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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]].

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.  

Presently, certain theorems have only been proved using nonconstructive methods. However, 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 [[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 number]]s is [[uncountable]] while the set of [[algebraic number]]s is [[countable]], and thus (many) transcendental numbers must exist. Of course, finding a specific example is a much more difficult endeavor.

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

[[Category:mathematics]]

[[Category:mathematics]]