A mathematician’s daughter in Erlangen
Amalie Emmy Noether was born on 23 March 1882 in Erlangen, a university town in Bavaria. Her father, Max Noether, was a respected mathematician at the university; her mother, Ida Amalia Kaufmann, came, like him, from a prosperous Jewish merchant family. Emmy was the eldest of four children. Her brother Fritz also became a mathematician; Alfred, a chemist, died young; Gustav Robert was chronically ill.
As a girl she did not stand out academically. She was near-sighted, had a slight lisp, learned to cook and clean and play the piano, loved to dance, and at school was one of the few pupils who attended classes in the Jewish religion. In 1900 she passed the Bavarian examinations to teach English and French in girls’ schools with the grade ‘very good’. A respectable career as a language teacher lay open. She chose mathematics instead.
Fighting her way into the university
It was an unconventional choice. Two years earlier the University of Erlangen’s senate had declared that mixed-sex education would ‘overthrow all academic order’. Noether was one of only two women among nearly a thousand students, allowed merely to audit lectures with each professor’s permission. In 1903–04 she spent a semester at Göttingen hearing David Hilbert, Felix Klein and Hermann Minkowski. When Bavaria finally allowed women to enrol, she returned to Erlangen in 1904 and wrote a doctoral dissertation under Paul Gordan on the invariants of ternary biquadratic forms, graduating summa cum laude in 1907. It ended with a list of more than three hundred computed invariants. Later she dismissed it, bluntly, as rubbish.
For the next seven years she worked at Erlangen’s Mathematical Institute without pay or title, sometimes lecturing in place of her ailing father and quietly supervising doctoral students who were officially his. Under Gordan’s successor Ernst Fischer she discovered Hilbert’s more abstract methods, and her work began to change.
Göttingen and the theorem
In 1915 Hilbert and Klein invited her to Göttingen, then the world’s leading centre of mathematics. They were wrestling with Einstein’s new general theory of relativity, in particular with how energy conservation worked in it, and needed her expertise in invariants. The philologists and historians of the faculty objected to a woman teaching; one asked what soldiers returning from the war would think on finding themselves at a woman’s feet. Hilbert is said to have retorted that a university faculty was not a bathhouse. For four years Noether’s lectures were advertised under Hilbert’s name.
Soon after arriving she proved the result now called Noether’s theorem, presented by Klein to the Göttingen Royal Society of Sciences on 26 July 1918 in her paper ‘Invariante Variationsprobleme’. It states that every continuous symmetry of a physical system corresponds to a conserved quantity. If the laws of physics are the same today as tomorrow, energy is conserved; if they are the same here as there, momentum is conserved; if they do not depend on direction, angular momentum is conserved. A companion theorem explained why general relativity’s energy laws behave differently. Einstein, reading her work, admired its generality.
What the theorem changed
Before Noether, conservation laws were treated as separate facts of nature, discovered one by one. After her, they became consequences of symmetry, and symmetry became the physicist’s master key. The physicists Leon Lederman and Christopher Hill have called her theorem one of the most important mathematical theorems ever proved in guiding the development of modern physics, possibly on a par with the Pythagorean theorem.
Its influence grew through the twentieth century. Quantum mechanics, quantum field theory and the Standard Model of particle physics are built on symmetry principles, and physicists routinely look for new symmetries to predict new conservation laws and particles. Every student of theoretical physics meets Noether’s theorem, often long before learning anything about the woman who proved it.
Remaking abstract algebra
Among mathematicians Noether is celebrated even more for what came next. In 1919 the post-war revolution finally allowed her habilitation, and in the 1920s she transformed algebra. Her 1921 paper ‘Idealtheorie in Ringbereichen’ developed a general theory of ideals in commutative rings, proving in full generality the decomposition theorem first established by the chess champion and mathematician Emanuel Lasker, now the Lasker–Noether theorem. Objects satisfying her ‘ascending chain condition’ are still called Noetherian.
Her style was radically conceptual: instead of calculating, she looked for the general structures, rings, ideals, modules, homomorphisms, that made calculations unnecessary. Later she united the representation theory of groups with the theory of modules and ideals, and her ideas fed into algebraic topology. Her student Bartel van der Waerden’s textbook Moderne Algebra (1930–31), built largely on her lectures, taught her approach to generations. Nathan Jacobson wrote that the development of abstract algebra, one of the most distinctive innovations of twentieth-century mathematics, was largely due to her.
Teacher, colleague, ‘der Noether’
Noether gathered a devoted circle in Göttingen, nicknamed the ‘Noether boys’, among them Max Deuring, Hans Fitting, Ernst Witt and Grete Hermann. Her lectures were fast, unplanned and often baffling to outsiders, but for her students they were workshops in which new mathematics was made; she gave her ideas away freely, and many appeared under others’ names. Even after being granted the title of professor in 1922 she remained unsalaried until a small teaching contract the following year, and she never received a full professorship or election to the Göttingen academy.
She lived simply, was indifferent to dress and manners, and talked mathematics through meals and walks. She spent a winter at Moscow State University in 1928–29 and sympathised openly with the political left. Her international stature was confirmed in 1932, when she shared the Ackermann–Teubner prize with Emil Artin and gave a plenary address to the International Congress of Mathematicians in Zürich.
Dismissed for being Jewish
Jewishness, gender and politics had always counted against her; colleagues recalled that faculty hostility to her appointment was partly aimed at her Jewish origins and her social-democratic sympathies. In 1933 the Nazis made it official. Under the Law for the Restoration of the Professional Civil Service, which purged Jews and political opponents from public posts, she received a notice in April withdrawing her right to teach at Göttingen. Students in the Nazi movement had already demanded ‘Aryan mathematics’; one of her own former students joined the campaign against Jewish colleagues.
Noether met the blow with remarkable calm. She kept teaching students privately in her apartment, reportedly unperturbed even when one arrived in a stormtrooper’s uniform. Hermann Weyl recalled that her courage, frankness and unconcern for her own fate were a moral solace amid the hatred around them.
The Noether school
Noether’s influence spread through people as much as papers. Her first doctoral student, Grete Hermann, later known for her work on the foundations of quantum mechanics, spoke reverently of her ‘dissertation-mother’. Others included Max Deuring, Hans Fitting, Ernst Witt, Chiungtze Tsen and Jakob Levitzki, who went on to teach at the Hebrew University of Jerusalem. Around them gathered a wider circle, Wolfgang Krull, Emil Artin, Helmut Hasse, Richard Brauer, who carried her methods into number theory and the theory of algebras.
Visitors came from across the world. The Russian topologist Pavel Alexandrov became a close friend and nicknamed her ‘der Noether’, using the masculine article as a mark of respect. Bartel van der Waerden, who arrived from Amsterdam in 1924, called her originality absolute beyond comparison. When the university building was closed for a holiday she led her class out to the woods and lectured in a coffee house; when she was barred from teaching, she taught at home.
She was also generous to a fault, letting students and colleagues publish ideas that were really hers, and seemed untroubled by it. Olga Taussky-Todd remembered a lunch at which Noether, absorbed in mathematics, gesticulated so wildly that she spilled food over her dress without noticing. In the United States she noted wryly that Princeton University was ‘the men’s university, where nothing female is admitted’, but found warm support among the young women and colleagues at Bryn Mawr.
Refuge and sudden death
With a grant arranged through the Rockefeller Foundation, Noether took a post at Bryn Mawr College in Pennsylvania in late 1933. There she taught a small group of women including Olga Taussky-Todd and her only American doctoral student, Ruth Stauffer, and from 1934 she also lectured at the Institute for Advanced Study in Princeton.
In April 1935 surgeons removed a large ovarian tumour. She seemed to be recovering when, on 14 April 1935, she lost consciousness and died, aged fifty-three. Her ashes rest beneath the cloisters of the Old Library at Bryn Mawr. Her brother Fritz, who had fled to the Soviet Union, was later arrested there and shot in 1941. In a letter to The New York Times, Albert Einstein called her ‘the most significant creative mathematical genius thus far produced since the higher education of women began’. Today the theorem that bears her name is taught in every physics department in the world, and ‘Noetherian’ is part of the everyday vocabulary of mathematics, her methods, as she predicted, everywhere.
Friends at Bryn Mawr held a small memorial service at the college president’s house a few days after her death; Hermann Weyl and Richard Brauer travelled from Princeton to deliver eulogies. In the months that followed, Einstein, van der Waerden, Weyl and Alexandrov all published tributes, and Weyl’s memorial address, with its tribute to her courage and warmth, is still widely quoted.
Sources & further reading
Emmy Noether, Wikipedia · Emmy Noether, MacTutor History of Mathematics (University of St Andrews) · Emmy Noether, Wikiquote · Emmy Noether: a creative mathematical genius, Plus Magazine (University of Cambridge) · Researched from the sources above and checked against them. Spotted an error? Tell us.
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