[page 52]
We saw in Chapter II. that two classes have the same number of terms when they are “similar,” i.e. when there is a one-one relation whose domain is the one class and whose converse domain is the other. In such a case we say that there is a “one-one correlation” between the two classes.
In the present chapter we have to define a relation between relations, which will play the same part for them that similarity of classes plays for classes. We will call this relation “similarity of relations,” or “likeness” when it seems desirable to use a different word from that which we use for classes. How is likeness to be defined?
We shall employ still the notion of correlation: we shall assume that the domain of the one relation can be correlated with the domain of the other, and the converse domain with the converse domain; but that is not enough for the sort of resemblance which we desire to have between our two relations. What we desire is that, whenever either relation holds between two terms, the other relation shall hold between the correlates of these two terms. The easiest example of the sort of thing we desire is a map. When one place is north of another, the place on the map corresponding to the one is above the place on the map corresponding to the other; when one place is west of another, the place on the map corresponding to the one is to the left of the place on the map corresponding to the other; and so on. The structure of the map corresponds with that of [page 53] the country of which it is a map. The space-relations in the map have “likeness” to the space-relations in the country mapped. It is this kind of connection between relations that we wish to define.
We may, in the first place, profitably introduce a certain restriction. We will confine ourselves, in defining likeness, to such relations as have “fields,” i.e. to such as permit of the formation of a single class out of the domain and the converse domain. This is not always the case. Take, for example, the relation “domain,” i.e. the relation which the domain of a relation has to the relation. This relation has all classes for its domain, since every class is the domain of some relation; and it has all relations for its converse domain, since every relation has a domain. But classes and relations cannot be added together to form a new single class, because they are of different logical “types.” We do not need to enter upon the difficult doctrine of types, but it is well to know when we are abstaining from entering upon it. We may say, without entering upon the grounds for the assertion, that a relation only has a “field” when it is what we call “homogeneous,” i.e. when its domain and converse domain are of the same logical type; and as a rough-and-ready indication of what we mean by a “type,” we may say that individuals, classes of individuals, relations between individuals, relations between classes, relations of classes to individuals, and so on, are different types. Now the notion of likeness is not very useful as applied to relations that are not homogeneous; we shall, therefore, in defining likeness, simplify our problem by speaking of the “field” of one of the relations concerned. This somewhat limits the generality of our definition, but the limitation is not of any practical importance. And having been stated, it need no longer be remembered.
A relation S is said to be a “correlator” or an “ordinal correlator” of two relations P and Q if S is one-one, has the field of Q for its converse domain, and is such that P is the relative product of S and Q and the converse of S.
Two relations P and Q are said to be “similar,” or to have “likeness,” when there is at least one correlator of P and Q.
These definitions will be found to yield what we above decided to be necessary.
It will be found that, when two relations are similar, they share all properties which do not depend upon the actual terms in their fields. For instance, if one implies diversity, so does the other; if one is transitive, so is the other; if one is connected, so is the other. Hence if one is serial, so is the other. Again, if one is one-many or one-one, the other is one-many [page 55] or one-one; and so on, through all the general properties of relations. Even statements involving the actual terms of the field of a relation, though they may not be true as they stand when applied to a similar relation, will always be capable of translation into statements that are analogous. We are led by such considerations to a problem which has, in mathematical philosophy, an importance by no means adequately recognised hitherto. Our problem may be stated as follows:—
Given some statement in a language of which we know the grammar and the syntax, but not the vocabulary, what are the possible meanings of such a statement, and what are the meanings of the unknown words that would make it true?
The reason that this question is important is that it represents, much more nearly than might be supposed, the state of our knowledge of nature. We know that certain scientific propositions—which, in the most advanced sciences, are expressed in mathematical symbols—are more or less true of the world, but we are very much at sea as to the interpretation to be put upon the terms which occur in these propositions. We know much more (to use, for a moment, an old-fashioned pair of terms) about the form of nature than about the matter. Accordingly, what we really know when we enunciate a law of nature is only that there is probably some interpretation of our terms which will make the law approximately true. Thus great importance attaches to the question: What are the possible meanings of a law expressed in terms of which we do not know the substantive meaning, but only the grammar and syntax? And this question is the one suggested above.
For the present we will ignore the general question, which will occupy us again at a later stage; the subject of likeness itself must first be further investigated.
Owing to the fact that, when two relations are similar, their properties are the same except when they depend upon the fields being composed of just the terms of which they are composed, it is desirable to have a nomenclature which collects [page 56] together all the relations that are similar to a given relation. Just as we called the set of those classes that are similar to a given class the “number” of that class, so we may call the set of all those relations that are similar to a given relation the “number” of that relation. But in order to avoid confusion with the numbers appropriate to classes, we will speak, in this case, of a “relation-number.” Thus we have the following definitions:—
The “relation-number” of a given relation is the class of all those relations that are similar to the given relation.
“Relation-numbers” are the set of all those classes of relations that are relation-numbers of various relations; or, what comes to the same thing, a relation-number is a class of relations consisting of all those relations that are similar to one member of the class.
When it is necessary to speak of the numbers of classes in a way which makes it impossible to confuse them with relation-numbers, we shall call them “cardinal numbers.” Thus cardinal numbers are the numbers appropriate to classes. These include the ordinary integers of daily life, and also certain infinite numbers, of which we shall speak later. When we speak of “numbers” without qualification, we are to be understood as meaning cardinal numbers. The definition of a cardinal number, it will be remembered, is as follows:—
The “cardinal number” of a given class is the set of all those classes that are similar to the given class.
The most obvious application of relation-numbers is to series. Two series may be regarded as equally long when they have the same relation-number. Two finite series will have the same relation-number when their fields have the same cardinal number of terms, and only then—i.e. a series of (say) 15 terms will have the same relation-number as any other series of fifteen terms, but will not have the same relation-number as a series of 14 or 16 terms, nor, of course, the same relation-number as a relation which is not serial. Thus, in the quite special case of finite series, there is parallelism between cardinal and relation-numbers. The relation-numbers applicable to series may be [page 57] called “serial numbers” (what are commonly called “ordinal numbers” are a sub-class of these); thus a finite serial number is determinate when we know the cardinal number of terms in the field of a series having the serial number in question. If n is a finite cardinal number, the relation-number of a series which has n terms is called the “ordinal” number n. (There are also infinite ordinal numbers, but of them we shall speak in a later chapter.) When the cardinal number of terms in the field of a series is infinite, the relation-number of the series is not determined merely by the cardinal number, indeed an infinite number of relation-numbers exist for one infinite cardinal number, as we shall see when we come to consider infinite series. When a series is infinite, what we may call its “length,” i.e. its relation-number, may vary without change in the cardinal number; but when a series is finite, this cannot happen.
We can define addition and multiplication for relation-numbers as well as for cardinal numbers, and a whole arithmetic of relation-numbers can be developed. The manner in which this is to be done is easily seen by considering the case of series. Suppose, for example, that we wish to define the sum of two non-overlapping series in such a way that the relation-number of the sum shall be capable of being defined as the sum of the relation-numbers of the two series. In the first place, it is clear that there is an order involved as between the two series: one of them must be placed before the other. Thus if P and Q are the generating relations of the two series, in the series which is their sum with P put before Q, every member of the field of P will precede every member of the field of Q. Thus the serial relation which is to be defined as the sum of P and Q is not “P or Q” simply, but “P or Q or the relation of any member of the field of P to any member of the field of Q.” Assuming that P and Q do not overlap, this relation is serial, but “P or Q” is not serial, being not connected, since it does not hold between a member of the field of P and a member of the field of Q. Thus the sum of P and Q, as above defined, is what we need in order [page 58] to define the sum of two relation-numbers. Similar modifications are needed for products and powers. The resulting arithmetic does not obey the commutative law: the sum or product of two relation-numbers generally depends upon the order in which they are taken. But it obeys the associative law, one form of the distributive law, and two of the formal laws for powers, not only as applied to serial numbers, but as applied to relation-numbers generally. Relation-arithmetic, in fact, though recent, is a thoroughly respectable branch of mathematics.
It must not be supposed, merely because series afford the most obvious application of the idea of likeness, that there are no other applications that are important. We have already mentioned maps, and we might extend our thoughts from this illustration to geometry generally. If the system of relations by which a geometry is applied to a certain set of terms can be brought fully into relations of likeness with a system applying to another set of terms, then the geometry of the two sets is indistinguishable from the mathematical point of view, i.e. all the propositions are the same, except for the fact that they are applied in one case to one set of terms and in the other to another. We may illustrate this by the relations of the sort that may be called “between,” which we considered in Chapter IV. We there saw that, provided a three-term relation has certain formal logical properties, it will give rise to series, and may be called a “between-relation.” Given any two points, we can use the between-relation to define the straight line determined by those two points; it consists of a and b together with all points x, such that the between-relation holds between the three points a, b, x in some order or other. It has been shown by O. Veblen that we may regard our whole space as the field of a three-term between-relation, and define our geometry by the properties we assign to our between-relation.1 Now likeness is just as easily [page 59] definable between three-term relations as between two-term relations. If B and B' are two between-relations, so that “xB(y, z)” means “x is between y and z with respect to B,” we shall call S a correlator of B and B' if it has the field of B' for its converse domain, and is such that the relation B holds between three terms when B' holds between their S-correlates, and only then. And we shall say that B is like B' when there is at least one correlator of B with B'. The reader can easily convince himself that, if B is like B' in this sense, there can be no difference between the geometry generated by B and that generated by B'.
It follows from this that the mathematician need not concern himself with the particular being or intrinsic nature of his points, lines, and planes, even when he is speculating as an applied mathematician. We may say that there is empirical evidence of the approximate truth of such parts of geometry as are not matters of definition. But there is no empirical evidence as to what a “point” is to be. It has to be something that as nearly as possible satisfies our axioms, but it does not have to be “very small” or “without parts.” Whether or not it is those things is a matter of indifference, so long as it satisfies the axioms. If we can, out of empirical material, construct a logical structure, no matter how complicated, which will satisfy our geometrical axioms, that structure may legitimately be called a “point.” We must not say that there is nothing else that could legitimately be called a “point”; we must only say: “This object we have constructed is sufficient for the geometer; it may be one of many objects, any of which would be sufficient, but that is no concern of ours, since this object is enough to vindicate the empirical truth of geometry, in so far as geometry is not a matter of definition.” This is only an illustration of the general principle that what matters in mathematics, and to a very great extent in physical science, is not the intrinsic nature of our terms, but the logical nature of their interrelations.
We may say, of two similar relations, that they have the same [page 60] “structure.” For mathematical purposes (though not for those of pure philosophy) the only thing of importance about a relation is the cases in which it holds, not its intrinsic nature. Just as a class may be defined by various different but co-extensive concepts—e.g. “man” and “featherless biped”—so two relations which are conceptually different may hold in the same set of instances. An “instance” in which a relation holds is to be conceived as a couple of terms, with an order, so that one of the terms comes first and the other second; the couple is to be, of course, such that its first term has the relation in question to its second. Take (say) the relation “father”: we can define what we may call the “extension” of this relation as the class of all ordered couples (x, y) which are such that x is the father of y. From the mathematical point of view, the only thing of importance about the relation “father” is that it defines this set of ordered couples. Speaking generally, we say:
The “extension” of a relation is the class of those ordered couples (x, y) which are such that x has the relation in question to y.
It is clear that the “structure” of the relation does not depend upon the particular terms that make up the field of the relation. The field may be changed without changing the structure, and the structure may be changed without changing the field—for [page 61] example, if we were to add the couple ae in the above illustration we should alter the structure but not the field. Two relations have the same “structure,” we shall say, when the same map will do for both—or, what comes to the same thing, when either can be a map for the other (since every relation can be its own map). And that, as a moment’s reflection shows, is the very same thing as what we have called “likeness.” That is to say, two relations have the same structure when they have likeness, i.e. when they have the same relation-number. Thus what we defined as the “relation-number” is the very same thing as is obscurely intended by the word “structure”—a word which, important as it is, is never (so far as we know) defined in precise terms by those who use it.
There has been a great deal of speculation in traditional philosophy which might have been avoided if the importance of structure, and the difficulty of getting behind it, had been realised. For example, it is often said that space and time are subjective, but they have objective counterparts; or that phenomena are subjective, but are caused by things in themselves, which must have differences inter se corresponding with the differences in the phenomena to which they give rise. Where such hypotheses are made, it is generally supposed that we can know very little about the objective counterparts. In actual fact, however, if the hypotheses as stated were correct, the objective counterparts would form a world having the same structure as the phenomenal world, and allowing us to infer from phenomena the truth of all propositions that can be stated in abstract terms and are known to be true of phenomena. If the phenomenal world has three dimensions, so must the world behind phenomena; if the phenomenal world is Euclidean, so must the other be; and so on. In short, every proposition having a communicable significance must be true of both worlds or of neither: the only difference must lie in just that essence of individuality which always eludes words and baffles description, but which, for that very reason, is irrelevant to science. Now the only purpose that philosophers [page 62] have in view in condemning phenomena is in order to persuade themselves and others that the real world is very different from the world of appearance. We can all sympathise with their wish to prove such a very desirable proposition, but we cannot congratulate them on their success. It is true that many of them do not assert objective counterparts to phenomena, and these escape from the above argument. Those who do assert counterparts are, as a rule, very reticent on the subject, probably because they feel instinctively that, if pursued, it will bring about too much of a rapprochement between the real and the phenomenal world. If they were to pursue the topic, they could hardly avoid the conclusions which we have been suggesting. In such ways, as well as in many others, the notion of structure or relation-number is important.
[Chapter VI notes]
1. This does not apply to elliptic space, but only to spaces in which the straight line is an open series. Modern Mathematics, edited by J. W. A. Young, pp. 3–51 (monograph by O. Veblen on “The Foundations of Geometry”).