an infinite process, and is both workable and useful; but if presented as an account of the way in which a completed all-embracing and perfectly harmonious experience of the whole of Reality is internally organised, it involves the completion of the infinite process, and is therefore self-contradictory and finally inadequate.[94] This reflection may serve to lead up to another which seems to take us into the heart of the matter. The researches upon which Prof. Royce’s defence of the relational scheme is based were in the first instance investigations into the significance of the number-series. As such they start with the conception of a system which is a whole of _parts_ external to one another[95] as the object of inquiry. Consequently, while such investigations are of the highest philosophical importance as bringing out the implications of this concept, they are only valid as an analysis of ultimate Reality, provided that the concept of whole and part is an adequate expression of the way in which the whole Reality is present in its constituents and they in it. But if, as we ourselves urged in a previous chapter, the conception of a whole of parts is entirely inadequate to express the intimate union between the absolute experience and finite experiences,[96] the proof that the indefinite process is logically implied in the relation of whole and part does not show it to belong to the structure of ultimate Reality. Rather, we should be inclined to urge, the fact that the relational scheme leads to the indefinite process proves that the conception of whole and part upon which it is based does not truly represent the mode of union between a completed experience and its components. And therefore the attempt to interpret this union in terms of the number-series cannot stand the test of criticism. At the same time, Professor Royce’s argument in any case throws considerable light upon the problem of relation. For it shows _why_ the attempt to construct the world as a system of qualities in relation leads to the indefinite regress. For a complete experience embodying at one stroke the whole of existence, such a construction would, as we have seen, because essentially incomplete, be impossible. But when _we_ try to piece together the data of our fragmentary experience into a connected whole, we inevitably have to start with more or less isolated facts as fixed terms and weld them together by a relation. In doing so we unavoidably put ourselves at the point of view from which the numerical series arises; we unavoidably treat existence _as if_ it were a whole of mutually external parts. And so the indefinite regress involved in the nature of the number-system inevitably parades the whole of our discursive and relational thinking about existence. But its presence is due to the inadequacy of the conception of Reality with which discursive thought has to work.