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SCIENCE. 



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



mere sport. On the contrary, he reports 

 a significant order of truth. As a fact, 

 the ideal systems of the pure mathematician 

 are customarily defined with an obvious, 

 even though often highly abstract and re- 

 mote, relation to the structure of our ordi- 

 nary empirical world. Thus the various 

 algebras which have been actually devel- 

 oped have, in the main, definite relations to 

 the structure of the space world of our 

 physical experience. The different systems 

 of ideal geometry, even in all their ideality, 

 still cluster, so to speak, about the sugges- 

 tions which our daily experience of space 

 and of matter give us. Yet I suppose that 

 no mathematician would be disposed, at the 

 present time, to accept any brief definition 

 of the degree of closeness or remoteness of 

 relation to ordinary experience which shall 

 serve to distinguish a trivial from a gen- 

 uinely significant branch of mathematical 

 theory. In general a mathematician who 

 is devoted to the theory of functions, or to 

 group theory, appears to spend little time 

 in attempting to show why the develop- 

 ment of the consequences of his postulates 

 is a significant enterprise. The concrete 

 mathematical interest of his inquiry sus- 

 tains him in his labors, and wins for him 

 the sympathy of his fellows. To the ques- 

 tions, 'Why consider the ideal structure of 

 just this system of object at all?' 'Why 

 study various sorts of numbers, or the prop- 

 erties of functions, or of groups, or the 

 system of points in projective geometry?' 

 — the pure mathematician in general, cares 

 to reply only, that the topic of his special 

 investigation appears to him to possess suffi- 

 cient mathematical interest. The freedom 

 of his science thus justifies his enterprise. 

 Yet, as I just pointed out, this freedom is 

 never mere caprice. This ideal interest is 

 not without a general relation to the con- 

 cerns even of common sense. In brief, as 

 it seems at once fair to say, the pure mathe- 

 matician is working under the influence of 



more or less clearly conscious philosophical 

 motives. He does not usually attempt to 

 define what distinguishes a significant from 

 a trivial system of postulates, or what con- 

 stitutes a problem worth attacking from the 

 point of view of pure mathematics. But 

 he practically recognizes such a distinction 

 between the trivial and the significant re- 

 gions of the world of ideal truth, and since 

 philosophy is concerned with the signifi- 

 cance of ideas, this recognition brings the 

 mathematician near in spirit to the philos- 

 opher. 



Such, then, is the position of the pure 

 mathematician. What, by way of contrast, 

 is that of the philosopher ? We may reply 

 that !o state the formal consequences of 

 exact assumptions is one thing; to reflect 

 upon the mutual relations, and the whole 

 sig-nificance of such assumptions, does in- 

 deed involve other interests; and these 

 other interests are the ones which directly 

 carry us over to the realm of philosophy. 

 If the theory of numbers belongs to pure 

 mathematics, the study of the place of the 

 number concept in the system of human 

 ideas belongs to philosophy. Like the 

 mathematician, the philosopher deals di- 

 rectly with a realm of ideal truth. But to 

 unify our knowledge, to comprehend its. 

 sources, its meaning, and its relations to the 

 whole of human life, these aims constitute 

 the proper goal of the philosopher. In 

 order, however, to accomplish his aims, the 

 philosopher must, indeed, take account of 

 the results of the special physical science; 

 but he must also turn from the world of 

 outer phenomena to an ideal world. For 

 the unity of things is never, for us mortals, 

 anything that we find given in our experi- 

 ence. You can not see the unity of knowl- 

 edge; you can not describe it as a phe- 

 nomenon. It is for us now, an ideal. And 

 precisely so, the meaning of things, the 

 relation of knowledge to life, the signifi- 

 cance of our ideals, their bearing upon one- 



