![]() 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. Purely formal proofs, written fully in symbolic language without the involvement of natural language, are considered in proof theory. In most mathematical literature, proofs are written in terms of rigorous informal logic. Proofs employ logic expressed in mathematical symbols, along with natural language which usually admits some ambiguity. 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. ![]() 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. 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". 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, along with the accepted rules of inference. Ī mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The diagram accompanies Book II, Proposition 5. ![]() ![]() 29, one of the oldest surviving fragments of Euclid's Elements, a textbook used for millennia to teach proof-writing techniques. ![]()
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