Editing Axiom (section)

===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. [[tautology (logic)|Tautologies]] excluded, nothing can be deduced if nothing is assumed. Axioms and postulates are thus the basic assumptions underlying a given body of deductive knowledge. They are accepted without demonstration. All other assertions ([[theorem]]s, in the case of 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]].<ref name=":0" />

The ancient Greeks considered [[geometry]] as just one of several [[science]]s, 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.<ref>{{Cite web |date=2024-10-08 |title=Aristotle {{!}} Biography, Works, Quotes, Philosophy, Ethics, & Facts {{!}} Britannica |url=https://www.britannica.com/biography/Aristotle |access-date=2024-11-14 |website=www.britannica.com |language=en}}</ref>

An "axiom", in classical terminology, referred to a [[self-evident]] assumption common to many branches of science. A good example would be the assertion that:
<blockquote>When an equal amount is taken from equals, an equal amount results.</blockquote>

At the foundation of the various sciences lay certain additional [[Hypothesis|hypotheses]] that 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. Aristotle warns that the content of a science cannot be successfully communicated if the learner is in doubt about the truth of the postulates.<ref>Aristotle, Metaphysics Bk IV, Chapter 3, 1005b "Physics also is a kind of Wisdom, but it is not the first kind. – And the attempts of some of those who discuss the terms on which truth should be accepted, are due to want of training in logic; for they should know these things already when they come to a special study, and not be inquiring into them while they are listening to lectures on it." W.D. Ross translation, in The Basic Works of Aristotle, ed. Richard McKeon, (Random House, New York, 1941)</ref>

The classical approach is well-illustrated{{efn|Although not complete; some of the stated results did not actually follow from the stated postulates and common notions.}} by [[Euclid's Elements|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 both directions.
:# It is possible to describe a [[circle]] with any center and any radius.
:# It is true that all [[right angle]]s are equal to one another.
:# ("[[Parallel postulate]]") It is true that, if a straight line falling on two straight lines make the [[polygon|interior angles]] on the same side less than two right angles, the two straight lines, if produced indefinitely, [[Line-line intersection|intersect]] on that side on which are the [[angle]]s less than the two right angles.

:;Common notions:
:# Things which are equal to the same thing are also equal to one another.
:# If equals are added to equals, the wholes are equal.
:# If equals are subtracted from equals, the remainders are equal.
:# Things which coincide with one another are equal to one another.
:# The whole is greater than the part.