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

122 MATHEMATICAL PHILOSOPHY
call àF and AF' identical if, and only if, every propositional function in AF or AF' is in the other, AF' or AF, it
being, of course, understood that some mere rewording
may be required to show that a statement in one is in
the other. You see, at once, that two postulate systems
are equivalent or non-equivalent according as the corresponding doctrinal functions are identical or non-identical;
and conversely. It is a fact of no little scientific and
philosophic interest-for it is far from "self-evident"that, within limits, the postulates and the theorems in a
doctrinal function may interchange their respective rôles
without destroying the function's identity. Some questions arise here which, so far as I know, no one has asked,
and which I am unable to answer. One of them is:
what are the "limits" within which the mentioned
interchange of rôles may occur?
The only way to know that two equivalent systems
are equivalent is to prove them equivalent. It would be
very enlightening and a lot of fun to illustrate the process,
but it would delay our course too much. Perhaps you
will try your hand at the game. Two extremely interesting systems which, I believe, though I have not proved
it in full detail, are equivalent to Hilbert's system and
consequently to one another are the systems devised
respectively by Professor O. Veblen and Professor Mario
Pieri. Veblen's system, called " A System of Axioms for
Geometry," is found in Volume V of The Transactions of
the American Mathematical Society (I904). This system,
in a modified form, was subsequently presented by its
author as the initial monograph in the Monographs on
Topics of Modern Mathematics (edited by Professor
J. W. A. Young)-a volume which, though its articles
fiffer Wi'eXy in aim, spirit anà excellence and though it