a) Isomorphisms and Models

Among the furniture of the world, there is the category of structure (“whole”). A structure is a complex entity which consists of parts in relations to each other. It differs from a set because of the relations which it contains. It differs from a fact because of the relations in which it can stand to other things. These differences between structures, on the one hand, and sets and facts, on the other, are well-known, and I shall merely remind you of their most important features. Two sets are the same if and only if their members are the same. Two structures are identical, by contrast, if and only if (a) their nonrelational parts are the same, (b) their relations are the same, and (c) corresponding parts stand in corresponding relations. This clearly shows the difference between sets and structures. Facts differ from structures most strikingly in that they stand in such relations to each other as conjunction, disjunction, etc. Furthermore, there are negative facts, while there are no negative structures. These distinctions between structures, on the one hand, and sets and facts, on the other, are important because in my ontology sets, facts, and structures are the only complex entities (see my The Categorial Structure of the World).

The most interesting feature of structures is that they can be isomorphic to each other. Again, I shall be brief. We say of two structures S₁ and S₂ that they are isomorphic to each other if and only if the following is the case: There exists a relation R₁ and a relation R₂ such that: (1) R₁ is a one-one relation between the relations of S₁ called U_n and the relations of S₂ called V_n, (2) R₂ is a one-one relation between the members of the fields of U_n and the members of the fields of V_n such that: (3) if any n members form an n-tuple that stands in one of the relations U_n, then the members related to them by R₂ form an n-tuple that stands in the corresponding relation from V_n. Less precisely, two structures are isomorphic to each other if and only if their nonrelational parts are correlated one-one to each other, their relations are coordinated one-one to each other, and corresponding parts are coordinated by corresponding relations.

As ontologists, we must be much more careful than logicians and mathematicians and must call attention to the following two points. Firstly, there may not exist the relations R₁ and R₂, but merely the corresponding sets of ordered couples (sets of structures!). Just as there are sets which are not determined by properties—recall the axiom of choice—so are there sets which are not determined by relations. In case R₁ and R₂ do not exist, but the corresponding sets do, our explanation of isomorphism must of course be changed. Secondly, there exists no such relation as being isomorphic to something. To say that two structures are isomorphic to each other is a mere abbreviation of a longer story.

Isomorphisms are the delight of logicians and mathematicians. It would not occur to us, even for a moment, to begrudge or belittle this pleasure. Who but an insensitive worshipper of the humanities could be indifferent to the beauty and elegance of algebra? And yet, we cannot help but add a word of caution: In the hands of philosophically-minded logicians and mathematicians, algebra—the theory of structures—has caused endless confusion. It is the purpose of this chapter to try to dispel some of these confusions. Alas, I have no reason to expect that I shall succeed any better than Russell, Frege, and many others have in the past. There seems to exist a “mathematical set of mind” which simply refuses to acknowledge the true philosophical relevance of isomorphisms. Because of this mind-set, I shall have to repeat the obvious in the next few paragraphs.

Consider any nonempty set of things, which we shall call “elements,” and certain relations between them such that the following ten forms are true:

(1) If a and b are elements, so too is a‡b.

(2) If a and b are elements, so too is a#b.

(3) There is an element x such that a‡x = a for every element a.

(4) There is an element y such that a#y = a for every element a.

(5) a‡b = b‡a for all elements a and b whose combinations are also elements.

(6) a#b = b#a for all elements a and b whose combinations are also elements.

(7) a‡(b#c) = (a‡b)#(a‡c) for all elements a, b, and c whose combinations are also elements.

(8) a#(b‡c) = (a#b)‡(a#c) for all elements a, b, and c whose combinations are also elements.

(9) There are at least two elements u and v such that u is not identical with v.

(10) If the elements x and y exist and are unique, then there is an element a such that (1) a‡ā = y and (2) a#ā = x.

A structure of this kind is called a “Boolean algebra.” Boolean algebras are so interesting because a certain structure of numbers as well as a certain structure of sets are Boolean algebras. If we think of our elements as natural numbers, of ‡ as addition, # as multiplication, and if we take x and y to be o and 1, respectively, then the forms (1) through (9) turn into true laws of arithmetic. But we can also interpret the forms in a different way. This time we think of the elements as sets, of ‡ as the union relation, of # as the intersection relation, and of x and y as the null set and the universal set, respectively. We get again nine true statements about sets. For sets, we can even add form (10), taking a to be the complement set of a. There is no straightforward interpretation of (10) for arithmetic. But, then, why should there be? Numbers are not sets. It is therefore not surprising that they do not behave in every respect like sets. It is remarkable enough that a structure formed from relations among numbers should be so much like a structure formed from relations among sets. You know, of course, that there is a third structure which is a Boolean algebra, namely, a structure formed from certain relations between states of affairs. We can interpret our elements to be states of affairs, ‡ is disjunction, # is conjunction, ā is the negation of a, and the two elements x and y are the two truth-values true and false, respectively. Upon this interpretation, the ten forms yield ten laws of “propositional logic.”

Now for the obvious:

Fortified with these truisms, let us go back and inspect some of the historical roots of the philosophical confusions which, in my opinion, still flourish.

b) Dedekind’s Notion of Number

The most common and serious confusion to which the discovery of isomorphisms has led is the belief that forms somehow “define” what they are about. Of course, they are not really about anything, since they are not sentences that make statements. How else, then, shall I put it? One believes that these forms somehow determine what the “place holders” represent. But place holders do not represent anything; that is why these shapes are called here “place holders.” You can see how hard it is even to formulate the view we are about to criticize. Let us try again: One believes that the forms somehow define things for the place holders which, by themselves, do not represent anything. The technical term for this kind of “definition” is ‘implicit definition’. The idea behind this kind of “definition” is at least as old as Poincare and as young as Quine. In between, we can read about it in Hilbert as well as in Schlick. As a matter of fact, implicit definitions have been so popular that one can hardly open a book about the philosophy of mathematics without making their acquaintance. But it all goes back to Dedekind (see R. Dedekind, “Was sind und was sollen die Zahlen”).

Dedekind gives four forms for what he calls “a singly infinite system N”:

The crucial place holders are ‘N’, ‘I’, and ‘R’. It is understood that N is a kind of entity, I is one of the things of kind N, and R is a one-one relation. If we take N to be the natural numbers, I to be the number one, and R to be the relation of immediate successor, we get four true sentences about natural numbers. The first sentence, for example, says then that the immediate successors of natural numbers are themselves natural numbers; and the third sentence states that one is not the immediate successor of any natural number. But Dedekind’s four forms can be interpreted in many other ways as well. For example, we may take I to be the number one as before, but interpret N to be just the odd natural numbers, and R to be the relation of being an immediate successor in the series of odd natural numbers. Once again, we get four true sentences. This time, these sentences are about the odd natural numbers. In a similar fashion, we can get four true sentences about all natural numbers divisible by ten. And so on.

One cannot but be impressed by Dedekind’s brilliant insight into the nature of the progression of natural numbers. But Dedekind goes on to claim that his four forms somehow describe the natural numbers:

If in the contemplation of a singly infinite system N, ordered by a mapping R, we disregard entirely the peculiar nature of the elements, retaining only the possibility of distinguishing them, and consider only the relations in which they are placed by the ordering mapping R, then these elements are called natural numbers or ordinal numbers or simply numbers, and the basic element I is called the basic number of R number series N.

(Dedekind, p. 360)

I take it that Dedekind thinks of the four expressions as forms rather than axioms (true sentences); for he does not identify the natural numbers with any particular series of entities. Nor does he say that just any kind of entity which satisfies the forms is the series of natural numbers. Rather, he thinks of the natural numbers as a product of a process of abstraction. By means of this process, we obliterate all of the distinguishing characteristics of the various kinds of entities which satisfy the forms. Since the numbers are arrived at by such a mental process of abstraction, Dedekind thinks of them as creations of the mind: “In regard to this liberation of the elements from any other content (abstraction), one can justifiedly call the numbers a free creation of the human mind” (Dedekind, p. 360).

Dedekind is wrong. By means of abstraction, we do not arrive at the notion of natural number, but rather at the notion of progression. If we consider the series of natural numbers, we may notice that there are other things which also form such a series, for example, the series of numbers divisible by ten. And this realization may lead us on to notice that all of these different kinds of things have something in common, namely, that they form progressions. By abstracting from what distinguishes one progression from another, we arrive, not at the notion of some sort of “indefinite kind of thing” called “a natural number,” but at the concept of a definite feature which definite kinds of entity share, namely, at the notion of (forming a) progression. By another process of abstraction, we may arrive at the notion of a Boolean algebra.

Dedekind, I said, is mistaken. But at least he does not claim that numbers are “whatever the forms say they are”. He does not claim that the forms “implicitly define” the natural numbers. He saw, it seems to me, that there are different models for the four forms. However, it is clear that Dedekind’s monumental achievement invites a conception of forms as definitions.

c) Implicit Definitions

1) POINCARÉ VERSUS RUSSELL

Our story of the enduring love affair between logicians and philosophers of science, on the one hand, and implicit definitions, on the other, begins with the classic dispute between Russell and Poincaré. In 1897, Russell published his Essays on the Foundations of Geometry; Poincare, in the Revue de Metaphysique et de Morale of 1889, wrote a long review of Russell’s work. Among other things, Poincare raised the question of the definition of geometric terms. We cannot go into the details of Poincaré’s criticism and Russell’s reply to it. What is at stake, it must suffice to say, is the notion of definition. While Poincaré seems to request a definition of the primitive terms of the geometric axioms, Russell considers such a request absurd. The very notion of a primitive term precludes its definition. In particular, it would be absurd to think of the geometric axioms as definitions of the geometric terms which they contain.

I think there existed in that debate a certain misunderstanding between Russell and Poincaré; a misunderstanding invited by Poincaré’s unfortunate use of the term ‘definition’ and by Russell’s stubborn refusal to look beyond this use. What Poincaré really was asking for, it seems, was an interpretation of the primitive terms. What, he asked, is a straight line? Russell, of course, had an answer to this kind of question: What those terms represent, is given to us by acquaintance. Russell believes that it is undoubtedly by analysis of perceived objects that we obtain acquaintance with what is meant by a straight line in actual space. What made Russell reluctant to give this straightforward answer to Poincaré, we may surmise, is the fact that it was customary at that time to think of a straight line as an “ideal object,” and of geometry as being about such “ideal objects.” Kant’s shadow still loomed menacingly over the philosophical scene. Ideal objects, not being a part of the perceptual world around us, were supposed to be given to “pure intuition.” Poincaré put his finger on this sore spot when he guessed what Russell’s answer to his question would be:

that there is no need to define [the primitive terms] because these things are directly known through intuition. I find it difficult to talk to those who claim to have a direct intuition of equality of two distances or of two time lapses; we speak very different languages. I can only admire them, since I am thoroughly deprived of this intuition.

(J. A. Coffa, To the Vienna Station, chap. VII)

We may agree with Poincaré that there exists no special faculty of intuition, no Kantian pure intuition, which acquaints us with those properties and relations. But this does not imply, as Poincaré seems to think, that we are not acquainted with them at all. What Russell should have said in reply to Poincaré is that we perceive with our very eyes that two lines are equal or unequal in length. Nor are these lines “ideal,” whatever that may mean. We are talking about lines drawn with pencil or chalk on paper or a blackboard. We are talking about ordinary perceptual objects. The primitive terms of (Euclidian) geometry represent perceptual properties and relations, and geometry is about the individual things which have these properties and stand in these relations. What is involved is neither a mysterious faculty of pure intuition nor an obscure ideal object.

This straightforward answer to Poincaré’s question, however, has seldom been given. The discovery of non-Euclidian geometry seems to have clouded rather than enlightened the minds of many philosophers. What could be more obvious than that ‘straight line’ means straight line, and that we explain to a child what property we have in mind by drawing a straight line with a pencil on a piece of paper? Why does anyone believe that this case is any different from explaining what property olive green is by showing the child an olive green piece of cardboard? Yet, Coffa says in his otherwise insightful book:

What was no longer a serious possibility circa 1900 was to conceive of acquaintance as playing the specific semantic explanatory role that it was supposed to play in the atomist picture of knowledge, whereby the construction of geometric theory would start with acquaintance, then proceed to a construction of claims and perhaps conclude with the testing of these claims.

(Coffa, chap. VII)

But this is precisely how geometry does proceed. In Goedel’s words: “In geometry, e.g., the question as to whether Euclid’s fifth postulate is true retains its meaning if the primitive terms are taken in a definite sense, i.e., as referring to the behavior of rigid bodies, rays of light, etc.” (K. Goedel, “What is Cantor’s Continuum Problem,” p. 271).

What does Coffa think is the correct alternative to our view? In praise of Poincaré, Coffa says:

Poincaré’s conventionalism is based on the idea that in order to understand geometery, one must stand Russell’s argument on its head: since geometric primitives do not acquire their meaning prior to their incorporation into the axiomatic claims, such axioms do not express propositions in Frege’s or Russell’s sense.

(Coffa, chap. VII)

The so-called axioms of geometry, according to this view, are really implicit definitions of the primitive terms which they contain. Coffa sums up this view in these rousing words:

Poincaré’s point is, therefore, that all we can say about the meanings of geometric primitives is what geometric axioms say. Under these circumstances, the thesis of semantic atomism prevents those axioms from conveying any sort of factual (non-semantic) information. No wonder that they are neither analytic (in Kant’s first sense) nor synthetic, since they are not propositions. No wonder either that they had always been regarded as extraordinary claims, endowed with a particular strong sort of truth. The error was to think that they convey a privileged sort of information, or information about some extraordinary domain. Their distinguishing feature is that they determine, to the extent needed in geometry, the meanings of geometric primitives; and the conviction that they are necessary emerges from the fact that we would be talking about something else or, better yet, meaning something different from what is intended, if we denied them. Geometric axioms are definitions disguised as claims, and what they define is the indefinables.

(Coffa, chap. VII)

When we look at this approving description of Poincaré’s view, the shortcomings of this view become all too obvious. Let us agree, for the moment, to use ‘axiom’ in a neutral fashion for whatever geometers call “axioms.” Let us further assume that one of these axioms contains the expression ‘straight line’. There are only two possibilities: Either this expression is the ordinary English phrase which means straight line, or else it is not. In the latter case, it is a mere “place holder,” and we could have in its stead any other arbitrary expression. In the former case, the axiom is a true sentence of English. It says something about straight lines. In the second case, on the other hand, it is a mere form. In this case, it is not a sentence; it is neither true nor false. Nor does the form say anything about straight lines. Now, turn to Coffa’s very first sentence. Coffa claims that the axioms say something about the meanings of geometric primitives, that is, in our example, about straight line. As we have just seen, this can only be true if we think of the axioms as English sentences. If we think of them as mere forms, then they do not say anything about anything and, in particular, they do not say anything about straight lines or the property of being a straight line. But turn now to Coffa’s second sentence and you will find the exact opposite conception of the axioms. Coffa this time maintains that the axioms do not convey any kind of factual information. Now the axioms are treated as mere forms. Coffa seems to be unaware that he is working at one and the same time with two incompatible conceptions of the axioms. I think that we have here the basic confusion which spawns the notion of implicit definitions: Axioms are treated at the same time both as saying something about their primitives and also as not saying something about them. They are treated at the same time as axioms proper and also as mere forms. The charm of this confusion is that it allows one to switch back and forth between these two incompatible conceptions, just as the philosophical argument may require. When one argues that the axioms do not express propositions, one treats them as mere forms. When one argues that they define their primitive terms, that they say something about the meanings of these terms, one treats them as expressing propositions.

This confusion between treating an axiom as a mere form and treating it as a declarative sentence (an axiom) is neatly illustrated in a letter from Hugo Dingler to Frege:

‘2’ and ‘3’, too, are for me mere signs which acquire a “sense” only through the assumptions or axioms which one presupposes for them. The sentence ‘3 > 2’ merely seems to have an independent sense, namely, when I work with the “presentation” [Vorstellung], as is the case on the pre-logical or pre-axiomatic level. Then ‘3’, ‘>’, ‘2’ are . . . “popular concepts”, that is, [they] are presentations abstracted from practical life with which the understanding works intuitively, as actually is the case in ordinary life for everyone everytime. For science, however, in the form to be aspired to, this condition cannot be ideal. The “presentations” which someone has when he sees the signs ‘2’ and ‘3’ should not, it seems to me, form the basis of science, but must be considered in their logically analyzed form. But then the sentence 3 > 2 is not an independent sentence, but merely a part of a—as you put it so nicely—sentence structure [Satzgefuege] which is based on a series of irreducible presuppositions or axioms. But then the difference between a > b and 3 > 2 is merely a matter of degree and not a matter of principle: both times, two signs are connected by ‘>’, only that in one case there exist a few additional presuppositions about these signs, a > b thus constitutes only one presupposition, while 3 > 2 carries a whole group of presuppositions with it . . . Without these presuppositions, 3 > 2 is in no way different from a > b, for then ‘3’ and ‘2’ are completely arbitrary signs, just like any other.

(My translation from the German. The letter appears in G. Frege, Philosophical

and Mathematical Correspondence, pp. 23-25)

If one is allowed to treat the axioms of geometry at the same time both as forms and as true sentences, a number of traditional problems disappear, as Coffa indicates. Are the axioms of geometry analytic or synthetic? Well, they are not analytic like the truths of arithmetic, someone may reason. Are they then synthetic? How can they be? Surely they are different from the truths of botany! So, they cannot be analytic and they cannot be synthetic. Well, if they are mere forms, then it is perfectly obvious how we can escape from the threatening dilemma: Mere forms, since they do not represent propositions, are neither analytic nor synthetic. Are the axioms of geometry necessary? Well, they are not analytic, so the reasoning goes on. But they are obviously necessary in some sense. If the axioms are really definitions that say something about the primitives of geometry, then it is perfectly understandable how they can be necessary without being analytic: If we denied the axioms, we would not be talking about the primitives of geometry but about other things. For example, we could not mean straight line by ‘straight line’ if we denied that two points determine a straight line.

Wittgenstein, by the way, adds as usual his own twist to this confused conception. In a letter to Schlick, he writes:

Does geometry talk about cubes? Does it say that the cube-form has certain properties? . . . Geometry does not talk about cubes but, rather, it constitutes the meaning of the word cube, etc. Geometry says, e.g., that the sides of a cube are of equal length, and nothing is easier than to confuse the grammar of this sentence with that of the sentence ‘the sides of a wooden cube are of equal length’. And yet, one is an arbitrary grammatical rule whereas the other is an empirical sentence.

(Coffa, chap. XIV)

Wittgenstein here supplants the notion of implicit definition by that of a grammatical rule. That is his peculiar twist.

2) HILBERT VERSUS FREGE

The second chapter of our story is about Hilbert and Frege: Hilbert conceives of the axioms of his celebrated Grundlagen der Geometrie as implicit definitions. Frege objects. Again, there can be no doubt in my mind that Frege is on the side of the angels. And, again, many contemporary philosophers of science have sided with Hilbert. Witness, for example, H. Scholz’s assessment of the controversy:

. . . no one doubts nowadays that while Frege himself created much that was radically new on the basis of the classical conception of science, he was no longer able to grasp Hilbert’s radical transformation of this conception of science, with the result that his critical remarks, though very acute in themselves and still worth reading today, must nevertheless be regarded as essentially beside the point.

(G. Frege, 1980, p. 31)

Contrary to Scholz, I think that Hilbert is confused and that Frege’s remarks are precisely on target. (F. Kambartel is one of the few contemporary philosophers who reaches the same conclusion in his Erfahrung und Struktur: Bausteine zu einer Kritik des Empirismus und Formalismus.) In a letter to Hilbert of December 27, 1899, Frege makes all of the important distinctions between sentences, axioms, definitions, etc. and objects that the axioms of Hilbert’s system do not, as Hilbert claims, define the geometric properties (point, line, plane) and relations (lies, between, parallel, congruent, etc.). In his reply, Hilbert mentions the three points on which he most fundamentally disagrees with Frege:

The topics of this dispute have been discussed by quite a few philosophers. I shall merely add a few comments. Firstly, I think, as I said already, that Frege is correct on all three points. Secondly, it seems obvious to me that consistency neither implies truth nor existence. Thirdly, it is of course true that Hilbert’s axioms, when treated as forms, allow for many different interpretations. But this very fact conflicts with Hilbert’s first point, namely, that the axioms are definitions. We see that Hilbert treats the axioms sometimes as sentences (definitions) and sometimes as mere forms, thus creating the confusion which I pointed out earlier.

It must be mentioned that Frege clearly saw that there is still another way of looking at Dedekind’s and Hilbert’s axiomatizations of arithmetic and geometry, respectively (see Frege’s letter to Hilbert of January 6, 1900; Frege, 1980, p. 43, and, especially, Frege’s “Ueber die Grundlagen der Geometry”). We replace the axioms by the corresponding forms, but with variables in place of the place holders. Then we consider those expressions which consist of a conjunction of the “axiom forms” as antecedent and a “theorem form” as consequent. If we close these expressions for the variables, we get a set of true sentences, namely, sentences of logic. If these truths of logic are properly instantiated, we arrive at true conditionals whose antecedents are the axioms of the theory. We have before us now a set of logical truths from which we can get, say, geometric truths, if we properly instantiate and separate the antecedents from the consequents. Needless to say, this whole process has nothing to do with definition. Nor does it turn geometry or arithmetic, properly understood, into logic.

Reading Hilbert’s replies to Frege, one can fairly feel Hilbert’s impatience with Frege’s “philosophical quibbles.” He shows an attitude quite common among philosophically inclined scientists. Things are really much simpler than philosophers tend to make them. We really need not worry about the nature of geometric points, lines, etc. and how we are acquainted with them. These objects, whatever they may be, are implicitly defined by the axioms of geometry. Nor need we torment ourselves about the nature of truth and existence. Consistency is truth and existence. Finally, the nature of geometry is an open book. Geometry is really a branch of logic; it consists of the set of propositions which state the implications from the axioms to the theorems (compare Russell’s first sentence of the first chapter of the Principles of Mathematics: “Pure mathematics is the class of all propositions of the form ‘p implies q’ ” . . .)

3) QUINE’S RESURRECTION OF IMPLICIT DEFINITIONS

After Frege’s incisive criticism, one should have hoped that implicit definitions had been laid to rest forever. Alas, this hope is in vain. They still play the role of a deus ex machina on the philosophical stage. If your philosophical play is hopelessly entangled, implicit definitions to the rescue! (cf., for example, Schlick’s attempt to stave off idealism by appealing to implicit definitions. M. Schlick, Allgemeine Erkenntnislehre, pp. 29-38).

Quine has recently argued that the axiomatic development of every synthetic theory can be replaced by an axiomatic development of the same theory whose true statements follow from the truths of arithmetic. He then claims that this alleged fact vindicates the view that axioms are implicit definitions (W. V. Quine, “Implicit Definitions Sustained”).

Consider a set of chemical properties, C_i, of a certain theory of chemistry and let ‘A(C_i)’ abbreviate the conjunction of the axioms of this theory. It is well known that one can also give an arithmetic interpretation of the axiom forms of this theory in terms of certain arithmetic properties K_i (see D. Hilbert and P. Bernays, Grundlagen der Mathematik, vol. 2, p. 253). Upon such an interpretation, we get the axioms A(K_i). Quine now constructs a third interpretation in terms of properties F_i allegedly defined on the basis of both C_i and K_i. This construction proceeds in two steps. (1) The individual variables of the chemical and arithmetic theories must be allowed to range over both physical objects and natural numbers. This is achieved by what Quine calls “hidden inflation.” We take an arbitrary element a of, say, the physical universe of discourse and extend the original interpretation of the chemical predicates to natural numbers by stipulating that the chemical properties belong to them if and only if they belong to a. Then we reinterpret the arithmetic predicates in a similar way. (2) The predicates ‘_iF’ of the third interpretation are defined as follows: ‘F_i(x)’ is short for ‘(A(C_i) and C_i(x)) or (not-A(C_i) and K_i(x))’. (For the sake of simplicity, I only consider one-place predicates of the first level.) Now one can show that ‘A(F_i)’ follows from the arithmetic truths ‘A(K_i)’ alone. Hence it may be said that ‘A(F_i)’ is true as a matter of arithmetic, that is, that it is an arithmetic truth. (Quine makes this point in terms of analyticity: If arithmetic is analytic, then ‘A(F_i)’ is analytic.)

There are several things wrong with this construction and with Quine’s conclusion. First of all, it makes no sense to stipulate, for example, that the inflated chemical predicates represent properties which belong to numbers if and only if they belong to an arbitrary physical object a. One cannot create properties as one wishes. One cannot just say: “Let there be such and such properties” and expect them to come into being. A contemplated property either exists or it does not exist. We can at best find out which is the case. There are certain chemical properties which may or may not belong to a. But these properties most certainly do not belong to natural numbers, and there is nothing Quine, or Dreben, or anyone else can do about it. At best, we may discover some properties—but we would hardly call them “chemical properties”—which belong to physical objects and which also belong to natural numbers if and only if they belong to a given physical object a. But I have no idea what these properties could be and seriously doubt that there are any.

There are also several things wrong with the definition of the predicate ‘F_i’. Firstly, there is no reason at all to assume that ‘F_i’ represents a property (or some properties). From the fact that we agree to abbreviate an expression by a shorter one, it does not follow that the completely arbitrary sign ‘F_i’ is a predicate. Secondly, there is no guarantee whatsoever that the definiens represents a property (or properties). Quite to the contrary, since the definiens is a form, it could not possibly represent a property, as I have argued elsewhere (see my Ontological Reduction; and also “Structures, Functions, and Forms,” pp. 11-32). In addition to these two objections, there is another difficulty. Quine’s definition assures that the properties F_i—assuming that there are such properties—are coextensive with the properties C_i. But these two sets of properties are not the same. If we hold, as seems reasonable, that two theories may be distinguished, among other things, by the properties they attribute to things, then it follows that, contrary to Quine, ‘A(C_i)’ is the axiomatic development of a chemical theory, but ‘A(F_i)’ is not; for it mentions very queer properties—if we continue to assume for the sake of making a point that there are such properties—which are not chemical properties at all, but which happen to be coextensive with chemical properties.

Finally, that F_i and C_i have the same extension is not a mere matter of abbreviation, as one may be inclined to think from the definition given above. It depends on the chemical fact that ‘A(C_i)’ happens to be true. For if we assume that ‘A(C_i)’ is false, F_i will have the same extension as K_i instead. Thus we have in ‘F_i’ a group of predicates representing properties whose extensions depend on whether or not a certain chemical theory is true. And this fact clearly distinguishes the properties C_i from the properties F_i.

But what does all of this have to do with implicit definitions? It is not entirely clear to me why Quine thinks that his construction shows that implicit definitions are vindicated. Perhaps he reasoned as follows. To say that axioms are implicit definitions is to say, at least in part, that they are analytic, like ordinary kinds of definition. But we can show by means of the construction that axioms can be turned into analytic statements. Hence the conception of axioms as definitions is vindicated.

I think that Quine’s construction shows, if it shows that much, that the “theory” ‘A(F_i)’ can be deduced from ‘A(K_i)’. Thus, if the former were really a theory about certain things and their properties, it would be true as a matter of arithmetic. Let us agree, for the sake of the argument, that it would be an arithmetic theory. But how does this show that ‘A(C_i)’ is an arithmetic theory? Of course, it does not show this if we are correct in our claim, contrary to Quine, that the chemical properties C_i are not the same as the properties F_i. But even if we assume, contrary to fact, that ‘A(F_i)’ is an axiomatization of the same theory as ‘A(C_i)’, this would merely mean that the chemical theory is true as a matter of arithmetic (whatever that may mean), but not that it is true as a matter of definition. The axioms ‘A(F_i)’ would be comparable to arithmetic truths that follow from other arithmetic truths.

4) IMPLICIT DEFINITIONS AND TRUTH BY CONVENTION

Even if Quine’s construction were sound, the desired analogy between definitions and axioms can only be drawn if we use the notion of analyticity as a verbal bridge. What is analytic, one must hold, is true by definition. Since Quine’s construction seems to show that the axioms of chemistry, for example, are analytic, so one may reason, they are true by definition, that is, they are definitions. And this brings us back to the main point of our investigation, namely, the alleged analyticity of arithmetic. It is taken for granted by most recent and contemporary philosophers that arithmetic is analytic. What gives rise to this conviction is the “felt necessity” of arithmetic truths. To show that arithmetic is analytic, therefore, one must explain this necessity. Up to this point, there is general agreement. But now the views diverge. Frege and the logicists claim that the necessity of arithmetic is the necessity of logic, whatever that may be. A moment ago, we encountered a different strategy. Analyticity means, not reducibility to logic, but truth by definition. And since we can adopt any definition we feel like proposing, analyticity becomes truth by fiat. And since we can count on general agreement about our stipulations, analyticity turns out to be truth by convention. This is the refrain one hears time and again from analytic philosophers: Arithmetic is necessary, that is, analytic, because arithmetic truths are true by definitions, that is, by convention.

Quine’s construction aims at making even the axioms of chemistry analytic. This, surely, is not in the spirit of making a sharp distinction between arithmetic, on the one hand, and the sciences, on the other. We must remind ourselves that Quine rejects the traditional and widely accepted distinction between analytic and synthetic truths. His defense of implicit definitions, therefore, kills two birds with one stone: It vindicates the conception of axioms as implicit definitions and, at the same time, breaks down the wall between arithmetic and the sciences. But if the axioms of chemistry are just as “analytic” as the truths of arithmetic, what happens to the “felt necessity” of the latter? Here is Quine’s answer in a nutshell:

There are statements which we choose to surrender last, if at all, in the course of revamping our sciences in the face of new discoveries; and among these there are some which we will not surrender at all, so basic are they to our whole conceptual scheme. Among the latter are to be counted the so-called truths of logic and mathematics, regardless of what further we may have to say of their status in the course of a subsequent sophisticated philosophy.

(W. V. O. Quine, “Truth by Convention,” p. 342)

In part, we agree with Quine. We, too, reject the traditional positivistic analytic-synthetic distinction which rests on the conviction that while science is a matter of fact, mathematics and logic are not. We share Quine’s view that science as well as mathematics and logic are made from one cloth. But Quine thinks that the cloth is woven from conventions, while we believe that it is woven from facts. While Quine argues, in the article about implicit definitions, that even chemistry can be made a matter of convention, we insist that even mathematics and logic are a matter of fact. Perhaps a better way of describing our difference is to say that while we both agree that mathematics and logic are as “empirical” as the sciences, ‘empirical’ means something quite different to Quine from what it means to us. The reason why we do not easily surrender the truths of mathematics and logic is that they are fundamental truths about the structure of the world.

d) Three Grades of Model Mania

1) MODELS AND CONSTRUCTIONAL DEFINITIONS

Axioms are not definitions; they do not define anything. But they may be said to describe the things which they mention. For example, Dedekind’s third axiom says that the number I is not the successor of any natural number. It says something true about this number. It states one of the characteristics of I. In this sense, it may be said to describe a feature of the number. But it does not define what ‘I’ stands for. Quite to the contrary, we must know what ‘I’ stands for in order to understand that the axiom says something about the number 1. To the question: What are numbers? one may therefore reply: Whatever the axioms say they are. But we must not confuse this answer with the quite different answer: Whatever the forms say they are. As I emphasized earlier, forms, in distinction from axioms, do not “say” anything.

But forms have models, and this fact has given birth to a whole branch of the philosophy of mathematics. It has even given rise to a new way of doing ontology. Quine, one of the main proponents of this approach, holds that one kind of entity is reduced to another kind of entity, if these two kinds yield isomorphic models for the same set of forms (see Quine’s “Ontological Reduction and the World of Numbers” in W. V. Quine, The Ways of Paradox, New York: Random House, 1966; And see also N. Goodman, “Constructional Definitions” in The Structure of Appearance).

To be more specific, Quine holds that an ontological reduction has been achieved if one can specify a correlator between the individuals of the original theory and the individuals of the model such that all the properties (and relations) of the model are isomorphic to the properties (and relations) of the original theory. For arithmetic, represented by Dedekind’s or Peano’s axioms, there are many models which fulfill this condition. We mentioned earlier that the even natural numbers and the odd natural numbers yield models for the respective forms. But there are also progressions of sets which are models of those forms. For example, there is the “Zermelo progression” consisting of the empty set, the set whose only member is the empty set, the set whose only member is the set whose only member is the empty set, and so on. And there is also the “von Neumann progression,” consisting of the empty set, the set whose only member is the empty set, the set consisting of the empty set and the set whose only member is the empty set, and so on. It follows from Quine’s conception that the even natural numbers can be reduced to the odd natural numbers. It also follows that the progression of natural numbers can be reduced to either the Zermelo progression or to the von Neumann progression. But this shows clearly that Quine’s notion of ontological reduction is an odd one. It does not bear on the question of what numbers are. Obviously, the natural numbers are not identical with the odd natural numbers. Nor are they identical with the Zermelo sets. Nor are they identical with the von Neumann sets. If we assume that Quine’s method of reduction answers the question of what the natural numbers are, then the answer would be that they are both Zermelo sets and von Neumann sets, and since these two kinds of sets are not identical, the method is reduced to absurdity. According to our view, of course, the natural numbers are not identical with any of those other progressions.

If one admits, and Quine seems to admit it, that the different models for the Peano forms involve different domains of different entities, then there can be no question of “reducing” one of these domains to another. What philosophical sense could it possibly make to speak of “reducing” a line of ten elephants to the first ten natural numbers? Model construction reveals that very different kinds of entity can yield models for the same forms. It shows that very different kinds of thing can form isomorphic structures.

Quine and his followers concede that the natural numbers are not the Zermelo sets. But they are loath to part with the idea that model construction has something to do with ontological reduction. So they add a new wrinkle to the dialectic. Zermelo sets are not identical with the natural numbers, they claim, but they can serve the same purpose as the natural numbers: “Any objects will serve as numbers so long as the arithmetical operations are defined for them and the laws of arithmetic are preserved” (W. V. O. Quine, Set Theory and Its Logic, p. 81). And on the same page he claims that “We are free to take o as anything we like, and construe S as any function we like, so long merely as the function is one that, when applied in iteration to o, yields something different on every further application”. In another paper, Quine puts it this way: “Just so, we might say, Frege and von Neumann showed how to skip the natural numbers and get by with what we may for the moment call Frege classes and von Neumann classes.” These classes, he continues, “Simulate the behavior of the natural numbers to the point where it is convenient to call them natural numbers . . .” (“Ontological Reduction and the World of Numbers,” in The Ways of Paradox, pp. 199-207, p. 200).

But in what sense of “free” are we free, as Quine claims, to take 0 as anything we like and to construe S as any function we like, as long as the repeated application of S yields a progression? It seems to me that we are free to pick any first element and any one of several possible relations as long as we are not interested in the progression of natural numbers, but merely in a progression of whatever kind. Similarly, to say, as Quine does, that the progression of even numbers serves as a version of number can only mean that since the even numbers and the natural numbers both form progressions, it does not matter which one we consider, so long as we are only interested in a progression, but do not care what particular progression it is. And to say that we can skip the natural numbers and get by with Zermelo sets can only mean that we can pick the progression of Zermelo sets rather than the progression of natural numbers, as long as we are only interested in considering a progression of whatever sort. In short, as long as we are not interested in a particular progression, but only in the feature of being a progression, any example of a progression will do for the purpose at hand, just as any human being will do as an example, if we are merely interested in the property of being a human being. But just as there are many different human beings, irrespective of which one we select as our example, so there are many different progressions, irrespective of which one we consider. Zermelo sets and Frege sets both form progressions. They are alike in this respect. But this is the only sense in which the entities in the one domain “simulate” the behavior of the entities in the other domain. There is no reason whatsoever why we should deliberately confuse the one domain with the other. Zermelo is like Frege in that both are human beings, but we do not believe for a moment that this is a sufficient reason for confusing them with each other. Nor would it be convenient to call Frege “Zermelo”, or conversely. Why, then, should it ever be convenient to call Zermelo sets anything but “Zermelo sets,” and to call natural numbers anything but “natural numbers”?

Natural numbers form a progression, and so do many other kinds of things. To believe that for this reason alone the natural numbers should be reduced to or replaced by some other progression is as absurd as to believe that Frege should be reduced to or replaced by Zermelo, just because they are both human beings.

2) MODELS AND SEMANTIC ROLES

So far in this chapter, we have looked at two answers to the question of what numbers are. According to the first, numbers are whatever the axioms of arithmetic define them to be. According to the second, they are whatever we want them to be as long as we stick to things that satisfy the axiom forms of arithmetic. The next attempt to wring some kind of answer out of the fact that there are isomorphic models for the arithmetic forms is even more peculiar. It leads to the answer that numbers are not anything in particular:

Any system of objects, whether sets or not, that forms a recursive progression must be adequate . . . That any recursive sequence whatever would do suggests that what is important is not the individuality of each element but the structure which they jointly exhibit. . . I therefore argue . . . that numbers could not be objects at all; for there is no more reason to identify any individual number with any one particular object than with any other (not already known to be a number) . . .

The properties of being numbers which do not stem from the relations they bear to one another in virtue of being arranged in a progression are of no consequence whatsoever. But it would be only these properties that would single a number as this object or that.

(P. Benacerraf, “What Numbers Could Not Be”, pp. 69-70, my italics)

As I understand it, Benacerraf’s argument has two parts. The first part is a straightforward argument from the existence of isomorphic models, and it goes like this. Any progression, no matter of what things it consists, “must be adequate” for the purpose of arithmetic. But that every progression “would do for the purpose of arithmetic” shows that what is important is the progression, not the elements of the progression. Therefore, arithmetic deals with progressions in general, and the numbers of arithmetic are not any individual, specific objects.

In this argument, we can distinguish three ingredients. Firstly, there is the pragmatic idea, which we have chastised before, that for the purposes of arithmetic any progression does equally well. This is simply not true. Any progression does equally well when our purpose is not to do arithmetic, but to consider the nature of progressions or, in other words, to consider progressions in general. Secondly, having conceived of arithmetic as being concerned with the nature of progressions rather than with the progression of natural numbers, Benacerraf concludes that arithmetic is concerned with the nature of an element in general, rather than with natural numbers. Thirdly and lastly, Benacerraf identifies the notion of an element of a progression with the notion of a number. The property of being a number is but the property of being an element of a progression. A number is anything that occurs in a progression. These numbers, of course, are definite things. What else could they be? But they are not elements of just one particular progression. They are not, for example, Zermelo sets, although Zermelo sets are numbers; and they are not von Neumann sets, although von Neumann sets are also numbers. But what about the progression of natural numbers, are they not numbers? Benacerraf’s view seems to imply that while there are many progressions, there is no progression of natural numbers.

At this point, I think, the second of his two arguments is relevant. This argument rests on the quite amazing ontological fact that numbers have no properties, with the exception of the categorial property of being a number. What characterizes an individual number is not a set of properties, but rather a set of relations in which it stands to other numbers. What properties does the number two have? Well, it is an even number. But this means, of course, that it is divisible by two, that is, that it stands in the division relation to numbers. And similarly for all other alleged properties: They are one and all “relational properties.”

But this does not mean, as Benacerraf seems to go on to argue, that numbers cannot be distinguished from other kinds of entities. First of all, there is the fact, for example, that elephants have properties which numbers do not have. Elephants eat peanuts, numbers do not. Secondly, and more importantly, the arithmetic relations among numbers suffice to distinguish numbers from all other kinds of things. What uniquely characterizes the numbers are the relations in which they stand to each other and to nothing else. Numbers and only numbers stand in the sum relation to each other; numbers and only numbers stand in the “successor relation” to each other, and so on. Neither elephants nor sets are sums of each other; neither elephants nor sets are “successors” of each other. Of course, sets stand in relations to each other which are isomorphic to the so-called successor relation. But these relations are not identical with the successor relation. Consider the Zermelo progression and the von Neumann progression of sets. The relations R₁ and R₂ which order these progressions are isomorphic to the successor relation. But they are different from each other and different from the successor relation. R₁ is the relation between a set and its singleton; R₂ can perhaps best be described as the relation which an element has to the set which has exactly it and all of its elements as elements. Neither one of these two relations is the relation which a natural number has to the next natural number. Thus what distinguishes between the Zermelo progression and the von Neumann progression; and between these two, on the one hand, and the progression of natural numbers, on the other, are the different, though isomorphic, relations that obtain between the elements of the progressions. For the rest, it is simply not true, as Benacerraf claims, that only properties could single out numbers from other kinds of things.

Let me quote another passage from Benacerraf:

That a system of objects exhibits the structure of the integers implies that the elements of that system have some properties not dependent on structure. It must be possible to individuate those objects independently of the role they play in that structure. But this is precisely what cannot be done with the numbers. To be the number 3 is no more and no less than to be preceded by 2, 1, and possibly 0, and to be followed by . . . Any object can play the role of 3; that is, any object can be the third element in some progression.

(Benacerraf, p. 70)

According to Benacerraf, the system of the natural numbers cannot exhibit the structure of the natural numbers; for the natural numbers cannot be individuated independently of the role they play in that structure. I take this to imply that there is no such thing as the system of natural numbers. In other words, Benacerraf argues that in regard to the Peano forms, for example, there exist all kinds of models except the model of the natural numbers. And this is so because numbers, in distinction from other things, have no properties independent of structure.

We hold that numbers are distinguished from all other kinds of things by the relations that hold among them. The question therefore becomes: Do numbers stand in certain characteristic relations which are independent of structure? And the answer, I submit, is clearly affirmative. Numbers and only numbers stand in the “successor relation” to each other. Is this not a feature which is independent of structure? What distinguishes one structure from another structure, when both have the same structure, may well be the fact that they contain different though isomorphic relations. From the fact that two structures share the same structure, it does not follow that they contain the same relations. And indeed, none of the structures which are isomorphic to the progression of the natural numbers contain the so-called successor relation. We see that Benacerraf’s argument would be sound if isomorphic relations were identical. But they need not be.

So much for the exotic view that while there are many progressions, there is no progression of the natural numbers. In a recent article, it appears to me, Resnik has taken this view to its limits:

The second [problem] arises from the fact that no mathematical theory can do more than determine its objects up to isomorphism. Thus the platonist seems to be in the paradoxical position of claiming that a given mathematical theory is about certain things and yet be unable to make any definite statement of what these things are.

(M. D. Resnik, “Mathematics as a Science of Patterns: Ontology,” p. 529)

How does Resnik arrive at the thesis that “no mathematical theory can do more than determine its objects up to isomorphism”? He does not tell us. One thing is certain, though; this thesis does not follow from the admitted fact that there are many different models for the axiom forms of arithmetic. Nor does it follow from any other set of facts, for it is false. From this false thesis, Resnik arrives at the view that: “The objects of mathematics, that is, the entities which our mathematical constants and quantifiers denote, are structureless points or positions in structures. As positions in structures, they have no identity or feature outside of structure” (Resnik, p. 530). What is a position in a structure? Perhaps Resnik holds the following view. The number three, for example, is the third position in many different structures. There are different structures, and these structures do consist of certain things. For example, there is the Zermelo progression and there is the von Neumann progression. In the Zermelo progression, the third position is occupied by the set whose only member is the set whose only member is the empty set; in the von Neumann progression, that position is occupied by quite a different set, namely, by the set which has two members: the empty set, and the set whose only member is the empty set. Although these two elements are quite different, they nevertheless occupy “the same position.” A position, therefore, is some kind of property or feature which different things can share (relative to the structures in which they occur). But the number three is not just any position. It is the third position. Here we are confronted with the unanalyzed and unexplained (ordinal) notion of three. Why do we attribute this feature to the two sets mentioned earlier? Obviously, we do so by reference to a third progression, namely, the progression of natural numbers. The Zermelo set turns out to be that part of the Zermelo progression which is coordinated to the number three in the progression of natural numbers. And the same holds for the von Neumann set. What is common to these sets is not some kind of undefined property or feature that may be called “the third position,” but rather that both sets are coordinated to the number three. But this explication of the notion of the third position implies, apparently contrary to what Resnik wishes to hold, that there exists the progression of natural numbers.

3) MODELS AND SEMANTIC NIHILISM

Resnik claims “that no mathematical theory can do more than determine its objects up to isomorphism.” I assumed that this claim was based on the fact that there are isomorphic models for the arithmetic axiom forms, and then I pointed out that the claim simply does not follow from the fact. Let us now follow another, though connected, chain of thought. Consider the true sentence: This pen is blue at t_j’ and its associated form: This f is g at t_n’. Obviously, there are interpretations of this form which are different from the original one and yield true sentences. For example, This book is green at t₁’ happens to be true. Assume that we conclude that since there are many different “models” for this sentence form, the sentence determines its objects only “up to isomorphism.” Would it not follow that we do not really know what the sentence is about “up to isomorphism”? And this would imply that we do not really know at all what the sentence is about. Hence we would have to conclude that we do not really know what any of our (declarative, true) sentences is about. I think that we have here a clear case of a reductio argument: If this is the consequence of our assumption that where there are models, there is indeterminacy up to isomorphism, then the assumption must be false.

I suspect, however, that there are philosophers who do not share my view and are willing to accept the conclusion. I suspect, furthermore, that this conclusion leads to a bundle of loosely connected thoughts which have been dubbed “the model theoretic argument against realism” (see H. Putnam, “Models and Reality,”). Whatever arguments there may be against realism, it seems obvious to me that they cannot possibly be based on the fact that certain forms allow for different models. But be that as it may, it is at any rate mistaken to believe that the sentence This pen is blue at t₁’ “determines its objects only up to isomorphism.” This sentence is about this particular pen, not about that book, and it says about the pen that it is blue (at t₁) rather than that it is green, or round, or sweet. Similarly for the axioms of arithmetic. The third Dedekind axiom reads: I is not the successor of any natural number. This axiom is about the number I, not about o or Paris, and it says about this particular number that it is not the successor of any number, not that it is not the brother of any Spanish king.

Of course, the associated forms for the sentence about the pen and the axioms for arithmetic may be said to “determine their objects only up to isomorphism.” Obviously, the form ‘This f is g at t_n’ does not tell us what it is about, for it is really not about anything. It is a form of something, of a sentence, that can be about something. It is a mere skeleton that needs to be fleshed out. Those who claim that the axioms of arithmetic do not determine their objects make the simple mistake, chastised in an earlier section, of confusing the axioms with their forms.

There is no reason why we should stop with the form ‘This f is g at t_n’. Why not replace the rest of the words in this form by appropriate schematic letters? We may proceed, for example, to get the “pure” form: ‘T f R g S t_n’. And this expression does not even determine its objects (the f’s and g’s) up to isomorphism. If one does not distinguish between the axioms of arithmetic, on the one hand, and all of the associated forms derivable from those axioms, on the other, one might as well conclude that the axioms of arithmetic do not even determine their objects up to isomorphism. One might as well conclude that they do not determine any objects at all.

In conclusion, let us look at one last argument for this sort of semantic nihilism (see Hockney’s review of my Ontological Reduction, in Philosophical Books). We are to imagine a civilization whose entire arithmetic vocabulary consists of set-theoretical and logical items. These people speak a version of Zermelo arithmetic. Among them, there exists a person with my philosophical views called “Gro.” Gro comes to hold that there is a special subcategory of the category of sets which he calls “sumbers” and which consist of the Zermelo sets arranged in the familiar way. Now, Gro assigns sumbers either (1) to the category of set, or (2) to the category of quantifier. According to Hockney, I cannot say in regard to alternative (1) that sumbers are really numbers and belong to the category of quantifier, since I hold that numbers are not sets. Quite so! On the other hand, if I claim that sumbers are just sets, and that there are other entities called “numbers,” then Gro will reply, according to Hockney, that any fool can see that numbers are fictions, for no such entities are constituents of the states of affairs we perceive. The debate about alternative (2) proceeds presumably along the same lines. From this Hockney concludes that there are no means by which this argument can be resolved without giving up Gro’s and my common principles; and these principles depend for their application on the view that it is clear what the constituents of states of affairs are. And from this he concludes further that these principles are useless, that there are no ultimate criteria which tell us to what our terms really refer, and that there is no unique answer to the question: What are numbers? The philosophical significance of constructional definitions consists, according to Hockney, in that they may be employed to display the range of legitimate ontological alternatives.

Hockney’s chain of reasoning has several puzzling links. Firstly, it is not clear to me what he means by saying that the imagined civilization has an arithmetical vocabulary. Does this mean that those people use shapes like ‘0’, ‘1’, etc.—and that they make the corresponding noises—when they talk about Zermelo sets? I do not think so; for then a tribe of aborigines may be said, with equal justification, to be using a quantum theoretical vocabulary if the members of the tribe happen to use certain shapes and noises in order to talk about gum trees, rocks, and dingos. Rather, what must be meant is that the imagined civilization uses some other kinds of shapes and noises in order to talk about Zermelo sets. But since there is an isomorphism between the progression of natural numbers and the progression of Zermelo sets, one conceives of these shapes and noises as constituting an arithmetical vocabulary. However, this conception is odd at best and misleading at worst. Imagine a civilization that has an axiomatized theory of zoology, but no arithmetic. (If you cannot conceive of such a civilization, then I doubt that you can conceive of Hockney’s civilization either.) Assume further that there is an arithmetical interpretation of the forms of that zoological theory. We could then say, following Hockney’s strange usage, that the entire arithmetical vocabulary of this civilization consists of zoological terms. From our point of view, we would protest, of course, that this civilization has no arithmetical vocabulary at all, since its members never talk or even think about numbers. But we cannot prevent Hockney from using ‘arithmetical vocabulary’ in this bizarre fashion. And no philosophical harm need be done, as long as we constantly keep in mind just how strangely he uses this expression.

The civilization of zoologists, in my opinion, talks about animals, not about numbers. But of course there are numbers in addition to animals, and the animals of the theory merely “simulate the behavior of natural numbers” in that unproblematic sense which is explicated in terms of the arithmetic model of the forms of the zoological theory. Similarly, in regard to Hockney’s civilization: Those people talk about Zermelo sets, not about numbers. There are numbers in addition and distinction to Zermelo sets, and Zermelo sets merely “simulate the behavior of the natural numbers.” To this, Gro presumably replies that numbers are nothing but fictions, since they are never perceived as constituents of states of affairs. At this point, I find a second unclarity in Hockney’s argument. We are not told whether or not the members of the imagined civilization perceive such states of affairs as that there are two tigers in the cage and that there are four pencils on the desk. Without this information, we simply do not know what to make of Gro’s reply.

Let us assume that those people perceive, just as we do, that there are two tigers in the cage and that there are four pencils on the desk. Gro is then simply and plainly mistaken. He and his friends do perceive numbers, no matter how vigorously and stubbornly he may deny it. On the other hand, if we assume that Gro and his friends do not perceive such states of affairs, then we understand immediately why he does not believe us when we tell him that there are states of affairs involving numbers and, hence, that there are numbers. But what does this difference in our respective beliefs show? Surely not that there are no numbers. Otherwise, we could as easily prove that there are no tigers by merely imagining a civilization that does not perceive tigers, or that there are no philosophers by imagining a civilization that does not perceive philosophers.

What Hockney’s argument is supposed to show, to remind ourselves, is that there is no way in which the debate between Gro and me can be resolved. This is the third obscure point. Does this mean that Gro may stick to his guns no matter what, even though he does see “numerical states of affairs”? Of course Gro may. But that would not prove anything. Does it mean that Gro’s reply is just as reasonable as my view, provided that Gro cannot perceive numerical states of affairs? Gro is then certainly justified in being skeptical about the existence of numbers, just as someone who does not perceive tigers is justified in being skeptical about the existence of tigers. But this does not show that Gro’s reply is just as reasonable, just as correct, as my view. If it did, then we could prove by the same sort of argument that it is just as reasonable to deny the existence of tigers, or of anything else, as it is to assert their existence. No debate about the existence of anything whatsoever could then be resolved.

This brings us to what Hockney calls the moral of his story, namely, that there are no ultimate criteria which tell us to what our terms refer, so that there is no unique answer to the question: What are numbers? Now, I do not care about ultimate criteria and unique answers. Ordinary criteria and plain answers will do. From this vantage point, I find Hockney’s moral not just unclear, but downright incoherent. If we do not have criteria that tell us to what our terms refer, how do we know to what they refer? And if Hockney means to imply that we do not know to what they refer, how does he know that his imagined civilization talks about Zermelo sets? How does he know, for that matter, that he himself is talking about Zermelo sets rather than, say, von Neumann sets? How can he possibly distinguish between different (though “equally legitimate”) models for the same axiom forms? And if there is no answer to the question: What are numbers? because we do not know to what our words refer, how could there be an answer to the question: What are tigers? Or does Hockney believe that there are no answers to any of these questions?

Looking back, it must surprise a philosopher of the next century that the discovery of isomorphic models gave rise to so much bad philosophy about the nature of mathematics. We just traced a path of misconceptions from implicit definitions to semantic nihilism.