Axiom definitions and example sentences on Wordnik

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noun A self-evident or universally recognized truth; a maxim: "It is an economic axiom as old as the hills that goods and services can be paid for only with goods and services” (Albert Jay Nock).
noun An established rule, principle, or law.
noun A self-evident principle or one that is accepted as true without proof as the basis for argument; a postulate.

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A self-evident, undemonstrable, theoretical, and general proposition to which every one who apprehends its meaning must assent. The Greek word was probably applied by Plato (though it does not occur in his dialogues in this sense) to certain first premises of mathematics; and this continues to be the ordinary use of the term. It was extended by Aristotle to similar principles supposed to underlie other branches of knowledge. The axioms or “common notions” of Euclid, as given in English translations, are twelve in number, viz.: (1) Things which are equal to the same are equal to one another. (2) If equals be added to equals, the wholes are equal. (3) If equals be taken from equals, the remainders are equal. (4) If equals be added to unequals, the wholes are unequal. (5) If equals be taken from unequals, the remainders are unequal. (6) Things which are double of the same are equal to one another. (7) Things which are halves of the same are equal to one another. (8) Magnitudes which coincide with one another, that is, which exactly fill the same space, are equal to one another. (9) The whole is greater than its part. (10) Two straight lines cannot inclose a space. (11) All right angles are equal to one another. (12) If a straight line meets two straight lines, so as to make the two interior angles on the same side of it taken together less than two right angles, these straight lines, being continually produced, shall at length meet upon that side on which are the angles which are less than two right angles. Only the first three of these are universally acknowledged to be authentic, though the latest editor, Heiberg, allows the eighth and ninth also. Euclid gives besides a list of postulates, which, as given in English translations, are: (1) Let it be granted that a straight line may be drawn from any one point to any other point. (2) That a terminated straight line may be produced to any length in a straight line. (3) And that a circle may be described from any center, at any distance from that center. What the English editions give as the eleventh and twelfth axioms formed originally the fourth and fifth postulates, and in the best MS. the tenth axiom appears as the sixth postulate. It would thus seem that he understood by “postulate” a geometrical premise which was asked to be taken for granted, and by “axiom” or “common notion,” a not specially geometrical principle with the use of which the learner would be already familiar. This agrees with Aristotle's definition of an axiom as a principle which he who would learn must bring of himself. The Leibnitzians distinguish a postulate as a self-evident practical principle from an axiom as a self-evident theoretical principle. According to Kant, an axiom is a necessary and general synthetical proposition which declares a property of pure space or time and rests directly on intuition, and is thus self-evident. He refused the name to the genuine “common notions” of Euclid, holding these to be analytical propositions. Modern mathematicians seem to regard the axioms of geometry as an analysis of the independent properties of space, so that the longer the list, provided the propositions are really independent, the more perfectly has the design been fulfilled. Many eminent mathematicians hold that there is no reason to think these axioms to be exactly true, but that they must be assumed to be slightly erroneous one way or the other; although experience shows that they approximate so nearly to the truth that it may be doubted whether it will ever be possible to measure the amount of their error. A similar doctrine is held by some thinkers concerning metaphysical axioms, such as the axiom that every event is determined by causes.
Any higher proposition, obtained by generalization and induction from the observation of individual instances; the enunciation of a general fact; an empirical law. This use originated with Bacon, influenced probably by the employment of axiom by the Stoics to mean any proposition.
In logic, a proposition, whether true or false: a use of the term which originated with Zeno the Stoic. Synonyms Maxim, Truism, etc. See aphorism.

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The first argument that naturally occurs to an opponent of this axiom is the difficulty of actually carrying it out. — My Philosophical Development
Thus if the axiom is to be adhered to, we must suppose that a term is not other than its nature. — My Philosophical Development
The reason for accepting an axiom, as for accepting any other proposition, is always largely inductive, namely that many propositions which are nearly indubitable can be deduced from it, and that no equally plausible way is known by which these propositions could be true if the axiom were false, and nothing which is probably false can be deduced from it. — My Philosophical Development
If the axiom is apparently self-evident, that only means, practically, that it is nearly indubitable; for things have been thought to be self-evident and have yet turned out to be false. — My Philosophical Development
It is possible that the use of the vicious-circle principle, as embodied in the above hierarchy of types, is more drastic than it need be, and that by a less drastic use the necessity for the axiom might be avoided. — My Philosophical Development

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