Axiom

Axiom, a Greek word meaning a 'decision' or 'assumption,' is commonly used to signify a general proposition which the understanding recognises as true, as soon as the import of the words conveying it is apprehended. Such a proposition is therefore known directly, and does not need to be deduced from any other. Of this kind, for example, are all propositions whose predicate is a property essential to our notion of the subject. Every rational science requires such fundamental propositions, from which all the truths composing it are derived; the whole of geometry, for instance, rests on comparatively a very few axioms. Whether there is, for the whole of human knowledge, any single, absolutely first axiom, from which all else that is known may be deduced, is a question that has given rise to much disputation; but the fact that human knowledge may have various starting-points answers it in the negative. Mathematicians use the word axiom to denote those propositions which they must assume as known from some other source than deductive reasoning, and employ in proving all the other truths of the science. The rigour of method requires that no more be assumed than are absolutely necessary. Every self-evident proposition, therefore, is not an axiom in this sense, though, of course, it is desirable that every axiom be self-evident; thus, Euclid rests the whole of geometry on fifteen assumptions, but he proves propositions that are at least as self-evident as some that he takes for granted. That 'any two sides of a triangle are greater than the third,' is as self-evident as that 'all right angles are equal to one another,' and much more so than his assumption about parallels (see PARALLELS). Euclid's assumptions are divided into three 'postulates' or demands, and twelve 'common notions,' the term axiom is of later introduction. The distinction between axioms and postulates is usually stated in this way: an axiom is 'a theorem granted without demonstration;' a postulate is 'a problem granted without construction'—as, To draw a straight line between two given points.