Editing Existence theorem (section)

{{short description|Theorem which asserts the existence of an object}}
[[File:Sqrt2 is irrational.svg|thumb|Geometrical proof that an irrational number exists: If the isosceles right triangle ABC had integer side lengths, so had the strictly smaller triangle A'B'C. Repeating this construction would obtain an infinitely descending sequence of integer side lengths.]]
In [[mathematics]], an '''existence theorem''' is a [[theorem]] which asserts the existence of a certain object.<ref>{{Cite web|url=https://www.dictionary.com/browse/existence-theorem|title=Definition of existence theorem {{!}} Dictionary.com|website=www.dictionary.com|language=en|access-date=2019-11-29}}</ref> It might be a statement which begins with the phrase "[[there exists|there exist(s)]]", or it might be a universal statement whose last [[Quantifier (logic)|quantifier]] is [[Existential quantification|existential]] (e.g., "for all {{math|''x''}}, {{math|''y''}}, ... there exist(s) ..."). In the formal terms of [[First-order logic|symbolic logic]], an existence theorem is a theorem with a [[prenex normal form]] involving the [[existential quantifier]], even though in practice, such theorems are usually stated in standard mathematical language. For example, the statement that the [[sine]] function is [[continuous function|continuous]] everywhere, or any theorem written in [[big O notation]], can be considered as theorems which are existential by nature—since the quantification can be found in the definitions of the concepts used.

A controversy that goes back to the early twentieth century concerns the issue of purely theoretic existence theorems, that is, theorems which depend on non-constructive foundational material such as the [[axiom of infinity]], the [[axiom of choice]] or the [[law of excluded middle]]. Such theorems provide no indication as to how to construct (or exhibit) the object whose existence is being claimed. From a [[constructivism (mathematics)|constructivist]] viewpoint, such approaches are not viable as it leads to mathematics losing its concrete applicability,<ref>See the section on [[Constructive proof#Non-constructive proofs|nonconstructive proofs]] of the entry "''[[Constructive proof]]''".</ref> while the opposing viewpoint is that abstract methods are far-reaching,{{explain|date=March 2021}} in a way that [[numerical analysis]] cannot be.