latter case, are the two spaces coincident[1]; or does the spiritual include the other as that includes a given plane or line; or is every line and every plane of one space perpendicular to every line and plane of the other? The third alternative seems untenable; for, admitting it, how could a spiritual force produce even a deflective effect in physical space? But the other two remain, and the second may fall in with such purely physical speculations as W. W. Rouse Ball's, who seeks to obtain both Newton's law and certain results of spectrum analysis from the virtual hypothesis that our known space, lying like a plane in a larger space, can vibrate in the fourth direction. [Messenger of Mathematics, June, 1891.]
It remains now to notice briefly, from the standpoint of mathematics, certain supposed difficulties of the Free Will hypothesis. We are told that the Determinist theory is forced upon us:
(1) By the principle of conservation of energy;
(2) By the law of causation;
(3) By statistics and the known effect of environment upon conduct and character; and even
(4) By consciousness.
(1) The first difficulty disappears with the admission of quasi-perpendicularity; and, as I have tried to show, some such admission is almost necessary in order to explain other phenomena than those of our supposed freedom.
(2) If this means that all causation is physical, in the sense that excludes independent contributions to the stream, then it simply begs the question. If it means that an infinite mind, knowing perfectly well the present, could with certainty trace thence the stream of events both upwards and downwards, this again is an assumption, made plausible by hereditary ideas as to Fate, and the Divine Decrees and Foreknowledge, and perhaps also by two fallacies as to the fixity of the Past. Doubtless the Past is fixed, if only because it is past; but since the same fixity would result if the Past were wholly inferrible from the
- ↑ Yet not necessarily coextensive. Our familiar space, if "positively curved," may be said to coincide absolutely with another having just half or double its extent, and yet each space to be complete and perfectly symmetric, two points of one space being at every one point of the other. The like is true if the second alternative be chosen.
