Editing Existence theorem

{{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.

=='Pure' existence results==
In mathematics, an existence theorem is purely theoretical if the proof given for it does not indicate a construction of the object whose existence is asserted. Such a proof is non-constructive,<ref>{{Cite web|url=http://mathworld.wolfram.com/ExistenceTheorem.html|title=Existence Theorem|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-11-29}}</ref> since the whole approach may not lend itself to construction.<ref>{{cite book|author=Dennis E. Hesseling|title=Gnomes in the Fog: The Reception of Brouwer's Intuitionism in the 1920s|url=https://books.google.com/books?id=6CXyBwAAQBAJ&pg=PA376|date=6 December 2012|publisher=Birkhäuser|isbn=978-3-0348-7989-7|page=376}}</ref> In terms of [[algorithm]]s, purely theoretical existence theorems bypass all algorithms for finding what is asserted to exist. These are to be contrasted with the so-called "constructive" existence theorems,<ref name="RubinsteinRubinstein1998">{{cite book|author1=Isaak Rubinstein|author2=Lev Rubinstein|title=Partial Differential Equations in Classical Mathematical Physics|url=https://books.google.com/books?id=pLjLf0_yvqsC&pg=PA246|date=28 April 1998|publisher=Cambridge University Press|isbn=978-0-521-55846-4|page=246}}</ref> which many constructivist mathematicians working in extended logics (such as [[intuitionistic logic]]) believe to be intrinsically stronger than their non-constructive counterparts.

Despite that, the purely theoretical existence results are nevertheless ubiquitous in contemporary mathematics. For example, [[John Forbes Nash, Jr.|John Nash]]'s original proof of the existence of a [[Nash equilibrium]] in 1951 was such an existence theorem. An approach which is constructive was also later found in 1962.<ref>{{cite book|author=Schaefer, Uwe|title=From Sperner's Lemma to Differential Equations in Banach Spaces : An Introduction to Fixed Point Theorems and their Applications|url=https://books.google.com/books?id=1B2yBQAAQBAJ&pg=PA31|date=3 December 2014|publisher=KIT Scientific Publishing|isbn=978-3-7315-0260-9|page=31}}</ref>

==Constructivist ideas==
From the other direction, there has been considerable clarification of what [[constructive mathematics]] is—without the emergence of a 'master theory'. For example, according to [[Errett Bishop]]'s definitions, the continuity of a function such as {{math|sin(''x'')}} should be proved as a constructive bound on the [[modulus of continuity]], meaning that the existential content of the assertion of continuity is a promise that can always be kept. Accordingly, Bishop rejects the standard idea of pointwise continuity, and proposed that continuity should be defined in terms of "local uniform continuity".<ref>{{Cite web|url=https://ncatlab.org/nlab/show/Bishop's+constructive+mathematics|title=Bishop's constructive mathematics in nLab|website=ncatlab.org|access-date=2019-11-29}}</ref> One could get another explanation of existence theorem from [[type theory]], in which a proof of an existential statement can come only from a ''term'' (which one can see as the computational content).

== See also ==
*[[Constructive proof]]
*[[Constructivism (philosophy of mathematics)]]
*[[Uniqueness theorem]]

==Notes==
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{{DEFAULTSORT:Existence Theorem}}
[[Category:Mathematical theorems]]
[[Category:Mathematical and quantitative methods (economics)]]