Constructive mathematics

Mathematics that requires existence proofs to provide or justify a construction of the object asserted to exist. Its proofs can often yield algorithms or explicit witnesses.

Constructive proof: A proof that establishes a claim using methods that provide witnesses or explicit procedures when existence is asserted. This is the central proof style through which constructive mathematics establishes existence.
Existence theorem: A theorem asserting that at least one object satisfies specified conditions. Constructive mathematics imposes requirements on how such theorems are proved.
L. E. J. Brouwer: A Dutch mathematician who founded intuitionism and challenged classical assumptions about mathematical proof. His foundational program shaped modern constructive mathematics.
Proofs as programs: The correspondence in which constructive proofs can be interpreted as programs and propositions as types. It turns constructive existence proofs into procedures that can compute witnesses.
Classical mathematics: Mathematics practiced with classical logic, including unrestricted use of the law of excluded middle. It permits proofs of existence that need not construct or identify a witness.
Brouwer–Heyting–Kolmogorov interpretation: An interpretation of logical propositions as types of evidence or constructions that establish them. It explains constructive meanings for connectives and quantifiers in terms of proofs.
Quantifier: A logical operator, such as “for all” or “there exists,” that specifies how many objects satisfy a predicate. Constructive interpretations give existential and universal quantifiers proof-theoretic content.
Errett Bishop: An American mathematician who developed a systematic constructive analysis in the twentieth century. His work demonstrated that substantial analysis could be developed constructively.
Program extraction: The process of deriving executable programs from formal proofs. Constructive proofs make extracted programs available from mathematical existence arguments.
Proof by contradiction: A proof method that establishes a claim by deriving a contradiction from its negation. Some classical uses establish existence without producing the object required constructively.

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