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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 ?
Thirty tester cultivars with known Yr genes were used for gene postulation, including the world differentials, European differentials, partly Chinese differentials and others.
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".
Should Brazil avoid what would be the biggest shock of the lot in this World Cup and clip Turkey's wings for a second time in a fortnight, then the paranoid postulations of Italy's Gazetta Dello Sport will prove truly prophetic.
that theories of nationalism, postulations of "routes to modernity" and the modelling of national identity have almost without exception been based on Western precedents, the authors seek to import into such tropes a recognition that while there is no specifically Asian form of the nation, Asian experience of engendering nationhood offers fruitful comparative data that can enrich general theoretical discussion.
The intermediate principles accepted include "skeptical epistemic standards, a realist interpretation of theories, restrictive scruples on ontological postulations, strong demands on explanation, and Hume's principle" (pp.
There is little doubt that people within the field of rehabilitation have not forgotten the days of forced institutionalization and therefore, the postulations of Gallagher may not be historically exclusive.
I'll be billboarding your specific criticisms, comments and postulations on the media in general and those in St.
As is well known, Jacques Derrida repudiates any "transcendental signified" or "center," and this repudiation co-determines his postulations concerning "differance.
Every hitting coach should familiarize himself with the postulations of the masters: Lau, Hriniak, and Williams.
A susceptibility to infinitely diverse postulations and a fair degree of latitude towards the contortions of transatlantic critical diction are keys to a positive response to this volume.
The four thousand rules and postulations were codified and arranged by subject by Rabbi Judah I (c135 - 220).
They are postulations for the building of a just and sovereign state that can endure today and tomorrow as long as Lebanon and its free and sovereign people can prevent coups, domination, tyranny and suppression," it said.