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 quantitative transformation that mathematics and the mathematical sciences begin. We may ask, then, why should there be any other than mathematical 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 quantitative; 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 knowledge in connection with a fickle world like that of qualitative change. It is here that reflection finds its problem, and noticing that the tendency 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 _b_ 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 a basis that is constituted by a _second act of reflection_; one that translates our world into a system of phenomena causally inter-related and connected with their underlying grounds. [Footnote 1: I do not raise the question of qualitative mathematics at all. It is clear that the first mathematical reflection will be quantitative.] [Footnote 2: By natural causation I mean such a relationship between _a_ and _b_ in a phenomenal system as enables _a_ through its connection with its ground to determine _b_.]