248 METAPHYSICS 



that results from considering it under the concepts and categories 

 of number. In short, to mathematical reflection of this simple sort, 

 the things of observation will resolve themselves into a plurality of 

 countable things, which the numbering reflection becoming explicit 

 in its ordinal and cardinal moments will translate into a system that 

 will be regarded as a whole made up of the sum of its parts. The very 

 first step, then, in the reflective transformation of things resolves 

 them into a dual system, the world conceived as a cardinal whole that 

 is made up of its ordinal parts, and exactly equal to them. This 

 mathematical conception is moreover purely quantitative; involving 

 the exact and stable equivalence of its parts or units and that of the 

 sum of the parts with the whole. Now it is with this purely quantita- 

 tive transformation that mathematics and the mathematical sciences 

 begin. We may ask, then, why should there be any other than mathe- 

 matical science, 1 and what ground can non-mathematical science point 

 to as substantiating its claims? I confess I can see no other final 

 reason than this, that mathematical science does not meet the whole 

 demand we feel obliged to make on our world. If mathematics were 

 asked to vindicate itself, it no doubt would do so by claiming that 

 things present quantitative aspects on which it founds its procedure. 

 In like manner non-mathematical, or, as we may call it, physical or 

 natural science, will seek to substantiate its claims by pointing to 

 certain ultra-quantitative or qualitative aspects of things. It is true 

 that, so far as things are merely numerable, they are purely quantita- 

 tive; but mathematics abstracts from the content and character of its 

 units and aggregates, which may and do change, so that a relation 

 of stable equivalence is not maintained among them. In fact, the 

 basis of these sciences is found in the tendency of things to be always 

 changing and becoming different from what they were before. The 

 problem of these sciences is how to ground a rational scheme of know- 

 ledge in connection with a fickle world like that of qualitative change. 

 It is here that reflection finds its problem, and noticing that the tend- 

 ency of this world of change is for a to pass into b and thus to lose 

 its own identity, the act of reflection that rationalizes the situation is 

 one that connects a and 6 by relating them to a common ground x of 

 which they stand as successive manifestations or symbols. X thus 

 supplies the thread of identity that binds the two changes a and b into 

 a relation to which the name causation may be applied. And just as 

 quantitative equivalence is the principle of relationship among the 

 parts of the simple mathematical world, so here in the world of the 

 dynamic or natural sciences, the principle of relation is natural 

 causation. 2 We find, then, that the non-mathematical sciences rest on 



1 I do not raise the question of qualitative mathematics at all. It is clear that 

 the first mathematical reflection will be quantitative. 



2 By natural causation I mean such a relationship between o and b in a phenom- 

 enal system as enables a through its connection with its ground to determine b. 



