**Emmy Noether (1882-1935)**

**By Else Hoyrup**

**Introduction **

This is a non-technical introduction to Emmy Noether, a German mathematician and physicist; one of the greatest mathematicians and physicists of the 20^{th} Century. I first heard her name while an undergraduate studying modern abstract algebra. She was one of the principal founders of this discipline.

She was the only woman I heard of when I studied mathematics in the mid 1960’s. As a young person, my specialty besides algebra was algebraic topology, a field where Noether also exerted great influence. This has also contributed to my interest in her. (Now my field is history of science).

But during the last years, I have learned that today Noether is also considered one of the most central theoretical physicists for her discovery of the link between symmetries and conservation laws, called the Noether theorem. I find this fascinating: She was a theoretical mathematician, who has become extremely famous today for her physics, many years after her death.

**Biography**

When Emmy Noether was young, the German universities were closed to women, but they gradually opened up during Noether’s time.

But she never got a regular job, probably because she was a woman and a Jew. Since other Jews did become professors, for instance Noether’s own father and brother, it was probably her being a woman which barred her from a regular job. For many years she even earned no money at all, lecturing under the great mathematician David Hilbert’s name in Göttingen.

**“Habilitation” **

Hilbert did the best he could to get her a regular and paid job at the University of Göttingen. The first time was in 1915, where he met with fierce opposition from the humanists at the university, who – in contrast to the mathematicians – were fiercely against women at the university. Hilbert argued:

“Aber die Fakultät ist doch keine Badeanstalt!” (“After all, this is a university, not a public baths”).

But the humanists would not give in. – Remember, in 1915 the First World War had just started. – The humanists argued that if women were allowed to teach at the university, they would become competitors to the male soldiers, when they returned from the war.

As we shall see in the passage about Emmy Noether’s physics, her results in two papers from 1918 helped David Hilbert, the great mathematician Felix Klein, who was also in Göttingen, and Einstein, who was not, in their work with mathematical problems in the general theory of relativity. Therefore all three of them wanted to help Noether to get a paid job at the University of Göttingen. Einstein was especially impressed by Noether’s results.

In 1919, after the war, Klein finally succeeded in getting Emmy Noether, not a job, but the permission to teach at the University of Göttingen (see below). Time had changed after the war in the young Weimar Republic and given women more opportunities.

In order to get the right to teach at a university in Germany, the candidate must have a degree higher than the equivalent of the PhD, called in German “Habilitation”. But it was not enough to write a good thesis, you must also give an oral test lecture.

She wrote her habilitation thesis in 1918 on some important questions in mathematical physics, relating to the general theory of relativity. It was called:

*Invariante Variationsprobleme.* (“Invariant Variation Problems”).

In my passage “Physics”, I write more about this important article, which has become a hot theme among modern physicists. Her results are called the **Noether theorem. **

So in 1919, she was finally given her Habilitation and with it the right to teach at the University of Göttingen. She was now a “Privatdozent”, which was not a job, and which was without any payment! In 1922, she was given the title “Ausserordentlicher Professor”, which still did not give her any salary.

In 1923, she at last got a little money for her work. She got what was called a “Lehrauftrag”, that is a remuneration for her teaching hours. But this Lehrauftrag should be renewed each semester.

___________________________

But against all odds, she managed to live a life which was satisfying to her. She did not worry about worldly wealth. She got accustomed to poverty, which she managed by having a frugal lifestyle and living in an inexpensive boarding house. She never married and never had children.

When Hitler came to power in 1933, Noether and other Jews were fired from their university jobs. She was totally obnoxious to the Nazis: She was a Jew, she was an academic woman, she was a social democrat and she was a pacifist.

So she was forced to emigrate to America in 1933. Unfortunately, she died in 1935, aged 53.

In a eulogy after her death, her fellow mathematician from Germany and US, Hermann Weyl, remarked on Noether’s extremely fine, generous and unselfish personality: “During the Nazi terror against German Jews, she was never concerned about her own fate, but the more concerned about her fellow mathematicians and her students”.

**Mathematics**

Noether was the creator of a whole new scientific school, which became one of the most brilliant schools of mathematics: The school of abstract algebra. In her hands algebra was directed away from cumbersome calculations to operating of abstract, general concepts like (mathematical) groups, (mathematical) rings and (mathematical) ideals. She proved some beautiful theorems, for instance in 1921 in her article in Mathematische Annalen: *Idealtheorie in Ringbereichen *(“The Theory of Ideals in Ring Domains”).

But you cannot measure her influence only by her own publications: She was most generous in giving away her ideas to her students and colleagues, and she was extremely inspiring. She also inspired work in other mathematical fields, like algebraic topology and algebraic geometry. Her way of working consisted among other things in working in a close knit network of followers, the so called “Noether boys”. Mathematics was her only passion, she worked with it all the time, and she was happy with it. On Sundays, she went for long walks in the countryside with her Noether boys. She was a great and pleasant personality.

Her colleague and friend, the Russian algebraic topologist Alexandrov gave a fine eulogy about her after her premature death in 1935:

*The fundamental characteristic of her mathematical talent was the striving for general formulations of mathematical problems and the ability to find the formulation which reveals the essential logical nature of the question, stripped of any incidental peculiarities which complicate matters and obscure the fundamental point. *

And further:

*It was she who taught us to think in terms of simple and general algebraic concepts – homeomorphic mappings, groups and rings with operators, ideals – and not in terms of cumbersome algebraic computations; and thereby opened up the path to finding algebraic principles in places where such principles had been obscured by some complicated special situation which was not at all suited for the accustomed approach of the classical algebraists. *

Although Noether preferred to think in abstract terms, she did not make abstractions for the sake of abstractions themselves, but because abstractions in her hand became more fruitful.

As already mentioned, an era in her life ended, when the Nazis came to power in 1933 and she was dismissed from her small university job. She emigrated to US and landed at Bryn Mawr College in Pennsylvania, a women’s college, which had a fine reputation. Here she created a new school of women mathematicians. She also traveled by train to nearby Princeton University to lecture once a week at the Institute for Advanced Study. Among her listeners was Albert Einstein, who was a great admirer of Noether.

You can read more about Noether at the website *Mac Tutor History of Mathematics Archive*:

http://www-history.mcs.st-and.ac.uk/Mathematicians/Noether_Emmy.html

**Physics**

In 1915 the mathematicians David Hilbert and Felix Klein invited Noether to the famous University of Göttingen in Germany. Hilbert and Klein were working with Einstein’s general theorem of relativity, but they had all three encountered a seeming paradox: Under the general relativity there seemed to be no local energy conservation. Under the special relativity there was both local and global energy conservation. Emmy Noether was invited to help the three involved solve their problem, because she was already famous for her work on invariants.

She solved the problem quickly and elegantly and on top of that, she made further generalizations. Her paper on the matter has the title *Invariante Variationsprobleme *(“Invariant Variation Problems”). It was written and published in 1918. Just before this paper, also in 1918, she had published a related paper: *Invarianten beliebiger Differentialausdrücke *(“Invariants of Arbitrary Differential Expressions”). This paper was more mathematical, whereas the other paper was more physical.

Einstein was very impressed by her results and in May 1918 he wrote to Hilbert after reading the paper “Invarianten beliebiger Differentialausdrücke”:

*Yesterday I received from Miss Noether a very interesting paper on invariants. I’m impressed that such things can be understood in such a general way. It would have done no harm to the soldiers returning to Göttingen from the trenches to be sent to school under Miss Noether! She seems to know her stuff. *

Note: The quotation in the original German version uses Einstein’s own word *Feldgrauen*. In most English translations, this is rendered as *the Old Guard (of Göttingen). *The Old Guard means the professors in Göttingen, among them Hilbert and Klein themselves. But this is a linguistic misunderstanding of some consequence. The word *Feldgrauen *means the returning soldiers from the battlefields in The First World War. Literary, the word means soldiers with their gray uniforms. But even among younger people in Germany, many are not familiar with this expression. My thanks to Cordula Tollmien for pointing out this misunderstanding and for sending me an extremely interesting article in German about the matter. The author, Reinhard Siegmund-Schultze, furthermore writes that it is difficult to translate Einstein’s irony in his letters. Einstein was a pacifist, contrary to most Germans, including Klein himself. So he did not want men to go to war. Instead, he writes that he wanted them to go to the university and listen to Emmy Noether, who was also a pacifist. – The above English translation of Einstein’s praise is Reinhard Siegmund-Schultzes’ wording.

But she did not herself attach much importance to her work on physics. Her now famous paper, *Invariante Variationsprobleme, *deals with theoretical physics and especially mathematical physics and both she herself and her contemporary mathematics circle saw her first and foremost as a mathematician. For many years, not many mathematicians or physicists did study her article carefully, maybe because it was rather technical and on the border between physics and mathematics, maybe because of gender discrimination. Therefore it was not cited very often for many years. But today, many years after her death, her results are on every physicist’s lips! This fascinates me!

A bit technically speaking, the article is about problems in the calculus of variations with differential invariants. She proves two theorems and their converses and together the four theorems are called collectively the **Noether theorem. **(Sometimes they are called the Noether theorem**s**).

Physically speaking, they deal with the fundamental connection between symmetry, invariance and conservation laws:

Time translational invariance gives conservation of energy.

Space translational invariance gives conservation of momentum.

Isotropy (invariance to different directions) gives conservation of angular momentum.

Time invariance and spatial invariance are absolutely fundamental to scientific thinking, in that they guarantee that an experiment done at another time or at another place gives the same results. Without it one could not have science and scientific theory!

Today Noether’s results are the cornerstone in the basis of modern theoretical physics. On top of that they have proved to be central to areas of physics, which were new, for instance elementary particle physics.

According to physics professor Jeppe Dyre: *The* *Noether theorem is the most beautiful and most central result in physics! *

For further reading, see the website *Contributions of 20*^{th}* Century Women to Physics: *

http://cwp.library.ucla.edu/Phase2/Noether,_Amalie_Emmy@861234567.html

**Conclusion **

Emmy Noether and her work are exceptions in the history of science and most interesting to me too. I hope that some other people will also find her interesting. Today she is remembered in several ways by things bearing her name: Noetherian rings in mathematics. The Noether theorem in physics described above. And also a crater on the Moon.

In her own lifetime, the high point of recognition was in 1932, at the International Mathematical Congress in Zürich, where she gave a plenary lecture. But although she was much esteemed in mathematical circles, her work as an unofficial editor of the important journal Mathematische Annalen was never credited officially on paper. Why I don’t know. This may have caused her some grief. But she was never bitter about the obstructions she met. She just concentrated on her mathematics and her mathematician friends.

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It’s important that we recognize these who taught us as well as those who inspire/motivate us, despite being a math teacher, Mrs Giles ironically teaches the importance of history