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: