prove

Prove is a verb meaning to establish the truth of a statement through evidence or argument. In everyday language it can mean demonstrating something convincingly, even if absolute certainty is not claimed. In mathematics and formal disciplines, proof has a more stringent, precise meaning.

In mathematics, a proof is a logically valid argument that derives a conclusion from accepted axioms and previously proven statements. A proof must be rigorous within a specified formal system and provide a transparent chain of reasoning that others can verify. The conclusion follows necessarily from the premises if the reasoning is sound.

Common proof techniques include direct proofs, proofs by contrapositive, proofs by contradiction (reductio ad absurdum), and mathematical induction. Some proofs are constructive, yielding an explicit example, while others are nonconstructive but still valid within the system.

Outside mathematics, proof usually means evidence or justification for a claim. Scientific claims rely on converging evidence rather than absolute proofs, and repeatable results strengthen belief in a hypothesis. In law, the standard of proof—such as beyond a reasonable doubt or preponderance of the evidence—governs outcomes and varies by context.

Etymology and history: prove derives from Latin probare, through Old French prover; the term has long indicated testing and verification. Over time, its sense broadened from practical testing to rigorous mathematical justification, while remaining rooted in the idea of establishing truth.