Questions tagged [constructive-mathematics]

The term "constructive mathematics" refers to the discipline in mathematics in which one proves the existence of mathematical objects only by presenting a construction that provides such an object. Indirect proofs involving proof by contradiction and law of excluded middle are considered nonconstructive. Constructivism is the philosophical stance that the only "true" mathematics is constructive mathematics.

The term "constructive mathematics" refers to the discipline in mathematics in which one proves the existence of mathematical objects only by presenting a construction that provides such an object. Indirect proofs involving proof by contradiction are considered nonconstructive. Construvtivism is the philosophical stance that the only "true" mathematics as constructive mathematics.

In constructivism, an existence proof is not accepted, unless the object in question is constructed. As an example of a nonconstructive proof, consider the following classical proof of the fact that there are irrational numbers $ a $ and $ b $ such that $ a ^ b $ is rational:

Either $ { \sqrt 2 } ^ { \sqrt 2 } $ is rational, in which case we take $ a = b = \sqrt 2 $; or else $ { \sqrt 2 } ^ { \sqrt 2 } $ is irrational, in which case we take $ a = { \sqrt 2 } ^ { \sqrt 2 } $ and $ b = \sqrt 2 $.

The above argument is nonconstructive, because as it stands, it does not enable us to pinpoint which of the two choices of the pair $ ( a , b ) $ has the required property. An alternative proof for the same theorem which is constructive, goes like:

Take $ a = \sqrt 2 $ and $ b = \log _ 2 9 $.

Also, the law of excluded middle is typically not accepted as an axiom. That's because it can result in nonconstructive reasoning, as the above example illustrates. Therefore classical logic is rejected by constructivists, and instead they use intuitionistic logic, which is essentially classical logic without the law of the excluded middle. There are also mathematical axioms like the axiom of choice rejected by constructivists, as they have nonconstructive consequences.

As some of classical methods are not constructively valid, there are classically valid sentences that don't have constructive proofs. As an example there is no constructive proof for the following sentence:

For every real number $ x $, either $ x < 0 $, $ x = 0 $ or $ x > 0 $.

There is a suitable replacement for this which is constrcutively valid. In many applications this alternative is sufficient, although it's slightly weaker than the classical sentence:

For every real number $ x $ and every positive real number $ \epsilon $, either $ x < 0 $, $ | x | < \epsilon $ or $ x > 0 $.

Constructivism has different varieties, among which the most famous are:

  1. , a formal basis for the theory of intuitionism founded by L. E. J. Brouwer
  2. Recursive constructive mathematics, a.k.a russian construve mathematics, founded by A. A. Markov
  3. Bishop's constructive mathematics, founded by E. Bishop
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Construct a real $x$ such that ZF does not prove whether $x\in\mathbb{Q}$

I have little knowledge of formal logic or set theory. Nevertheless it occurred to me to wonder whether one can give an explicit construction of a real number, such that it does not follow from the usual ZF axioms of set theory whether or not it is…
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Why an understandable open problem In mathematics can not solvable?

If we check the standard open problems in Mathematics which they are understandable even for a student in Midlle school as example if we take this problem : " Is there an odd perfect number" ? it's understandable at a least for high school level or…
zeraoulia rafik
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Is there a sense in which a formal theory can be more or less "metamathematically aware"?

For example, in classical logic, because of the law of the excluded middle, all propositions must be true or false, so in order to prove that a proposition is actually independent of the theory, we need to use some metamathematical ideas. However,…
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Numerical integration of functions over computable Cauchy sequences

I'm interested in exact real arithmetic (and by extension constructive analysis). A nice representation of real numbers is via Cauchy Sequences. The basic idea being that you have a function which, given a rational error bound, returns a rational…
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Finding a formula in intuitionist logic

I am looking for a formula which is true semantically but not syntactically in propositional intuitionist logic. Does it exist? If yes what's that? Thanks for your help.
fateme jl
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Constructing real numbers and why every real is not computable

Constructive vs computable real numbers asked two questions : Why isn't every real number computable? How is it possible to construct an uncountable set? What is wrong with following answers? 1.Because there is at least one real number that is not…
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