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A rigorous mathematical argument which unequivocally demonstrates the truth of a given Proposition. A mathematical statement which has been proven is called a Theorem.
There is some debate among mathematicians as to just what constitutes a proof. The Four-Color Theorem is an example of this debate, since its ``proof'' relies on an exhaustive computer testing of many individual cases which cannot be verified ``by hand.'' While many mathematicians regard computer-assisted proofs as valid, some purists do not.
See also Paradox, Proposition, Theorem
References
Garnier, R. and Taylor, J. 100% Mathematical Proof. New York: Wiley, 1996.
Solow, D. How to Read and Do Proofs: An Introduction to Mathematical Thought Process, 2nd ed. New York: Wiley, 1990.