Cardinal and Ordinal Numbers

Since the meaning of a number like '2', and the truth of an expression like '2+2=4', seem completely independent of personal or empirical experience, Mathematics has long been considered to exist in an abstract realm. Accordingly, any debate over the Ontological status of Mathematical entities has concerned just whether that abstract realm is the infrastructure of reality itself, as per, e. g., Pythagoras, or is in the Intellect, as per, e. g., Russell. In either case counting and measuring are considered applied modes of Mathematics, and, likewise, Ordinal Numbers are construed as derived from Cardinals. But, 2+2=4 is easily refuted by 2 apples + 2 oranges=? In other words, presumably 'pure' Mathematical formulas implicitly quantify homogenous items, e. g. '2+2=4' is actually '(x)(2x+2x=4x)'. So, Numbers cannot escape being quantities, which means that they originate in Counting. Likewise, Ordinal Numbers are original, and Cardinals derivative, which is reinforced by the consideration that the latter lack an ordering principle, i. e. a principle that places e. g. the distinct entity 1 before, not after, the distinct entity 2. In other words, Counting is not '1, 2, 3, . . .' but 'first, second, third, . . .,' from which the former are abstracted. Russell's mistake is to believe that his 'Successor Function' generates the Cardinals, whereas it is the Ordinals that are successive. That the Successor Function is recursive, i. e. is applied, in turn, to each new successor that is generated, shows that Counting is cumulative, and, hence, is an Evolvemental process. Or, to put it more familiarly, Counting involves the introduction of a new item, and its subsequent integration into the given group, which is the Evolvemental process of Diversification without loss of Unity.