# axiom

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Related to Axioms: Euclid's axioms

## axiom,

in mathematics and logic, general statement accepted without proofproof,
in mathematics, finite sequence of propositions each of which is either an axiom or follows from preceding propositions by one of the rules of logical inference (see symbolic logic).
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as the basis for logically deducing other statements (theorems). Examples of axioms used widely in mathematics are those related to equality (e.g., "Two things equal to the same thing are equal to each other"; "If equals are added to equals, the sums are equal") and those related to operations (e.g., the associative lawassociative law,
in mathematics, law holding that for a given operation combining three quantities, two at a time, the initial pairing is arbitrary; e.g., using the operation of addition, the numbers 2, 3, and 4 may be combined (2+3)+4=5+4=9 or 2+(3+4)=2+7=9.
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and the commutative lawcommutative law,
in mathematics, law holding that for a given binary operation (combining two quantities) the order of the quantities is arbitrary; e.g., in addition, the numbers 2 and 5 can be combined as 2+5=7 or as 5+2=7.
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). A postulate, like an axiom, is a statement that is accepted without proof; however, it deals with specific subject matter (e.g., properties of geometrical figures) and thus is not so general as an axiom. It is sometimes said that an axiom or postulate is a "self-evident" statement, but the truth of the statement need not be evident and may in some cases even seem to contradict common sense. Moreover, a statement may be an axiom or postulate in one deductive system and may instead be derived from other statements in another system. A set of axioms on which a system is based is often wished to be independent; i.e., no one of its members can be deduced from any combination of the others. (Historically, the development of non-Euclidean geometry grew out of attempts to prove or disprove the independence of the parallel postulate of Euclid.) The axioms should also be consistent; i.e., it should not be possible to deduce contradictory statements from them. Completeness is another property sometimes mentioned in connection with a set of axioms; if the set is complete, then any true statement within the system described by the axioms may be deduced from them.

## axiom

control of the LABOUR PROCESS is a further topic of general importance. See also INTELLECTUAL LABOUR.

## axiom

(as in geometry, but also in social theory) the taken-for-gr anted assumption or postulate of a model or theory from which other propositions can be derived. See also FORMAL THEORY AND FORMALIZATION OF THEORY.

## Axiom

in a given theory, a proposition that is not proved in the deductive construction of the theory but is accepted as a basic starting point in proving the theory’s other propositions. Usually the propositions of the theory under examination chosen as axioms are known to be true or can be considered true within the framework of this theory.

Originating in ancient Greece, the term “axiom” is first encountered in Aristotle. It then entered geometry through the works of Euclid’s followers and commentators. Because of the popularity of Aristotelian philosophy during the Middle Ages, the term came to be used in other areas of science and then in everyday life. A general proposition which, being completely apparent, does not require proof came to be called an axiom. The nature of this obviousness was seen, according to views going back to Plato, that such fundamental truths as mathematical axioms were innate to man. I. Kant’s doctrine of the a priori quality of axioms—that they precede all experience and do not depend upon it—was the culmination of such views. The first strong blow to the view of axioms as eternal and immutable a priori truths was the construction of a non-Euclidean geometry by N. I. Lobachevskii.

In criticizing Hegel’s views on logical axioms, the figures of Aristotelian syllogisms, V. I. Lenin wrote that “man’s practical activity had to lead man’s consciousness billions of times to the repetition of various logical figures, so that these figures could attain the significance of an axiom” (Filosofskie tetradi, 1969, p. 172). The obviousness of axioms, which are considered as truths not requiring proof, is caused by the conditionality of centuries-old human experience and practice, including experimentation, and the development of science.

In addition, the decline of the concept of an axiom as an a priori truth led to the bifurcation of the concept of axioms. The need to experiment in the sphere of construction of new theories, ever-increasing in connection with the investigation of practice; the need to exchange one axiom for another, as well as their relativity; their dependence on the previously encountered concrete conditions of experience and on the level of the development of science; and the impossibility of selecting once and for all as axioms such propositions to be true absolutely under all conditions —all this caused the appearance of a concept of axioms in a sense which differed somewhat from the traditional. This concept of axioms depends on which theory is being examined and how it is being pursued. Thus, axioms of a given theory become those propositions that in the deductive construction of the theory are accepted as initial, no matter how simple and obvious they are. Moreover, from experience, for example, from the construction of various non-Euclidean geometries and their later interpretation and practical application, it is apparent that to require a theory’s axioms to be true while constructing that theory is impossible.

The creation of a developed apparatus of mathematical logic is connected with the subsequent development of the concept of axioms. In formal calculus, an axiom is no longer an assumption of some inclusive scientific theory but simply one of those formulas from which, according to the rules of deduction in this calculus, are deduced the remaining formulas (theorems) demonstrated in it.

A. V. KUZNETSOV

## axiom

[′ak·sē·əm]
(mathematics)
Any of the assumptions upon which a mathematical theory (such as geometry, ring theory, and the real numbers) is based. Also known as postulate.

## axiom

1. a generally accepted proposition or principle, sanctioned by experience; maxim
2. a self-evident statement
3. Logic maths a statement or formula that is stipulated to be true for the purpose of a chain of reasoning: the foundation of a formal deductive system

## AXIOM

(language)
A commercially available subset of Scratchpad, from IBM.

["Axiom - The Scientific Computing System", R. Jenks et al, Springer 1992].

## axiom

(logic)
A well-formed formula which is taken to be true without proof in the construction of a theory.

Compare: lemma.
References in classic literature ?
It is an algebraic axiom, which makes us proceed from a known to an unknown quantity, and not from an unknown to a known; but sit down, sir, I beg of you.
It has long been an axiom of mine that the little things are infinitely the most important.
We shall not even take the trouble to demonstrate this, for it is an axiom in morals, as in physics.
He has an axiom which carries the thought-kernel that what man has done, man can do, and it doesn't cut any figure with Perry whether a fellow knows how to do it or not.
And quite enchanted with his joke, the ferocious Orangeman took his cresset and his keys to conduct Cornelius to the cell, which on that very morning Cornelius de Witt had left to go into exile, or what in revolutionary times is meant instead by those sublime philosophers who lay it down as an axiom of high policy, "It is the dead only who do not return.
Just as swiftly flashed through his brain two thoughts: the first, the axiom, WHENEVER AND WHEREVER YOU FIND A LEPER, LOOK FOR THE OTHER LEPER; the second, the desired Irish terrier, who was owned by Daughtry, with whom Kwaque had been long associated.
It obeyed no known laws of physics, and overthrew the hoary axiom that like things performed to like things produce like results.
It may be further remarked, that Miss Knag still aimed at youth, although she had shot beyond it, years ago; and that she was weak and vain, and one of those people who are best described by the axiom, that you may trust them as far as you can see them, and no farther.
It is only implied that whatever particular motion I consider, whether it be swift or slow, perpendicular, horizontal, or oblique, or in whatever object, the axiom concerning it holds equally true.
I believe it to be infinitely the truer axiom of the two that innocence can look like guilt.
In the first place, they were vaguely persuaded that every foreigner had a knife about him; in the second, they held it to be a sound constitutional national axiom that he ought to go home to his own country.
Although Oliver had been brought up by philosophers, he was not theoretically acquainted with the beautiful axiom that self-preservation is the first law of nature.

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