constructive (redirected from constructively)

(mathematics)

constructive - A proof that something exists is "constructive" if it provides a method for actually constructing it. Cantor's proof that the real numbers are uncountable can be thought of as a *non-constructive* proof that irrational numbers exist. (There are easy constructive proofs, too; but there are existence theorems with no known constructive proof). Obviously, all else being equal, constructive proofs are better than non-constructive proofs. A few mathematicians actually reject *all* non-constructive arguments as invalid; this means, for instance, that the law of the excluded middle (either P or not-P must hold, whatever P is) has to go; this makes proof by contradiction invalid. See intuitionistic logic for more information on this. Most mathematicians are perfectly happy with non-constructive proofs; however, the constructive approach is popular in theoretical computer science, both because computer scientists are less given to abstraction than mathematicians and because intuitionistic logic turns out to be the right theory for a theoretical treatment of the foundations of computer science.