October 7, 1904.] 



SCIENCE. 



459 



which enables you in your reasonings about 

 these relations to skip middle terms, and 

 so to perform some operation of elimina- 

 tion, is the property which is meant when 

 one calls relations of this type transitive. 

 But, on the other hand, these two classes 

 of relations differ from each other in that 

 the leveling relations are, while the serial 

 relations are not, symmetrical or reciprocal. 

 Thus, if A is equal to B, B is equal to A. 

 But if X is greater than Y, then Y is not 

 greater than X, but less than X. So the 

 leveling relations are symmetrical transi- 

 tive relations. But the serial relations are 

 transitive relations which are not sym- 

 metrical. 



All this is now well known. It is no- 

 table, however, that nearly all the processes 

 of our exact sciences, as at present devel- 

 oped, can be said to be essentially such as 

 lead either to the placing of sets or classes 

 of objects on the same level, by means of 

 the use of symmetrical transitive relations, 

 or else to the arranging of objects in or- 

 derly rows or series, by means of the use 

 of transitive relations which are not sym- 

 metrical. This holds also of all the appli- 

 cations of the exact sciences. Whatever 

 else you do in science (or, for that matter, 

 in art), you always lead, in the end, either 

 to the arranging of objects, or of ideas, or 

 of acts, or of movements, in rows or series, 

 or else to the placing of objects or ideas of 

 some sort on the same level, by virtue of 

 some equivalence, or of some invariant 

 character. Thus numbers, functions, lines 

 in geometry, give you examples of serial 

 relations. Equations in mathematics are 

 classic instances of leveling relations. So, 

 of course, are invariants. Thus, again, 

 the whole modern theory of energy consists 

 of two parts, one of which has to do with 

 levels of energy, in so far as the quantity 

 of energy of a closed system remains in- 

 variant through all the transformations of 

 the system, while the other part has to do 



with the irreversible serial order of the 

 transformations of energy themselves, 

 which follow a set of unsymmetrical rela- 

 tions, in so far as energy tends to fall 

 from higher to lower levels of intensity 

 within the same system. 



The entire conceivable universe then, and 

 all of our present exact science, can be 

 viewed, if you choose, as a collection of 

 objects or of ideas that, whatever other 

 types of relations may exist, are at least 

 largely characterized either by the leveling 

 relations, or by the serial relations, or by 

 complexes of both sorts of relations. Here, 

 then, we are plainly dealing with very 

 fundamental categories. The 'between' 

 relations of geometry can of course be de- 

 fined, if you choose, in terms of transitive 

 relations that are not symmetrical. There 

 are, to be sure, some other relations pres- 

 ent in exact science, but the two types, the 

 serial and leveling relations, are especially 

 notable. 



So far the modern logicians have for 

 some time been in substantial agreement. 

 Russell's brilliant book is a development of 

 the logic of mathematics very largely in 

 terms of the two types of relations which, 

 in my own way, I have just characterized ; 

 although Russell gives due regard, of 

 course, to certain other types of relations. 



But hereupon the question arises, 'Are 

 these two types of relations what Russell 

 holds them to be, viz., ultimate and irre- 

 ducible logical facts, unanalyzable cate- 

 gories—mere data for the thinker? Or 

 can we reduce them still further, and thus 

 simplify yet again our view of the cate- 

 gories 1 



Here is where Kempe's generalization 

 begins to come into sight. These two cate- 

 gories, in at least one very fundamental 

 realm of exact thought, can be reduced to 

 one. There is, namely, a world of ideal 

 objects which especially interest the logi- 

 cian. It is the world of a totality of pas- 



