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

54 MATHEMATICAL PHILOSOPHY
albeit of a different type from the variables it containshaving for its values the true propositions derivable from
it by means of its verifiers.
With the foregoing ideas and distinctions in mind,
iet us return to the Hilbert postulates and ask: Are they
propositions or propositional functions? To answer, it is
necessary and sufficient to ascertain whether or not they
contain variables. We observe at once the presence in
them of certain substantive terms-"point," "straight
line," "plane," and "space"-which seem to denote the
things about which the postulates talk, their subjectmatter-and certain relational terms-"between" and
"congruent"-which have the air of denoting definite
fundamental relations among the "points" or figures
composed of them. We must now ask: Do these terms
denote constants-things of unique and definite meaning
-or do they play the rôle of variables? Euclid does indeed, as you know, give what he calls "definitions" of
point, line and plane, but in his proofs and constructions
he makes no use whatever of the so-called definitions,
which he ought to have called descriptions designed merely
to indicate what he meant by the terms; or, better, he
ought to have omitted the definitions as logically useless.
As to the term, space, it does not, as it should not, occur
in Euclid's Elements. By examining Hilbert's book, you
will find that he does not attempt either to define or to
describe any of the above-mentioned six terms, except,
of course, in so far as they are defined-restricted in their
possible meanings-by having to satisfy, or verify, the
postulates. The omission of all other definition of them
is deliberate. And so our question is reduced to this:
Does the requirement that the things denoted by the
six terms-"point," "straight line," etc.-make the terms