construction

n 1: the act of constructing or building something; "during the

construction we had to take a detour"; "his hobby was

the building of boats" [syn: {building}]

2: the commercial activity involved in constructing buildings;

"their main business is home construction"; "workers in

the building trades" [syn: {building}]

3: a thing constructed; a complex construction or entity; "the

structure consisted of a series of arches"; "she wore her

hair in an amazing construction of whirls and ribbons"

[syn: {structure}]

4: a group of words that form a constituent of a sentence and

are considered as a single unit; "I concluded from his

awkward constructions that he was a foreigner" [syn: {expression}]

5: the creation of a construct; the process of combining ideas

into a congruous object of thought [syn: {mental synthesis}]

6: an interpretation of a text or action; "they put an

unsympathetic construction on his conduct" [syn: {twist}]

7: drawing a figure satisfying certain conditions as part of

solving a problem or proving a theorem; "the assignment

was to make a construction that could be used in proving

the Pythagorean theorem"

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)