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Dictionary of the History of Ideas

Studies of Selected Pivotal Ideas
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7 occurrences of Dictionary_of_the_History_of_Ideas
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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


163

“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.