axiom(redirected from Axioms)
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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).
..... Click the link for more information. 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.
..... Click the link for more information. 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.
..... Click the link for more information. ). 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.
axiomcontrol 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.
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 - The Scientific Computing System", R. Jenks et al, Springer 1992].