Re: [HM] How much did H. know of Italian work (ca. 1899)?
Jan Mycielski (jmyciel@euclid.Colorado.EDU)
Tue, 21 Jul 1998 13:02:15 -0600 (MDT)
DEFINITIONS versus AXIOMS
In a recent set of letters in HM there is a discussion of the question if
Hilbert was a Euclidean or not. By a Euclidean the correspondents mean a
person who proves things from definitions and not from axioms.
However, what is the difference between definitions and axioms?
I believe there is none. Indeed every definition in an axiomatic theory is
an axiom about a new term or relation symbol. While a system of axioms is
a definition of a class of models.
This may explain why mathematicians are not interested in the
distinction between Euclideans and non-Euclidians. And, I suspect that the
question if Hilbert was a Euclidean is spurious.
Am I right?
Jan Mycielski