Editing David Hilbert (section)

===Solving Gordan's Problem<!-- predominantly capitalized in the literature; "Gordan's problem" and "Gordan's Problem" redirect here-->===
Hilbert's first work on invariant functions led him to the demonstration in 1888 of his famous ''finiteness theorem''. Twenty years earlier, [[Paul Gordan]] had demonstrated the [[theorem]] of the finiteness of generators for binary forms using a complex computational approach. Attempts to generalize his method to functions with more than two variables failed because of the enormous difficulty of the calculations involved. To solve what had become known in some circles as ''Gordan's Problem'', Hilbert realized that it was necessary to take a completely different path. As a result, he demonstrated ''[[Hilbert's basis theorem]]'', showing the existence of a finite set of generators, for the invariants of [[algebraic form|quantics]] in any number of variables, but in an abstract form. That is, while demonstrating the existence of such a set, it was not a [[constructive proof]]—it did not display "an object"—but rather, it was an [[existence proof]]{{Sfn|Reid|1996|p=[https://books.google.com/books?id=mR4SdJGD7tEC&pg=PA36 36–37]}} and relied on use of the [[law of excluded middle]] in an infinite extension.

Hilbert sent his results to the ''[[Mathematische Annalen]]''. Gordan, the house expert on the theory of invariants for the ''Mathematische Annalen'', could not appreciate the revolutionary nature of Hilbert's theorem and rejected the article, criticizing the exposition because it was insufficiently comprehensive. His comment was:

{{verse translation|lang=ger|
Das ist nicht Mathematik. Das ist Theologie.
|
This is not Mathematics. This is Theology.{{Sfn|Reid|1996|p=34}}}}

[[Felix Klein|Klein]], on the other hand, recognized the importance of the work, and guaranteed that it would be published without any alterations. Encouraged by Klein, Hilbert extended his method in a second article, providing estimations on the maximum degree of the minimum set of generators, and he sent it once more to the ''Annalen''. After having read the manuscript, Klein wrote to him, saying:

{{blockquote|Without doubt this is the most important work on general algebra that the ''Annalen'' has ever published.{{Sfn|Reid|1996|p=195}}}}

Later, after the usefulness of Hilbert's method was universally recognized, Gordan himself would say:

{{blockquote|I have convinced myself that even theology has its merits.<ref name=":0">{{harvnb|Reid|1996|p=[https://books.google.com/books?id=mR4SdJGD7tEC&pg=PA37 37].}}</ref>}}

For all his successes, the nature of his proof created more trouble than Hilbert could have imagined. Although [[Leopold Kronecker|Kronecker]] had conceded, Hilbert would later respond to others' similar criticisms that "many different constructions are subsumed under one fundamental idea"—in other words (to quote Reid): "Through a proof of existence, Hilbert had been able to obtain a construction"; "the proof" (i.e. the symbols on the page) ''was'' "the object".<ref name=":0" /> Not all were convinced. While [[Leopold Kronecker|Kronecker]] would die soon afterwards, his [[Constructivism (mathematics)|constructivist]] philosophy would continue with the young [[Luitzen Egbertus Jan Brouwer|Brouwer]] and his developing [[intuitionist]] "school", much to Hilbert's torment in his later years.<ref>cf. {{harvnb|Reid|1996|pp=148–149.}}</ref> Indeed, Hilbert would lose his "gifted pupil" [[Hermann Weyl|Weyl]] to intuitionism—"Hilbert was disturbed by his former student's fascination with the ideas of Brouwer, which aroused in Hilbert the memory of Kronecker".{{Sfn|Reid|1996|p=148}} Brouwer the intuitionist in particular opposed the use of the Law of Excluded Middle over infinite sets (as Hilbert had used it). Hilbert responded:

{{blockquote|Taking the Principle of the Excluded Middle from the mathematician ... is the same as ... prohibiting the boxer the use of his fists.{{Sfn|Reid|1996|p=150}}}}