Note 188 — Chapter V · MAKROKOSMOS

It has not hitherto been seen that this fact is implicit in Euclid’s famous parallel axiom (“through a point only one parallel to a straight line is possible”).

This was the only one of the Classical theorems which remained unproved, and as we know now, it is incapable of proof. But it was just that which made it into a dogma (as opposed to any experience) and therefore the metaphysical centre and main girder of that geometrical system. Everything else, axiom or postulate, is merely introductory or corollary to this. This one proposition is necessary and universally-valid for the Classical intellect, and yet not deducible. What does this signify?

It signifies that the statement is a symbol of the first rank. It contains the structure of Classical corporeality. It is just this proposition, theoretically the weakest link in the Classical geometry (objections began to be raised to it as early as Hellenistic times), that reveals its soul, and it was just this proposition, self-evident within the limits of routine experience, that the Faustian number-thinking, derived from incorporeal spatial distances, fastened upon as the centre of doubt. It is one of the deepest symbols of our being that we have opposed to the Euclidean geometry not one but several other geometries all of which for us are equally true and self-consistent. The specific tendency of the anti-Euclidean group of geometries—in which there may be no parallel or two parallels or several parallels to a line through a point—lies in the fact that by their very plurality the corporeal sense of extension, which Euclid canonized by his principle, is entirely got rid of; for what they reject is that which all corporeal postulates but all spatial denies. The question of which of the three Non-Euclidean geometries is the “correct” one (i.e., that which underlies actuality)—although Gauss himself gave it earnest consideration—is in respect of world-feeling entirely Classical and therefore it should not have been asked by a thinker of our sphere. Indeed it prevents us from seeing the true and deep meaning implicit in the plurality of these geometries. The specifically Western symbol resides not in the reality of one or of another, but in the true plurality of equally possible geometries. It is the group of space-structures—in the abundance of which the classical system is a mere particular case—that has dissolved the last residuum of the corporeal into the pure space-feeling.