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see axiomaxiom,
in mathematics and logic, general statement accepted without proof as the basis for logically deducing other statements (theorems). Examples of axioms used widely in mathematics are those related to equality (e.g.
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(from the Latin postulatum, “demand”), a proposition, condition, assumption, or rule that is accepted without proof because of certain reasons. Generally, however, a postulate is accepted with some substantiation, and this substantiation usually serves as an argument in favor of accepting the postulate. The nature of the acceptance may vary. A proposition is accepted as true, as in meaningful axiomatic theories, or as provable, as in formal axiomatic systems. Some instructions are adopted “for execution” as rules governing the formation of the formulas of some calculus or as rules of inference of the calculus that make it possible to derive theorems from the axioms. Some principles abstracted from the data of repeated experience are made the foundation for physical and other natural scientific theories; examples include principles of the type “laws of conservation.” Some positions, prescripts, and norms (legal, for example) are accorded, as a result of other positions, the status of laws. Some religious, philosophical and ideological dogmas are made the foundation for certain systems of belief. For all the diversity of these examples, there is something common to them all: without sparing arguments designed to convince us of the rationality, or legitimacy, of the postulates that we propose, we ultimately simply demand—hence the origin of the word “postulate”— acceptance. In such cases we say that the propositions advanced are postulated.

Such a broad concept, so rich in shades of meaning, naturally has many concrete, more specialized, and therefore extremely varied realizations. The following is a list of some of the most common realizations.

(1) Euclid, who gave the first known systematic axiomatic description of geometry, distinguished between postulates (αιτηματα), which assert the feasibility of certain geometrical constructions, and axioms proper, which affirm (postulate) that the results of these constructions have certain properties. Moreover, he defined axioms as propositions of a purely logical (and not geometrical) character that he accepted without proof, such as “the part is less than the whole.” The dual and not clearly drawn line of delimitation between these similar concepts persisted beyond Euclid.

(2) The terms “axiom” and “postulate” were and are often used synonymously. In particular, Euclid’s well-known fifth postulate of parallel lines is called the parallelism axiom in Hilbert’s axiomatics.

(3) At the same time, the term “axiom” is used by many authors to denote “purely logical” propositions accepted in a given theory without proof. (See, for example, A. Church, Vvedenie ν matematicheskuiu logiku, vol. 1, subsecs. 07 and 55, Moscow, 1960 [translated from English].) By contrast, the term “postulate” is used in reference to specific concepts of a given (usually mathematical) theory.

(4) According to another tradition in mathematical logic, postulates of a formal system (calculus) include axioms written in the language proper (“subjective” language) of the system and the rules of inference formulated in the metalanguage of the given theory (and therefore belonging to its metatheory). (See, for example, S. C. Kleene, Vvedenie ν metamatematiku, subsecs. 19 and 77, Moscow, 1957 [translated from English].)

(5) “Postulates” is the name given to assertions of deductive and (especially) semideductive sciences that cannot be proved, if only because the arguments and facts supporting them are exclusively experimental and inductive in character. In many such cases we speak of the assertion of the equivalency of some intuitively clear but not clearly formulated assertion or concept that is an explication (refinement) of the former and therefore formulable at a fundamentally higher level of abstraction. Examples of the first type are the fundamental principles of thermodynamics and the principle of the constancy of the speed of light and its limiting character; an example of the second type is the Church thesis in the theory of algorithms.




Logic Maths an unproved and indemonstrable statement that should be taken for granted: used as an initial premise or underlying hypothesis in a process of reasoning
References in periodicals archive ?
If the first and second postulate requirements are met, the third postulate follows that interventions targeting the causal event should significantly lower the incidence of AD.
Most physicists expect that as triple-slit experiments are conducted with other particles, such as electrons and bucky-balls, the case for Born's postulate will get stronger, Dowker says.
Sequence-based identification of microbial pathogens: a reconsideration of Koch's postulates.
The author postulates that the clot in the left internal jugular vein in this patient altered the blood flow dynamics, which led to increased venous pressure and the resultant venous insufficiency in the VPVC.
We will see how certain postulates defined on the knowledge level must be transformed into "less restrictive" postulates with the objective of respecting the basic principles of the AGM model: minimum change, syntax irrelevancy and adequacity of knowledge representation.
A young boy watches people skip stones across the water, and postulates that his island home is shrinking.
Then, he postulates that the demands of modern life are causing a surge in cases of attention-deficit disorder.
The network of laboratories in 17 countries organized by the World Health Organization (WHO) coordinated information sharing (David Heymann, WHO) and was instrumental in rapidly identifying the etiologic agent of SARS (1) and in fulfilling Koch's postulates (2) (Albert Osterhaus, Erasmus University, Rotterdam).
Significantly, he was schooled in Vienna, in an atmosphere that also produced Karl Krauss' barbed witticisms and Adolf Loos' anti-ornamentalism; the teachings he received there perhaps allowed him in later years to open himself to the abstract postulates of the Neoplasticists, as well as the strident world of Surrealism and Dada.
Since I discovered the linguistic postulates of the Jaqi languages of South America to be quite different from those of English, I have found it worthwhile to compare the major Jaqi and English postulates--ones that require realization in virtually every sentence of each.
The gold standard for proving that something is infectious is Koch's postulates, and the company will validate earlier findings on Koch's postulates with calcifying nanoparticles in laboratory animals, including testing whether the infection can be prevented or treated with a proprietary drug combination.
Examining the ancient Greek moral theory that happiness is desired as the ultimate end of life, and that a life of virtue is a happy life, in comparison to modern postulates that measure the happiness of life by self-determined appraisals rather than through adherence to ethical theory, Eudaimonia And Well-Being is a well reasoned and superbly presented account.