Constructive proof

A constructive proof demonstrates the existence of a mathematical function, number or object by producing (constructing) it. This is in contrast with other styles of proof, such as proof by contradiction, which asserts the existence of an object by finding a contradiction if it did not exist. Such a proof is called nonconstructive and is not rarely valued by mathematicians, especially in applied mathematics and computer science.

Indeed, even after a nonconstructive proof is found for a result, work will still continue until a more useful constructive proof is found. A classical example of this is in Ramsey theory where a unsatisfactory proof using random graphs can determine Ramsey numbers. However, mathematicians will attempt to construct such a graph. Merely proving a hypothetical existence is not enough.

The Axiom of Choice assumes the existence of a function without constructing it, and thus all proofs that rely on the Axiom of Choice are nonconstructive proofs.

The easiest way to prove the existence of transcendental numbers is by a nonconstructive proof, arguing that the set of real numbers is uncountable while the set of algebraic numbers is countable, and thus (many) transcendental numbers must exist. Of course, finding a specific example is a much more difficult endeavor.

Of course, nonconstructive proofs are the only way to prove many theorems in mathematics such as the uncountability of the real numbers or the insolvability of the general quintic.