Hinton - The Fourth Dimension.pdf

102
THE FOURTH DIMENSION
in the third representative cube is the one in the second
wall to the right for the major premiss, the third wall
from the front for the minor premiss, and the top layer
for the conclusion.
It is easily seen that in the diagram this cube is
marked, and so with all the valid conclusions. The
regions marked in the total region show which combinations
of the four variables, major premiss, minor
premiss, figure, and conclusion exist.
That is to say, we objectify all possible conclusions, and
build up an ideal manifold, containing all possible conbinations
of them with the premisses, and then out of
this we eliminate all that do not satisfy the laws of logic.
The residue is the syllogism, considered as a canon of
reasoning.
Looking at the shape which represents the totality
of the valid conclusions, it does not present any obvious
symmetry, or easily characterisable nature. A striking
configuration, however, is obtained, if we project the fourdimensional
figure obtained into a three-dimensional one;
that is, if we take in the base cube all those cubes which
have a marked space anywhere in the series of four
regions which start from that cube.
This corresponds to making abstraction of the figures,
giving all the conclusions which are valid whatever the
figure may be.
Proceeding in this way we obtain the arrangement of
marked cubes shown in fig. 57. We see
that the valid conclusions are arranged
almost symmetrically round one cube—the
one on the top of the column starting from
AAA. There is one breach of continuity
Fig. 57.
however in this scheme. One cube is
unmarked, which if marked would give
symmetry. It is the one which would be denoted by the