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

66 MATHEMATICAL PHILOSOPHY
Before considering another postulate, let us illustrate
the following theorem (a propositional function in the doctrinal function HAF): Any given v2 separates the remaining
vl's of the v3 into two classes such that, if Vi' and vi" are one
of them in one of the classes and the other in the other, the
segment vi' vi" contains a vi of the v2; and that, if v1' and
vl" are both in one of the classes, the segment does not
contain a vi of the V2. (It is theorem 5 of Hilbert's book.)
A fairly careful examination of Fig. 5 will suffice to conA''
FIG. 5.
vince you that that theorem is verified in our new interpretation. One of the two classes of lines is composed
of all the lines of c that go between O and a; all the
other lines compose the other class. Segments AA' and
BB' contain no line of the pathopencil a, but any such
segment as AB contains a line of a.
You should not fail to compare Fig. 5 with Hilbert's
figure for the corresponding proposition in doctrine D1,
the old familiar interpretation. The two figures are the
same logically but very different psychologically: in the