THE SCIENCES OF THE IDEAL 153 



to which he conforms are eternal. But his method is that of an inner 

 considerateness rather than of a curiosity about external phenomena. 

 His objective world is at the same time an essentially ideal world, 

 and the eternal verity in whose light he seeks to live has, throughout 

 his undertakings, a peculiarly intimate relation to the purposes of 

 his own constructive will. 



One may then sum up the difference of attitude which is here in 

 question by saying that, while the student of outer nature is ex- 

 plicitly conforming his plans of action, his ideas, his ideals, to an 

 order of truth which he takes to be foreign to himself the student 

 of the other sort of truth, here especially in question, is attempting 

 to understand his own plans of action, that is, to develop his ideas, 

 or to define his ideals, or else to do both these things. 



Now it is not hard to see that this search for some sort of ideal 

 truth is indeed characteristic of every one of the investigations 

 which have been grouped together in our division of the normative 

 sciences. Pure mathematics shares in common with philosophy 

 this type of scientific interest in ideal, as distinct from physical or 

 phenomenal truth. There is, to be sure, a marked contrast between 

 the ways in which the mathematician and the philosopher approach, 

 select, and elaborate their respective sorts of problems. But there 

 is also a close relation between the two types of investigation in 

 question. Let us next consider both the contrast and the analogy in 

 some of their other most general features. 



Pure mathematics is concerned with the investigation of the logical 

 consequences of certain exactly stateable postulates or hypotheses 

 such, for instance, as the postulates upon which arithmetic and analy- 

 sis are founded, or such as the postulates that lie at the basis of any 

 type of geometry. For the pure mathematician, the truth of these 

 hypotheses or postulates depends, not upon the fact that physical 

 nature contains phenomena answering to the postulates, but solely 

 upon the fact that the mathematician is able, with rational consist- 

 ency, to state these assumed first principles, and to develop their 

 consequences. Dedekind, in his famous essay, " Was Sind und Was 

 Sollen die Zahlen," called the whole numbers " freie Schopfungen des 

 Menschlichen Geistes; " and, in fact, we need not enter into any dis- 

 cussion of the psychology of our number concept in order to be able 

 to assert that, however we men first came by our conception of the 

 whole numbers, for the mathematician the theory of numerical truth 

 must appear simply as the logical development of the consequences 

 of a few fundamental first principles, such as those which Dedekind 

 himself, or Peano, or other recent writers upon this topic, have, in 

 various forms, stated. A similar formal freedom marks the develop- 

 ment of any other theory in the realm qf pure mathematics. Pure 

 geometry, from the modern point of view, is neither a doctrine forced 



