466 MATHEMATICS 



Furthermore, it is easy to see that when we speak of objects of 

 different kinds, as, for instance, the points and lines of geometry, we 

 are introducing a notion which can very readily be expressed in our 

 relational notation. For this purpose we need merely to introduce 

 a further relation which is satisfied by two or more objects when and 

 only when they are of the same "kind." 



Let us turn finally to the relations themselves. It is customary 

 to distinguish here between dyadic relations, triadic relations, etc., 

 according as the relation in question connects two objects, three 

 objects, etc. There are, however, relations which may connect any 

 number of objects, as, for instance, the relation of collinearity which 

 may hold between any number of points. Any relation holds for 

 certain ordered groups of objects but not for others, and it is in no 

 way necessary for us to fix our attention on the fact, if it be true, 

 that the number of objects in all the groups for which a particular 

 relation holds is the same. This is the point of view we shall adopt, 

 and we shall relegate the property that a relation is dyadic, triadic, 

 etc., to the background along with the various other properties 

 relations may have, 1 all of which must be taken account of in the 

 proper place. 



We are thus concerned in any mathematical investigation, from 

 our present point of view, with just two conceptions: first a set, or 

 as the logicians say, a class of objects a, b, c, . . .; and secondly a 

 class of relations R, S, T , . . . . We may suppose these objects 

 divested of any qualitative, quantitative, spatial, or other attributes 

 which they may have had, and regard them merely as satisfying or not 

 satisfying the relations in question, where, again, we are wholly 

 indifferent to the nature which these relations originally had. And 

 now we are in a position to state what I conceive to be really the 

 essential point in Kempe's definition of mathematics; although I 

 have omitted one of the points on which he insists most strongly, 2 

 by saying: 



If we have a certain class of objects and a certain class of relations, 

 and if the only questions which we investigate are whether ordered 

 groups of these objects do or do not satisfy the relations, the results 

 of the investigation are called mathematics. 



theory of abstract groups (cf., for example, Huntington, Bulletin of the Ameri- 

 can Mathematical Society, June, 1902), where the postulate: 

 If a and b belong to the class, a o b belongs to the class, 



which in this form looks indecomposable, immediately breaks up, when stated in 

 the relational form, into the following two: 



1. If a and b belong to the class, there exists an element c of the class such that 

 R(a, b, c). 



2. If a, b, c, d belong to the class, and if R(a, b, c) and R(a, b, d), then c = d. 



1 For instance, the property of symmetry. A relation is said to be symmetrical 

 if it holds or fails to hold independently of the order in which the objects are taken. 



2 Namely, that the only relation that need be considered is that of being " in- 

 distinguishable," t. e., a symmetrical and transitive relation between two groups 

 of objects. 



