constructive
A proof that something exists is "constructive"
if it provides a method for actually constructing it.
{Cantor}'s proof that the {real number}s are {uncountable} can
be thought of as a ▫non-constructive▫ proof that {irrational
number}s 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.
(1995-04-13)