fuse together in this one part, A’. If the map of England, before discussed, be an inexact and summary map, such as we actually always make, it need contain no part that visibly, or exactly, presents the place or the form of the map itself, as a part of the surface of England. But the Kette is constructed in such wise that the part is in exact correspondence to the whole when, as in Dedekind’s definition of the Infinite, the correspondence is ähnlich, so that any different elements in the object have different elements corresponding to them in the image, while every element has its own uniquely determined corresponding image. It will be observed that in case of inexact or dissimilar self-representation, we have a failure or external limitation of our self-representative purpose. Only exact self-representation is free from such external interference.
Yet even an exact self-correspondence can be brought to pass, within a system, by making it correspond not to a true portion of itself, but, member for member, to the whole of itself. Thus the system ɑbcd, consisting of the already distinguishable elements ɑ, b, etc., may be put in exact correspondence to itself by making b correspond to ɑ, and so represent ɑ, while, in similar fashion, c corresponds to b, d to c, and, finally, ɑ itself to d. In this case the system is, in a particular way, “transformed” into the image bcdɑ, in such wise as to be exactly self-representative. But the system ɑbcd might also be represented, element for element, by the system cbdɑ, where the order of the elements was again different, but where c now corresponded to the original ɑ, b to itself, d to c, and ɑ to d. Such “substitutions,” as they are called, give rise to self-representative systems of a type different from the one that we have heretofore had in mind. But in the general mathematical theory of “transformations,” and of “groups of operations,” self-representation of such types plays a great part. And in cases of such a type, to be sure, exact self-representation, and finitude of the system, are capable of perfect combination Such self-representations need not be endless, and can be exact. There are many remarkable instances known to
