October 7, 1904.] 



SCIENCE. 



455 



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 criti- 

 cism of the value of a theory of pure mathe- 

 matics. The value of an idea, for the 

 sciences which constitute our division, is 

 dependent upon the place which this idea 

 occupies in the whole organized scheme or 

 system of human ideas. The idea of num- 

 ber, for instance, familiar as its applica- 

 tions 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 collections of objects viewed in general, 

 and to the metaphysical concept of the 

 self. Relations of this sort, which the dis- 

 cussions of the number concept by Dede- 

 kind, Cantor, Peano and Russell have re- 

 cently brought to light — such relations, I 

 say, constitute what truly justitied 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 light the 

 unity of truth, either, as in mathematics, by 

 developing systems of truth which are sig- 

 nificant by virtue of their actual relations 

 to this unity, or, as in philosophy, by ex- 

 plicitly seeking the central idea about 

 which all the many ideas cluster. 



Now, an ancient and fundamental prob- 

 lem for the philosophers is that which has 

 been called the problem of the categories. 

 This problem of the categories is simply 



the more formal aspect of the whole philo- 

 sophical problem just defined. The philos- 

 opher aims to comprehend the unity of the 

 system of human ideas and ideals. Well, 

 then, what are the primal ideas? Upon 

 what group of concepts do the other 

 concepts of human science logically de- 

 pend? About what central interests is 

 the system of human ideals clustered? In 

 ancient thought Aristotle already ap- 

 proached this problem in one way. Kant, 

 in the eighteenth century, dealt with it in 

 another. We students of philosophy are 

 accustomed to regret what we call the ex- 

 cessive formalism of Kant, to lament that 

 Kant was so much the slave of his own 

 relatively superficial and accidental table 

 of categories, and that he made the treat- 

 ment of every sort of philosophical prob- 

 lem turn upon his own schematism. Yet 

 we can not doubt that Kant was right in 

 maintaining that philosophy needs, for the 

 successful development of every one of its 

 departments, a well-devised and substan- 

 tially complete system of categories. Our 

 objection to Kant's over-confidence in the 

 virtues of his own schematism is due to the 

 fact that we do not now accept his table 

 of categories as an adequate view of the 

 fundamental concepts. The efforts of 

 philosopher since Kant have been repeat- 

 edly devoted to the task of replacing his 

 scheme of categories by a more adequate 

 one. I am far from regarding these purely 

 philosophical efforts made since Kant as 

 fruitless, but they have remained, so far, 

 very incomplete, and they have been held 

 back from their due fulness of success by 

 the lack of a sufficiently careful survey and 

 analysis of the processes of thought as these 

 have come to be embodied in the living sci- 

 ences. Such concepts as number, quantity, 

 space, time, cause, continuity, have been 

 dealt with by the pure philosophers far 

 too siimmarily and superficially. A more 

 thoroughgoing analysis has been needed. 



