Search results
Results From The WOW.Com Content Network
However, there is a non-constructive proof that shows that linkedness is decidable in polynomial time. The proof relies on the following facts: The set of graphs for which the answer is "yes" is closed under taking minors. I. e., if a graph G can be embedded linklessly in 3-d space, then every minor of G can also be embedded linklessly.
A non-constructive proof might show a solution exists without specifying either an algorithm to obtain it or a specific bound. Even if the proof is constructive, showing an explicit bounding polynomial and algorithmic details, if the polynomial is not very low-order the algorithm might not be sufficiently efficient in practice.
(Constructive proofs of the specific example above are not hard to produce; for example = and = are both easily shown to be irrational, and =; a proof allowed by intuitionists). By non-constructive Davis means that "a proof that there actually are mathematic entities satisfying certain conditions would not have to provide a method to ...
In proof by exhaustion, the conclusion is established by dividing it into a finite number of cases and proving each one separately. The number of cases sometimes can become very large. For example, the first proof of the four color theorem was a proof by exhaustion with 1,936 cases. This proof was controversial because the majority of the cases ...
At its core, this proof is non-constructive because it relies on the statement "Either q is rational or it is irrational"—an instance of the law of excluded middle, which is not valid within a constructive proof. The non-constructive proof does not construct an example a and b; it merely gives a number of possibilities (in this case, two ...
Cantor's correspondence with Richard Dedekind shows the development of his ideas and reveals that he had a choice between two proofs: a non-constructive proof that uses the uncountability of the real numbers and a constructive proof that does not use uncountability. Historians of mathematics have examined Cantor's article and the circumstances ...
For example, any theorem of classical propositional logic of the form has a proof consisting of an intuitionistic proof of followed by one application of double-negation elimination. Intuitionistic logic can thus be seen as a means of extending classical logic with constructive semantics.
From the other direction, there has been considerable clarification of what constructive mathematics is—without the emergence of a 'master theory'. For example, according to Errett Bishop's definitions, the continuity of a function such as sin(x) should be proved as a constructive bound on the modulus of continuity, meaning that the existential content of the assertion of continuity is a ...