# constructive

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constructive
A constructive proof is one that enables one to give an example, or give a rule for finding an example, of a mathematical object with some property. A nonconstructive proof might result in us knowing that an example exists, but having no idea how to define it. The axiom of choice in set theory is the classical non-constructive existence axiom: it tells us that a certain set exists, whether or not there is any prospect of finding a condition defining membership in it. Similarly the definition of a function to take the value 0 if every even number is the sum of two primes, and 1 if this is not so, is classically a definition enabling us to assert that there is a number that is the value of the function, although we cannot identify which. The view that such a definition is inadmissible and that mathematics should confine itself to constructive proofs and definitions is known as constructivism. Constructivism will be suspicious of indirect existence proofs, because showing that a contradiction follows from denying that some object exists need not of itself show us how to identify the object. Constructivism frequently involves suspicion of the idea of a completed infinite set, thought of as a self-standing object of investigation, as a finite set would be.

Philosophy dictionary. . 2011.

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• constructive — con·struc·tive /kən strək tiv/ adj: created by a legal fiction: as a: inferred by a judicial construction or interpretation b: not actual but implied by operation of the law made a constructive entry when he refused to take the opportunity for a… …   Law dictionary

• Constructive — Con*struct ive, a. [Cf. F. constructif.] 1. Having ability to construct or form; employed in construction; as, to exhibit constructive power. [1913 Webster] The constructive fingers of Watts. Emerson. [1913 Webster] 2. Derived from, or depending… …   The Collaborative International Dictionary of English

• constructive — UK US /kənˈstrʌktɪv/ adjective ► useful and likely or intended to improve something: »a constructive dialogue/discussion/meeting »Listening to constructive criticism of certain aspects of your work should help you to make the most of your… …   Financial and business terms

• constructive — in general use means ‘helpful, positive’, as in constructive criticism. In this meaning it is the opposite of destructive. In legal language it is often applied to ‘what in the eye of the law amounts to the act or condition specified’ (OED), and… …   Modern English usage

• constructive — [kən struk′tiv] adj. [ML constructivus] 1. helping to construct; leading to improvements or advances; formative; positive [constructive criticism] 2. of construction or structure 3. inferred or implied by legal or judicial interpretation… …   English World dictionary

• constructive — early 15c., derived by interpretation, from M.Fr. constructif or from M.L. constructivus, from L. construct , pp. stem of construere to heap up (see CONSTRUCTION (Cf. construction)). Meaning pertaining to construction is from 1817; having the… …   Etymology dictionary

• constructive — *implicit, virtual Analogous words: inferential, ratiocinative (see under INFERENCE): implied, involved (see INCLUDE) Antonyms: manifest Contrasted words: express, *explicit, definite: *evident, patent, obvious …   New Dictionary of Synonyms

• constructive — [adj] helpful effective, positive, practical, productive, useful, valuable; concept 401 Ant. destructive, hurting, injurious, negative, unhelpful …   New thesaurus

• constructive — ► ADJECTIVE 1) serving a useful purpose. 2) Law derived by inference; not stated explicitly. DERIVATIVES constructively adverb constructiveness noun …   English terms dictionary

• constructive — That which is established by the mind of the law in its act of construing facts, conduct, circumstances, or instruments. That which has not the character assigned to it in its own essential nature, but acquires such character in consequence of… …   Black's law dictionary

• constructive — That which is established by the mind of the law in its act of construing facts, conduct, circumstances, or instruments. That which has not the character assigned to it in its own essential nature, but acquires such character in consequence of… …   Black's law dictionary