maps, of the type just described. The whole infinite series, possessing no last member, would be asserted as a fact of existence. I need not observe that Mr. Bradley would at once reject such an assertion as a self-contradiction. It would be a typical instance of the sort of endlessness of structure that makes him reject Space, Time, and the rest, as mere Appearance. But I am still interested in pointing out that whether we continued faithful to our supposed revelation, or, upon second thought, followed Mr. Bradley in rejecting it as impossible, our faith, or our doubt, would equally involve seeing that the one plan of mapping in question necessarily implies just this infinite variety of internal constitution. We should, moreover, see how and why the one and the infinitely many are here, at least within thought’s realm, conceptually linked. Our map and England, taken as mere physical existences, would indeed belong to that realm of “bare external conjunctions.” Yet the one thing not externally given, but internally self-evident, would be that the one plan or purpose in question, namely, the plan fulfilled by the perfect map of England, drawn within the limits of England, and upon a part of its surface, would, if really expressed, involve, in its necessary structure, the series of maps within maps such that no one of the maps was the last in the series.
This way of viewing the case suggests that, as a mere matter
of definition, we are not obliged to deal solely with processes
of construction as successive, in order to define endless series.
A recurrent operation of thought can be characterized as one
that, if once finally expressed, would involve, in the region
where it had received expression, an infinite variety of serially
arranged facts, corresponding to the purpose in question. This
consideration leads us back from our trivial illustration to the
realm of general theory.
II. Definition of a Type of Self-Representative Systems
Let there be, then, any recurrent operation of thought, or any meaning in mind whose expression, if attempted, involves such a recurrent operation. That is, let there be
