Reference
An axiom, postulate or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek axí?ma (??????) 'that which is thought worthy or fit' or 'that which commends itself as evident.'[1][2]
The term has subtle differences in definition when used in the context of different fields of study. As defined in classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question.[3] As used in modern logic, an axiom is a premise or starting point for reasoning.[4]
As used in mathematics, the term axiom is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". Logical axioms are usually statements that are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (A and B) implies A), while non-logical axioms (e.g., a + b = b + a) are actually substantive assertions about the elements of the domain of a specific mathematical theory (such as arithmetic).
When used in the latter sense, "axiom", "postulate", and "assumption" may be used interchangeably. In most cases, a non-logical axiom is simply a formal logical expression used in deduction to build a mathematical theory, and might or might not be self-evident in nature (e.g., parallel postulate in Euclidean geometry).[5] To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms), and there may be multiple ways to axiomatize a given mathematical domain.
Any axiom is a statement that serves as a starting point from which other statements are logically derived. Whether it is meaningful (and, if so, what it means) for an axiom to be "true" is a subject of debate in the philosophy of mathematics.[6]
https://en.wikipedia.org/wiki/Axiom
A way of arriving at a scientific theory in which certain primitive assumptions, the so-called axioms (cf. Axiom), are postulated as the basis of the theory, while the remaining propositions of the theory are obtained as logical consequences of these axioms.
In mathematics, the axiomatic method originated in the works of the ancient Greeks on geometry. The most brilliant example of the application of the axiomatic method — which remained unique up to the 19th century — was the geometric system known as Euclid's Elements (ca. 300 B.C.). At the time the problem of the description of the logical tools employed to derive the consequences of an axiom had not yet been posed, but the Euclidean system was a very clear attempt to obtain all the basic statements of geometry by pure derivation based on a relatively small number of postulates — axioms — whose truth was considered to be self-evident.
The discovery of a non-Euclidean geometry by N.I. Lobachevskii and J. Bolyai at the beginning of the 19th century stimulated further development of the axiomatic method. They showed that if the traditional, and apparently the only "objectively true" , fifth postulate of Euclid concerning parallel lines is replaced by its negation, then it is possible to develop in a purely logical manner a geometry which is just as elegant and meaningful as is Euclidean geometry. The attention of mathematicians of the 19th century was thus drawn to the deductive manner of constructing mathematical theories; this in turn gave rise to a new problem, connected with the concept of the axiomatic method itself and with the formal (axiomatic) mathematical theory. With the gradually increasing number of mathematical theories which had been axiomatically derived — one can, in particular, mention the axiomatic derivation of elementary geometry by M. Pasch, G. Peano and D. Hilbert which, unlike Euclid's Elements is logically unobjectionable, and Peano's first attempt at the axiomatization of arithmetic — the concept of a formal axiomatic system became more rigorous (see below), resulting in a class of specific problems which eventually established proof theory as one of the main chapters of modern mathematical logic.
https://encyclopediaofmath.org/index.php?title=Axiomatic_method
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