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

88 MATHEMATICAL PHILOSOPHY
are distinct unless xi X2 and y =y2. You see that the
field is non-spatial, non-geometric, for numbers and number dyads have no essential reference to space and would
continue to be perfectly good objects, or subjects, of
thought if all spatial sense and all conception of space
were to vanish; symbols for numbers and for dyads do
indeed occupy room, but numbers themselves and dyads
do not.
And now it is time to say that our non-geometric
interpretation of HAF arises from assigning to the ppstulate variables constant values, or meanings, as follows:
vi will mean a dyad of N; V2 will mean a system of dyads,
i.e., the dyads satisfying an equation of the form Ax +
By+C=O, where either A or B is not zero, i.e., f!0
or B#O; Ri will mean between in the sense that, if
(x1, yi), (x2, y2) and (X3, y3) are three dyads of a same
system, (x2, y2) will be said to be between (xi, yi) and
(x3, y3) if and only if x2 is between xi and X3 or y2 is
between yi and y3; and R2 will mean congruent in the
sense that two dyadic pairs (xi, yi), (x2, y2) and (X3, y3),
(X4, y4),-that is, two segments (xi, yi,)(x2, y2), (X3, y3)
(X4, y4),-will be said to be congruent when and only
when V/(x -X2)2 +(yi -y2)2 (x3 -x4)2 + (y2 -y4)2;
with a like meaning for congruence of angles to be give
later.
Are the postulates in HAF verified by the meanings
thus assigned? It will be very instructive to examine
the matter somewhat carefully.
Postulate (I).-Let Ax +By+C=O be an undetermined system s; d1, d2, any two dyads (xi, yi), (x2, y2)
of field N; di and d2 will belong to s when and only when
Axi +By +C =0,
(I) Axq+By2+C=0;