mathematical concept of the series of ordinal numbers; Cantor's modern theory of infinite collections leads to a complete revision of the argument of the Kantian mathematical antinomies; and the mathematical theory of groups seems to shed new light on the psychological nature of the æsthetic object. Further developments along these lines may be expected in all branches of philosophy. In short, the problem of the categories, fundamental to all the mathematical and philosophical sciences, can be solved only by the coöperation of the philosophers and the mathematicians.
We may describe external objects from either of two mutually exclusive points of view: idealism, for which they are representations, or realism, for which they are things. The hypothesis of psychophysical parallelism, that to a given cerebral state corresponds a parallel or equivalent psychic state, cannot be consistently stated from either of these points of view; it involves a confusion of the two, an unconscious passing from one to the other. For consistent idealism the brain state is itself a representation, and hence cannot be the source of the other representations, or 'equivalent' or 'parallel' to them. For realism, on the other hand, external objects form a continuous system of relations, of interacting powers and forces; here, again, the cerebral state is but a part, and cannot cause the whole, or be equivalent to it. The reconciliation of the two points of view in the parallelistjc hypothesis is merely apparent. The illusion is favored by several assumptions, including (1) the idea of a cerebral soul, i.e., a concentration of the faculty of representation in the cortex, (2) the idea that all causality is of the mathematico-mechanical type, (3) the idea that a simple process of abstraction will suffice to change our concrete representations into colorless things-in-themselves, and (4) the idea that if two wholes are connected, every part of the one is connected with some definite part of the other.
In its perfect form parallelism has not yet been defined. All of the many and contradictory varieties rest, however, on the theories of automatism and epiphenomenalism. All of its forms fall into one of three main divisions. The first attributes consciousness to all phenomena of life; wherever there is life there is consciousness. The second supposes consciousness to appear only when certain conditions are present; e.g., when adaptation to new conditions is required, as opposed to the simple reflex. The last form reduces consciousness to a single zone, clear but limited, as represented by an act of choice. Regarded in one way, mind cannot be regarded other than as a force, and this leads back to the theory of con-
