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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Bijection between natural numbers and the real numbers in constructive mathematics

In this video: https://www.youtube.com/watch?v=zmhd8clDd_Y&t=1025s (called Five Stages of Accepting Constructive Mathematics) at exactly 24:09 the lecturer says that it is possible to find a bijection between natural numbers and the real numbers in…
NotSure
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In what way intuitionism is unique in the constructive approach

I am writing a paper on the subjects of constructivism and intuitionism. While I do know that intuitionism is a part of constructivism; it is also written that a lot of logic in intuitionism is unique and not shared with other constructivist…
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contraposition in intuitionistic logic

I know that a sentence does not imply its contrapositive in institutionistic logic. I tried very hard to come up with a counter model to prove that but I failed. Can someone help me please? Any hints will be appreciated. Thanks so much.
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Help with proving $f^\ast R^\times = \operatorname{Spec}A_f$ where $f\in A$ and $R= \operatorname{Spec} k[x]$

Suppose $\mathcal E$ is the large Zariski site of $k$-algebras for $k$ a (commutative unitary) ring. Let $R=\operatorname{Spec}k[x]$ be the line object of $\mathcal E$. Let $A$ be a $k$-algebra and $f\in A$. Naturally identifying $f\in A$ with…
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An intuition connected with Heyting implication

Suppose $L$ is a bounded lattice and let $\Rightarrow$ be its Heyting implication, i.e. the operation defined in the standard way: $x\Rightarrow y$ is the largest object of the set $\{u\in L\mid u\cdot x\leqslant y\}$. I have a slight problem with…
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Give a "constructive" proof of the fact that in a metric space the intersection of two open balls is open

Main Question Can someone give a "constructive" proof of the fact that, Let $(X,d)$ be a metric space and $x,y\in X$. Let $B_d(x,r_x)$ and $B_d(y,r_y)$ be two open balls centered respectively at $x$ and $y$ for $r_x,r_y\in \mathbb{R}^+$. Suppose…
user170039
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Elements of bounded distributive lattice belonging to same prime ideals are equal?

I have read in a paper that by an easy application of Zorn's lemma one may show that two elements of a bounded distributive lattice are equal iff they are contained in exactly the same prime ideals of the lattice. What is the intuition behind this…
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Division algorithm proof without well ordering principle

Is there a constructive proof of the division algorithm that doesn't invoke the well ordering principle?
apricity
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Definition of all real numbers in terms of four arithmetic operations on 1? Reading material?

Is there a (more or less complete) theory of real numbers, where every number except for $1$ is defined by a combination of arithmetic operations acting on $1$. I know we can build any whole number by using addition and subtraction of 1s, and any…
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What is a Cauchy sequence of definable reals whose limit is undefinable?

Wikipedia claims that there exist Cauchy sequences of definable numbers whose limit is not definable. Are there constructive proofs of this? If so, what is an example of a Cauchy sequence of definable reals whose limit is undefinable? Also, does…
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Constructing the Integers from the Naturals

I'm watching a video right now about the construction of the Integers from the Naturals. The way to do so was to define the equivalence relation $$(a,b)\text{ is equivalent to }(c,d)\text{ if }a+d=c+b$$ It was then said that "with a little algebra,…
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How to construct a polynomial from a radix-term?

A term only composed of the following operatings shall henceforth be called a radix term because I don't know how these terms are called. A radix term $t$ is either an integer or a sum of two radix terms, a product of two radix terms, or a radix…
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how can I publish my log approximation formula

I've successfully found out a formula which can give log value of any base till 4-5 places after decimal I want to know whether it can get published because I've seen some journals which have published approximation of log for different basses.but…
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Are below sentences held in intuitionistic logic?

$(p \to (q \to r)) \to ((p \to r) \lor (q \to r))$ $(p \to (q \lor r)) \to ((p \to q) \lor (p \to r))$ are holded in intuitionistic logics? I could not found the model of intuitional logic that aboves are not true. So, if these can be proved, give…
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Annihilator in dual space

Let U and W are subspaces of a vector space V. If U is subset of W, inh(W) is subset of inh(U). Is the Converse true? How?
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