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

POSTULATE PROPERTIES 125
Other Farieties of Postulate Systems and Doctrinal
Functions.-By "other" varieties I mean such as are
not equivalent to the foregoing systems. Before citing
them, or rather some of them-for I am far from intending
to list all that have been devised-a word of caution
seems desirable. In inventing a postulate system the
inventor is never, or almost never, aiming at the establishment of what we have been calling a doctrinal function.
He is aiming at establishing autonomously a particular
one of the many doctrines which, as we have seen, a
doctrinal function has for its values. Such special interest
of the inventor, guiding and controlling him, is nearly
always betrayed by the air or color of his speech: often
by his giving his system a kind of name indicating that
the system has a specific subject-matter, which it has not;
nearly always by calling the postulates propositions, which
they are not, instead of propositional functions, which
they are; and usually by denoting the undefined terms by
names instead of variable-symbols as if the undefined
terms were constants instead of variables. This precaution will, I trust, help to keep you from gaining a false
impression from the following citations. The systems in
the list are almost random selections, and the list is far
from exhaustive but, by help of the numerous references
in the systems cited, it will afford you a clue to all or
nearly all extant systems.
The Axioms of Projective Geometry, by A. N. Whitehead, Cambridge University Press, I906.
The Axioms of Descriptive Geometry, by A. N. Whitehead, Cambridge University Press, I907.
" A Set of Axioms for Line Geometry," by E. R. Hedrick and Louis Ingold, Transactions of the American
Mathematical Society, Vol. XV, I9I4.