Editing Axiom (section)

==Etymology==
The word ''axiom'' comes from the [[Greek language|Greek]] word {{lang|grc|ἀξίωμα}} (''axíōma''), a [[verbal noun]] from the verb {{lang|grc|ἀξιόειν}} (''axioein''), meaning "to deem worthy", but also "to require", which in turn comes from {{lang|grc|ἄξιος}} (''áxios''), meaning "being in balance", and hence "having (the same) value (as)", "worthy", "proper". Among the [[ancient Greece|ancient Greek]] [[philosopher]]s and [[Greek mathematics|mathematicians]], axioms were taken to be immediately evident propositions, foundational and common to many fields of investigation, and self-evidently true without any further argument or proof.<ref name=":0">{{Cite web|url=http://www.ptta.pl/pef/haslaen/a/axiom.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.ptta.pl/pef/haslaen/a/axiom.pdf |archive-date=2022-10-09 |url-status=live|title=Axiom — Powszechna Encyklopedia Filozofii|website=Polskie Towarzystwo Tomasza z Akwinu}}</ref>

The root meaning of the word ''postulate'' is to "demand"; for instance, [[Euclid]] demands that one agree that some things can be done (e.g., any two points can be joined by a straight line).<ref>Wolff, P. ''Breakthroughs in Mathematics'', 1963, New York: New American Library, pp&nbsp;47–48</ref>

Ancient geometers maintained some distinction between axioms and postulates. While commenting on 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."<ref>{{cite book |author-link=T. L. Heath |last=Heath |first=T. L. |date=1956 |title=The Thirteen Books of Euclid's Elements |location=New York |publisher=Dover |page=200}}</ref> [[Boethius]] translated 'postulate' as ''petitio'' and called the axioms  ''notiones communes'' but in later manuscripts this usage was not always strictly kept.{{citation needed|date=April 2023}}