162 NORMATIVE SCIENCE 



found to be of such significance as to be worthy of exhaustive treat- 

 ment. Thus the instinctive sense for significant truth, which has all 

 along been guiding the development of mathematics, comes at least 

 to a clear and philosophical consciousness. And meanwhile the es- 

 sential categories of thought are seen in a new light. 



The second result still more directly concerns a philosophical logic. 

 It is this: Since the few types of relations which this sort of ana- 

 lysis reveals as the fundamental ones in exact science are of such 

 importance, the logic of the present day is especially required to face 

 the questions : What is the nature of our concept of relations ? What 

 are the various possible types of relations? Upon what does the 

 variety of these types depend? What unity lies beneath the variety? 



As a fact, logic, in its modern forms, namely, first that symbolic 

 logic which Boole first formulated, which Mr. Charles S. Peirce and 

 his pupils have in this country already so highly developed, and 

 which Schroeder in Germany, Peano's school in Italy, and a num- 

 ber of recent English writers have so effectively furthered and 

 secondly, the logic of scientific method, which is now so actively 

 pursued, in France, in Germany, and in the English-speaking coun- 

 tries -- this whole movement in modern logic, as I hold, is rapidly 

 approaching new solutions of the problem of the fundamental nature 

 and the logic of relations. The problem is one in which we are all 

 equally interested. To De Morgan in England, in an earlier genera- 

 tion, and, in our time, to Charles Peirce in this country, very im- 

 portant stages in the growth of these problems are due. Russell, in 

 his work on the Principles of Mathematics has very lately under- 

 taken to sum up the results of the logic of relations, as thus far 

 developed, and to add his own interpretations. Yet I think that 

 Russell has failed to get as near to the foundations of the theory 

 of relations as the present state of the discussion permits. For 

 Russell has failed to take account of what I hold to be the most 

 fundamentally important generalization yet reached in the general 

 theory of relations. This is the generalization set forth as early as 

 1890, by Mr. A. B. Kempe, of London, in a pair of wonderful but 

 too much neglected, papers, entitled, respectively, The Theory of 

 Mathematical Form, and The Analogy between the Logical Theory 

 of Classes and the Geometrical Theory of Points. A mere hint first 

 as to the more precise formulation of the problem at issue, and then 

 later as to Kempe's special contribution to that problem, may be in 

 order here, despite the impossibility of any adequate statement. 



Ill 



The two most obviously and universally important kinds of rela- 

 tions known to the exact sciences, as these sciences at present exist, 

 are: (1) The relations of the type of equality or equivalence; and 



