Hilbert's early carrier: encounters with allies and rivals
The article refers to the history of invariant theory; it covers the five years between 1888 and 1893, when Hilbert worked on this field. Hilbert' allies were Felix Klein and Hermann Minkowski, his rivals were Leopold Kronecker and Paul Gordan. It all began with a journey to Berlin, Leipzig and Göttingen from March 9 to April 7, 1888; now Hilbert had caught fire. Though Hilbert's invariant theory was deeply influenced by Kronecker, he later developed a deep antipathy toward Kronecker. Kronecker expressed his own missgivings with the sentence ``Hilbert's approach to invariant theory was theology not mathematics''. Soon Hilbert was able to extend Gordan's finiteness theorem from systems of binary forms over real or complex numbers to forms in any number of variables. Gordan, however, found Hilbert's proof entirely correct but he noticed a gap. Instead, Hilbert began his ``Siegeszug''. In 1893 Hilbert presented his final contribution which was based on five fundamental principles. As far as history is concerned, Hilbert distinguished three periods: 1. the naive period, that was Cayley and Sylvester, 2. the formal period, that was Clebsch and Gordan, 3. and the cirtical period, that was Hilbert. After Hilbert, invariant theory began to wither. ``Invariant theory thus entered the annals of mathematics, its history already sketched by the man who wrote its epitaph'' (p. 81). Hilbert now turned to a new field, number theory that is.
scientific article; zbMATH DE number 2170561