HILBERT, DAVID.

Ueber die Vollen Invariantensysteme.

Leipzig, B.G. Teubner, 1893. 8vo. Original printed wrappers, no backstrip. In "Mathematische Annalen. Begründet 1893 durch Alfred Clebsch und Carl Neumann. 42. Band. 3. Heft." Entire issue offered. [Hilbert:] Pp. 314-73. [Entire issue: Pp. 314-604].

First printing of Hilbert's fundamental landmark paper in which he "INTRODUCED STUNNING NEW IDEAS WHICH HAVE DEEPLY INFLUENCED THE DEVELOPMENT OF MODERN ALGEBRA AND ALGEBRAIC GEOMETRY." (Buchberger. Gröbner bases and applications. P. 63). The ideas presented in the present paper was introduced in his 1890-paper, but here he "called attention to the fact that his earlier results failed to give any idea of how a finite basis for a system of invariants could actually be construted. [...] To show how these drawbacks could be overcome, Hilbert thus adopted an even more general standpoint [...]. He described the guiding idea of this culminating paper of 1893 as invariants could actually be constructed". (Hendricks. Proof theory: history and philosophical significance. P. 59)

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 exceedingly complicated calculations involved. Hilbert realized that it was necessary to take a completely different path.
Hilbert sent his results to the Mathematische Annalen. Gordan, the expert on the theory of invariants for the Mathematische Annalen, did not appreciate the revolutionary nature of Hilbert's theorem and rejected the article. His - now famous - comment was: Das ist nicht Mathematik. Das ist Theologie. (i.e. This is not Mathematics. This is Theology).
Klein, on the other hand, recognized the importance of the work immediately, and guaranteed that it would be published without the slightest alterations. Encouraged by Klein and by the comments of Gordan, 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: "WITHOUT DOUBT THIS IS THE MOST IMPORTANT WORK ON GENERAL ALGEBRA that the Annalen has ever published."
Later, after the usefulness of Hilbert's method was universally recognized, Gordan himself said: "I have convinced myself that even theology has its merits".(Klein. Development of mathematics in the 19th century. P. 311).

Order-nr.: 44428