ON THE THEORY OF SCIENCE 343 



impossibility of thinking its opposite is an impossible undertaking, 

 because every sort of nonsense can be thought : where the proof was 

 thought to have been given, there has always been a confusion of 

 thought and intuition, proof or inspection. 



With this one proposition of course the theory of order is not 

 exhausted, for here it is not a question of the development of this 

 theory, but of an example of the nature of the problems of science. 

 Of the further questions we shall briefly discuss the problem of 

 association. 



If we have two groups A and B given, one can associate with every 

 member of A one of B; that is, we determine that certain operations 

 which can be carried on with the members of A are also to be carried 

 on with those of B. Now we can begin by simply carrying out the 

 association, member for member. Then we shall have one of three 

 results: A will be exhausted while there are still members of B left, 

 or B will be exhausted first, or finally A and B will be exhausted at 

 the same time. In the first case we call A poorer than B; in the second 

 B poorer than A; in the third both quantities are alike. 



Here for the first time we come upon the scientific concept of 

 equality, which calls for discussion. There can be no question of a 

 complete identity of the two groups which have been denominated 

 equal, for we have made the assumption that the members of both 

 groups can be of any nature whatever. They can then be as different 

 as possible, considered singly, but they are alike as groups. However 

 I may arrange the members of A, I can make a similar arrangement 

 of the members of B, since every member of A has one of B associated 

 with it; and with reference to the property of arrangement there is no 

 difference to be observed between A and B. If, however, A is poorer 

 or richer than B, this possibility ceases, for then one of the groups 

 has members to which none of the members in the other group cor- 

 responds; so that the operations carried out with these members 

 cannot be carried out with those of the other group. 



Equality in the scientific sense, therefore, means equivalence, 

 or the possibility of substitution in quite definite operations or for 

 quite definite relations. Beyond this the things which are called 

 like may show any differences whatever. The general scientific 

 process of abstraction is again easily seen in this special case. 



On the basis of the definitions just given, we can establish further 

 propositions. If group A equals B, and B equals C, then A also 

 equals C. The proof of this is that we can relate every member 

 of A to a corresponding member of B and by hypothesis no 

 member will be left. Then C is arranged with reference to B, and 

 here also no member is left. By this process every member of A, 

 through the connecting link of a member of B, is associated with 

 a member of C, and this association is preserved even if we cut out 



