# Noether, Emmy (1882–1935)

# Noether, Emmy (1882–1935)

**German theoretical mathematician who pioneered the study of cross product and abstract algebra, and who contributed to the discovery of the theory of relativity.** Name variations: Amalie Emmy Noether. Born in Erlangen, Germany, on March 23, 1882; died in Bryn Mawr, Pennsylvania, on April 14, 1935; daughter of Ida (Kaufman) Noether and Max Noether; attended Städtischenen Höheren Töchterschule, University of Göttingen, University of Erlangen; never married; no children.

*Awarded the Alfred Achermann-Teubner Memorial Prize for the Advancement of Mathematical Studies (1932).*

*Passed the Bavarian state teacher's examination (1900); enrolled at the University of Erlangen (1900); passed the Bavarian state high school teacher's examination (1903); enrolled at the University of Göttingen (1903); withdrew from the University of Göttingen (1904); enrolled at the University of Erlangen (October 1904); awarded Ph.D. by the University of Erlangen (December 1907); invited to teach at the University of Göttingen (1915); awarded the honorary position of Associate Professor without Tenure (1919); granted a lectureship in algebra (1923); was a visiting professor, University of Moscow (1928); was a visiting professor, University of Frankfurt (1930); attended the International Mathematical Congress, Zurich, Switzerland (September 1932); dismissed from the University of Göttingen (April 7, 1933); was a visiting professor, Bryn Mawr College, Pennsylvania (1933); lectured regularly at the Institute for Advanced Study, Princeton University (1934); joined the American Mathematical Society (1934).*

## Selected writings:

*(Edited by N. Jacobson)* Gesammelte Abhaudlungen (Collected Papers, *NY: Springer-Verlag, 1983*).

The oldest of four children, Emmy Noether was born in Erlangen, Germany, in 1882, the daughter of **Ida Kaufman Noether** and Max Noether. She experienced a typical childhood for a girl of her class: finishing schools and musical instruction were followed by lessons in French and English. In 1900, Noether took the Bavarian state teacher's examination, and passed with a grade of "very good." She seemed poised for a traditional career teaching languages in an all-girls' school.

Although women were admitted to the universities of France, England, and Italy, such was not the case in Germany. The historian Heinrich von Treitschke exemplified the conservatism of the era:

Many sensible men these days are talking about surrendering our universities to the invasion of women, and thereby falsifying their entire character. This is a shameful display of moral weakness. They are only giving way to the noisy demands of the press. The intellectual weakness of their position is unbelievable…. The universities are surely more than mere institutions for teaching science and scholarship. The small universities offer the students a comradeship which in the freedom of its nature is of inestimable value for the building of a young man's character.

Max Noether was a respected mathematician at the University of Erlangen. Known as "the Father of Algebraic Geometry," he exposed his daughter to theoretical mathematics from an early age. Two of her three brothers also went on to become distinguished scientists; Alfred Noether received a doctorate in chemistry, while Emmy's youngest brother Fritz became a physicist.

Albert Einstein">

In the judgment of the most competent living mathematicians, fraülein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began.

In 1900, German universities began to admit female students. They did not, however, grant degrees to women, except under exceptional circumstances. Encouraged by her father, Noether enrolled at the University of Erlangen, becoming one of only two female students there. At the same time, she also studied for the high school certification examination, which she passed in 1903.

Though certified to teach high school, Noether was determined to pursue a career in mathematics, so she enrolled at the University of Göttingen. Göttingen, which figured so prominently in the mathematics of the 18th and 19th centuries, was home to Felix Klein, Otto Blumental, Hermann Minkowski, and David Hilbert. Within a few months, however, Noether returned to the University of Erlangen when it announced that it would grant formal degrees to women. Paul Gordan, a colleague of Max Noether, acted as Emmy's supervisor, and strongly influenced her views on algebraic invariants. For many years afterwards, his photograph hung in her study. In December 1907, Emmy Noether passed her oral examination *summa cum laude* and was awarded a Ph.D.

Noether's doctoral thesis "On the Complete Systems of Invariants for Ternary Biquadratic Forms" was considered to be an uninspired effort by contemporary mathematicians. Indeed, Noether herself dismissed the work as a "jungle of formulas." However, her thesis was widely applied by physicists, who dubbed it "Noether's theorem." As Peter G. Bergmann noted: "Noether's theorem, so-called, forms one of the corner stones of work in general relativity as well as certain aspects of elementary particle physics." For the next several years, Noether worked as an unpaid research assistant at the University of Erlangen, delivering lectures for her father, whose health was in an increasingly fragile state.

By 1915, Noether's reputation as a mathematician had spread to Göttingen, where David Hilbert and Felix Klein were tackling the tricky problem of the theory of relativity. Noether was invited to contribute her expertise. Her elegantly simple formulations of several of Einstein's concepts greatly impressed the eminent scientist. Albert Einstein wrote to Hilbert on May 24, 1918:

Yesterday I received from Miss Noether a very interesting paper on invariant forms. I am impressed that one can comprehend these matters from so general a viewpoint. It would not have done the Old Guard at Göttingen any harm, had they picked up a thing or two from her. She certainly knows what she is doing.

Noether was already the author of several articles by the time she reached Göttingen. Although she was beginning to emerge as one of Germany's top mathematicians, a suitable position was difficult to secure. As **Teri Perl** noted, "Göttingen now gave doctoral degrees to women, jobs, however, were another matter."

Both Klein and Hilbert supported Noether's fight for a permanent teaching position. However, strong opposition emerged from the non-scientists of the university. An unidentified opponent is reported to have argued:

How can it be allowed that a women become a Privatdozent (lecturer without pay)? Having become a Privatdozent, she can then become a professor and a member of the university senate. … What will our soldiers think when they return and find that they are expected to learn at the feet of a woman?

Exasperated by the obdurate comments of his colleagues, Hilbert is said to have exclaimed, "I do not see that the sex of the candidate is an argument against her admission as a Privatdozent. After all, we are a university, not a bathing establishment!" Hilbert's role as Noether's patron illustrates the dependent position women still found themselves in when the question of career advancement arose. Defying his colleagues, he arranged for Noether to lecture frequently in his classes.

The attempted Communist revolt in Germany in 1918 profoundly influenced Noether's political ideals. While she had never before been aligned with a political faction, the failure of the revolution led to Noether's membership in the Social Democratic Party. In latter years, she expressed strong pacifists sentiments. But the proclamation of the Wiemar Republic in 1919, greatly enhanced the legal status of women in Germany. In the same year, Noether was awarded the title of "Associate Professor without Tenure." The honorary position required no teaching and did not include a salary. Nevertheless, Noether was treated like a full professor, a recognition of her superior mathematical abilities. In 1923, she was granted a lectureship in algebra, which provided a tiny income. Writes Noether's nephew Gottfried:

Unquestionably, she had the mathematical ability to compete with the best. Even though the legal position of women had changed under the German Republic, old prejudices continued to persist. That there was prejudice against women in academic circles was clearly demonstrated by the fact that Noether was denied election to the Göttingen Gesellschaft de Wissenschaften [Göttingen Academy of Sciences]…. But additional reasons quite likely played a role too. She was a Jew; during the early 1920s, she was a member of the Social Democratic Party in Germany; she was a lifelong pacifist.

Noether's lectures were characterized by a relaxed and informal approach, which contrasted markedly with that of her male colleagues. She enjoyed considerable popularity with the student body. Her scholars were nicknamed "the Noether Boys," and Norbert Wiener wrote that "her many students flocked around her like a clutch of ducklings about a motherly hen."

While most contemporary mathematicians were obsessed with the computational analysis of algebra, the axiomatic approach had only recently been explored in the works of Karl Weierstrass, Friedrich Dedekind, **Sophia Kovalevskaya** , and Hilbert. The publication of Bertrand Russell's *Principia Mathematica* delved into the symbolic aspect of mathematics. This novel approach opened new horizons for abstract algebra, and propelled it into the forefront of contemporary mathematical research.

While at Göttingen, Noether began to work on the general theory of ideals, which owed much to Max Noether's Residual theorem. The breadth of her talents became increasingly evident with the joint publication in 1920 of an article on differential operators. The paper also revealed Noether's new interest in the axiomatic analysis of algebra. Her research into ideals formed the basis for the application of axiomatic methodology to mathematical research.

Noether also worked closely with Helmut Hasse and Richard Brauer. Hasse expanded Noether's theory of cross products connected with the theory of cyclic algebras, thus proving that the common algebraic number field moved in cycles. Herman Weyl called the Brauer-Hasse-Noether theorem "a high water mark in the history of algebra." As Lynn Osen pointed out, some of Noether's "importance to algebra has roots in her work with others, in instances where her original ideas took final shape in the work of her students or collaborators."

By 1924, the Mathematical Institute of Göttingen was split from the faculty of Philosophy. The institute received a new building in 1929, and for the first time in her career Noether had her own office. The establishment of the Mathematical Institute fulfilled Felix Klein's dream that one day Göttingen would become the "Mecca of Mathematics." In the minds of many, the university was elevated to the status of one of the foremost mathematical institutes in the world, along with Moscow, Paris, and Berlin.

Noether was invited to spend a semester at the University of Moscow in 1928. Though the series of lectures which she delivered brought her additional acclaim, it was still insufficient to capsize the burdensome weight of prejudice at Göttingen. Her host, Paul Alexandrov, noted that she adapted well to our "Moscow ways." She was also invited to lecture at the University of Frankfurt in 1930.

In 1932, the quality of Noether's research was finally acknowledged with the Alfred Achermann-Teubner Memorial Prize for the Advancement of the Mathematical Sciences. Presented as a tribute to all of Noether's research, the award came with an honorarium of 5,000 reichmarks. In September, Noether attended the International Mathematical Congress in Zurich, where she was the only woman to be given a plenary session. A summary of her work, read at the conference, was the highlight of the event. Among the delegates attending were Herman Weyl, Helmut Hasse, Edmund Landau, and Richard Courant.

The election of the National Socialist Party in 1933 spelled the end of academic freedom in Germany. Though most scholars remained neutral, universities were among Hitler's first victims. At the University of Göttingen, brown shirts and swastikas were soon seen everywhere on the campus. Noether received the following notification from the Ministry of Sciences, Arts, and Public Education: "On the basis of paragraph 3 of the civil service code of April 7, 1933, I hereby withdraw from you the right to teach at the University of Göttingen."

On April 26, 1933, the newspaper *Göttingen Tageblatt* announced that six Jewish professors had been fired, including Noether and Courant. More firings would follow, the newspaper warned ominously. As Courant, then head of the Mathematical Institute, noted, "Aryan students want Aryan mathematics and not Jewish mathematics." Wrote Saunders MacLane, a former Noether student:

The institute did continue to operate during the spring semester, but under evident conditions of strain with many faculty searching for suitable positions in other countries and many students hurrying to get theses done. By the following semester over the half the people… were elsewhere and the great days of Göttingen were fatally interrupted.

The German nation suffered an intellectual hemorrhage, as members of the intelligentsia sought refuge in the liberal democracies of the West and the Soviet Union. The persecution of Jewish academics forced them to emigrate. Like many of her colleagues, Noether chose the United States, where, with the help of **Anna Johnson Pell Wheeler** , she was offered the position of visiting professor at Bryn Mawr College. Richard Courant secured a position at New York University. Fritz Noether, however, resigned his position at the University of Breslau and moved to a research institute in Tomsk, Siberia.

For the first time in her career, Noether enjoyed a modicum of employment security. At Bryn Mawr, as at Göttingen, she was a favorite of the students. "She loved to walk," notes Clark Kimberling:

She would take her students for a jaunt on a Saturday afternoon. On these trips she would become so absorbed in her conversations on mathematics that she would forget about the traffic and her students would need to protect her.

Noether also lectured frequently at the Institute for Advanced Study at Princeton University, where Albert Einstein was now teaching. In 1934, Bryn Mawr created a fellowship and a scholarship in her honor. As well, she joined the American Mathematical Society. While efforts were underway to secure her a tenured position, she died suddenly while undergoing surgery to remove a tumor. Abraham Flexner, director of the Institute for Advanced Study, wrote on April 25, 1935: "The last year and a half had been the happiest in her whole life, for she was appreciated at Bryn Mawr and Princeton as she had never been in her own country."

Emmy Noether was at the height of her intellectual power, and her sudden death at the age of 53 shocked most of her colleagues. Unlike many mathematicians, Noether's most productive years occurred in later life. Her theoretical work in the field of algebra was greatly facilitated by her legendary ability to abstract ideas. Wrote Einstein:

In the realm of algebra in which the most gifted mathematicians have been busy for centuries, [Noether] discovered methods which have proved of enormous importance in the development of the present day younger generation of mathematicians.

The impact of Emmy Noether's work was felt far beyond the realm of mathematics. Her research on invariants contributed to the discovery of the theory of relativity, and to the advancement of particle physics. "Noether's theorem" is still used by physicists today.

Although celebrated academically, Noether never secured appropriate employment in Germany. The victim of discrimination as a woman and as a Jew, Noether bore her circumstances philosophically. When one of her students attended a meeting in Nazi garb, she is reported to have laughed. Nevertheless, many of Noether's colleagues deplored the lack of recognition accorded to her. Paul Alexandrov rebuked the "high-ranking Prussian academic bureaucracy" for refusing to acknowledge Noether's talents by granting her a permanent academic appointment.

Despite continual discrimination, Noether devoted her life to research and to teaching, and the impact of her teaching was profound. Among her most distinguished students were Max Deuring, Hans Fitting, W. Krull, and **Olga Taussky Todd** . All made notable contributions to the study of mathematics. Emmy Noether rightly deserves to be known as the most influential female mathematician to have lived thus far, and her pioneering efforts in the fields of cross product algebra and abstract algebra earned her a place among the most distinguished mathematicians of the 20th century.

## Todd, Olga Taussky (1906–1995)

**Austrian-American mathematician.** Name variations: Olga Taussky-Todd. Born on August 30, 1906; died in Pasadena, California, on October 7, 1995.

Trained in number theory, Olga Taussky Todd worked with David Hilbert at the University of Göttingen; she emigrated to America around the same time (during the Nazis' early years in power) as other members of the Hilbert school, including **Emmy Noether** .

## sources:

Bell, E.T. *The Development of Mathematics.* NY: McGraw-Hill, 1945.

Dick, A. *Emmy Noether, 1882–1935.* Basel: Birckhäuser, 1981.

Jones, Lorella, M. "Intellectual Contributions of Women to Physics," in *Women of Science: Righting the Record.* G. Kass-Simon and Patricia Farnes, eds. Bloomington: Indiana University Press, 1990.

Osen, Lynn M. *Women in Mathematics.* Cambridge, MA: MIT Press, 1974.

Perl, Teri. *Math Equals.* Menlo Park, CA: Addison-Wesley, 1978.

Srinivasan, B., and J. Sally, eds. *Emmy Noether in Bryn Mawr: Proceedings of a Symposium Sponsored by the Association for Women in Mathematics in Honour of Emmy Noether's 100th Birthday.* NY: Springer-Verlag, 1983.

## suggested reading:

Brewer, J.W., and Martha K. Smith, eds. *Emmy Noether: A Tribute to Her Life and Work.* NY: Dekker, 1981.

**Hugh** **A.** **Stewart** , M.A., Guelph, Ontario, Canada

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