proof, in law: see evidence.
proof, in mathematics, finite sequence of propositions each of which is either an axiom or follows from preceding propositions by one of the rules of logical inference (see symbolic logic). Mathematical proofs are quite distinct from inductive, statistical, heuristic, analogical, and other types of reasoning or persuasion that are sometimes accepted as proofs in other fields of science or human affairs. Proof theory has developed into one of the important branches of modern mathematical logic. Some schools of mathematical logic reject certain methods in proofs, such as use of the law of excluded middle (either p is true or p is false) or of mathematical definitions involving properties that are not effectively verifiable.
proof, in printing, a trial impression for inspection. Proofreading is the inspection and marking of proof for correction of errors and imperfections. Proofreaders' marks are included in dictionaries. Directions for proofreading are given in several sources including The Chicago Manual of Style, published by the Univ. of Chicago Press (15th ed. 2003); Words into Type, by M. E. Skillin and R. M. Gay (3d ed. 1974); and The Fine Art of Copyediting, by E. M. Stainton (1991).

In logic and mathematics, an argument that establishes a proposition's validity. Formally, it is a finite sequence of formulas generated according to accepted rules. Each formula either is an axiom or is derived from a previously established theorem, and the last formula is the statement that is to be proven. The essence of deductive reasoning (see deduction), this is the basis of Euclidean geometry and all scientific methods inspired by it. An alternative form of proof, called mathematical induction, applies to propositions defined through processes based on the counting numbers. If the proposition holds for math.n = 1 and can be shown to hold for math.n = math.k + 1 whenever math.n = math.k (a constant) is also true, then it holds for all values of math.n. An example is the assertion that the sum of the first math.n counting numbers is math.n(math.n + 1)/2.

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