Proof by deduction is the primary method of proof used in classical mathematics. Deductive proof is the process of deriving conclusions from logical premises without resort to empirical evidence. A deductive mathematical system typically consists of some definitions, some assumptions, called axioms or postulates, some rules of inference, and theorems. The proving of theorems is almost the definition of what a pure mathematician does for a living. Although the results of the mathematician's labor may be applicable to problems in the physical world, the mathematician's proofs of these results must come from within the structure of the deductive system in which the mathematics is being done. In such a system a deductive proof is a step by step procedure following the rules of inference and using only the definitions, axioms, and previously proved theorems. A simple example of a deductive proof schema goes back...