Beweis

Beweis is the German word for proof. In mathematics and logic, a Beweis is a rigorous argument that establishes the truth of a proposition beyond doubt within a given system of axioms and inference rules. A proof proceeds from accepted definitions and previously proven statements, using accepted methods of deduction to derive the statement as a necessary consequence.

Common proof techniques include direct proof, where a statement follows immediately from assumptions; proof by contradiction, where the negation leads to inconsistency; proof by contrapositive, equivalent reformulation; and mathematical induction for statements about integers. There are constructive proofs, which actually build the object in question, and non-constructive proofs, which assert existence without explicit construction. Probabilistic proofs use randomness to establish truth with high probability.

Formal proofs aim to show a sequence of derivations in a precise formal language, whereas informal proofs emphasize clarity and insight. In mathematics, a proof is supposed to be checkable by others; gaps or unjustified steps undermine validity. The acceptability of a proof can depend on the underlying axioms and accepted conventions of the subfield.

Outside mathematics, Beweis in German also denotes evidence or demonstration in legal and everyday language; such Beweise support a conclusion but do not always have the universal certainty of mathematical proofs. In science, a theory is supported by evidence and experiments, but remains open to revision rather than being proved in the mathematical sense.