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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Aren't constructive math proofs more "sound"?

Since constructive mathematics allows us to avoid things like Russell's Paradox, then why don't they replace traditional proofs? How do we know the "regular" kind of mathematics are free of paradox without a proof construction?
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a "natural" real number that is not computable

Most of the examples of non-computable real numbers use some kind of a diagonalization construction over some turing computable model of computation. See Are there any examples of non-computable real numbers?. I want to know if there are "natural"…
Abhishek Anand
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How far is it true that statements dependent on Axiom of Choice are not constructive.

Axiom of Choice is often used in mathematics to construct various objects, such as basis of $\mathbb{R}$ as a vector space over $\mathbb{Q}$, unmeasurable subset of $\mathbb{R}$, or a non-principal ultrafilter on $\mathbb{N}$. It is a popular…
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Does Tychonoff's Theorem imply Excluded Middle?

It is well-known that using excluded middle, we can prove that Tychonoff's Theorem implies the axiom of choice. This was proved by Kelley in 1950. However, the standard proof requires excluded middle in several places. It requires excluded middle to…
Mark Saving
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Double negation elimination in constructive logic

How can I prove that the double negation elimination is not provable in constructive logic? To clarify, double negation elimination is the following statement: $$\neg\neg q \rightarrow q$$
Ameen
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Nonnegative linear functionals over $l^\infty$

My purpose is a clarification of the role of the axiom of choice in constructing limits for bounded sequences. Namely, we want a linear functional of norm 1 defined on the space of all bounded complex sequences that takes nonnegative sequences to…
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Understanding a proof of Diaconescu's theorem

I am trying to walk through the proof of Diaconescu's theorem that the axiom of choice implies the law of excluded middle at http://plato.stanford.edu/entries/intuitionism/#ChoAxi. To paraphrase: Let $A$ be a statement (which we will think of as…
asmeurer
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Countable choice and term extraction

The constructive Axiom of Countable Choice (ACC) is widely accepted due to its computational content. It states that: $$ \forall n\in \mathbb{N} . \exists x \in X . \varphi [n, x] \implies \exists f: \mathbb{N} \longrightarrow X . \forall n \in…
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Equivalence between middle excluded law and double negation elimination in Heyting algebra

It's well-know that in intuitionistic logic, middle excluded law and double negation elimination are equivalent. For example, in Johnstone - Topo theory, I read that, in a Heyting algebra, $p\vee\neg p=\top$ holds if and only if $\neg\neg p=p$ for…
Fabio Lucchini
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A group is not the union of two subgroups, constructively.

Let $G$ be a group, and let $H$ and $K$ be subgroups of $G$. The following is well-known: Proposition 1. If $H \cup K = G$, then $H = G$ or $K = G$. See, for instance, this answer. Question. Is proposition 1 provable in intuitionistic first-order…
Zhen Lin
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Online tools for checking validity of classical, intuitionistic, ... logic formulas?

What online tools are available, where one can enter a formula of (first order) propositional or predicate logic, and have it check whether it is valid classically, intuitionistically, or even minimally or in some other logical framework? I found…
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Is there such a thing as 'overtification' (dual to compactification)?

The dual to the notion of compactness for spaces is overtness. This duality is not manifest in the category of spaces but rather in the quantifiers used to define these notions. Is there a process of 'overtification' corresponding to the process of…
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How or why does intutionistic logic proof negations from within the theory, constructively?

I'm having a little of a cognitive dissonance why, in intuitionistic logic, it's possible to show stentences like $(\neg A \land \neg B) \implies \neg(A\lor B).$ In plain text: If 'A isn't true' as well as 'B isn't true', then neither is 'A or…
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Examples of co-implication (a.k.a co-exponential)

In Dual Intuitionistic Logic and a Variety of Negations: The Logic of Scientific Research Yaroslav Shramko, inspired by Popper, makes an interesting case that co-constructive logic as the logic of refutation is the logic of empirical science. In a…
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Is there any result that has applications that can't be proved in constructive mathematics?

Constructive mathematics is distinguished from its traditional counterpart, classical mathematics, by the strict interpretation of the phrase “there exists” as “ we can construct”. Is there any result in classical mathematics that is extensively…
Julien__
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