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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Is there a clean non-contrived theorem that can only be proven by contradiction?

I know (see Can every proof by contradiction also be shown without contradiction? that there are some theorems that can be proven by contradiction (relying on the law of the excluded middle, that for any proposition $A$, the axiom "either $A$ or not…
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Is there a “nice” “constructive” field of numbers?

I am wondering about this. I've had some interest in “constructive” mathematics, although also some rather strong opinions against those who want to insist that everything else is “wrong” in favor of it. One constructive object is the “computable…
The_Sympathizer
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Approximate spectral decomposition

I am interested in effective and computations for finding approximate spectral decompositions in some suitable format. Namely, let $A: H \rightarrow H$ be a Hermitian operator on an $n-$dimensional Hilbert space $H$ with the spectrum $\{\lambda_1,…
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Law of Excluded Middle Controversy

I was reading an introductory book on logic and it mentioned in passing that the Law of Excluded Middle is somewhat controversial. I looked into this and what I got was the intuistionists did not accept it in contrasts to the formalists. I'm curious…
Red
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Can the principle of explosion be removed from constructive logic?

Classical logic has the theorem ($p\wedge\lnot p)\rightarrow q$, which I will call EFQ ("ex falso quodlibet"). Constructive logic often has the principle built in, in the form of an axiom $\bot\rightarrow q$ which one can use to prove EFQ via…
MJD
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What is a constructive proof of $\lnot\lnot(P\vee\lnot P)$?

Glivenko's theorem says that $\lnot\lnot P$ is a theorem of intuitionistic logic whenever $P$ is a theorem of classical logic. Is it closely related to the so-called Gödel–Gentzen negative translation which embeds classical logic into intuitionistic…
MJD
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Examples of non-constructive results

I'm giving a talk on constructive mathematics, and I'd like some snappy examples of weird things that happen in non-constructive math. For example, it would be great if there were some theorem claiming $\neg \forall x. \neg P(x)$, where no $x$…
luqui
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Is there a simple example of how the law of the excluded middle can be inapplicable?

Why does a logic system not use the law of the excluded middle? I studied non-classical logic (intuitionistic and modal) where double negation can't be removed and the law of excluded middle can't be used. But what is a simple example that…
Niklas Rosencrantz
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What is the dual of implication?

You may divide Intuitionistic Propositional Logic into the negative and positive fragments. The negative fragment includes truth, conjunction, and implication while the positive fragment includes falsity and disjunction. There is an obvious duality…
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Axiom of Choice - Type Theory (Proof)

Background In Intuitionistic Type Theory (p. 27-28), Martin Löf provides a proof of the axiom of choice that is constructively valid. This version is considerably weaker than the ordinary set theory version, since there are no quotient types. Now,…
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About Gödel-Gentzen negative translation

I have a question about Gödel-Gentzen negative translation. According to the Wikipedia article for negative translations, "a sentence $\phi$ may not imply its negative translation $\phi^{\rm N}$". I am not sure if I understand correctly this…
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Are there still mathematicians who don't accept proof by contradiction?

When I was a kid, I read popular scientific texts about the different philosophies of mathematics; formalism, intuitionism, constructivism and many others. I learned that there existed mathematicians who did not accept proofs by contradiction and…
mathreadler
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Dense topology <=> double negation operator in a constructive metatheory?

The dense topology for a category $\mathbb{C}$ can be defined as follows, writing $\mathscr{H}:\mathbb{C}\to\widehat{\mathbb{C}}$ for the Yoneda embedding (considering sieves on $c$ as subfunctors of $\mathscr{H}(c)$): $$J_{\mathsf{dense}}(c)…
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Is the Knaster-Tarski Fixed Point Theorem constructive?

According to Tarski's Fixed Point Theorem, for a complete lattice $L$, and monotone function $f:L \rightarrow L$, the set of fixed points of $f$ forms complete lattice. Definition of $lfp(f)$ and $gfp(f)$ is given in terms of set $\text{Red}(f)$…
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Algebraically, What Does $\Bbb R$ get us?

In terms of the basic algebraic operations -- addition, negation, multiplication, division, and exponentiation -- is there any gain moving from $\Bbb Q$ to $\Bbb R$? Say we start with $\Bbb N$: $\Bbb N$ is closed under addition and multiplication. …
user137731
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