two characters.[1] It may be that which a given process so approaches that we ourselves are able to get and to remain near at will to, — that is, less than any predesignated distance from, — the limit, although the process in question, by itself, never reaches the limit. So we can get as near as we choose to 2, by adding terms of the series 1 + ½ + ¼, etc. Or, again, in the second place, the limit may be defined as that which, never attained by the process in question, is demonstrably a finality that occupies, in order, the first place immediately beyond the whole series of incomplete stages which the endless process in question defines. Thus, 2 is the least number that lies beyond, or that is greater than all possible fractions, of the form 1 + ½, 1 + ½ + ¼, 1 + ½ + ¼ + ⅛, etc. Usually, in mathematics, both senses of limit are combined (as they are in the example just used). But not so in the case here before us. Being is not an object that we men come near at will to finally observing, so that while we never get it wholly present in our internal meanings, we can come as near as we like to telling all that it is. But the Real, as our judgments and empirical investigations seek it, is that determinate object which all our ideas and experiences try to decide upon, and to bring within the range of our internal meanings; while, by the very nature of our fragmentary hypotheses and of our particular experiences, it always lies Beyond.
Yet if we could reach that limit of determination which
- ↑ See Georg Cantor, in the Zeitschrift f. Philosophie und Philosophische Kritik, Bd. 91, p. 110. The finite limit of a “convergent series” has both characters. But the “determinate infinite,” viewed as the limit of the whole-number-series, has only the latter of the two characters.
