Help
Index

 Maarten Maartensz:    Philosophical Dictionary | Filosofisch Woordenboek                      

 A - Axiom

 

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 self-evident 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 "self-evident" 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 self-evident (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.
 


See also: Axiomatic method, Logical Terms, Primitive


Literature:

Carnap, Euclid, Halmos, Quine, Stegmüller, Suppes
 

 Original: Sep 15, 2004                                                Last edited:12 December 2011.   Top