Postulate
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The term postulate, or axiom, indicates a statement or assumption that is agreed by everyone to be so obvious or self-evident that no proof is necessary, and which can be used to prove other statements or theorems. Neither axioms nor postulates can be proven (within a system) using more basic statements. However, in many elementary textbooks, where the student does not have the training to understand a more rigorous approach, many otherwise-provable statements are accepted as postulates to allow further development of the subject.
Although ideally one would construct a mathematical or logical system with no assumed statements, this obviously is not possible. Just as any subject must have a set of specific (undefined) words to whose definition everyone agrees, any mathematical or logical system must start from a set of agreed-upon statements from which all further statements (theorems) can be built. These agreed-upon statements are called axioms and postulates.
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:
- Two points determine a line.
- Any line segment can be extended in a straight line as far as desired, in either direction.
- Given any length and any point, a circle can be drawn having the length as radius and that point as center.
- All right angles are congruent.
- 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.
Characteristics
A set of postulates should have several characteristics.
- They should be self-evident and easily understood, involving as few undefined terms as possible.
- They should be as few in number as possible.
- 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.
- 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
Keep in mind that the discovery of postulates is based on what man sees as common sense, and 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.