THE SCIENCES OF THE IDEAL 155 



enterprise. The concrete mathematical interest of his inquiry sustains 

 him in his labors, and wins for him the sympathy of his fellows. To 

 the questions, " Why consider the ideal structure of just this system 

 of object at all? " " Why study various sorts of numbers, or the 

 properties 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 sufficient 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 concerns even of common sense. In brief, as 

 it seems at once fair to say, the pure mathematician is working under 

 the influence of more or less clearly conscious philosophical motives. 

 He does not usually attempt to define what distinguishes a signi- 

 ficant from a trivial system of postulates, or what constitutes a pro- 

 blem worth attacking from the point of view of pure mathematics. 

 But he practically recognizes such a distinction between the trivial 

 and the significant regions of the world of ideal truth, and since 

 philosophy is concerned with the significance of ideas, this recogni- 

 tion brings the mathematician near in spirit to the philosopher. 



Such, then, is the position of the pure mathematician. What, by 

 way of contrast, is that of the philosopher? We may reply that to 

 state the formal consequences of exact assumptions is one thing; to 

 reflect upon the mutual relations, and the whole significance of such 

 assumptions, does indeed 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 directly with a realm of ideal truth. But to unify 

 our knowledge, to comprehend its sources, its meaning, and its re- 

 lations 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 experience. You 

 cannot see the unity of knowledge; you cannot describe it as a phe- 

 nomenon. It is for us now, an ideal. And precisely so, the mean- 

 ing of things, the relation of knowledge to life, the significance of 

 our ideals, their bearing upon one another --these are never, for us 

 men, phenomenally present data. Hence the philosopher, however 

 much he ought, as indeed he ought, to take account of phenomena, 

 and of the results of the special physical sciences, is quite as deeply 

 interested in his own way, as the mathematician is interested in his 



