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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 ?
The postulation method of resistance genes was designed to study wheat genetics and is a fast, useful and simple method to identify genes in cultivars at the seedling stage in a greenhouse.
The information mentioned above has been summarized by Lepori and colleagues, (43) and is consistent with the postulation that acetylcholinergic mechanisms play an important hitherto unrecognized role in offsetting the hypertension and cardiac sympathetic activation caused by NOS inhibition in humans.
The identificative copulative particle that occurs as "ngi" with the pronouns of all persons and noun classes as complement in Southern Ndebele, also serves as language external evidence for the postulation of the identificative copulative particle of Zulu as "ngi".
The explanation was made based on a postulation that the preservice teacher may have had much difficulty in teaching science during the teaching practice.
there is proof supporting the fundamental postulation that the role of a great manager cannot be placed in a box, a company policy, or applied across companies or organizations
Later in contemplation of an immune response, Burnet added, "The failure of antibody production against autologous cells demands the postulation of an active ability of the reticulo-endothelial cells to recognize 'self' patterns from 'non-self' patterns in organic material taken into their substance" (3).
Given that ``W'' is the first president since Herbert Hoover with no Jews in his cabinet, that he showed no interest in the Middle East peace process until after 9-11, I don't buy Tracie Breiter's postulation that Bush is better for Israel than Kerry (who has two Jewish grandparents).
She shows how the emergence of these philosophical concepts relates to literary theory and allows for the postulation of a theoretical foundation for the modern novel.
The EPPM'S theoretical postulation rests on its two key constructs: perceived threat and perceived efficacy.
The postulation of a world outside was at best heuristic, not metaphysical.
He bases that postulation on the fact that the drug chain put less than $50,000 into developing the service and it is probably the only chain drug web site that has a positive return on capital.
He dove in and quickly distinguished himself winning important faculty assignments and working with many of the great postwar physicists, among them Feynman and George Zweig, who shared with Gell-Mann the initial postulation of the quark.