[FOM] hilbert's proof

josef@us.es josef at us.es
Tue Mar 9 14:11:05 EST 2010

Yes, the method was non-constructive, although Hilbert was able to write another paper offering a constructive procedure. In one of his papers he remarks that his method was based on subsuming the problem of invariant forms under the theory of algebraic functions in several variables. As happens with other of his early results, Hilbert was here influenced by the work of Richard Dedekind.

Dedekind and Heinrich Weber had published in 1882 a seminal paper with a purely algebraic treatment of the theory of functions in one variable. They relied on an ideal theory for fields of algebraic functions. Weber was one of Hilbert's professors at Koenigsberg in the mid-1880s.

As one can see, the "totally new method" was not absolutely new.
Incidentally, the Wikipedia entry is not very reliable, especially what it says towards the end mixing Kronecker and Brouwer, etc. One detail: Gordan's remark (This is not Mathematics. This is Theology.) was certainly humorous -- a pure existence proof reminds one of those proofs of the existence of God that had been criticized by Kant...


Jose Ferreiros

Date: Sun, 7 Mar 2010 20:17:27 -0800
From: Robert Knighten <RLK at knighten.org>
Subject: [FOM]  hilbert's proof
To: <addamo at wp.pl>

 > What was an essence of the "totally new method" by which Hilbert could
 > prove his the existence of a finite basis for the invariant forms?
 > with reagards, adam

It was non-constructive.  There is a reasonable brief discussion of this on
Wikipedia: http://en.wikipedia.org/wiki/David_Hilbert#The_finiteness_theorem

-- Bob

Jose Ferreiros
Instituto de Filosofia, CCHS - CSIC
Albasanz, 26-28. Madrid 28037

Dpto. de Filosofia y Logica, Univ. de Sevilla
Camilo J. Cela, s/n. 41018 Sevilla, Spain

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