Editing Mathematical proof (section)

{{Short description|Reasoning for mathematical statements}}
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[[File:P. Oxy. I 29.jpg|thumb|upright=1.3|[[Papyrus Oxyrhynchus 29|P. Oxy. 29]], one of the oldest surviving fragments of [[Euclid]]'s ''[[Euclid's Elements|Elements]]'', a textbook used for millennia to teach proof-writing techniques. The diagram accompanies Book II, Proposition 5.<ref>{{cite web
|url=http://www.math.ubc.ca/~cass/Euclid/papyrus/papyrus.html
|title=One of the Oldest Extant Diagrams from Euclid
|author=Bill Casselman
|author-link=Bill Casselman (mathematician)
|publisher=University of British Columbia
|access-date=September 26, 2008
}}</ref>]]
A '''mathematical proof''' is a [[deductive reasoning|deductive]] [[Argument-deduction-proof distinctions|argument]] for a [[Proposition|mathematical statement]], showing that the stated assumptions [[logic]]ally guarantee the conclusion. The argument may use other previously established statements, such as [[theorem]]s; but every proof can, in principle, be constructed using only certain basic or original assumptions known as [[axiom]]s,<ref>{{cite book |author1=Clapham, C.  |author2=Nicholson, J.N.  |name-list-style=amp | title = The Concise Oxford Dictionary of Mathematics, Fourth edition |quote = A statement whose truth is either to be taken as self-evident or to be assumed. Certain areas of mathematics involve choosing a set of axioms and discovering what results can be derived from them, providing proofs for the theorems that are obtained.}}</ref><ref name="nutsandbolts">{{cite book|title=The Nuts and Bolts of Proofs: An Introduction to Mathematical Proofs |last=Cupillari |first=Antonella|author-link= Antonella Cupillari |edition=Third |year=2005 |orig-year=2001 |publisher=[[Academic Press]] |isbn=978-0-12-088509-1 |page=3}}</ref><ref>{{cite book|title=Discrete Mathematics with Proof |date=July 2009 |first=Eric |last=Gossett |page=86 |quote=Definition 3.1. Proof: An Informal Definition |publisher=[[Wiley (publisher)|John Wiley & Sons]] |isbn=978-0470457931}}</ref> along with the accepted rules of [[inference]]. Proofs are examples of exhaustive [[deductive reasoning]] that establish logical certainty, to be distinguished from [[empirical evidence|empirical]] arguments or non-exhaustive [[inductive reasoning]] that establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in ''all'' possible cases. A proposition that has not been proved but is believed to be true is known as a [[conjecture]], or a hypothesis if frequently used as an assumption for further mathematical work.

Proofs employ [[logic]] expressed in mathematical symbols, along with [[natural language]] that usually admits some ambiguity. In most mathematical literature, proofs are written in terms of [[Rigour#Mathematics|rigorous]] [[informal logic]]. Purely [[formal proof]]s, written fully in [[Symbolic language (mathematics)|symbolic language]] without the involvement of natural language, are considered in [[proof theory]]. The distinction between [[Proof theory#Formal and informal proof|formal and informal proofs]] has led to much examination of current and historical [[mathematical practice]], [[quasi-empiricism in mathematics]], and so-called [[Mathematical folklore|folk mathematics]], oral traditions in the mainstream mathematical community or in other cultures. The [[philosophy of mathematics]] is concerned with the role of language and logic in proofs, and [[mathematics as a language]].