P. Oxy. 29, one of the oldest surviving fragments of Euclid's Elements, a textbook used for millennia to teach proof-writing techniques. The diagram accompanies Book II, Proposition 5.[1]
A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms,[2][3][4] along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which 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 which usually admits some ambiguity. In most mathematical literature, proofs are written in terms of rigorous informal logic. Purely formal proofs, written fully in symbolic language without the involvement of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called 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.
^Bill Casselman. "One of the Oldest Extant Diagrams from Euclid". University of British Columbia. Retrieved September 26, 2008.
^Clapham, C. & Nicholson, J.N. The Concise Oxford Dictionary of Mathematics, Fourth edition. 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.
^Cupillari, Antonella (2005) [2001]. The Nuts and Bolts of Proofs: An Introduction to Mathematical Proofs (Third ed.). Academic Press. p. 3. ISBN 978-0-12-088509-1.
^Gossett, Eric (July 2009). Discrete Mathematics with Proof. John Wiley & Sons. p. 86. ISBN 978-0470457931. Definition 3.1. Proof: An Informal Definition
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