constructive

mathematics 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 irrationalnumbers 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 excludedmiddle (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.

(1995-04-13)