structure consisting of two pebbles, and it also denotes the temporal structure consisting of two tones. It is a common name of these two agglomerations as well as of many others. It names “indifferently” any one of these quite diverse wholes. I think that this view is false: ‘two’ represents one and only one thing, namely, the number two. But ‘two’ does not only denote indifferently different things, according to Mill’s view, it also connotes indifferently many different “properties,” many different characteristic ways in which those structures are formed. Thus ‘two’ connotes the way in which the pebbles are connected to form a spatial “heap,” but it also connotes the way in which the two tones are conjoined in order to form the temporal whole. I think that this view as well is mistaken. Two’ does not connote anything. How would one “add” one pebble to the two pebbles? I am not sure what Mill would say, but it seems to me that the only thing he could say is: By forming a new structure consisting of three pebbles; by creating a new “heap.” But this answer implies that the “operation” of addition consists in this case of bringing another pebble into spatial contiguity with the already existing heap of two pebbles. Of course, such “addition” is not possible in the case of tones. Here, I assume, addition would consist in playing a third tone after the other two. Thus addition of tones would involve an operation quite different from the one appropriate for pebbles. And similarly for other kinds of structure. In short, addition, too, would mean quite different things for different agglomerations. And again, I think this view is false: addition, the sum relation, is one thing and not many.

This is Mill’s predicament: His empiricism demands that numbers be “sensible” things. We must be acquainted with them through the senses. No mysterious “Platonic” faculty of contemplation or “Husserlian” power of eidetic intuition exists. Mill must therefore look for something that (a) is sensible and (b) can reasonably be called a number. What he comes up with are his so-called “aggregates,” that is, structures of certain sorts: spatial structures, temporal structures, etc. These wholes are “sensible”; we can perceive them. But these structures are not numbers, and their characteristic relations are not the sum relation. What leads Mill astray is the powerful dogma that has ruled the minds of almost all philosophers, that the only things that can be perceived are things in space and time, that is, “physical phenomena.” Frege, as we shall see in the next section, concludes from Mill’s failure that numbers cannot be sensible and therefore abandons empiricism. As Mill and Frege both see it, numbers either can be perceived or they cannot. If they can be perceived, then they must be spatio-temporal things of the sort Mill talks about. If they cannot be perceived, then there must be some other “rational” faculty which acquaints us with them. Mill cannot bring himself to abandon empiricism. He therefore embraces the first horn of the dilemma. Frege, on the other hand, clearly sees that Mill’s view of numbers is untenable. He concludes that numbers cannot be spatio-temporal structures. But since he subscribes to the same dogma as Mill concerning the objects of perception, he cannot but reject empiricism. It is a measure of Frege’s genius that he came very close to finding a way out of the dilemma.

Before we leave Mill, let us cast a quick glance at a recent attempt to improve on Mill’s conception of number. Kessler has proposed the following view:

On the model I am suggesting, a number is to be understood as a special sort of relation which holds between aggregates and properties that pick out parts of those aggregates. For example, in claiming that a certain aggregate x contains 52 cards we are claiming that the numerical relation 52 obtains between the aggregate x and the property of being a card.

(G. Kessler, “Frege, Mill, and the Foundations of Arithmetic,” p. 69)

There is a problem with this account concerning what spatial structures, for example, are to be counted as “one-square” aggregates or as “two-square” aggregates, and so on (see P. M. Simons, “Against the Aggregate Theory of Number”). But I wish to voice a different kind of criticism. Consider any structure S consisting of n parts. According to Kessler, n is a relation between S and a property P, namely, the property that individuates the parts of S. For example, the deck of cards consists of 52 cards, the series of natural numbers from 1 to 10 consists of 10 numbers, and so on. Kessler considers any kind of structure, S, and any kind of consists-of relation, so that the general schema is: S consists of n P’s. And then he simply re-writes ‘S consists of n P’s’ as ‘n (S, P)’. But surely the relation between the deck of cards and the property of being a card is not identical with the number 52. Rather, this relation is the relation of “consisting of 52 things which have that property.” And here 52 is not a relation, but only part of a relation. Here 52 is not a relation, but something that quantifies things which are. The situation is analogous to the following. Someone (Cantor, for example) may hold that the property of being a card of the deck has the property 52, so that numbers turn out to be properties of properties. But what is true is not that P has the property 52, that is, 52 (P), but rather that the property of being a card of the deck has the property of having 52 things exemplify it. Here the number appears as a part of the relevant property, just as in Kessler’s example it occurs as part of the relevant relation. Here, the “property” which the property of being a card of the deck may be said to have is not 52, but the property of being exemplified by 52 things. There, the relation which the deck may be said to have to the property of being a card of the deck is not 52, but the relation of consisting of 52 things with that property.