Editing Axiom (section)

{{short description|Statement that is taken to be true}}
{{distinguish|axion|axon}}
{{redirect-several|dab=no|Axiom (disambiguation)|Axiomatic (disambiguation)|Postulation (algebraic geometry)}}
{{Use dmy dates|date=December 2020}}

An '''axiom''', '''postulate''', or '''assumption''' is a [[statement (logic)|statement]] that is taken to be [[truth|true]], to serve as a [[premise]] or starting point for further reasoning and arguments. The word comes from the [[Ancient Greek]] word {{wikt-lang|grc|ἀξίωμα}} ({{grc-transl|ἀξίωμα}}), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'.<ref>Cf. axiom, n., etymology. ''Oxford English Dictionary'', accessed 2012-04-28.</ref><ref>{{Cite book |url=https://www.oxfordreference.com/display/10.1093/acref/9780195392883.001.0001/m_en_us1224100 |title=New Oxford American Dictionary |publisher=Oxford University Press |date=2015 |isbn=9780199891535 |editor-last=Stevenson |editor-first=Angus |edition=3rd |doi=10.1093/acref/9780195392883.001.0001 |quote=a statement or proposition that is regarded as being established, accepted, or self-evidently true |editor-last2=Lindberg |editor-first2=Christine A. |url-access=subscription}}</ref>

The precise [[definition]] varies across fields of study. In [[classic philosophy]], an axiom is a statement that is so [[Self-evidence|evident]] or well-established, that it is accepted without controversy or question.<ref>"A proposition that commends itself to general acceptance; a well-established or universally conceded principle; a maxim, rule, law" axiom, n., definition 1a. ''Oxford English Dictionary'' Online, accessed 2012-04-28. Cf. Aristotle, ''[[Posterior Analytics]]'' I.2.72a18-b4.</ref><!-- HIDDEN UNTIL SOURCED —it is better known and more firmly believed than the conclusion.{{citation needed|date=May 2012}}--> In modern [[logic]], an axiom is a premise or starting point for reasoning.<ref>"A proposition (whether true or false)" axiom, n., definition 2. ''Oxford English Dictionary'' Online, accessed 2012-04-28.</ref>

In [[mathematics]], an ''axiom'' may be a "[[#Logical axioms|logical axiom]]" or a "[[#Non-logical axioms|non-logical axiom]]". Logical axioms are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (''A'' and ''B'') implies ''A''), while non-logical axioms are substantive assertions about the elements of the domain of a specific mathematical theory, for example ''a''&nbsp;+&nbsp;0&nbsp;=&nbsp;''a'' in integer arithmetic.

Non-logical axioms may also be called "postulates", "assumptions" or "proper axioms".<ref name="properaxioms" /> In most cases, a non-logical axiom is simply a formal logical expression used in deduction to build a mathematical theory, and might or might not be self-evident in nature (e.g., the [[parallel postulate]] in [[Euclidean geometry]]). 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), and there are typically many ways to axiomatize a given mathematical domain.

Any axiom is a statement that serves as a starting point from which other statements are logically derived. Whether it is meaningful (and, if so, what it means) for an axiom to be "true" is a subject of debate in the [[philosophy of mathematics]].<ref>See for example {{cite journal|first=Penelope|last=Maddy|journal=Journal of Symbolic Logic|title=Believing the Axioms, I|volume=53|issue=2|date=Jun 1988|pages=481–511|doi=10.2307/2274520|jstor=2274520}} for a [[mathematical realism|realist]] view.</ref>