XVIII
From this grand intuition of symbolic space-worlds came the last and conclusive creation of Western mathematic—the expansion and subtilizing of the function theory in that of groups. Groups are aggregates or sets of homogeneous mathematical images—e.g., the totality of all differential equations of a certain type—which in structure and ordering are analogous to the Dedekind number-bodies. Here are worlds, we feel, of perfectly new numbers, which are nevertheless not utterly sense-transcendent for the inner eye of the adept; and the problem now is to discover in those vast abstract form-systems certain elements which, relatively to a particular group of operations (viz., of transformations of the system), remain unaffected thereby, that is, possess invariance. In mathematical language, the problem, as stated generally by Klein, is—given an n-dimensional manifold (“space”) and a group of transformations, it is required to examine the forms belonging to the manifold in respect of such properties as are not altered by transformation of the group.
And with this culmination our Western mathematic, having exhausted every inward possibility and fulfilled its destiny as the copy and purest expression of the idea of the Faustian soul, closes its development in the same way as the mathematic of the Classical Culture concluded in the third century. Both those sciences (the only ones of which the organic structure can even to-day be examined historically) arose out of a wholly new idea of number, in the one case Pythagoras’s, in the other Descartes’. Both, expanding in all beauty, reached their maturity one hundred years later; and both, after flourishing for three centuries, completed the structure of their ideas at the same moment as the Cultures to which they respectively belonged passed over into the phase of megalopolitan Civilization. The deep significance of this interdependence will be made clear in due course. It is enough for the moment that for us the time of the great mathematicians is past. Our tasks to-day are those of preserving, rounding off, refining, selection—in place of big dynamic creation, the same clever detail-work which characterized the Alexandrian mathematic of late Hellenism.
A historical paradigm will make this clearer.
column 1
column two
CHAPTER III
THE PROBLEM OF WORLD-HISTORY
I
PHYSIOGNOMIC AND SYSTEMATIC