The term postulate, or axiom, indicates a starting assumption from which other statements are logically derived. It does not have to be self-evident (say, constancy of speed of light is not self-evident). Some axioms are experimental facts, but some are just assumptions not based on anything. This article does not cite its references or sources. ...

Obviously a chain of logical or mathematical derivations with no beginning is not possible (it would be infinite or circular otherwise). Some initial statements not following from anything (or brought from other fields - say, from experiment) thus are needed to build a logical or mathematical system - and they are called axioms and postulates. Logic (from ancient Greek λόγος (logos), meaning reason) is the study of arguments. ... Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. ... There are several meanings of derivation: A derivation in abstract algebra is a linear map that satisfies Leibniz law. ... A circular may be: the adjective form of circle an advertisement which is circulated a Pastoral letter, Encyclic, or Papal bull that is circulated between churches a circular argument is a term for a type of logical fallacy where the very thing that is trying to be proved is assumed...

Postulates and axioms do not have to be self-evident or intuitively correct, or majority approved. For example, second postulate of special relativity - constancy of speed of light - is not self evident nor intuitively correct, and when first proposed by Einstein was contrary to majority's opinion. In epistemology, a self-evident proposition is one that can be understood only by one who knows that it is true. ... The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest... A line showing the speed of light on a scale model of Earth and the Moon The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness. It is the speed of all electromagnetic radiation... Einstein redirects here. ...

Postulate vs. Axiom

The terms “postulate” and “axiom” are frequently used interchangeably as synonyms for each other (although there is a modern tendency to avoid using the word axiom, replacing it with property or postulate). But there is a difference in connotation that gives a shade of exactness to the definitions.

The term "axiom" has been applied historically to those statements that are applicable to a variety of fields of knowledge; for example: equivalence properties (reflexive, symmetric, and transitive); properties of equality and inequality (addition, subtraction, division, multiplication, and substitution); the whole is equal to the sum of its parts and is greater than any of its parts; etc. The general applicability of these properties to a wide variety of fields is obvious.

On the other hand, postulates apply to one, more specific field of knowledge. Probably the most famous set of postulates is Euclid's five postulates of plane geometry: Euclid(Greek: ), also known as Euclid of Alexandria, was a Greek mathematician who flourished in Alexandria, Egypt, almost certainly during the reign of Ptolemy I (323–283 BC). ...

1. Two points determine a line.
2. Any line segment can be extended in a straight line as far as desired, in either direction.
3. Given any length and any point, a circle can be drawn having the length as radius and that point as center.
4. All right angles are congruent.
5. Parallel postulate. If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side, if extended far enough.

a and b are parallel, the transversal t produces congruent angles. ...

Characteristics

A set of postulates should have several characteristics.

1. They should be self-evident and easily understood, involving as few undefined terms as possible.
2. They should be as few in number as possible.
3. The set should show consistency. A set of postulates is consistent if all the postulates (and the theorems derived from them) lead to no contradictions.
4. The set should show independence. A postulate is independent if no one of them can be shown to be a consequence of any of the other postulates – postulates are not provable.

Limitations

One should keep in mind that the discovery of postulates is based on what man sees as common sense, and that postulates are dependent on the limitations and fallibilities of man's senses, reasoning, and imagination. A mathematical or logical system will be defined by the set of postulates used. For example, several different (but entirely consistent) systems of geometry have been created using different sets of postulates. Variations of Euclid’s Parallel Postulate have given rise to such systems as Euclidean, Hyperbolic, or Elliptic geometries. Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. ... Lines through a given point P and hyperparallel to line l. ... Elliptic geometry (sometimes known as Riemannian geometry) is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. ...

See also

• Assertion
• Assumption
• Conjecture
• Corollary
• Guessology
• Lemma
• Porism
• Premise
• Proposition
• Scholium
• Theorem