Axiomatization as the organization of a deductive
system in a strictly axiomatic form dates from the last
part of the last
century. The very use of the term
“Axiomatics” as a noun is even more recent; it
is not
to be found in recent editions of the
Encyclopedia
Britannica (1962). It is mentioned in the
Enciclopedia
Italiana (1949); there it is defined
as the “name adopted
recently to signify that branch of
mathematical science
which deals with the ordering of
principles” (F. En-
riques). If we
adhered rigidly to this narrow definition,
the history of
“axiomatics” and of axiomatization
would be a brief
one, and its domain would be confined
to mathematics alone. Here we must
adopt the broader
interpretation in which these terms are often under-
stood and in which the very word
“axiom” is included:
an axiomatic system is one
composed of propositions
deducible from a small number of initial
propositions
posited as axioms. But what then is an
“axiom”?
“There is,” Leibniz says, “a class of
propositions
which, under the name of maxims or axioms, pass as
the principles of the sciences.... The scholastic phi-
losophers said that these propositions were self-evident
ex terminis, that is, as soon as the terms in them
are
understood” (New Essays IV, vii, 1).
And Bossuet de-
clares: “Those
propositions which are clear and intelli-
gible by themselves are called axioms or first princi-
ples” (Connaissance de
Dieu I, 13). Thus, in its classical
usage—with
various modifications which we shall see
later—an axiom is
characterized as combining two
features: as a principle it is the beginning or the basis
of a group of propositions
which it serves to demon-
strate; as a self-evident truth known immediately as
such, it
compels conviction without the aid of any
proof. It is, therefore, at one
and the same time a
certainty by itself and the basis of our certainty
with
respect to the propositions following from it.
Axiomatization will then consist in organizing a body
of propositions into a
deductive system such that the
principles of this system appear indubitable
by virtue
of their own self-evidence; the result is that the deduc-
tive apparatus performs the functions of
communi-
cating or transmitting to the
group of propositions of
the system the evidence and consequently the
certainty
of the axioms; this produces what has been called a
“transfer of evidence.” The deduction is in such a
case
categorical; it is demonstrative in the sense
in which
Aristotle defines demonstration as the “syllogism of
the
necessary,” the necessity residing both in the connec-
tion of the propositions and in the
very positing of
the initial propositions. Such should be the ideal
form
of scientific exposition, according to Aristotle: “it
is
necessary that scientific demonstration start from
premisses which
are true, primitive, immediate and
more evident than the conclusions, being
prior to them
as their cause” (Posterior
Analytic I, 2). This ideal was
to be perpetuated, with few
exceptions, until the be-
ginning of the
modern era.
