Note 87 — Introductory material

Admitting, then, in Lotze's sense, the subjectivity of space, the above possibility does not seem so empty as he imagines. He discusses it briefly, however, in order to pass on to what he regards as the real meaning of Metageometry. In this he is guilty of a mathematical mistake, which causes much irrelevant reasoning. For he believes that Metageometry constructs its spaces out of straight lines and angles in all respects similar to Euclid's, whence he derives an easy victory in proving that these elements can lead only to the one space. In this he has been misled by the phraseology of non-Euclideans, as well as by Euclid's separation of definitions and axioms. For the fact is, of course, that straight lines are only fully defined when we add to the formal definition the axioms of the straight line and of parallels. Within Euclidean space, Euclid's definition suffices to distinguish the straight line from all other curves; the two axioms referred to are then absorbed into the definition of space. But apart from the restriction to Euclidean space, the definition has to be supplemented by the two axioms, in order to define completely the Euclidean straight line. Thus Lotze has misconceived the bearing of non-Euclidean constructions, and has simply missed the point in arguing as he does. The possibility contemplated by a non-Euclidean, if it fell under any of Lotze's cases, would fall under the second case discussed above.