View source for Axiom

{{also|Axiom}}
==English==
{{was wotd|2017|March|16}}

===Etymology===
{{root|en|ine-pro|*h₂eǵ-}}
[[File:Euclid statue, Oxford University Museum of Natural History, UK - 20080315.jpg|thumb|A statue honouring the Greek [[mathematician]] {{w|Euclid}} (''[[floruit|fl.]]'' 300 [[BCE|{{smallcaps|b.c.e.}}]]) at the {{w|Oxford University Museum of Natural History}} in [[Oxford]], [[Oxfordshire]], [[England]], UK]]
[[File:Parallel postulate en.svg|thumb|One of Euclid’s [[distinctive]] [[postulate]]s, or axioms, is the [[parallel postulate]], which states that if a [[line segment]] [[intersect]]s two straight lines forming two [[interior angle]]s ''(α and β in the diagram above)'' on the same side that sum to less than two [[right angle]]s, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles]]
From {{der|en|frm|axiome}} in the 15th century, from {{der|en|la|axiōma||axiom; [[principle]]}}, from {{der|en|grc|ἀξίωμα||that which is [[think|thought]] to [[fit#Verb|fit]], a [[requisite]], that which a [[pupil]] is required to [[know]] [[beforehand]], a [[self-evident]] principle}}, from {{m|grc|ἀξιόω||to think fit or [[worthy]], to [[require]], to [[demand#Verb|demand]]}}, from {{m|grc|ἄξιος||fit, worthy|lit=[[weigh]]ing as much as; of like [[value#Noun|value]]}}, from {{m|grc|ἄγω||to weigh (down)}}.

===Pronunciation===
* {{IPA|en|/ˈak.sɪ.əm/|a=RP}}
* {{enPR|ăk'sēəm|a=GA}}, {{IPA|en|/ˈæk.si.əm/}}
* {{q|obsolete}} {{enPR|ăk'shəm}}, {{IPA|en|/ˈæk.ʃəm/}}<ref>{{R:Critical Pronouncing Dictionary|Axiom|124|ppage=1}}</ref>
* {{audio|en|en-us-axiom.ogg|a=GA}}
* {{audio|en|en-au-axiom.ogg|a=AU}}
* {{hyphenation|en|ax|i|om}}

===Noun===
{{examples|sense=mathematics|examples=Through a pair of [[distinct]] [[point#Noun|points]] there passes exactly one [[straight line]].<br />All [[right angle]]s are [[congruent]].}}
{{en-noun|s|axiomata}} (the latter is becoming less common and is sometimes considered archaic)

# {{lb|en|philosophy}} A [[seemingly]] [[self-evident]] or [[necessary]] [[truth]] which is based on [[assumption]]; a [[principle]] or [[proposition]] which cannot actually be [[prove]]d or [[disprove]]d.<ref>{{R:Century 1911}}</ref><ref>{{R:Webster 1913}}</ref>
#* {{quote-journal|en|author=R. M.|editor=“Sylvanus Urban” [pseudonym; {{w|Edward Cave}}]|title=To the Gent. who Signs Verax, V[olume] 17 p[age] 573. In Answer to His Defence of Mr Lyttelton's Expression, that Matter is not Inherent in the Deity.|magazine=[[w:The Gentleman's Magazine|The Gentleman's Magazine, and Historical Chronicle]]|location=London|publisher=Printed by Edw[ard] Cave, at [[w:St John's Gate, Clerkenwell|St John's Gate]]|month=January|year=1748|volume=XVIII|page=15, column 2|pageurl=https://books.google.com/books?id=DHJIAAAAYAAJ&pg=PA15|oclc=192374019|passage=Neither can I reconcile this opinion of yours, with your argument brought from reaſon; if the '''axiom''' there laid down by you be true, it follows that, when matter began to exiſt in the divine mind, either matter became of the nature of the divine mind, ''i.e.'' active and intelligent, or elſe the divine mind became of the nature of matter, ''i.e.'' inert and unintelligent: this is a hard dilemma; have we not reaſon to ſuſpect that '''axiom'''?}}
#* {{quote-book|en|author=w:William Enfield|chapter=Chapter VIII. Of the Academic Sect. Section I. Of Plato and His Philosophy.|title=The History of Philosophy, from the Earliest Periods: Drawn from [[w:Johann Jakob Brucker|Brucker]]'s Historia Critica Philosophiæ|location=London|publisher=Printed for {{w|Thomas Tegg}} and Son, 73, {{w|Cheapside}}; [[w:Richard Griffin and Company|R[ichard] Griffin and Co.]], [[Glasgow]]; Tegg and Co., [[Dublin]]; also, J. and S. A. Tegg, [[Sydney]] and [[Hobart|Hobart Town]]|year=1837|section=book II|pages=128–129|pageurl=https://books.google.com/books?id=LYYuAAAAYAAJ&pg=PA|oclc=867600514|passage={{smallcaps|Theoretical}} philosophy [[Plato]] divides into three branches, {{smallcaps|Theological, Physical}}, and {{smallcaps|Mathematical}}. On {{smallcaps|Theology}}, the fundamental doctrine of Plato, as of all other ancient philosophers, is, that from nothing nothing can proceed. This universal '''axiom''', applied not only to the infinite efficient, but to the material cause, Plato, in his [[w:Timaeus (dialogue)|Timæus]], lays down as the ground of his reasoning concerning the origin of the world.}}
#* {{quote-journal|en|year=1963|month=February|title=Nobody runs this railway, mate|journal=Modern Railways|page=73|text=Any publicity, runs the '''axiom''', is good publicity.}}
#* {{quote-book|1=en|year=1981|author=w:William Irwin Thompson|title=The Time Falling Bodies Take to Light: Mythology, Sexuality and the Origins of Culture|publisher=Rider/Hutchinson & Co.|location=London|page=16|passage="As above, so below" is an '''axiom''' from Hermetic mysticism, and in this Hermetic vision of physiology the tongue is connected through the spinal column to the penis.}}
#* {{quote-book|en|author=w:Bertrand Russell|editor=Charles R. Pigden|title=Russell on Ethics: Selections from the Writings of Bertrand Russell|location=London; New York, N.Y.|publisher=w:Routledge|year=1999|isbn=978-0-415-15659-2|passage=Can we then find '''axioms''' as self-evident as those of Arithmetic, on which we can build as on a sure foundation, which could be shaken only by a scepticism which should attack the whole fabric of our knowledge?}}
# {{lb|en|logic|mathematics|proof theory}} A [[fundamental]] [[assumption]] that serves as a basis for [[deduction]] of [[theorem]]s; a [[postulate]] (sometimes distinguished from postulates as being [[universal]]ly applicable, whereas postulates are [[particular]] to a certain science or context).
#* {{quote-book|en|author=“Philalethes Cantabrigiensis” [pseudonym; {{w|James Jurin}}]|title=Geometry No Friend to Infidelity: Or, A Defence of Sir {{w|Isaac Newton}} and the British Mathematicians, in a Letter to the Author of {{w|The Analyst}}|location=London|publisher=Printed for T. Cooper at the Globe in Ivy-Lane|date=10 April 1734|page=28|pageurl=https://books.google.com/books?id=OydcAAAAQAAJ&pg=PA28|oclc=745184450|passage={{...}} Geometry, ''an excellent Logic,'' as you obſerve, ''where the definitions are clear, where the Poſtulata cannot be refuſed, nor the '''Axioms''' denied''; {{...}}}}
#* {{quote-book|en|author=w:Colin McLarty|chapter=Rudimentary Structures in a Category|title=Elementary Categories, Elementary Toposes|series=Oxford Logic Guides|seriesvolume=21|location=Oxford|publisher=[[w:Oxford University Press|Clarendon Press]]; New York, N.Y.: {{w|Oxford University Press}}|year=1992|page=13|pageurl=https://books.google.com/books?id=V8cON1x39bIC&pg=PA13|isbn=978-0-19-853392-4|passage=The '''axioms''' read as follows. For every composable pair ''f'' and ''g'' the composite <math>f \circ g</math> goes from the domain of ''g'' to the codomain of ''f''. For each object ''A'' the identity arrow <math>1_A</math> goes from ''A'' to ''A''. Composing any arrow with an identity arrow (supposing that the two are composable) gives the original arrow. And composition is associative.}}
# An [[established]] [[principle]] in some [[artistic]] [[practice]] or [[science]] that is [[universally]] [[receive]]d.
#: {{ux|en|The '''axioms''' of political economy cannot be considered absolute truths.}}
#* {{quote-book|en|author=w:Giovanni Bianchi|title=A Dissertation against Blisters, Delivered in a Speech, before the [[w:Accademia dei Lincei|Lyncean Academy]] at Rimino, in June 1746|location=London|publisher=Printed by M. Cooper, at the Globe in [[w:Paternoster Row|Paternoster-Row]], M. Sheepy, under the [[w:Royal Exchange, London|Royal Exchange]] [[w:Cornhill, London|Cornhill]]; and J. Swan, opposite to [[w:Northumberland House|Northumberland-House]] by [[w:Charing Cross|Charing-Cross]]|year=1751|page=40|pageurl=https://books.google.com/books?id=cfCSKYJ3zVoC&pg=PA40|oclc=915390042|passage=But these innovating Medicaſters have introduced a Practice not only very precarious, but in many Reſpects extremely dangerous, and quite devoid of any one of the Qualities which conſtitute a good Remedy, ''viz.'' to cure the Patient, as the '''Axiom''' has it, ''cito, tuto, & jucunde, i.e.'' ſpeedily, ſafely, and pleaſantly.}}
#* {{quote-journal|en|title=To the Christian Judge Bailey|newspaper=The Republican|publisher=Printed and published by [[w:Richard Carlile|R[ichard] Carlile]], 55, {{w|Fleet Street}}|date=18 January 1822|volume=V|issue=3|page=337|pageurl=https://books.google.com/books?id=4hsrAAAAYAAJ&pg=PA337|oclc=7129024|passage=That there is an incomprehended power in Nature, is an '''axiom''' to which all must assent: but what that power is must be reduced to an '''axiom''' likewise, before any defence of prophecy, miracle, or any kind of superstition, can be made on solid grounds.}}
#* {{quote-book|en|author=[[w:Alexander Campbell (clergyman)|A[lexander] Campbell]]|chapter=Remission of Sins|title=A Connected View of the Principles and Rules by which the {{w|Living Oracles}} may be Intelligibly and Certainly Interpreted: of the Foundation on which All Christians may Form One Communion: and of the Capital Positions Sustained in the Attempt to Restore the Original Gospel and Order of Things; Containing the Principal Extras of the {{w|Millennial Harbinger}}, Revised and Corrected|location=Bethany, Va.|publisher=Printed and published by M'Vay and Ewing|year=1835|pages=252–253|pageurl=https://books.google.com/books?id=rmhBAQAAMAAJ&pg=PA253|oclc=3867659|passage=We proceed upon these as our '''axiomata''' in all our reasonings, preachings, writings—1st. unfeigned faith; 2d. a good conscience; 3d. a pure heart; 4th. love. The testimony of God apprehended produces unfeigned or genuine faith; faith obeyed, produces a good conscience. This [[Saint Peter|Peter]] defines to be the use of baptism, the answer of a good conscience. This produces a pure heart, and then the consummation is love—love to God and man.}}
#* {{quote-book|en|author=&#91;[[w:Catherine Gore|Catherine Grace Frances Gore]]&#93;|chapter=chapter IV|title=The Cabinet Minister. [...] In Three Volumes|location=London|publisher=[[w:Richard Bentley (publisher)|Richard Bentley]], {{w|New Burlington Street}}|year=1839|volume=I|pages=50–51|pageurl=https://books.google.com/books?id=K69xbo_6hU4C&pg=PA51|oclc=3574003|passage=For a moment Frank recoiled, with a young man's antipathy, from the idea of his sister turning out a ''femme savante''; but having fortunately retained the '''axiom''' that "there is no offence in [[w:bluestocking|blue stockings]] provided the petticoats are long enough to hide them," {{...}} he rejoiced that, doomed to live with a foolish old woman like her aunt, and a knot of stupid country neighbours, his sister had provided for herself in the old library a host of invaluable acquaintances, with whom she could live, and move, and have her being.}}

====Synonyms====
* {{sense|philosophy, mathematics}} {{l|en|axioma}} {{qualifier|now rare}}
* {{sense|logic, mathematics}} {{l|en|postulate}}

====Hypernyms====
* {{sense|logic}} {{l|en|well-formed formula}}, {{l|en|wff}}, {{l|en|WFF}}

====Hyponyms====
* {{sense|mathematics}} {{l|en|axiom of choice}}, {{l|en|axiom of infinity}}, {{l|en|axiom of pairing}}, {{l|en|axiom of power set}}, {{l|en|axiom of regularity}}, {{l|en|axiom of union}}, {{l|en|completeness axiom}}, {{l|en|parallel axiom}},

====Holonyms====
* {{sense|logic}} {{l|en|formal system}}

====Derived terms====
{{col|en|axiomic|axiomless|axiomlike|axiomatic|axiomatical|axiomatically|axiomatise,axiomatize|axiomatisation,axiomatization|Pasch's axiom|Peano axiom|Wightman axiom|axiom of countable choice|axiom of extensionality|axiom schema|axiom scheme|axiom system|field axiom|non-aggression axiom|nonaggression axiom}}

====Translations====
{{trans-top|philosophy: supposed self‐evident or necessary truth}}
* Albanian: {{t+|sq|aksiomë|f}}
* Arabic: {{t|ar|بَدِيهِيَّة|f}}
* Armenian: {{t+|hy|աքսիոմ}}
* Azerbaijani: {{t+|az|aksiom}}
* Belarusian: {{t|be|аксіо́ма|f}}, {{t|be|аксіёма|f}}
* Bulgarian: {{t+|bg|аксио́ма|f}}
* Catalan: {{t+|ca|axioma|m}}
* Chinese:
*: Mandarin: {{t+|cmn|公理|tr=gōnglǐ}}
* Czech: {{t+|cs|axiom|m}}
* Danish: {{t|da|aksiom|n}}
* Dutch: {{t+|nl|axioma|n}}
* Esperanto: {{t+|eo|aksiomo}}
* Estonian: {{t|et|aksioom}}
* Finnish: {{t|fi|perusväite}}, {{t|fi|aksiomi}}
* French: {{t+|fr|axiome|m}}
* Galician: {{t|gl|axioma|m}}
* Georgian: {{t|ka|აქსიომა}}
* German: {{t+|de|Axiom|n}}, {{t+|de|Annahme|f}}
* Greek: {{t+|el|αξίωμα|n}}
*: Ancient: {{t|grc|ἀξίωμα|n}}
* Hebrew: {{t+|he|אַקְסְיוֹמָה|f|tr=aksyóma}}
* Hindi: {{t+|hi|अभिगृहीत|f}}, {{t+|hi|स्वयंसिद्ध|f}}
* Hungarian: {{t+|hu|axióma}}
* Icelandic: {{t|is|frumsannindi|n-p}}, {{t|is|grundvallarregla|f}}, {{t|is|óræk vissa}}
* Ido: {{t+|io|axiomo}}
* Indonesian: {{t+|id|aksioma}}
* Irish: {{t|ga|aicsím|f}}
* Italian: {{t+|it|assioma|m}}
* Japanese: {{t+|ja|公理|tr=こうり, kōri}}
* Kazakh: {{t|kk|аксиома}}
* Khmer: {{t|km|ស្វ័យសត្យ}}
* Korean: {{t+|ko|공리(公理)}}
* Kyrgyz: {{t+|ky|аксиома}}
* Lao: {{t|lo|ກົດຄວາມຈິງ}}
* Latvian: {{t|lv|aksioma|f}}
* Lithuanian: {{t|lt|aksioma|f}}
* Malay: {{t+|ms|aksiom}}
* Mongolian:
*: Cyrillic: {{t|mn|аксиом}}
* Norwegian:
*: Bokmål: {{t+|nb|aksiom|n}}
*: Nynorsk: {{t|nn|aksiom|n}}
* Polish: {{t+|pl|aksjomat|m}}, {{t+|pl|pewnik|m}}
* Portuguese: {{t+|pt|axioma|m}}
* Romanian: {{t+|ro|axiomă|f}}
* Russian: {{t+|ru|аксио́ма|f}}
* Serbo-Croatian:
*: Cyrillic: {{t|sh|аксѝо̄м|m}}
*: Roman: {{t+|sh|aksìōm|m}}
* Slovak: {{t|sk|axióma|f}}
* Slovene: {{t+|sl|aksiom|m}}
* Spanish: {{t+|es|axioma|m}}
* Swedish: {{t+|sv|axiom|n}}
* Tagalog: {{t|tl|batlain}}
* Thai: {{t+|th|สัจพจน์}}, {{t|th|ฐานบท}}
* Turkish: {{t+|tr|aksiyom}}
* Ukrainian: {{t+|uk|аксіо́ма|f}}
* Uzbek: {{t+|uz|aksioma}}
* Vietnamese: {{t+|vi|chân lý}}, {{t+|vi|sự thật}}, {{t+|vi|chân lí}}
* Volapük: {{t+|vo|xiom}}
{{trans-bottom}}

{{trans-top|mathematics: fundamental assumption}}
* Armenian: {{t+|hy|աքսիոմ}}
* Catalan: {{t+|ca|axioma|m}}
* Chinese:
*: Mandarin: {{t+|cmn|公理|tr=gōnglǐ}}
* Czech: {{t+|cs|axiom|m}}
* Danish: {{t|da|aksiom|n}}
* Dutch: {{t+|nl|axioma|n}}
* Finnish: {{t|fi|aksiomi}}, {{t+|fi|peruslause}}
* French: {{t+|fr|axiome|m}}
* German: {{t+|de|Axiom|n}}, {{t+|de|Postulat|n}}, {{t+|de|Grundannahme|f}}, {{t+|de|Ausgangspunkt|m}}
* Greek: {{t+|el|αξίωμα|n}}
*: Ancient: {{t|grc|ἀξίωμα|n}}
* Hungarian: {{t+|hu|axióma}}
* Icelandic: {{t|is|frumsenda|f}}, {{t|is|frumsetning|f}}
* Ido: {{t+|io|axiomo}}
* Indonesian: {{t+|id|aksioma}}
* Irish: {{t|ga|aicsím|f}}
* Italian: {{t+|it|assioma|m}}
* Japanese: {{t+|ja|公理|tr=こうり, kōri}}
* Korean: {{t+|ko|공리(公理)}}
* Latin: {{t|la|axiōma|n}}
* Middle English: {{t|enm|axiom}}
* Middle French: {{t|frm|axiome}}
* Norwegian:
*: Bokmål: {{t+|nb|aksiom|n}}
*: Nynorsk: {{t|nn|aksiom|n}}
* Old French: {{t|fro|axiome}}
* Polish: {{t+|pl|aksjomat|m}}, {{t+|pl|pewnik|m}}
* Portuguese: {{t+|pt|axioma|m}}
* Russian: {{t+|ru|аксио́ма|f}}
* Serbo-Croatian:
*: Cyrillic: {{t|sh|аксѝо̄м|m}}
*: Roman: {{t+|sh|aksìōm|m}}
* Slovak: {{t|sk|axióma|f}}
* Slovene: {{t+|sl|aksiom|m}}
* Spanish: {{t+|es|axioma|m}}
* Swedish: {{t+|sv|axiom|n}}
* Tagalog: {{t|tl|batlain}}
* Thai: {{t+|th|สัจพจน์}}
{{trans-bottom}}

{{trans-top|established principle in art or science}}
* Catalan: {{t+|ca|axioma|m}}
* Danish: {{t|da|aksiom|n}}
* Finnish: {{t+|fi|aksiooma}}
* German: {{t+|de|Grundannahme|f}}, {{t+|de|Grundsatz|m}}, {{t+|de|Leitsatz|m}}
* Irish: {{t|ga|aicsím|f}}
* Norwegian:
*: Bokmål: {{t+|nb|aksiom|n}}
*: Nynorsk: {{t|nn|aksiom|n}}
* Polish: {{t+|pl|pewnik|m}}
* Portuguese: {{t+|pt|axioma|m}}
* Serbo-Croatian:
*: Cyrillic: {{t|sh|аксѝо̄м|m}}
*: Roman: {{t+|sh|aksìōm|m}}
* Spanish: {{t+|es|axioma|m}}
* Swedish: {{t+|sv|axiom|n}}
{{trans-bottom}}

====See also====
{{col|en|title=other terms of interest
|conjecture
|corollary
|demonstration
|hypothesis
|law
|lemma
|porism
|postulate
|premise
|principle
|proof
|proposition
|theorem
|theory
|truism
}}

===References===
<references/>

===Further reading===
* {{pedia}}

===Anagrams===
* {{anagrams|en|a=aimox|amoxi}}

==Czech==

===Pronunciation===
* {{cs-IPA}}

===Noun===
{{cs-noun|m-in}}

# {{l|en|axiom}}

====Declension====
{{cs-ndecl|m}}

====Derived terms====
{{col|cs
|axiom výběru<g:m>
}}

====Related terms====
{{col|cs|axiomatický|axiomatizovat|axiomatizace}}

{{C|cs|Logic|Mathematics}}

==Swedish==

===Noun===
{{sv-noun|n}}

# {{l|en|axiom}}

====Declension====
{{sv-infl-noun-n-zero}}

====Related terms====
* {{l|sv|axiomatisk}}