Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

420 MATREMATICAL PHILOSOPHY
point (bundle vertex) of the line (axis of plane-pencil)
determined by B' and B"; while I(u) and I(x) are imagination's correspondents of the foregoing concepts.
Finally, let us join with (6) and (7) a third equation
(9) E"'=0, independent of them and representing a
plane r"' or a point (plane bundle) B"'. Consider the
equation (io) E'+ xE"+uE"' =0. What are its T(u) and
T(x)? The former is a conceptual variable plane of the
point (or bundle) determined by 7r', r" and r"'; the latter
is a conceptual variable point (bundle vertex) of the
plane determined by B', B" and B"'; while I(u) and
I(x) are the imitating " images " of the same.
Let us now pass to n=4; our field of operation is
S4, a four- dimensional space of points. Consider
(II) UlX+U2X2+U3X3+U4X4+I =0. Its T(u) is a conceptual lineoid (an Sa) of points, and its T(x) is a conceptual hypersheaf of lineoids (a point enveloped by the
o 3 lineoids containing it). Now scrutinize carefully the
results, I(u) and I(x), of imagination's effort to imitate
or represent pictorially the concepts T(u) and T(x).
You observe at once the following facts: (a) both I(u)
and I(x) are inferior to their analogues for n=3 or 2;
(b) the defect of I(u) differs in kind from that of I(x).
Indeed the two kinds of defect are, in a sense, reciprocal;
for I(u), in trying to match T(u), though it succeeds in
imaging points and point configurations interior to the
lineoid or locus, presents no image of the lineoid itself
or the locus as a whole; while, on the other hand, I(x),
in trying to match T(x), presents an image corresponding
to the point or envelope but no image to match the
enveloping lineoids. The contrast may be vividly seen
as follows: Note that, in the one case, the lineoid is the
bond or lien of the elements,-points,-of which it is the
locus, and that, in the other case, the point is the bond or