Editing Mathematical proof (section)

==History and etymology==
{{See also|History of logic}}

The word ''proof'' derives from the Latin {{lang|la|probare}} 'to test'; related words include English ''probe'', ''probation'', and ''probability'', as well as Spanish {{lang|es|probar}} 'to taste' (sometimes 'to touch' or 'to test'),<ref>"proof" New Shorter Oxford English Dictionary, 1993, OUP, Oxford.</ref> Italian {{tlit|it|provare}} 'to try', and German {{lang|de|probieren}} 'to try'. The legal term ''probity'' means authority or credibility, the power of testimony to prove facts when given by persons of reputation or status.<ref>{{cite book|title = The Emergence of Probability: A Philosophical Study of Early Ideas about Probability, Induction and Statistical Inference|first = Ian|last = Hacking|author-link=Ian Hacking |publisher = [[Cambridge University Press]] |year= 1984|orig-year=1975 |isbn=978-0-521-31803-7}}</ref>

Plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof.<ref name="Krantz"/> It is likely that the idea of demonstrating a conclusion first arose in connection with [[geometry]], which originated in practical problems of land measurement.<ref>{{cite book|title=The development of logic |first1=William |last1=Kneale |first2=Martha |last2=Kneale |author-link1=William Kneale (logician) |date=May 1985 |orig-year=1962 |page=3 |edition=New |publisher=[[Oxford University Press]] |isbn=978-0-19-824773-9}}</ref> The development of mathematical proof is primarily the product of [[Greek mathematics|ancient Greek mathematics]], and one of its greatest achievements.<ref>{{Cite web|url=https://hal.archives-ouvertes.fr/hal-01281050/document|title=The genesis of proof in ancient Greece The pedagogical implications of a Husserlian reading|last1=Moutsios-Rentzos|first1=Andreas|last2=Spyrou|first2=Panagiotis|date=February 2015|website=Archive ouverte HAL|access-date=October 20, 2019}}</ref> [[Thales]] (624–546&nbsp;BCE) and [[Hippocrates of Chios]] (c.&nbsp;470–410 BCE) gave some of the first known proofs of theorems in geometry. [[Eudoxus of Cnidus|Eudoxus]] (408–355&nbsp;BCE) and [[Theaetetus (mathematician)|Theaetetus]] (417–369&nbsp;BCE) formulated theorems but did not prove them. [[Aristotle]] (384–322&nbsp;BCE) said definitions should describe the concept being defined in terms of other concepts already known.

Mathematical proof was revolutionized by [[Euclid]] (300&nbsp;BCE), who introduced the [[axiomatic method]] still in use today. It starts with [[undefined term]]s and [[axiom]]s, propositions concerning the undefined terms which are assumed to be self-evidently true (from Greek {{tlit|grc|axios}} 'something worthy'). From this basis, the method proves theorems using [[deductive logic]]. ''[[Euclid's Elements]]'' was read by anyone who was considered educated in the West until the middle of the 20th century.<ref>{{cite book|title=An Introduction to the History of Mathematics (Saunders Series) |first=Howard W. |last=Eves |author-link=Howard Eves |edition=6th |date=January 1990 |orig-year=1962 |page=141 |quote=No work, except The Bible, has been more widely used... |publisher=Cengage |isbn=978-0030295584}}</ref> In addition to theorems of geometry, such as the [[Pythagorean theorem]], the ''Elements'' also covers [[number theory]], including a proof that the [[square root of two]] is [[irrational number|irrational]] and a proof that there are infinitely many [[prime number]]s.

Further advances also took place in [[Mathematics in medieval Islam|medieval Islamic mathematics]]. In the 10th century, the Iraqi mathematician [[Abu Ali Muhammad ibn Abd al-Aziz al-Hashimi|Al-Hashimi]] worked with numbers as such, called "lines" but not necessarily considered as measurements of geometric objects, to prove algebraic propositions concerning multiplication, division, etc., including the existence of [[irrational number]]s.<ref>{{citation|last=Matvievskaya|first=Galina|year=1987|title=The Theory of Quadratic Irrationals in Medieval Oriental Mathematics|journal=[[New York Academy of Sciences|Annals of the New York Academy of Sciences]]|volume=500|issue=1|pages=253–277 [260]|doi=10.1111/j.1749-6632.1987.tb37206.x|bibcode=1987NYASA.500..253M|s2cid=121416910}}</ref> An [[Mathematical induction|inductive proof]] for [[arithmetic progression]]s was introduced in the ''Al-Fakhri'' (1000) by [[Al-Karaji]], who used it to prove the [[binomial theorem]] and properties of [[Pascal's triangle]].

Modern [[proof theory]] treats proofs as inductively defined [[data structure]]s, not requiring an assumption that axioms are "true" in any sense. This allows parallel mathematical theories as formal models of a given intuitive concept, based on alternate sets of axioms, for example [[axiomatic set theory]] and [[non-Euclidean geometry]].