the thinking processes concerned, the number series itself is endless; the results of addition and multiplication, the sums and products of the various numbers, are not only endless, but capable of endless combinations; and, in general, the properties of numbers are themselves infinitely infinite in number. But in this case the mathematician does not mourn over the “endless fission” to which the number concepts are indeed due, but he regards the numbers as a storehouse of positive and often very beautiful novelties, which his science studies for their intrinsic interest.
If mathematical science thus begins, in the simplest construction, with the outcome of a recurrent process, it is no wonder that the later development of the science, as exemplified by the theories of negative and of fractional numbers, of irrational and of complex numbers, of infinite series and of infinite products, and of all that, in Analysis and in the Theory of Functions, depends upon these more elementary theories, is everywhere full of conceptions and methods that result from observing what happens when an operation of thought is recurrent, or is such as to reinstate, in its expressions, the occasion for new expressions. Without such recurrence, and without such infinite processes, mathematical science would be reduced to a very minute fraction of its present range and importance.
But we are here primarily concerned with the metaphysical aspect of the recurrent processes of thought. Important as are the countless mathematical instances of our type of operations, we must so deal with their general theory as to be able to identify the results of recurrent thinking whenever they occur, whether in mathematics or in other regions of our reflection.
I propose here, then, first to illustrate, and then to discuss theoretically, the nature and ideal outcome of any recurrent operation of thought, and to develope, in this connection, what one may call the positive nature of the concept of Infinite
