Introduction to Mathematical Philosophy

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 [Pg 52] 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. We may define two relations and as "similar," or as having "likeness," when there is a one-one relation whose domain is the field of and whose converse domain is the field of , and which is such that, if one term has the relation [Pg 53] to another, the correlate of the one has the relation to the correlate of the other, and vice versa. A figure will make this clearer. Let and be two terms having the relation . Then there are to be two terms , , such that has the relation to , has the relation to , and has the relation to . If this happens with every pair of terms such as and , and if the converse happens with every pair of terms such as and , it is clear that for every instance in which the relation holds there is a corresponding instance in which the relation holds, and vice versa; and this is what we desire to secure by our definition. We can eliminate some redundancies in the above sketch of a definition, by observing that, when the above conditions are realised, the relation is the same as the relative product of and and the converse of , i.e. the -step from to may be replaced by the succession of the -step from to , the -step from to , and the backward -step from to . Thus we may set up the following definitions:— A relation is said to be a "correlator" or an "ordinal correlator" of two relations and if is one-one, has the field of for its converse domain, and is such that is the relative product of and and the converse of . Two relations and are said to be "similar," or to have "likeness," when there is at least one correlator of and . 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 [Pg 54] 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 [Pg 55] 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 [Pg 56] 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 is a finite cardinal number, the relation-number of a series which has terms is called the "ordinal" number . (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 and are the generating relations of the two series, in the series which is their sum with put before , every member of the field of will precede every member of