The fourth dimension

APPLICATION TO KANT’S THEORY OF EXPERIENCE 119

the points and no order in the turnings. <A being with an infinite number of axes mutually at right angles, with a definite sequence between them and no sequence between the points on the axes, would be in a condition formally indistinguishable from that of a creature who, according to an assumption more natural to us, had on each axis an infinite number of ordered points and no order of priority amongst the axes. A being in such a constituted world would not be able to tell which was turning and which was length along an axis, in order to distinguish between them. Thus to take a pertinent illustration, we may be in a world of an infinite number of dimensions, with three arbitrary points on each—three points whose order is indifferent, or in a world of three axes of arbitary sequence with an infinite number of ordered points on each. We can't tell which is which, to distinguish it from the other.

Thus it appears the mode of illustration which we have used is not an artificial one. There really exists in nature a duality of the kind which is necessary to explain the origin of order out of no order—the duality, namely, of dimension and position. Let us use the term group for that system of points which remains unchanged, whatever arbitrary change of its constituents takes place. We notice that a group involves a duality, is incouceivable without a duality.

Thus, according to Kant, the primary element of experience is the group, and the theory of groups would be the most fundamental branch of science. Owing to an expression in the critique the authority of Kant is sometimes adduced against the assumption of more than three dimensions to space. It seems to me, however, that the whole tendency of his theory lies in the opposite direction, and points to a perfect duality between dimensiot «nd position in a dimension.