THE SCIENCES OF THE IDEAL 161 



mathematics, so far as it has yet been erected, actually rests upon 

 a very few fundamental concepts and postulates, however you may 

 formulate them. What was not observed, however, by the earlier, 

 and especially by the philosophical, students of the categories, is 

 the form which these postulates tend to assume when they are 

 rigidly analyzed. 



This form depends upon the precise definition and classification 

 of certain types of relations. The whole of geometry, for instance, 

 including metrical geometry, can be developed from a set of postu- 

 lates which demand the existence of points that stand in certain 

 ordinal relationships. The ordinal relationships can be reduced, 

 according as the series of points considered is open or closed, either 

 to the well-known relationship in which three points stand when 

 one is between the other two upon a right line, or else to the ordinal 

 relationship in which four points stand when they are separated by 

 pairs; and these two ordinal relationships, by means of various log- 

 ical devices, can be regarded as variations of a single fundamental 

 form. Cayley and Klein founded the logical theory of geometry here 

 in question. Russell, and in another way Dr. Veblen, have given 

 it its most recent expressions. In the same way, the theory of whole 

 numbers can be reduced to sets of principles which demand the exist- 

 ence of certain ideal objects in certain simple ordinal relations. Dede- 

 kind and Peano have worked out such ordinal theories of the num- 

 ber concept. In another development of the theory of the cardinal 

 whole numbers, which Russell and Whitehead have worked out, 

 ordinal concepts are introduced only secondarily, and the theory 

 depends upon the fundamental relation of the equivalence or non- 

 equivalence of collections of objects. But here also a certain simple 

 type of relation determines the definitions and the development of 

 the whole theory. 



Two results follow from such a fashion of logically analyzing the 

 first principles of mathematical science. In the first place, as just 

 pointed out, we learn how few and simple are the conceptions and pos- 

 tulates upon which the actual edifice of exact science rests. Pure 

 mathematics, we have said, is free to assume what it chooses. Yet 

 the assumptions whose presence as the foundation principles of the 

 actually existent pure mathematics an exhaustive examination thus 

 reveals, show by their fewness that the ideal freedom of the mathe- 

 matician to assume and to construct what he pleases, is indeed, in 

 practice, a very decidedly limited freedom. The limitation is, as we 

 have already seen, a limitation which has to do with the essential 

 significance of the fundamental concepts in question. And so the 

 result of this analysis of the bases of the actually developed and 

 significant branches of mathematics, constitutes a sort of empirical 

 revelation of what categories the exact sciences have practically 



