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Mathematical proof

A mathematical proof is a step-by-step demonstration of the truth of a mathematical theorem. Proofs build on axioms, which are statements that are assumed to be true without proof, as well as previously-proved theorems.

Several types of proofs are widely used, such as proof by contradiction and proof by induction. Proofs that rely only on certain simple forms of reasoning are sometimes called elementary proofs.

Scientific proof

Unlike the theorems of mathematics, science does not seek to prove that its theories are true. Instead, the scientific method seeks to check whether the predictions implied by a theory are observed in nature. Therefore, as philosopher of science Karl Popper argued, science can only hope to show that a theory is false. But scientists recognize that science can never prove that a theory is true in the same sense that a mathematical theorem is true. Therefore scientists never claim that their theories are facts. Instead, science searches for theories that are not disproved by currently-known experimental observations. Insofar as theories are consistent with nature, they may serve as a guide to improve technology.

Legal Proof

In American courts, crimes are proved "beyond reasonable doubt" to a jury, based on the jury's own analysis of the admissible evidence. Other legal issues may be decided by clear and convincing evidence or by a preponderance of the evidence.