In the precise condition in which the posits are, as described above, it does not seem to be possible. But if we imagine a duality to exist in the manifold, a function of consciousness can be easily discovered which will produce order out of no order.
Let us imagine each posit, then, as having, a dual aspect. Let a be 1a in which the dual aspect is represented by the combination of symbols. And similarly let b be 2b, c be 3c, in which 2 and b represent the dual aspects of b, 3 and c those of c.
Since a can arbitrarily change into b, or into c, and so on, the particular combinations written above cannot be kept. We have to assume the equally possible occurrence of form such as 2a, 2b, and so on; and in order to get a representation of all those combinations out of which any set is alternatively possible, we must take every aspect with every aspect. We must, that is, have every letter with every number.
Let us now apply the method of space representation.
Note.—At the beginning of the next chapter the same structures as those which follow are exhibited in more detail and a reference to them will remove any obscurity which may be found in the immediately following passages. They are there carried on to a greater multiplicity of dimensions, and the significance of the process here briefly explained becomes more apparent.
Fig. 59.
Take three mutually rectangular axes in space 1, 2, 3 ([fig. 59]), and on each mark three points, the common meeting point being the first on each axis. Then by means of these three points on each axis we define 27 positions, 27 points in a cubical cluster, shown in [fig. 60], the same method of co-ordination being used as has been described before. Each of these positions can be named by means of the axes and the points combined.
Fig. 60.