166 NORMATIVE SCIENCE 



classes, of statements, or of decisions, are marvelously simple. All 

 the relations present may be viewed as variations of the mere con- 

 ception of opposition as distinct from non-opposition. 



All this holds, of course, so far, merely for the logician's world of 

 classes or of decisions. There, at least, all serial order can actually 

 be derived from wholly symmetrical relations. But Kempe now 

 very beautifully shows (and here lies his great and original contri- 

 bution to our topic) - - he shows, I say, that the ordinal relations 

 of geometry, as well as of the number-system, can all be regarded 

 as indistinguishable from mere variations of those relations which, 

 in pure logic, one finds to be the symmetrical relations obtaining within 

 pairs or triads of classes or of statements. The formal identity of the 

 geometrical relation called "between" with a purely logical relation 

 which one can define as existing or as not existing amongst the mem- 

 bers of a given triad of logical classes, or of logical statements, is 

 shown by Kempe in a fashion that I cannot here attempt to expound. 

 But Kempe's result thus enables one, as I believe, to simplify the 

 theory of relations far beyond the point which Russell in his brilliant 

 book has reached. For Kempe's triadic relation in question can be 

 stated, in what he calls its obverse form, in perfectly symmetrical 

 terms. And he proves very exactly that the resulting logical rela- 

 tion is precisely identical, in all its properties, with the fundamental 

 ordinal relation of geometry. 



Thus the order-systems of geometry and analysis appear simply 

 as special cases of the more general order-system of pure logic. The 

 whole, both of analysis and of geometry, can be regarded as a de- 

 scription of certain selected groups of entities, which are chosen, 

 according to special rules, from a single ideal world. This general 

 and inclusive ideal world consists simply of all the objects which can 

 stand to one another in those symmetrical relations wherein the pure lo- 

 gician finds various statements, or various decisions inevitably standing. 

 "Let me," says in substance Kempe, "choose from the logician's 

 ideal world of classes or decisions, what entities I will; and I will 

 show you a collection of objects that are in their relational structure, 

 precisely identical with the points of a geometer's space of n dimen- 

 sions." In other words, all of the geometer's figures and relations can 

 be precisely pictured by the relational structure of a selected system 

 of classes or of statements, whose relations are wholly and explicitly 

 logical relations, such as opposition, and whose relations may all 

 be regarded, accordingly, as reducible to a single type of purely 

 symmetrical relation. 



Thus, for all exact science, and not merely for the logician's special 

 realm, the contrast between symmetrical and unsymmetrical rela- 

 tions proves to be, after all, superficial and derived. The purely 

 logical categories, such as opposition, and such as hold within the 



