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However, based on the different meaning of the words in constructive mathematics, if there is a constructive proof that "α = 0 or α ≠ 0" then this would mean that there is a constructive proof of Goldbach's conjecture (in the former case) or a constructive proof that Goldbach's conjecture is false (in the latter case).
Such a proof by contradiction might be called non-constructive, and a constructivist might reject it. The constructive viewpoint involves a verificational interpretation of the existential quantifier, which is at odds with its classical interpretation. There are many forms of constructivism. [1]
Informally, this means that if there is a constructive proof that an object exists, that constructive proof may be used as an algorithm for generating an example of that object, a principle known as the Curry–Howard correspondence between proofs and algorithms.
In contrast, a constructive proof establishes that a particular object exists by providing a method of finding it. The following famous example of a nonconstructive proof shows that there exist two irrational numbers a and b such that a b {\displaystyle a^{b}} is a rational number .
constructive proof A proof that demonstrates the existence of a mathematical object by providing a method to construct it explicitly, as opposed to proving indirectly by contradiction. contextualism The theory that the context in which an assertion is made is essential for understanding its truth conditions and meaning. contingency
The base logic of constructive analysis is intuitionistic logic, which means that the principle of excluded middle is not automatically assumed for every proposition.If a proposition . is provable, this exactly means that the non-existence claim . being provable would be absurd, and so the latter cannot also be provable in a consistent theory.
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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 ...