Noether's position and the writing of the paper

Her position at Göttingen

Emmy Noether held no post at Göttingen in 1918. She had come at the invitation of Hilbert and Klein in 1915, and the faculty had declined to let her habilitate. Her lectures were announced under Hilbert's name.

She was an algebraist. The work that made her reputation, and that mathematicians mean when they say her name, is in ring theory and ideal theory and comes later. The 1918 paper sits outside that line.

It does not sit alone, though. She published a second paper in the Göttingen Nachrichten in the same year, Invarianten beliebiger Differentialausdrücke, at pages 37 to 44, presented by Klein at the session of 25 January 1918 — the same session at which Klein presented his first note. That paper is on the invariants of arbitrary differential expressions, and it is the bridge between her algebraic work and the July paper. Any account that treats Invariante Variationsprobleme as an isolated excursion has overlooked the January paper.

The writing

Noether spent part of the spring of 1918 in Erlangen, visiting her widowed and ailing father. Because she was away, her correspondence with Klein survives, and it records the work in progress.

A postcard of 15 February already sketches what became the second theorem, though only in a particular case. A letter of 12 March gives a preliminary formulation of the essential consequence of it, for a variational problem invariant under a subgroup of an infinite-dimensional group. By then the shape of the result was fixed.

On 23 July she presented her results herself, to the Göttingen Mathematical Society. On 26 July Klein presented the paper to the Göttingen Scientific Society, which was the more consequential venue because its audience was not restricted to pure mathematicians. The printed paper carries the line recording this, naming Klein and the session of 26 July.

The dedication is to Klein, on the fiftieth anniversary of his doctorate.

A caution about the July sessions

Neither published text is the text that was presented.

Noether's paper carries a footnote stating that the final version of the manuscript was not submitted until the end of September. Klein's second note carries a footnote saying that his manuscript did not reach final form until the middle of September.

Both papers were therefore revised for two months after their presentations, in a period when each author was reading the other. What was said at either session is not recoverable from the printed papers, and this volume makes no claim about it.

Klein's account of her theorems

When Klein reprinted his second note in his collected works, he added a footnote that states both of Noether's theorems and gives her formulation of Hilbert's assertion about energy.

The footnote is worth its own attention. It restates the first theorem: that invariance of an integral under a group with a finite number of essential parameters makes that many linearly independent combinations of the Lagrange expressions into divergences. It restates the second: that invariance under a group depending on arbitrary functions, entering to some order of derivative, produces that many identical relations among the Lagrange expressions and their derivatives to that order. And it gives the precise form of Hilbert's claim — the energy relations become improper if and only if the integral is invariant under an infinite group containing the translation group as a subgroup.

This matters for provenance. The most consequential result in the paper is attested twice from primary text: in Noether's own final section, and in Klein's footnote to his own paper. Two authors, two documents, one statement. Very little in the history of this subject is that well anchored.

Provenance: the German original of Noether's paper was read in full and is public domain. Her prose is paraphrased throughout and marked as paraphrased; no English wording of hers is transcribed anywhere in this volume. The correspondence dates and the account of the Erlangen visit are taken from the secondary literature, not from the letters. Klein's paper was read in a modern English translation whose terms of reuse have not been established, and is summarised, not quoted. See Sources.