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

DOCTRINAL INTERPRETATIONS 71
I will close this lecture by indicating,-merely indicating,-the analogous new interpretation of the Doctrinal
function ILaF', which, you remember, includes the entire
list of Hilbert postulates in their restated form. I shall
denote the new doctrine, or interpretation, by D2'. Let
S denote a projective space of three dimensions. We have
already formed the concept of such a space. All the lines
(or planes) of S that have in common point P are together
called a sheaf, or bundle, of lines (or planes); all the planes
having a common line constitute an axial pencil of planes.
Let O be a chosen point of S. Call the sheaf of lines (or
planes) having O for vertex the O-sheaf of lines (or planes).
In thought remove from S the 0-sheaf of lines and the
O-sheaf of planes. We thus remove from every other
line sheaf one line, from every other plane sheaf an axial
pencil and from every axial pencil (not contained in the
O-sheaf of planes) one plane. The ensembles, thus rendered defective, may be respectively called a pathosheaf
of lines, a pathosheaf of planes and a pathopencil of
planes, or plane pathopencil. Analogous to the polepolar transformation as to a circle,-which we have
already explained and used,-there is for S a pole-polar
transformation with respect to any given sphere converting each point into a polar plane and each plane into pole
point. Our field of operation-1/'-is S bereft of the two
O-sheaves. As you may have by this time surmised, our
new interpretation, or doctrine D2', arises on giving the
variable symbols in the postulates of HàF' meanings as
follows: vi will mean a plane of q'; v2, a pathopencil of
planes; V3, a pathosheaf of planes; V4, '; R1, between
in the sense that, if A, B, C are planes of a pathopencil,
B will be said to be between A and C if either of the latter
must rotate through the position of B to coincide with