The Teacher who Changed Mathematics

“In the judgment of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began.” Thus the great physicist Albert Einstein eulogized mathematician Emmy Noether, who died in 1935. Einstein’s eulogy was joined by tributes from mathematicians and physicists that describe Noether as the most important woman in the history of mathematics. 

Auditing student

Amalie Noether (or Nöther) was born on March 23, 1882, in Erlangen, Germany, to a traditional Jewish family. Her father, Max Noether, was a renowned professor of mathematics at the local university, and one of her three younger brothers also became a mathematician. 

In school, Noether did not stand out as a particularly bright student. Her peers and teachers described her as a four-eyed, smart, likeable, and affable. In high school, she began going by her middle name, Emmy. Excelling in high-school English and French, she decided to become a language teacher. She also learned to play the piano and enjoyed dancing in parties. 

In 1897 she graduated from high school and pursued the education path she chose for herself. She graduated in 1900, receiving top grades in the examination for teachers of English and French of the state of Bavaria.

Despite her excellent grades, Noether was never to teach in a classroom. At some point during her studies, she decided to change direction and chose to study mathematics. In those days, however, women were not permitted to study in German universities. Their only option was to take courses as auditing students, without receiving credit or a grade. And even this required the consent of each professor – not an obstacle for Noether with her father's colleagues, many of whom even allowed her to eventually take the exam.  

In the years to come, Noether took many courses at the University of Erlangen in mathematics and in other fields, such as classical studies, and was one of the only two women to study at the university, among 984 men. In 1903, she successfully passed the entrance exam to the university, although she was still not permitted to be officially enrolled. Following the exam, she took one semester at the University of Göttingen, which was a world-class center for excellence in mathematics and physics at the time. As an auditor, she took courses given by some of the most prominent scientists of the time, including physicist Karl Schwarzschild and mathematician David Hilbert.

Mathematics without salary

Upon completing her studies, Noether returned to Erlangen, and then her luck changed for the better: Germany changed its law, allowing women to study in the university. She seized the opportunity and set to study only mathematics this time. Receiving credit for her previous studies, she completed her PhD with highest honors in 1907 under the supervision of Paul Gordan.

Noether's dissertation was in the field of differential and algebraic invariants, a branch of algebra dealing with actions of groups or vector spaces. Hilbert had already identified the fundamentals of these invariants in the late 19th century, but Noether developed more efficient and practical methods to solve the problems that Hilbert solved in theory only. Though her dissertation was very well received, she described it as "rubbish" in later years.

After completing her PhD, the natural course was to pursue an academic position; but PhD or not, Noether was still a woman – and securing such position in Erlangen was out of the question. To be exact, there was an available position, but it was without pay, credit, or any formal recognition. In practice, Noether taught courses in which she was officially a "teaching assistant" and even mentored graduate students, with her father registered as the official supervisor, but she did not receive pay for doing so. Nevertheless, she stayed in Erlangen, partially to assist her father, who was in poor health.

Along with her teaching and mentoring, Noether continued her research work, which was deeply influenced by Ernst Fischer, who replaced Gordan after his retirement. She would later write that his encouragement to study abstract algebra from an arithmetic point of view shaped her entire future body of work. She continued to publish research papers, building a reputation for herself in the mathematics departments, and was invited to participate and give talks at conferences throughout Europe.

A proof of great importance in physics. The title of the paper that contains Noether's theorem, which ties conservation laws with symmetry in physical systems

Relations and relativity 

In 1915, Hilbert and the world-renowned mathematician Felix Klein offered Noether a position in Göttingen. Hilbert had begun working on the mathematical aspects of Einstein's theory of relativity, and needed an expert on invariants. Noether could not pass on the opportunity to work with the best mathematicians in the world on problems at the cutting edge of science.

Almost as soon as she arrived in Göttingen, Noether solved two seminal problems that Hilbert had been working on. One dealt with Riemannian manifolds, which are used to mathematically describe multi-dimensional structures and were an important tool in Einstein's breakthrough theories. The second problem dealt with the law of energy conservation, as part of the general theory of relativity. Her solution deciphered the connection between symmetry in a physical system and the conservation laws that apply to it. Namely, Noether provided the mathematical explanation for conservation laws, such as conservation of energy, momentum, or electric charge, which are pivotal both in theoretical physics and in its applications – from vehicle engineering to calculating the routes of celestial objects. This proof is now called Noether's theorem, and it's publication in 1918 cemented her status as one of the most prominent mathematicians of her time.

Her immense success did not translate to an improved financial or employment situation. On the contrary – exactly as in Erlangen, she was officially Hilbert's teaching assistant, although she actually taught his courses without pay. Noether had little means to live by, and the fact that it was a time of war did not improve the situation. Poverty was probably what sparked her enthusiastic socialist views, and while expressing them often, she did not take part in any organized political activity. Noether was also a pacifist her whole life, and opposed war of any kind.

Hilbert and Klein’s repeating efforts to arrange a position for her were not successful, and even their appeal to the German Ministry of Education was rejected. One university faculty member, who opposed hiring women, asked what would the soldiers returning from the war would think, when they find out they have to learn from women? Hilbert replied to this question by saying that he does not see what gender has to do with it – “After all, we are a university, not a bath house,” he said.

Ironically, it was ultimately the war that Noether so opposed that facilitated the change in her status. Germany's defeat in WWI led to many legislative changes, including the law that prevented women from becoming university faculty. In 1919 she was formally recognized as an external lecturer, still with no tenure or social benefits, but at least she could teach courses under her name and mentor graduate students.

Abundant professional appreciation with no financial compensation. Noether at Göttingen | Source: Bar-Ilan University

Rings and collaborations 

In the next years in Göttingen, Noether would mainly work in the field of abstract algebra, and was one of its founders. She led new developments in the branch of “ring theory,” which defines combinations of groups of numbers and operations. Noether is responsible for some of the breakthroughs in the field, for example, her finding of a generalized method to define rings with certain properties, now called Noetherian rings.

In 1924, Dutch mathematician Bartel van der Waerden arrived to work with Noether, and later published his well-known textbook Moderne Algebra, a large section of which is based on her work. This story shows another side to Noether, who often collaborated with other mathematicians, and allowed them to receive credit for work she has done herself. She did this many times with the mentored students. Describing her impact on mathematics, many colleagues refer not only to her many published papers, but also to her prolific collaborations and the inspiration and assistance she gave her to collaborators.

By the 1920s and early 1930s, Noether was considered one of the most important mathematicians worldwide. She was invited to speak at important conferences throughout Europe, and was even invited to teach one semester at Moscow State University – where her socialist tendencies probably didn’t hurt. In 1932, she received the Ackermann–Teubner Award given by the University of Leipzig for considerable contribution to mathematics.

The end of an era: The letter ordering Noether's dismissal from Göttingen in September 1933 | Source: National Library of Israel

The greatest

In 1933, the Nazis rose to power in Germany, and all of Noether's mathematical achievements were of no importance to them, since she was Jewish. Shortly afterwards, the Law for the Restoration of the Professional Civil Service was passed, enabling the dismissal of Jews from government positions. Soon enough, the University of Göttingen received a letter ordering it to dismiss Noether. Expelling all Jewish mathematicians from Göttingen was detrimental to the university. In 1934, the Reich Minister of Culture met with Hilbert and asked him how the department of mathematics was doing, now that it was released from the burden of Jews. Hilbert, who opposed expelling Jews, simply replied that Göttingen no longer has mathematics.

After being expelled from University of Göttingen, Noether crossed the Atlantic Ocean. Albert Einstein, who was already in the US and was quite familiar with her work, assisted her in receiving a faculty position at Bryn Mawr College, a prestigious women’s college in Pennsylvania. For the first time in her life, she received, along from professional recognition, also a nice salary and decent working conditions. In 1934, she also gave a workshop at the Institute for Advanced Study at Princeton University, one of the world’s most important centers in mathematics and theoretical physics. 

But Noether's new path was a short-lived one. In April 1935, after only a year and a half in the US, she was diagnosed with a pelvic tumor. She underwent surgery for its removal, and, in the first days after the operation, seemed to be recovering well, but a number of days later her fever shot up and she died on April 14, shortly after her 53rd birthday.

The tributes following her death highlight her indisputable greatness. Van der Waerden wrote that Noether's originality was “absolutely beyond comparison,” and mathematician Hermann Weyl added that her work “changed the face of algebra.” Another mathematician, Norbert Wiener, wrote just weeks before her death that Noether is “the greatest woman mathematician who has ever lived; and the greatest woman scientist of any sort now living, and a scholar at least on the plane of Madame Curie.”

Noether's body was cremated, and her ashes were buried under the walkway around the library at Bryn Mawr, which was her home following Nazi persecution. A street in her home town, as well as scores of schools and academic departments, including the Department of Mathematics at Bar-Ilan University, are named after her. In addition, a crater on the moon and an asteroid bear her name, in honor of her eminent contribution to physics.

Translated by Elee Shimshoni