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

BASIC CONCEPTS 57
tween," "congruent,"-the symbols Ri and R2. Then
postulate (i) will read: Two distinct vl's determine a
V2. For another example, postulate (8) will read: If
vlt, vI", Vi"' are v1's of a v2 and vi" has the relation R1
to vi' and vl"' then vi" bas R1 to Vi"' and vi'. It is
obvious that all of the postulates and theorems admit of
such restatement. I strongly recommend that, as a very
enlightening exercise, you thus restate all of the postulates, a few of the theorems, and rewrite the proof of at
least one of the latter.
Having thus dragged into solar light the fact,-hitherto evident only in the psychic light of understanding,that our postulates and theorems involve variables, let
us now think of the postulates and theorems as constituting a Whole-a definite Body of logically related
propositional functions. Not one of them is true; not
one of them is false. What is true is that the postulates
imply the theorems. But this statement of implication,
though it is a proposition and is a true one-is not a part
of the Whole; it is not contained in the Body of functions;
were we to put it in, it would stand there alone as an
intruder, being neither one of the postulates nor one of
the theorems, neither a premise nor a conclusion, neither
an implier nor an implied; it is a philosophical proposition
about the Whole but is not a member of it; it is a critical
commentary upon it but not upon itself; it is a judgment,-a just and important judgment,-regarding the
Body of propositional functions, but is wholly external
to it.
This definite Body of logically compendent propositional functions, if one will but meditate upon it, is a
truly wonderful thing-a great indestructible shining
Form of forms-" poised in eternal calm" above the