October 7, 1904.] 



SCIENCE. 



457 



however, by the earlier, and especially by 

 the philosophical, students of the categories, 

 is the form which these postulates tend to 

 assume when they are rigidly analyzed. 



This form depends upon the precise 

 definition and classification of certain types 

 of relations. The whole of geometry, for 

 instance, including metrical geometry, can 

 be developed from a set of postulates which 

 demand the existence of points that stand 

 in certain ordinal relationships. The 

 ordinal relationships can be reduced, ac- 

 cording as the series of points considered 

 is open or closed, either to the well-known 

 relationship in which three points stand 

 when one is between the other two upon a 

 right line, or else to the ordinal relationship 

 in which four points stand when they are 

 separated by pairs; and these two ordinal 

 relationships, by means of various logical 

 devices, can be regarded as variations of a 

 single fundamental form. Cayley and Klein 

 founded the logical theory of geometry 

 here in question. Russell, and in another 

 way Dr. Veblen, have given it its most 

 recent expressions. In the same way, the 

 theory of whole numbers can be reduced 

 to sets of principles which demand the ex- 

 istence of certain ideal objects in certain 

 simple ordinal relations. Dedekind and 

 Peano have worked out such ordinal 

 theories of the number concept. In an- 

 other development of the theory of the 

 cardinal whole numbers, which Russell and 

 "Whitehead have worked out, ordinal con- 

 cepts are introduced only secondarily, and 

 the theory depends upon the fundamental 

 relation of the equivalence or non-equiva- 

 lence of collections of objects. But here 

 also a certain simple type of relation deter- 

 mines the definitions and the development 

 of the whole theory. 



Two results follow from such a fashion 

 of logically analyzing the first principles of 

 mathematical science. In the first place, 

 as just pointed out, we learn Jiotv feiv and 



simple are the conceptions and postulates 

 upon which the actual edifice of exact sci- 

 ence rests. Pure mathematics, we have 

 said, is free to assume what it chooses. Yet 

 the assumptions whose presence as the 

 foundation principles of the actually ex- 

 istent pure mathematics an exhaustive ex- 

 amination thus reveals, show by their few- 

 ness that the ideal freedom of the mathe- 

 matician to assume and to construct what 

 he pleases, is indeed, in practise, a very 

 decidedly limited freedom. The limitation 

 iw, as we have already seen, a limitation 

 which has to do with the essential sig- 

 nificance of the fundamental concepts in 

 question. And so the result of this anal- 

 ysis of the bases of the actually developed 

 and significant branches of mathematics, 

 constitutes a sort of empirical revelation 

 of what categories the exact sciences have 

 practically found to be of such significance 

 as to be worthy of exhaustive treatment. 

 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 essential categories of 

 thought are seen in a new light. 



The second result still more directly con- 

 cerns a philosophical logic. It is this: 

 Since the few types of relations which this 

 sort of analysis reveals as the fundamental 

 ones in exact science are of such impor- 

 tance, the logic of the present day is espe- 

 cially required to face the questions : What 

 is the nature of our C07icept of relations? 

 What are the various possible types of rela- 

 tions? Upon what does the variety of 

 these types depend? What unity lies be- 

 neath the variety ? 



As a fact, logic, in its modern forms, viz., 

 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 



