Proof

Proof is a reasoned argument intended to establish the truth of a proposition. In mathematics and logic, a proof is a deductive demonstration that, assuming the stated axioms and previously proven theorems, the conclusion necessarily follows. In everyday language, proof can refer to evidence or argument that persuades but does not guarantee truth.

Mathematical proofs are built from axioms by rules of inference. They are evaluated for logical validity and soundness; a proof must be valid (the conclusion follows from the premises) and the premises must be true within the given system. A proof therefore aims to show that the conclusion must be the case whenever the assumptions hold.

Common proof techniques include direct proof, proof by contradiction, proof by contrapositive, and mathematical induction. Constructive proofs provide an explicit example or construction, while non-constructive proofs establish existence without producing an explicit example.

Examples illustrate the idea. A direct proof shows that if a number n is even, then n^2 is even. An induction proof can establish that the sum 1 + 2 + ... + n equals n(n+1)/2 for all positive integers n.

Formal proofs and automated reasoning: proofs can be expressed in formal systems, and proof assistants such as Coq or Isabelle can verify their correctness. Gödel's incompleteness theorems show that in any sufficiently powerful system, there are true statements that cannot be proven within the system itself.

In other domains, proof can also mean evidence meeting a standard of certainty, such as proof beyond a reasonable doubt in law; in science, hypotheses are supported by evidence rather than proven with absolute certainty.