Editing Existence theorem (section)

=='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>