[181]. The want of perspective in children’s drawings is emphatically not perceptible to the children themselves.
[182]. His idea that the a priori-ness of space was proved by and through the unconditional validity of simple geometrical facts rests, as we have already remarked, on the all-too-popular notion that mathematics are either geometry or arithmetic. Now, even in Kant’s time the mathematic of the West had got far beyond this naïve scheme, which was a mere imitation of the Classical. Modern geometry bases itself not on space but on multiply-infinite number-manifolds—amongst which the three-dimensional is simply the undistinguished special case—and within these groups investigates functional formations with reference to their structure; that is, there is no longer any contact or even possibility of contact between any possible kind of sense-perception and mathematical facts in the domain of such extensions as these, and yet the demonstrability of the latter is in no wise impaired thereby. Mathematics, then, are independent of the perceived, and the question now is, how much of this famous demonstrability of the forms of perception is left when the artificiality of juxtaposing both in a supposedly single process of experience has been recognized.
[183]. It is true that a geometrical theorem may be proved, or rather demonstrated, by means of a drawing. But the theorem is differently constituted in every kind of geometry, and that being so, the drawing ceases to be a proof of anything whatever.
[184]. So much so that Gauss said nothing about his discovery until almost the end of his life for tear of “the clamour of the Bœotians.”
[185]. The distinction of right and left (see p. [169]) is only conceivable as the outcome of this directedness in the dispositions of the body. “In front” has no meaning whatever for the body of a plant.
[186]. It may not be out of place here to refer to the enormous importance attached in savage society to initiation-rites at adolescence.—Tr.
[187]. Either in Greek or in Latin, τόπος (= locus) means spot, locality, and also social position; χώρα (= spatium) means space-between, distance, rank, and also ground and soil (e.g., τὰ ἐκ τῆς χώρας, produce); τὸ κένον (vacuum) means quite unequivocally a hollow body, and the stress is emphatically on the envelope. The literature of the Roman Imperial Age, which attempted to render the Magian world-feeling through Classical words, was reduced to such clumsy versions as ὁρατὸς τόπος (sensible world) or spatium inane (“endless space,” but also “wide surface”—the root of the word “spatium” means to swell or grow fat). In the true Classical literature, the idea not being there, there was no necessity for a word to describe it.
[188]. 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.