our thought may chance to conceive as possible, there correspond just as many final facts for an Absolute Experience; nor yet can we, on the other hand, exclude from concrete presentation, as final facts, such wholes as include an infinite series, merely because, for us, if we do not take due account of mathematical truth, the series seems to involve the empty repetition of “one more” and “one more.” For, as Poincaré has so finely pointed out, in the article before cited, it is precisely the “reasoning by recurrence” which is, in mathematics, the endless source of new results. Hereby, in the combination of his previous results for the sake of new insight, the mathematician is preserved from mere “identities,” and gets novelties. The “reasoning by recurrence,” however, is that form of reasoning whereby one shows that if a given truth holds in n cases, it holds for the n + 1st case, and so for all cases. Such processes of passing to " one more " instance of a given type, are processes not of barren repetition, but of genuine progress to higher stages of knowledge.
Precisely so it is, too, if one takes account of that other aspect of ordered series which it has been one principal purpose of this paper to emphasize. The numbers have interested us, not from any Pythagorean bias, but because their Order is the expression, not only of a profoundly significant aspect of all law in the world, but of the very essence of Selfhood, when formally viewed. Now reflective selfhood, taken merely as the abstract series, I know, and I know that I know, etc., appears to be a vain repetition of the same over and over. But this it appears merely if you neglect the concrete content which every new reflection, when taken in synthesis with previous reflections, inevitably implies in case of every living subject-matter. A life that knows not itself differs from the same life conscious of itself, by lacking precisely the feature that distinguishes rational morality alike from innocence and from brutish naïveté. A knowledge that is self-possessed differs from an unreflective type of consciousness by having all the marks that separate insight from blind faith.
“Thus we see,” says Spinoza, in a most critical passage of
