In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existence proof or pure existence theorem) which proves the existence of a particular kind of object without providing an example.
Some non-constructive proofs show that if a certain proposition is false, a contradiction ensues; consequently the proposition must be true (proof by contradiction). However, the principle of explosion (ex falso quodlibet) has been accepted in some varieties of constructive mathematics, including intuitionism.
Constructivism is a mathematical philosophy that rejects all but constructive proofs in mathematics. This leads to a restriction on the proof methods allowed (prototypically, the law of the excluded middle is not accepted) and a different meaning of terminology (for example, the term "or" has a stronger meaning in constructive mathematics than in classical).
Constructive proofs can be seen as defining certified mathematical algorithms: this idea is explored in the Brouwer–Heyting–Kolmogorov interpretation of constructive logic, the Curry–Howard correspondence between proofs and programs, and such logical systems as Per Martin-Löf's Intuitionistic Type Theory, and Thierry Coquand and Gérard Huet's Calculus of Constructions.
Read more about Constructive Proof: Example, Brouwerian Counterexamples
Famous quotes containing the words constructive and/or proof:
“The desert is a natural extension of the inner silence of the body. If humanity’s language, technology, and buildings are an extension of its constructive faculties, the desert alone is an extension of its capacity for absence, the ideal schema of humanity’s disappearance.”
—Jean Baudrillard (b. 1929)
“The moment a man begins to talk about technique that’s proof that he is fresh out of ideas.”
—Raymond Chandler (1888–1959)