Axiom:
Basic or first assumption in a
theory. To explain anything one needs
assumptions of some kind, usually both about the topic one tries to explain
and about the logic one uses for explanations. These assumptions are often
called the axioms of a theory.
1. Vagueries with the term "axiom"
The assumptions from which one can explain
whatever a theory is intended to explain have often, indeed since Antiquity,
been called the axioms of the theory, and the historical paradigm of it
all is Euclid's "Elements", that presents geometry on the basis
of a handful of axioms and definitions, from which the rest of geometry then
is deduced (indeed almost without mistake, in Euclid's case).
Until the 19th Century, two common demands about such axioms as one used
were that they should be selfevident and that they should be
true. Then it was realized that neither demand
is necessary for a first assumption to be perfectly usable as a first
assumption, and that the meaning of "selfevident" is far from evident, while
the most one can hope for from axioms is often that they are mutually
consistent.
Since then, at least in logically or mathematically enlightened circles,
the term "axiom" tends to be simply used as "first or
primitive assumption of a theory". In the
same circles, it is a widely known fact that almost any
theory can be based on many distinct
sets of axioms, each set of which is sufficient to
deduce all statements of the theory.
Hence, the enlightened use of the term "axiom" does not imply it must be
selfevident (though it may be, to some at least) nor that it must be
necessarily true (though again it may be, as e.g. the common axioms for
propositional logic).
And the fundamental reason to adopt an axiom is that one knows it entails
consequences one desires to establish, and to assume the axiom is a known way or
perhaps convenient way or possibly the only known way to deduce these consequences.
