“dichotomy,” i.e. of an exhaustive and twofold division of the Universe of your Discourse, as at any time you conceive its Reality to be opened to your ideal inspection.
This general situation of our thought in all those branches of inquiry where, as very often in mathematics, we deal with universal truth, and reason out results about Being, while still viewing Reality as Another than thought, is a situation that stimulates us to manifold inquiries. In the first place, as you at once see, the limitations of all our merely abstract and universal reasoning about the world, when taken as a world of external meanings, are, at a stroke, laid bare by virtue of these very considerations. For by mere reasoning, in these universal terms, we never directly and determinately characterize the Being of things as it finally is. We at best, and even if we are quite sure of our universal truths, tell what external Reality is not, and add that, of the remaining abstractly possible and definable alternatives, it is doubtless determinately some one, and no other. But, apart from any scepticism, justified or not, regarding the validity of our universal judgments themselves, they at best carry us a certain way only in an undertaking that seems essentially endless, and, in fact, worse than endless. And that is the undertaking of exhausting all the possible alternatives, and so of making the finally valid possibility, that can alone remain, into something absolutely determinate. And where the sole principle is that Omnis Determinatio est Negatio, this task is indeed not only endless, but hopeless.
This, in fact, is why mathematical science, especially
