Constructivism

Constructivism is a philosophy of mathematics that accepts mathematical existence claims only when supported by constructions. It rejects proofs that establish existence without exhibiting a method to produce an instance.

Constructive proof: A proof that establishes a claim by providing an explicit construction or method. It is the proof form that directly satisfies constructivism’s demand for mathematical witnesses.
Intuitionistic logic: A logic that interprets proofs constructively and does not accept unrestricted excluded middle. It supplies a formal logic commonly used to express constructivist standards.
L. E. J. Brouwer: A Dutch mathematician who founded intuitionism and challenged classical mathematical logic. His intuitionist philosophy gave constructivism one of its most influential foundations.
Nonconstructive proof: A proof that establishes a mathematical claim without providing an explicit construction of the object asserted to exist. It exemplifies the kind of existence proof constructivism does not generally accept.
Type theory: A formal system that classifies expressions by types and can encode proofs as terms. Constructive type theories make proofs and the objects they construct part of one formal framework.
Existential quantification: A logical operator asserting that at least one object in a domain has a stated property. Constructivism interprets an existential claim as requiring a way to produce a witness.
Classical logic: The standard logic that accepts principles such as unrestricted excluded middle. Its accepted proof methods can establish existence without supplying constructive evidence.
Intuitionism: A philosophy of mathematics holding that mathematical objects are mental constructions and proof is central to truth. It is Brouwer’s distinctive constructivist program, with commitments beyond constructive proof alone.
Proof by contradiction: A proof method that derives a contradiction from assuming a claim is false. Classical versions can establish existence without producing a witness, unlike constructive contradiction arguments.
Curry–Howard correspondence: A correspondence linking propositions with types and proofs with programs. It turns constructive proofs into computational objects that can yield programs.

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