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From Wikipedia

Axiom

In traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject to necessary decision. Therefore, its truth is taken for granted, and serves as a starting point for deducing and inferring other (theory dependent) truths.

In mathematics, the term axiom is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". In both senses, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived. Unlike theorems, axioms (unless redundant) cannot be derived by principles of deduction, nor are they demonstrable by mathematical proofs, simply because they are starting points; there is nothing else from which they logically follow (otherwise they would be classified as theorems).

Logical axioms are usually statements that are taken to be universally true (e.g., A and B implies A), while non-logical axioms (e.g., ) are actually defining properties for 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 general, a non-logical axiom is not a self-evident truth, but rather a formal logical expression used in deduction to build a mathematical theory. 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). There are typically multiple ways to axiomatize a given mathematical domain.

Outside logic and mathematics, the term "axiom" is used loosely for any established principle of some field.

## Etymology

The word "axiom" comes from the Greek word á¼€Î¾Î¯Ï‰Î¼Î± (axioma), a verbal noun from the verb á¼€Î¾Î¹ÏŒÎµÎ¹Î½ (axioein), meaning "to deem worthy", but also "to require", which in turn comes from á¼„Î¾Î¹Î¿Ï‚ (axios), meaning "being in balance", and hence "having (the same) value (as)", "worthy", "proper". Among the ancient Greekphilosophers an axiom was a claim which could be seen to be true without any need for proof.

The root meaning of the word 'postulate' is to 'demand'; for instance, Euclid demands of us that we agree that some things can be done, e.g. any two points can be joined by a straight line, etc.

Ancient geometers maintained some distinction between axioms and postulates. While commenting Euclid's books Proclus remarks that "Geminus held that this [4th] Postulate should not be classed as a postulate but as an axiom, since it does not, like the first three Postulates, assert the possibility of some construction but expresses an essential property". Boethius translated 'postulate' as petitio and called the axioms notiones communes but in later manuscripts this usage was not always strictly kept.

## Historical development

### Early Greeks

The logico-deductive method whereby conclusions (new knowledge) follow from premises (old knowledge) through the application of sound arguments (syllogisms, rules of inference), was developed by the ancient Greeks, and has become the core principle of modern mathematics. Tautologies excluded, nothing can be deduced if nothing is assumed. Axioms and postulates are the basic assumptions underlying a given body of deductive knowledge. They are accepted without demonstration. All other assertions (theorems, if we are talking about mathematics) must be proven with the aid of these basic assumptions. However, the interpretation of mathematical knowledge has changed from ancient times to the modern, and consequently the terms axiom and postulate hold a slightly different meaning for the present day mathematician, than they did for Aristotle and Euclid.

The ancient Greeks considered geometry as just one of several sciences, and held the theorems of geometry on par with scientific facts. As such, they developed and used the logico-deductive method as a means of avoiding error, and for structuring and communicating knowledge. Aristotle's posterior analytics is a definitive exposition of the classical view.

An â€œaxiomâ€�, in classical terminology, referred to a self-evident assumption common to many branches of science. A good example would be the assertion that When an equal amount is taken from equals, an equal amount results.

At the foundation of the various sciences lay certain additional hypotheses which were accepted without proof. Such a hypothesis was termed a postulate. While the axioms were common to many sciences, the postulates of each particular science were different. Their validity had to be established by means of real-world experience. Indeed, Aristotle warns that the content of a science cannot be successfully communicated, if the learner is in doubt about the truth of the postulates.

The classical approach is well illustrated by Euclid's Elements, where a list of postulates is given (common-sensical geometric facts drawn from our experience), followed by a list of "common notions" (very basic, self-evident assertions).

;Postulates
# It is possible to draw a straight line from any point to any other point.
# It is possible to extend a line segment continuously in a straight line.
# It is possible to describe a circle with any center and any radius.
# It is true that all right angles are equal to one another.
# ("Parallel postulate") It is true that, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, From Yahoo Answers

Question:I would like to know an alternate way to illustrate the Angle Addition Postulate. I've already made a typical acute angle with points ABC, and with D sticking out diagonally from point B, the vertex. But I would like another one or two samples of different kinds of angles (not acute) Thank you.

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Answers:When two angles share a common ray, say AB, where A is the vertex of both angles, and the angles are on opposite sides of AB (for example angle CAB is opening on the left side of AB and angle DAB opening on the right side of AB), then angle CAD = angle CAB + angle ABD. If B is on segment AC and between A and B then the sum of the lengths AB and BC is equal to the length of segment AC. AB+BC+AC. Both are really very obvious results similar to saying if I split a pile of rocks into two piles of rocks then the the number of rocks in the original pile is equal to the sum of the number of rocks in the two piles created from the first pile.

Question:What is the definition of an addition postulate? I don't need examples of problems because I understand that part.

Answers:Please clarify with the type of addition postulate. For example, Definition of Angle Addition Postulate Angle Addition Postulate states that if a point S lies in the interior of PQR, then PQS + SQR = PQR. ----------------------- Segment Addition Postulate If B is between A and C, then AB + BC = AC. There are several others.

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