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

102 MATHEMATICAL PHILOSOPHY
increasing. Then x==X2-xl =X3-X2=... =x-~-X-._.
The sum nx =xx; we choose n so that nxi>x', then
xn> x', but x'> 0; hence x' is between O and x~ and hence
d' is between d and di; which was to be proved.
To prove the compatibility of postulates we have to
find a set of things regarding which the postulates make
true statements when the things are put in place of the
variables. The better the things are known, the better
is the test. Now number dyads and systems thereof
are the best known of things; and so, in showing that
they verify the Hilbert postulates, we have established
their compatability by the diamond test.
Let us denote the doctrine arising from the foregoing
interpretation of HzF by D4. D4, as 1 have said and as
you must now plainly see, is in all strictness non-geometric,
having no spatial content. It is purely algebraic or
numerical-a two-dimensional theory of dyads and systems of dyads of real numbers. In point of form it is
Euclidean, having the same form as Euclidean plane
geometry; but to say that is to say that Euclidean
geometry has the same form as the Dyad doctrine. If
the latter had happened, as it might have happened, to
be developed prior to Euclidean geometry and received
a name proper to it, there would be precisely as much
sense and propriety in calling Euclidean geometry by
that name as there now is in calling the Dyad doctrine
Euclidean geometry.
This lecture has grown, I fear, to a wearisome length.
Yet I must ask your permission to continue long enough
to indicate very briefly an interpretation of HàF' analogous to the foregoing interpretation of HzF. The field
N' of the interpretation in question is composed of all the
triads (x, y, z) of the real numbers. The interpretation