Mathematical proof

A finite argument that establishes a mathematical statement from accepted axioms, definitions, and previously proved results. Its validity depends on the logical connection between its premises and conclusion.

Axiom: A statement accepted as a starting assumption within a mathematical theory. Axioms supply premises from which proofs derive further statements.
Direct proof: A proof that derives a conclusion directly from its assumptions by valid steps. This is the basic route when the assumptions straightforwardly imply the claim.
Mathematical conjecture: A mathematical statement proposed as true but not yet established by proof. A conjecture remains unproved even when extensive evidence supports it.
Euclid's Elements: An ancient Greek mathematical treatise that presents geometry through definitions, postulates, and propositions. Its axiomatic organization became a lasting model for mathematical exposition.
Mathematical rigor: The demand that mathematical reasoning meet precise standards of definition and valid inference. Standards of rigor specify what counts as an acceptable proof.
Definition: A precise specification of the meaning of a mathematical term or symbol. Definitions let proofs replace concepts with exact conditions.
Proof by contradiction: A proof that assumes a claim is false and derives an impossibility. It establishes the claim by showing its negation cannot hold.
Mathematical counterexample: An example showing that a universal mathematical statement is false. One counterexample can defeat a claim that a proof must establish for every case.
Euclidean geometry: The geometry of points, lines, and planes governed by Euclid's parallel postulate. Its propositions illustrate systematic derivation from explicit geometric assumptions.
Proof assistant: Software that helps construct and check formal mathematical proofs. These systems can verify formal derivations while changing how proofs are produced.

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