axiom

A statue honouring the Greek mathematician Euclid (fl. 300 b.c.e.) at the Oxford University Museum of Natural History in Oxford, Oxfordshire, England, UK

One of Euclid’s distinctive postulates, or axioms, is the parallel postulate, which states that if a line segment intersects two straight lines forming two interior angles (α and β in the diagram above) on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles

From Middle French axiome, from Latin axiōma (“axiom; principle”), from Ancient Greek ἀξίωμα (axíōma, “that which is thought to fit, a requisite, that which a pupil is required to know beforehand, a self-evident principle”), from ἀξιόω (axióō, “to think fit or worthy, to require, to demand”), from ἄξιος (áxios, “fit, worthy”, literally “weighing as much as; of like value”), from ἄγω (ágō, “I drive”).

• (Received Pronunciation) IPA^(key): /ˈaks.ɪ.əm/
• (General American) enPR: ăk'sēəm, IPA^(key): /ˈæks.i.əm/

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• Hyphenation: ax‧i‧om

axiom (plural axioms or axiomata) (the latter is becoming less common and is sometimes considered archaic)

1. (philosophy) A seemingly self-evident or necessary truth which is based on assumption; a principle or proposition which cannot actually be proved or disproved.
   • 1748 January, R. M., “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.”, in “Sylvanus Urban” [pseudonym; Edward Cave], editor, The Gentleman's Magazine, and Historical Chronicle, volume XVIII, London: Printed by Edw[ard] Cave, at St John's Gate, OCLC 192374019, page 15, column 2:

     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?

   • 1837, William Enfield, “Chapter VIII. Of the Academic Sect. Section I. Of Plato and His Philosophy.”, in The History of Philosophy, from the Earliest Periods: Drawn from Brucker's Historia Critica Philosophiæ, London: Printed for Thomas Tegg and Son, 73, Cheapside; R[ichard] Griffin and Co., Glasgow; Tegg and Co., Dublin; also, J. and S. A. Tegg, Sydney and Hobart Town, OCLC 867600514, book II, pages 128–129:

     Theoretical philosophy Plato divides into three branches, Theological, Physical, and Mathematical. On 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 Timæus, lays down as the ground of his reasoning concerning the origin of the world.

   • 1999, Bertrand Russell, Charles R. Pigden, editor, Russell on Ethics: Selections from the Writings of Bertrand Russell, London; New York, N.Y.: Routledge, →ISBN:

     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?

2. (logic, mathematics, proof theory) A fundamental assumption that serves as a basis for deduction of theorems; a postulate (sometimes distinguished from postulates as being universally applicable, whereas postulates are particular to a certain science or context).
   • 1734 April 10, “Philalethes Cantabrigiensis” [pseudonym; James Jurin], Geometry No Friend to Infidelity: Or, A Defence of Sir Isaac Newton and the British Mathematicians, in a Letter to the Author of The Analyst, London: Printed for T. Cooper at the Globe in Ivy-Lane, OCLC 745184450, page 28:

     […] Geometry, an excellent Logic, as you obſerve, where the definitions are clear, where the Poſtulata cannot be refuſed, nor the Axioms denied; […]

   • 1992, Colin McLarty, “Rudimentary Structures in a Category”, in Elementary Categories, Elementary Toposes (Oxford Logic Guides; 21), Oxford: Clarendon Press; New York, N.Y.: Oxford University Press, →ISBN, page 13:

     The axioms read as follows. For every composable pair f and g the composite ${\displaystyle f\circ g}$ goes from the domain of g to the codomain of f. For each object A the identity arrow ${\displaystyle 1_{A}}$ 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.

3. An established principle in some artistic practice or science that is universally received.

   The axioms of political economy cannot be considered absolute truths.

   • 1751, Giovanni Bianchi, A Dissertation against Blisters, Delivered in a Speech, before the Lyncean Academy at Rimino, in June 1746, London: Printed by M. Cooper, at the Globe in Paternoster-Row, M. Sheepy, under the Royal Exchange Cornhill; and J. Swan, opposite to Northumberland-House by Charing-Cross, OCLC 915390042, page 40:

     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.

   • 1822 January 18, “To the Christian Judge Bailey”, in The Republican, volume V, number 3, Printed and published by R[ichard] Carlile, 55, Fleet Street, OCLC 7129024, page 337:

     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.

   • 1835, A[lexander] Campbell, “Remission of Sins”, in A Connected View of the Principles and Rules by which the 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 Millenial Harbinger, Revised and Corrected, Bethany, Va.: Printed and published by M'Vay and Ewing, OCLC 3867659, pages 252–253:

     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 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.

   • 1839, [Catherine Grace Frances Gore], “chapter IV”, in The Cabinet Minister. [...] In Three Volumes, volume I, London: Richard Bentley, New Burlington Street, OCLC 3574003, pages 50–51:

     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 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.

• (philosophy, mathematics): axioma (now rare)
• (logic, mathematics): postulate

• (logic): well-formed formula, wff, WFF

• (mathematics): axiom of choice, axiom of infinity, axiom of pairing, axiom of power set, axiom of regularity, axiom of union, completeness axiom, parallel axiom

• (logic): formal system

• axiomatic
• axiomatical
• axiomatically
• axiomatise, axiomatize
• axiomatisation, axiomatization

• axiom in The Century Dictionary, The Century Co., New York, 1911
• axiom in Webster’s Revised Unabridged Dictionary, G. & C. Merriam, 1913.

• axiom on Wikipedia.Wikipedia

axiom m

1. axiom

• axiom výběru m

• axiomatický
• axiomatizovat
• axiomatizace

axiom n

1. axiom

• axiomatisk