Axiom. This term taken by Aristotle from Mathematics, whether from the
language of others, or from his own lost writings on that subject, is
claimed by him for Metaphysics or Ontology as well. It denotes an
immediate self-evident proposition, which does not admit of proof, but
from which proof proceeds. Boethius and the schoolmen render it, out of
regard to its etymology, by dignitas.
Aristotle lays stress upon the distinction between the general axioms
which are common to all science, and those which are proper to any one.
The former as applied to geometry are called
κοιναὶ ἔννοιαι by Euclid, and
the rest of those named axioms in our versions are by him placed among
the postulates ἀίτήματα.
So far the Aristotelian notion of an axiom, which is perhaps the most
generally entertained. The term, however, has been variously applied.
The Stoics gave the name to every general proposition, as does Bacon,
who even speaks of arriving at or exciting axioms, and proving them.
Kant restricts the title to the fundamental axioms of geometry, our pure
intuitions of space. The Cartesians follow the Aristotelian use, i.e.
the use in the Analytics and Metaphysics. In the eighth book of
the word has a wider latitude, which seems the most adapted to general
Meaning then by axioms the fundamental, universal, and self-evident
judgments on which all thought must hinge, such as that if equals be
added to equals the sums will be equal, and the like, Aristotle resolves
them all into the great logical law of Contradiction, that the same
thing cannot be and not be at the same time. By denying therefore an
axiom truly such we subvert thought, and turn all exercise of it into
futility. Those who understand the distinction will see that in this
view they are to be regarded as analytic not synthetic judgments. Kant,
confining the term to geometrical intuitions, classes them with the
latter. There are several questions respecting axioms over and above
those which relate to the use of the term, on which I cannot enter here.
The most modern of these will be found in Stewart's works, and in Mill's