Editing Constructive proof (section)

==A historical example==

Until the end of 19th century, all mathematical proofs were essentially constructive. The first non-constructive constructions appeared with [[Georg Cantor]]’s theory of [[infinite set]]s, and the formal definition of [[real number]]s.

The first use of non-constructive proofs for solving previously considered problems seems to be [[Hilbert's Nullstellensatz]] and [[Hilbert's basis theorem]]. From a philosophical point of view, the former is especially interesting, as implying the existence of a well specified object.

The Nullstellensatz may be stated as follows: If <math>f_1,\ldots,f_k</math> are [[polynomial]]s in {{mvar|n}} indeterminates with [[complex number|complex]] coefficients, which have no common complex [[zero of a function|zeros]], then there are polynomials <math>g_1,\ldots, g_k</math> such that 
:<math>f_1g_1+\ldots +f_kg_k=1.</math>

Such a non-constructive existence theorem was such a surprise for mathematicians of that time that one of them, [[Paul Gordan]], wrote: ''"this is not mathematics, it is theology''".<ref>{{Cite book|title=Circles disturbed: the interplay of mathematics and narrative — Chapter 4. Hilbert on Theology and Its Discontents The Origin Myth of Modern Mathematics|last=McLarty|first=Colin|date=April 15, 2008|publisher=Princeton University Press|others=Doxiadēs, Apostolos K., 1953-, Mazur, Barry|s2cid=170826113|isbn=9781400842681|location=Princeton|doi=10.1515/9781400842681.105|oclc=775873004}}</ref>

Twenty five years later, [[Grete Hermann]] provided an algorithm for computing <math>g_1,\ldots, g_k,</math> which is not a constructive proof in the strong sense, as she used Hilbert's result. She proved that, if <math>g_1,\ldots, g_k</math> exist, they can be found with degrees less than 
<math>2^{2^n}</math>.<ref>{{Cite journal|last=Hermann|first=Grete|date=1926|title=Die Frage der endlich vielen Schritte in der Theorie der Polynomideale: Unter Benutzung nachgelassener Sätze von K. Hentzelt|journal=Mathematische Annalen|language=de|volume=95|issue=1|pages=736–788|doi=10.1007/BF01206635|s2cid=115897210|issn=0025-5831}}</ref>

This provides an algorithm, as the problem is reduced to solving a [[system of linear equations]], by considering as unknowns the finite number of coefficients of the <math>g_i.</math>