fhe thus logicized rendering of any other discipline Can be shown in ,ma logo us fashion to vield the sort of discrepancy observed ust now for geometry, provided onlv that the postulates of the discipline admit, like those of geometry, of alternative applications; and such multiple applicability to he expected of a nv postulate set: Definition of mathematical notions on the basis of logical ones thus a more a rduous undertaking than would appear from a consideration solely of the truths and falsehoods of pure mathematics.