Introduction to Mathematical Philosophy

A great part of the philosophy of mathematics is concerned with relations, and many different kinds of relations have different kinds of uses. It often happens that a property which belongs to all relations is only important as regards relations of certain sorts; in these cases the reader will not see the bearing of the proposition asserting such a property unless he has in mind the sorts of relations for which it is useful. For reasons of this description, as well as from the intrinsic interest of the subject, it is well to have in our minds a rough list of the more mathematically serviceable varieties of relations. We dealt in the preceding chapter with a supremely important class, namely, serial relations. Each of the three properties which we combined in defining series—namely, asymmetry, transitiveness, and connexity—has its own importance. We will begin by saying something on each of these three. Asymmetry, i.e. the property of being incompatible with the converse, is a characteristic of the very greatest interest and importance. In order to develop its functions, we will consider various examples. The relation husband is asymmetrical, and so is the relation wife; i.e. if is husband of , cannot be husband of , and similarly in the case of wife. On the other hand, the relation "spouse" is symmetrical: if is spouse of , then is spouse of . Suppose now we are given the relation spouse, and we wish to derive the relation husband. Husband is the same as male spouse or spouse of a female; thus the relation husband can [Pg 42] be derived from spouse either by limiting the domain to males or by limiting the converse to females. We see from this instance that, when a symmetrical relation is given, it is sometimes possible, without the help of any further relation, to separate it into two asymmetrical relations. But the cases where this is possible are rare and exceptional: they are cases where there are two mutually exclusive classes, say and , such that whenever the relation holds between two terms, one of the terms is a member of and the other is a member of —as, in the case of spouse, one term of the relation belongs to the class of males and one to the class of females. In such a case, the relation with its domain confined to will be asymmetrical, and so will the relation with its domain confined to . But such cases are not of the sort that occur when we are dealing with series of more than two terms; for in a series, all terms, except the first and last (if these exist), belong both to the domain and to the converse domain of the generating relation, so that a relation like husband, where the domain and converse domain do not overlap, is excluded. The question how to construct relations having some useful property by means of operations upon relations which only have rudiments of the property is one of considerable importance. Transitiveness and connexity are easily constructed in many cases where the originally given relation does not possess them: for example, if is any relation whatever, the ancestral relation derived from by generalised induction is transitive; and if is a many-one relation, the ancestral relation will be connected if confined to the posterity of a given term. But asymmetry is a much more difficult property to secure by construction. The method by which we derived husband from spouse is, as we have seen, not available in the most important cases, such as greater, before, to the right of, where domain and converse domain overlap. In all these cases, we can of course obtain a symmetrical relation by adding together the given relation and its converse, but we cannot pass back from this symmetrical relation to the original asymmetrical relation except by the help of some asymmetrical [Pg 43] relation. Take, for example, the relation greater: the relation greater or less—i.e. unequal—is symmetrical, but there is nothing in this relation to show that it is the sum of two asymmetrical relations. Take such a relation as "differing in shape." This is not the sum of an asymmetrical relation and its converse, since shapes do not form a single series; but there is nothing to show that it differs from "differing in magnitude" if we did not already know that magnitudes have relations of greater and less. This illustrates the fundamental character of asymmetry as a property of relations. From the point of view of the classification of relations, being asymmetrical is a much more important characteristic than implying diversity. Asymmetrical relations imply diversity, but the converse is not the case. "Unequal," for example, implies diversity, but is symmetrical. Broadly speaking, we may say that, if we wished as far as possible to dispense with relational propositions and replace them by such as ascribed predicates to subjects, we could succeed in this so long as we confined ourselves to symmetrical relations: those that do not imply diversity, if they are transitive, may be regarded as asserting a common predicate, while those that do imply diversity may be regarded as asserting incompatible predicates. For example, consider the relation of similarity between classes, by means of which we defined numbers. This relation is symmetrical and transitive and does not imply diversity. It would be possible, though less simple than the procedure we adopted, to regard the number of a collection as a predicate of the collection: then two similar classes will be two that have the same numerical predicate, while two that are not similar will be two that have different numerical predicates. Such a method of replacing relations by predicates is formally possible (though often very inconvenient) so long as the relations concerned are symmetrical; but it is formally impossible when the relations are asymmetrical, because both sameness and difference of predicates are symmetrical. Asymmetrical relations are, we may [Pg 44] say, the most characteristically relational of relations, and the most important to the philosopher who wishes to study the ultimate logical nature of relations. Another class of relations that is of the greatest use is the class of one-many relations, i.e. relations which at most one term can have to a given term. Such are father, mother, husband (except in Tibet), square of, sine of, and so on. But parent, square root, and so on, are not one-many. It is possible, formally, to replace all relations by one-many relations by means of a device. Take (say) the relation less among the inductive numbers. Given any number greater than 1, there will not be only one number having the relation less to , but we can form the whole class of numbers that are less than . This is one class, and its relation to is not shared by any other class. We may call the class of numbers that are less than the "proper ancestry" of , in the sense in which we spoke of ancestry and posterity in connection with mathematical induction. Then "proper ancestry" is a one-many relation (one-many will always be used so as to include one-one), since each number determines a single class of numbers as constituting its proper ancestry. Thus the relation less than can be replaced by being a member of the proper ancestry of. In this way a one-many relation in which the one is a class, together with membership of this class, can always formally replace a relation which is not one-many. Peano, who for some reason always instinctively conceives of a relation as one-many, deals in this way with those that are naturally not so. Reduction to one-many relations by this method, however, though possible as a matter of form, does not represent a technical simplification, and there is every reason to think that it does not represent a philosophical analysis, if only because classes must be regarded as "logical fictions." We shall therefore continue to regard one-many relations as a special kind of relations. One-many relations are involved in all phrases of the form "the so-and-so of such-and-such." "The King of England," [Pg 45] "the wife of Socrates," "the father of John Stuart Mill," and so on, all describe some person by means of a one-many relation to a given term. A person cannot have more than one father, therefore "the father of John Stuart Mill" described some one person, even if we did not know whom. There is much to say on the subject of descriptions, but for the present it is relations that we are concerned with, and descriptions are only relevant as exemplifying the uses of one-many relations. It should be observed that all mathematical functions result from one-many relations: the logarithm of , the cosine of , etc., are, like the father of , terms described by means of a one-many relation (logarithm, cosine, etc.) to a given term (). The notion of function need not be confined to numbers, or to the uses to which mathematicians have accustomed us; it can be extended to all cases of one-many relations, and "the father of " is just as legitimately a function of which is the argument as is "the logarithm of ." Functions in this sense are descriptive functions. As we shall see later, there are functions of a still more general and more fundamental sort, namely, propositional functions; but for the present we shall confine our attention to descriptive functions, i.e. "the term having the relation to ," or, for short, "the of ," where is any one-many relation. It will be observed that if "the of " is to describe a definite term, must be a term to which something has the relation , and there must not be more than one term having the relation to , since "the," correctly used, must imply uniqueness. Thus we may speak of "the father of " if is any human being except Adam and Eve; but we cannot speak of "the father of " if is a table or a chair or anything else that does not have a father. We shall say that the of "exists" when there is just one term, and no more, having the relation to . Thus if is a one-many relation, the of exists whenever belongs to the converse domain of , and not otherwise. Regarding "the of " as a function in the mathematical [Pg 46] sense, we say that is the "argument" of the function, and if is the term which has the relation to , i.e. if is the of , then is the "value" of the function for the argument . If is a one-many relation, the range of possible arguments to the function is the converse domain of , and the range of values is the domain. Thus the range of possible arguments to the function "the father of " is all who have fathers, i.e. the converse domain of the relation father, while the range of possible values for the function is all fathers, i.e. the domain of the relation. Many of the most important notions in the logic of relations are descriptive functions, for example: converse, domain, converse domain, field. Other examples will occur as we proceed. Among one-many relations, one-one relations are a specially important class. We have already had occasion to speak of one-one relations in connection with the definition of number, but it is necessary to be familiar with them, and not merely to know their formal definition. Their formal definition may be derived from that of one-many relations: they may be defined as one-many relations which are also the converses of one-many relations, i.e. as relations which are both one-many and many-one. One-many relations may be defined as relations such that, if has the relation in question to , there is no other term ' which also has the relation to . Or, again, they may be defined as follows: Given two terms and ', the terms to which has the given relation and those to which ' has it have no member in common. Or, again, they may be defined as relations such that the