wade-w

Several people have mentioned proof by contradiction. The basic idea is that in order to prove P, show that ~P leads to a contradiction. In mathematical philosophy, the intuitionist (also called the constructivist) school does not accept this as valid reasoning. I don't agree with them, but I think it's important to note that not all mathematicians agree on what constitutes a valid proof.

A famous example of this method is the standard proof that the square root of 2 is irrational:

Suppose that sqrt (2) is rational. Then there exists integers p and q, such that p and q are relatively prime (they have no common factors), q is non-zero, and p/q =sqrt(2) => p^2/q^2 = 2 => p^2 = 2q^2 => p^2 is even, and thus p is even. Therefore there exists an integer n such that 2n = p, hence (2n)^2 = 2q^2 => 2n^2 = q^2 => q^2 is even, and thus q is even. But p and q cannot both be even since they are relatively prime, a contradiction. Therefore sqrt (2) is irrational.

Undercurrent

Just to be a very broad pedant, it actaully says that any formal axiomatic system that is sophisticated enough to encode arithmetic either admits a statement, S, for which neither S nor not S is provable, or it admits a statement S for which both S and not S is provable.