Mathematical proof

A mathematical proof is a <a href="/facts/Deductive_reasoning/HadA1zWS">deductive</a> <a href="/facts/Argument-deduction-proof_distinctions/JwHLTCkd">argument</a> for a <a href="/facts/Proposition/0877MFMD">mathematical statement</a>, showing that the stated assumptions <a href="/facts/Logic/8JnpJLtt">logically</a> guarantee the conclusion. The argument may use other previously established statements, such as <a href="/facts/Theorem/f2Pe1FMD">theorems</a>; but every proof can, in principle, be constructed using only certain basic or original assumptions known as <a href="/facts/Axiom/mLjP14Mc">axioms</a>, along with the accepted rules of <a href="/facts/Inference/RBbe95bP">inference</a>. Proofs are examples of exhaustive <a href="/facts/Deductive_reasoning/HadA1zWS">deductive reasoning</a> that establish logical certainty, to be distinguished from <a href="/facts/Empirical_evidence/JUaYSGJc">empirical</a> arguments or non-exhaustive <a href="/facts/Inductive_reasoning/EJ992JJ9">inductive reasoning</a> 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 <a href="/facts/Conjecture/oPWUb7SF">conjecture</a>, or a hypothesis if frequently used as an assumption for further mathematical work.
Proofs employ <a href="/facts/Logic/8JnpJLtt">logic</a> expressed in mathematical symbols, along with <a href="/facts/Natural_language/B4gLS7Kd">natural language</a> that usually admits some ambiguity. In most mathematical literature, proofs are written in terms of <a href="/facts/Rigour/v23FKfhw">rigorous</a> <a href="/facts/Informal_logic/4R63HwsY">informal logic</a>. Purely <a href="/facts/Formal_proof/6nEhV4cw">formal proofs</a>, written fully in <a href="/facts/Symbolic_language_(mathematics)/Clsg8rbt">symbolic language</a> without the involvement of natural language, are considered in <a href="/facts/Proof_theory/kopBdQn9">proof theory</a>. The distinction between <a href="/facts/Proof_theory/kopBdQn9">formal and informal proofs</a> has led to much examination of current and historical <a href="/facts/Mathematical_practice/wFICd4rc">mathematical practice</a>, <a href="/facts/Quasi-empiricism_in_mathematics/WR4IBAW8">quasi-empiricism in mathematics</a>, and so-called <a href="/facts/Mathematical_folklore/pqRlBWJR">folk mathematics</a>, oral traditions in the mainstream mathematical community or in other cultures. The <a href="/facts/Philosophy_of_mathematics/z7xBcdWA">philosophy of mathematics</a> is concerned with the role of language and logic in proofs, and <a href="/facts/Mathematics_as_a_language/lrmnY5wk">mathematics as a language</a>.

Mathematical proof open-in-new

Mathematical proof