Introduction to Mathematical Philosophy

IN this chapter we have to consider an axiom which can be enunciated, but not proved, in terms of logic, and which is convenient, though not indispensable, in certain portions of mathematics. It is convenient, in the sense that many interesting propositions, which it seems natural to suppose true, cannot be proved without its help; but it is not indispensable, because even without those propositions the subjects in which they occur still exist, though in a somewhat mutilated form. Before enunciating the multiplicative axiom, we must first explain the theory of selections, and the definition of multiplication when the number of factors may be infinite. In defining the arithmetical operations, the only correct procedure is to construct an actual class (or relation, in the case of relation-numbers) having the required number of terms. This sometimes demands a certain amount of ingenuity, but it is essential in order to prove the existence of the number defined. Take, as the simplest example, the case of addition. Suppose we are given a cardinal number , and a class which has terms. How shall we define ? For this purpose we must have two classes having terms, and they must not overlap. We can construct such classes from in various ways, of which the following is perhaps the simplest: Form first all the ordered couples whose first term is a class consisting of a single member of , and whose second term is the null-class; then, secondly, form all the ordered couples whose first term is [Pg 117] the null-class and whose second term is a class consisting of a single member of . These two classes of couples have no member in common, and the logical sum of the two classes will have terms. Exactly analogously we can define , given that is the number of some class and is the number of some class . Such definitions, as a rule, are merely a question of a suitable technical device. But in the case of multiplication, where the number of factors may be infinite, important problems arise out of the definition. Multiplication when the number of factors is finite offers no difficulty. Given two classes and , of which the first has terms and the second terms, we can define as the number of ordered couples that can be formed by choosing the first term out of and the second out of . It will be seen that this definition does not require that and should not overlap; it even remains adequate when and are identical. For example, let be the class whose members are , , . Then the class which is used to define the product is the class of couples: This definition remains applicable when or or both are infinite, and it can be extended step by step to three or four or any finite number of factors. No difficulty arises as regards this definition, except that it cannot be extended to an infinite number of factors. The problem of multiplication when the number of factors may be infinite arises in this way: Suppose we have a class consisting of classes; suppose the number of terms in each of these classes is given. How shall we define the product of all these numbers? If we can frame our definition generally, it will be applicable whether is finite or infinite. It is to be observed that the problem is to be able to deal with the case when is infinite, not with the case when its members are. If [Pg 118] is not infinite, the method defined above is just as applicable when its members are infinite as when they are finite. It is the case when is infinite, even though its members may be finite, that we have to find a way of dealing with. The following method of defining multiplication generally is due to Dr Whitehead. It is explained and treated at length in Principia Mathematica, vol. I. * 80 ff., and vol. II. * 114. Let us suppose to begin with that is a class of classes no two of which overlap—say the constituencies in a country where there is no plural voting, each constituency being considered as a class of voters. Let us now set to work to choose one term out of each class to be its representative, as constituencies do when they elect members of Parliament, assuming that by law each constituency has to elect a man who is a voter in that constituency. We thus arrive at a class of representatives, who make up our Parliament, one being selected out of each constituency. How many different possible ways of choosing a Parliament are there? Each constituency can select any one of its voters, and therefore if there are voters in a constituency, it can make choices. The choices of the different constituencies are independent; thus it is obvious that, when the total number of constituencies is finite, the number of possible Parliaments is obtained by multiplying together the numbers of voters in the various constituencies. When we do not know whether the number of constituencies is finite or infinite, we may take the number of possible Parliaments as defining the product of the numbers of the separate constituencies. This is the method by which infinite products are defined. We must now drop our illustration, and proceed to exact statements. Let be a class of classes, and let us assume to begin with that no two members of overlap, i.e. that if and are two different members of , then no member of the one is a member of the other. We shall call a class a "selection" from when it consists of just one term from each member of ; i.e. is a "selection" from if every member of belongs to some member [Pg 119] of , and if be any member of , and have exactly one term in common. The class of all "selections" from we shall call the "multiplicative class" of . The number of terms in the multiplicative class of , i.e. the number of possible selections from , is defined as the product of the numbers of the members of . This definition is equally applicable whether is finite or infinite. Before we can be wholly satisfied with these definitions, we must remove the restriction that no two members of are to overlap. For this purpose, instead of defining first a class called a "selection," we will define first a relation which we will call a "selector." A relation will be called a "selector" from if, from every member of , it picks out one term as the representative of that member, i.e. if, given any member of , there is just one term which is a member of and has the relation to ; and this is to be all that does. The formal definition is: A "selector" from a class of classes is a one-many relation, having for its converse domain, and such that, if has the relation to , then is a member of . If is a selector from , and is a member of , and is the term which has the relation to , we call the "representative" of in respect of the relation . A "selection" from will now be defined as the domain of a selector; and the multiplicative class, as before, will be the class of selections. But when the members of overlap, there may be more selectors than selections, since a term which belongs to two classes and may be selected once to represent and once to represent , giving rise to different selectors in the two cases, but to the same selection. For purposes of defining multiplication, it is the selectors we require rather than the selections. Thus we define: "The product of the numbers of the members of a class of classes " is the number of selectors from . We can define exponentiation by an adaptation of the above [Pg 120] plan. We might, of course, define as the number of selectors from classes, each of which has terms. But there are objections to this definition, derived from the fact that the multiplicative axiom (of which we shall speak shortly) is unnecessarily involved if it is adopted. We adopt instead the following construction:— Let be a class having terms, and a class having terms. Let be a member of , and form the class of all ordered couples that have for their second term and a member of for their first term. There will be such couples for a given , since any member of may be chosen for the first term, and has members. If we now form all the classes of this sort that result from varying , we obtain altogether classes, since may be any member of , and has members. These classes are each of them a class of couples, namely, all the couples that can be formed of a variable member of and a fixed member of . We define as the number of selectors from the class consisting of these classes. Or we may equally well define as the number of selections, for, since our classes of couples are mutually exclusive, the number of selectors is the same as the number of selections. A selection from our class of classes will be a set of ordered couples, of which there will be exactly one having any given member of for its second term, and the first term may be any member of . Thus is defined by the selectors from a certain set of classes each having terms, but the set is one having a certain structure and a more manageable composition than is the case in general. The relevance of this to the multiplicative axiom will appear shortly. What applies to exponentiation applies also to the product of two cardinals. We might define "" as the sum of the numbers of classes each having terms, but we prefer to define it as the number of ordered couples to be formed consisting of a member of followed by a member of , where has terms and has terms. This definition, also, is designed to evade the necessity of assuming the multiplicative axiom. [Pg 121] With our definitions, we can prove the usual formal laws of multiplication and exponentiation. But there is one thing we cannot prove: we cannot prove that a product is only zero when one of its factors is zero. We can prove this when the number of factors is finite, but not when it is infinite. In other words, we cannot prove that, given a class of classes none of which is null, there must be selectors from them; or that, given a class of mutually exclusive classes, there must be at least one class consisting of one term out of each of the given classes. These things cannot be proved; and although, at first sight, they seem obviously true, yet reflection brings gradually increasing doubt, until at last we become content to register the assumption and its consequences, as we register the axiom of parallels, without assuming that we can know whether it is true or false. The assumption, loosely worded, is that selectors and selections exist when we should expect them. There are many equivalent ways of stating it precisely. We may begin with the following:— "Given any class of mutually exclusive classes, of which none is null, there is at least one class which has exactly one term in common with each of the given classes." This proposition we will call the "multiplicative axiom."[24] We will first give various equivalent forms of the proposition, and then consider certain ways in which its truth or falsehood is of interest to mathematics. [24]Principia Mathematica, vol. I. * 88. Also vol. III. * 257-258. The multiplicative axiom is equivalent to the proposition that a product is only zero when at least one of its factors is zero; i.e. that, if any number of cardinal numbers be multiplied together, the result cannot be 0 unless one of the numbers concerned is 0. The multiplicative axiom is equivalent to the proposition that, if be any relation, and any class contained in the converse domain of , then there