beweisen

Beweisen is the German verb meaning to prove, or to establish the truth of a statement. In mathematics and logic, a Beweis (proof) is a finite sequence of logically valid steps that derives a conclusion from axioms, definitions, and previously established results. A proof must be rigorous and deductive, so that the conclusion follows with necessity, not merely with probability or empirical support. The noun Beweis is commonly used in German mathematics to mean "proof."

Typical techniques include direct proofs, proofs by contrapositive, proofs by contradiction, mathematical induction, and constructive or non-constructive proofs. A proof aims to show that a statement is universally true within a given formal system, rather than just holding for a specific example. Some proofs appeal to analysis, geometry, or algebra, but all must maintain logical validity and explicit reasoning. In mathematical practice, proofs often incorporate definitions, lemmas, and theorems in a coherent chain of justification.

In broader usage, "beweisen" applies to evidence and argument beyond mathematics: in science, to be convincing; in law, to meet standards of proof. In German, you would say "Ich beweise das" (I prove that) or "Der Beweis" (the proof). The concept of proof has deep historical roots in Greek geometry and was formalized in modern times through axiomatic systems, with developments such as Hilbert's program; Gödel’s incompleteness theorem later clarified limits to what can be proven within any given formal framework.