454 



SCIENCE. 



[N. S. Vol. XX. No. 510. 



seek to reduce philosophy to the mere de- 

 velopment, in mathematical form, of the 

 consequences of certain arbitrary hypoth- 

 eses. He will distinguish between a re- 

 flection upon the unity of tlie system of 

 truth and an abstract development of this 

 or that selected aspect of the system. But 

 he will see more and more that, in so far 

 as he undertakes to be exact, he must aim 

 to become, in his own way, and with due 

 regard to his own purposes, mathematical; 

 and thus the union of mathematical and 

 philosophical inquiries, in the future, will 

 tend to become closer and closer. 

 II. 

 So far, then, I have dwelt upon extremely 

 general considerations relating to the unity 

 and the contrast of mathematical and 

 philosophical inquiries. I can well con- 

 ceive, however, that the individual worker 

 in any one of the numerous branches of in- 

 vestigation Avhich are represented by the 

 body of students Avhom I am privileged to 

 address, may at this point mentally inter- 

 pose the objection that all these considera- 

 tions are, indeed, far too general to be of 

 practical interest to any of us. Of course, 

 all we who study these so-called normative 

 sciences are, indeed, interested in ideas, for 

 their own sakes— in ideas so distinct from, 

 although of course also somehow related to, 

 phenomena. Of course some of us are 

 rather devoted to the development of the 

 consequences of exactly stated ideal hypoth- 

 eses, and others to reflecting as we can upon 

 what certain ideas and ideals are good for, 

 and upon what the unity is of all ideas and 

 ideals. Of course if we are wise enough 

 to do so, we have much to learn from one 

 another. But, you will say, the assertion 

 of all these things is a commonplace. The 

 expression of the desire for further mutual 

 cooperation is a pious wish. You will in- 

 sist upon asking further : " Is there just 

 now any concrete instance in a modern type 

 of research Avhich furnishes results such as 



are of interest to all of us? Are we 

 actually doing any productive work in com- 

 mon? Are the philosophers contributing 

 anything to human knowledge which has a 

 genuine bearing upon the interests of 

 mathematical science? Are the mathe- 

 maticians contributing anything to phi- 

 losophy 1 ' ' 



These questions are perfectly fair. More- 

 over, as it happens, they can be distinctly 

 answered in the affirmative. The present 

 age is one of a rapid advance in the actual 

 unification of the fields of investigation 

 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 gen- 

 eralities. I shall try to make these gen- 

 eralities definite enough to be not wholly 

 unfruitful. 



We have already emphasized one ques- 

 tion which may be said to interest, in a 

 very direct way, both the mathematician 

 and the philosopher. The ideal postulates, 

 whose consequences mathematical 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 correspond- 

 ence 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 sig- 

 nificant postulates, and are not the mere 

 conventions of an arbitrary game. But 

 now what constitutes the sig-nificance 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 theoi-y? No, the theory of functions, 

 the theory of numbers, group theory, have 



