Appearance or in Reality. And here, then, the relation of Unity and Variety is clear to us.
Our generalization, however, of the process upon which Mr. Bradley insists, enables us to make more fruitful and positive our result. There are recurrent operations of thought. Whenever they act, they imply, upon their face, endless processes. Do such processes inevitably lead us to results wholly vain and negative? Is the union of One and Many which they make explicit an insignificant union? Or, on the other hand, is this union typical of the general constitution of Reality?
The first answer is that, at all events in the special science of mathematics, processes of this type are familiar, and lie at the basis of highly and very positively significant researches. If we merely name a few such instances of endless processes, we shall see that iterative thinking, if once made an ideal, — a method of procedure, — and not merely dreaded as a failure to reach finality, becomes a very important part of the life of the exact sciences, and developes results which have a very significant grade of Reality.
The classic instance of the recurrent or iterative operations of thought is furnished, in elementary mathematics, by the Number Series. A recurrent operation first developes the terms of this series; and thereby makes the counting of external objects, and all that, in our human science, follows therefrom, possible. A secondary recurrent operation, based upon the primary operation, appears in the laws governing the process called the “Addition” of whole numbers. A tertiary and once more recurrent operation appears in the laws governing Multiplication.[1] In consequence of this recurrent nature of
- ↑ The precise sense in which the Number Series itself is the outcome of a recurrent operation of thought will be explained, in general accord with Dedekind’s theory, further on. Addition and Multiplication, in any particular instance, as in the adding or in the multiplying of 7 and 5, are of course operations terminated by the finding of the particular sum or product, and in so far they are finite and non-recurrent. But the laws of Addition and Multiplication (e.g., the Associative law), and the relation of both these operations to one another and to the number system, are dependent, in part, upon the fact that the result of every addition or multiplication of whole numbers is itself a whole number, uniquely determined, and, as a number, capable of entering into the formation of new sums and products.
