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con⋅struc⋅tive
[kuh
n-struhk-tiv] 
| 1. | constructing or tending to construct; helping to improve; promoting further development or advancement (opposed to destructive ): constructive criticism. |
| 2. | of, pertaining to, or of the nature of construction; structural. |
| 3. | deduced by inference or interpretation; inferential: constructive permission. |
| 4. | Law. denoting an act or condition not directly expressed but inferred from other acts or conditions. |
Based on the Random House Dictionary, © Random House, Inc. 2009.
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| con·struc·tive (kən-strŭk'tĭv) adj.
|
Copyright © 2009 by Houghton Mifflin Company.
Published by Houghton Mifflin Company. All rights reserved.
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Constructive
Con*struct"ive\, a. [Cf. F. constructif.]1. Having ability to construct or form; employed in construction; as, to exhibit constructive power. The constructive fingers of Watts. --Emerson. 2. Derived from, or depending on, construction or interpretation; not directly expressed, but inferred. Constructive crimes (Law), acts having effects analogous to those of some statutory or common law crimes; as, constructive treason. Constructive crimes are no longer recognized by the courts. Constructive notice, notice imputed by construction of law. Constructive trust, a trust which may be assumed to exist, though no actual mention of it be made.Cite This Source
Main Entry: con·struc·tive
Pronunciation: k&n-'str&k-tiv
Function: adjective
: created by a legal fiction: as a : inferred by a judicial construction or interpretation b : not actual but implied by operation of the law
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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 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.
(1995-04-13)
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