objects. Feelings, deeds, persons, lives, stellar spectra, chemical elements, processes of evolution, types of doctrine, modes of conduct, aesthetic values, in brief, beings of all grades, invite serial treatment as soon as they are compared. Various series, already conceived, can be combined in the most varied ways, so as to give us systems of objects that no longer can be arranged in any single serial order. We thus get Systems whose series are interwoven and interrelated in most manifold fashions. Mr. Kempe’s example of the classes in a single “Universe of Discourse,” while it by no means exhausts the complexity of the relations that are definable through conceiving various systems of series connected together, is so complex that the space of the geometer, we have seen, corresponds to one only of the special forms definable within that system.[1]
The conception of systems of facts such that any two members of the system may be viewed as linked by series of intermediaries, is thus indeed capable of application in
- ↑ Mr. Kempe’s system illustrates, amongst other things, very definitely the fact that the generalized conception of a series of intermediaries, linking two given objects, ɑ and b, is an infinitely variable concept. If two objects can be linked by one series, they can, in general, be linked by an infinity of other series of intermediaries. Thus all the classes in any Universe of Discourse are, by Mr. Kempe’s definition, contained between any class ɑ and the negative of that class, not-ɑ. Again between any class ɑ and a class i included within ɑ, you can establish an infinite number of different series of intermediate classes. It is thus also in space, if you consider the various curves by which two points can be connected. But the spatial relation of points on a line is inadequate to express all the possibilities of the generalized relation of between. In Mr. Kempe’s system the same object x can be defined as between ɑ and b, b and c, and c and ɑ, and can yet be different from all three.
