164 NORMATIVE SCIENCE 



orderly rows or series, by means of the use of transitive relations 

 which are not symmetrical. This holds also of all the applications 

 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 arrang- 

 ing 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 quan- 

 tity of energy of a closed system remains invariant through all the 

 transformations of the system, while the other part has to do with 

 the irreversible serial order of the transformations of energy them- 

 selves, which follow a set of unsymmetrical relations, 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 defined, if 

 you choose, in terms of transitive relations that are not symmet- 

 rical. There are, to be sure, some other relations present 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 substan- 

 tial agreement. Russell's brilliant book is a development of the 

 logic of mathematics very largely in terms of the two types of rela- 

 tions which, in my own way, I have just characterized; although 

 Russell gives due regard, of course, to certain other types of rela- 

 tions. 



But hereupon the question arises, "Are these two types of rela- 

 tions what Russell holds them to be, namely, ultimate and irre- 

 ducible logical facts, unanalyzable categories mere data for the 

 thinker? Or can we reduce them still further, and thus simplify 

 yet again our view of the categories? 



Here is where Kempe's generalization begins to come into sight. 

 These two categories, 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 logician. It is the 

 world of a totality of possible logical classes, or again, it is the ideal 



