158 NORMATIVE SCIENCE 



can be distinctly answered in the affirmative. The present age is one 

 of a rapid advance in the actual unification of the fields of investi- 

 gation which are included within the scope of this present division. 

 What little time remains to me must be devoted to indicating, as 

 well as I can, in what sense this is true. I shall have still to deal 

 in very broad generalities. I shall try to make these generalities 

 definite enough to be not wholly unfruitful. 



We have already emphasized one question which may be said to 

 interest, in a very direct way, both the mathematician and the 

 philosopher. The ideal postulates, whose consequences mathemat- 

 ical science undertakes to develop, must be, we have said, significant 

 postulates, involving ideas whose exact definition and exposition 

 repay the labor of scientific scrutiny. Number, space, continuity, 

 functional correspondence or dependence, group-structure these 

 are examples of such significant ideas; the postulates or ideal 

 assumptions upon which the theory of such ideas depends are signi- 

 ficant postulates, and are not the mere conventions of an arbitrary 

 game. But now what constitutes the significance of an idea, or 

 of an abstract mathematical theory? What gives an idea a worthy 

 place in the whole scheme of human ideas? Is it the possibility of 

 finding a physical application for a mathematical theory which 

 for us decides what is the value of the theory? No, the theory of 

 functions, the theory of numbers, group theory, have a significance 

 which no mathematician would consent to measure in terms of the 

 present applicability or non-applicability of these theories in physical 

 science? In vain, then, does one attempt to use the test of applied 

 mathematics as the main criticism of the value of a theory of pure 

 mathematics. The value of an idea, for the sciences which con- 

 stitute our division, is dependent upon the place which this idea 

 occupies in the whole organized scheme or system of human ideas. 

 The idea of number, for instance, familiar as its applications are, 

 does not derive its main value from the fact that eggs and dollars 

 and star-clusters can be counted, but rather from the fact that the 

 idea of numbers has those relations to other fundamental ideas 

 which recent logical theory has made prominent relations, for 

 instance, to the concept of order, to the theory of classes or collec- 

 tions of objects viewed in general, and to the metaphysical concept . 

 of the self. Relations of this sort, which the discussions of the num- 

 ber concept by Dedekind, Cantor, Peano, and Russell have recently 

 brought to light such relations, I say, constitute what truly justi- 

 fied Gauss in calling the theory of numbers a "divine science." .As 

 against such deeper relations, the countless applications of the 

 number concept in ordinary life, and in science, are, from the truly 

 philosophical point of view, of comparatively small moment. What 

 we want, in the work of our division of the sciences, is to bring to 



