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

114 MATHEMATICAL PHILOSOPHY
postulate was incapable of demonstration and many of
the best of them devoted their genius to vain attempts
at proving it. At length, however, mathematicians
learned better and began to produce geometries on the
basis of postulate systems in which Euclid's parallel
postulate was contradicted. That such geometries are
just as logically possible as Euclid's I will show in a
subsequent lecture. What I wish now to say is that any
geometry built upon a postulate system containing
Euclid's parallel-postulate, or its equivalent, is called
Euclidean, however widely it may differ in other respects
from Euclid's Elements; and, correspondingly, any
geometry, like that of Lobachevski or that of Riemann,
whose postulate system contains a contradictory of
Euclid's parallel-postulate, is said to be non-Euclidean,
no matter how much it may be like Euclid's Elements
in other respects. Such are the specific and more usual
senses in which these familiar adjectives are employed
in the literature of geometry.
It must occur to you at once that there is no good reason for confining the use of the terms, in the sense just
indicated, to geometry. For the Hilbert postulate (I3)
being in agreement with the parallel-postulate of Euclid,
it is evident that we may with evident and perfect propriety call Euclidean all of the infinitely many doctrines
having HAF or HAF' for matrix, whether the doctrines
be true or false, and whether they be geometric or algebraic or neither the one nor the other.
What Are the Properties that a Collection of Propositional Functions Must Have in Order to be a Postulate
System?-I will name three properties: pregnance, or
productiveness, or fertility; compendence, or connectedness; and compatibility, or consistency. In discussions