什么是“证明” The definition of Proof

据说，数学家的主要任务是“证明定理”，而不是创立新的数学理论。如果熟悉《集合论》、《范畴论》、《泛代数》等数学基础，创造一个新的数学理论，一般不是很困难的。据说一个重点大学毕业的数学专业的本科生，就可以像模像样地创造一个新的理论。
那么，什么是证明？

美苏两个超级大国的数学家们的定义：
Encyclopaedia of Mathematics (Edited by Michiel Hazewinkel, CWI, Amsterdam): The Online Encyclopaedia of Mathematics is the most up-to-date and comprehensive English-language graduate-level reference work in the field of mathematics today.

http://eom.springer.de/P/p075420.htm
Proof
A reasoning conducted according to certain rules in order to demonstrate some proposition (statement, theorem); it is based on initial statements (axioms). In practice, however, it may also be based on previously demonstrated propositions. Any proof is relative, since it is based on certain unprovable assumptions. Rules of conducting a reasoning and methods of proof form a main topic in logic. See Proof theory.
A.S. Kuzichev

大英百科全书 Encyclopædia Britannica
http://search.eb.com/eb/article-9061543
proof
in logic, an argument that establishes the validity of a proposition. Although proofs may be based on inductive logic, in general the term proof connotes a rigorous deduction. In formal axiomatic systems of logic and mathematics, a proof is a finite sequence of well-formed formulas (generated in accordance with accepted formation rules) in which: (1) each formula is either an axiom or is derived from some previous formula or formulas by a valid inference; and (2) the last formula is that which is to be proved. For proof by cases, see dilemma.

我英文太差，不敢翻译。
《中国大百科全书》、《Encyclopaedia of Mathematics》的介绍和书籍：
集合论jihelun (卷名：数学) set theory，见： http://202.112.118.40:918/web/index.htm
Set theory, naïve，见： http://eom.springer.de/S/s084750.htm
K. Kuratowski, A. Mostowski. Set theory. North-Holland, 1968.
N. Bourbaki. Elements of mathematics. Theory of sets. Addison-Wesley, 1968. (Translated from French).

范畴fanchou (卷名：数学) category， 见： http://202.112.118.40:918/web/index.htm
Category，见： http://eom.springer.de/C/c020740.htm
B.Mitchell. Theory of categories. Academic Press, New York, 1965.
G.M. Kelly. Basic concepts of enriched category theory. Cambridge Univ. Press, 1982.

泛代数fandaishu (卷名：数学) universal algebra，见： http://202.112.118.40:918/web/index.htm
Universal algebra，见： http://eom.springer.de/U/u095630.htm
G. Grätzer. Universal algebra. Springer, 1979.
P.M. Cohn. Universal algebra. Reidel, 1981.