# Emergence of Women at the Highest Levels of Mathematics

# Emergence of Women at the Highest Levels of Mathematics

*Overview*

Throughout history women have made important contributions to the field of mathematics. Ada Byron Lovelace (1815-1852) wrote the first computer program in 1844, and Florence Nightingale (1820-1910) invented the pie chart. Despite their accomplishments, however, women mathematicians have faced almost insurmountable odds; some were persecuted, and one was even martyred. Hypatia (c. 370-415), the first mathematician to formulate the idea of conic sections, was brutally killed in 415 A.D. because she was a powerful intellectual. Sophie Germain (1776-1831) has a theorem named after her, but was barred from classes at the Paris Polytechnique in eighteenth-century France because she was a woman. French mathematician Emilie de Breteuil (1706-1749) received her excellent education only because her family thought her too tall and ugly to get married. Sonya Kovalevskaya (1850-1891) also has a theorem named after her, but as a woman she could neither enroll in classes in Berlin nor later obtain a university position in Germany or her native Russia. In nineteenth-century England, Mary Somerville (1780-1872) needed her husband's approval just to write a book about mathematics; her outstanding research about violet light magnetizing a steel needle had to be presented to the Royal Society by her husband because women were not accepted as members. For centuries, women mathematicians faced incredible oppression. Nevertheless, their genius, determination, and courage paved the way for their successors in the twentieth century.

*Background*

The main obstacle facing female mathematicians at the turn of the century was that few universities accepted women as doctoral candidates. One exception to this, however, was Göttingen Mathematical Institute in Germany. An important mathematical center, Göttingen was a magnet for both male and female mathematicians. Grace Chisholm Young (1868-1944) chose to study there because graduate schools in her native England did not yet admit women. In 1896, Young became the first woman to receive an official doctorate in Germany. (Sonya Kovalevskaya's 1874 doctorate was not official because she was not enrolled in classes at the time.) Young published her own book on geometry in 1905; it included patterns for geometric figures that are still used in math classes. The next year she and her husband, mathematician William Young (1863-1942), published the first book to provide comprehensive applications of problems in mathematical analysis and set theory.

Emmy Amalie Noether (1882-1935), a brilliant abstract algebraist, also attended lectures at Göttingen. When German law finally allowed women to be regular university students, Noether transferred to Erlangen where her father, a famous mathematics professor, taught. There she received her Ph.D. in 1907, *summa cum laude*, her thesis listing systems of more than 300 covariant forms. After her degree, Emmy Noether faced the second major obstacle that confronted women mathematicians in the first half of the twentieth century: few universities accepted
them as research professors. Noether was not allowed to teach at the university in Erlangen because, professors argued, young German soldiers returning from World War I would be shocked to find a woman lecturing to them at their university classes. David Hilbert (1862-1943) convinced Noether to return to Göttingen in 1915. There she and Hilbert collaborated on the general theory of relativity, with Noether providing the mathematical formulations that became the basis of several concepts of Albert Einstein's general theory of relativity. A paper on differential equations that she coauthored in 1920 firmly established her reputation as a mathematical genius; the paper explained the axiomatic approach and made an invaluable contribution to the field of mathematics, specifically abstract algebra. By 1930, Noether was a vital part of the Göttingen mathematical team, a stimulating lecturer, and a prolific publisher.

The German political climate, however, forced Noether's dismissal from the university because she was Jewish. Colleagues found her a position in 1933 at Bryn Mawr, a women's college in Pennsylvania. Unfortunately Noether died there only two years later. Albert Einstein wrote in the* New York Times* that Emmy Noether "...was the most significant creative mathematical genius thus far produced since the higher education of women began." Today mathematicians study structures called Noetherian rings, named to honor Emmy Noether, who had developed them.

American Anna Pell Wheeler (1883-1966) was given a fellowship to study mathematics at Göttingen. She completed all the coursework but left before receiving her doctorate because of a dispute with Hilbert. Wheeler was nonetheless awarded her Ph.D. from the University of Chicago in 1909, using the thesis she had begun in Germany. She, too, taught at Bryn Mawr where she later became full professor and head of mathematics. The focus of her work was integral equations and infinite dimensional linear spaces. Wheeler was the first woman to give colloquium lectures at meetings of the American Mathematical Society.

In 1915, American mathematician Olive Clio Hazlett (1890-1974) also received her Ph.D. from the University of Chicago where she was her thesis advisor's second female student. She spent her career at the University of Illinois and was considered one of only two important women in America in mathematics. Hazlett wrote 17 research papers on nilpotent algebras, division algebras, modular invariants, and the arithmetic of algebras, more than any other pre-1940 female American mathematician. She was authored an article about quaternions for the *Encyclopedia Britannica* and was the editor of the *Transactions of the American Mathematical Society *from 1923-1935.

Mary Lucy Cartwright (b. 1900) attended Oxford from 1923 to 1930, after Oxford agreed in 1921 to allow women to take final degrees. Cartwright received her Ph.D. in mathematics from Oxford in 1930 and then accepted a fellowship at Girton College, the women's college at Cambridge University. Cartwright discovered the phenomena subsequently known as chaos. She received innumerable honors and was the first female mathematician elected as Fellow of the Royal Society in 1947. She was also elected President of the London Mathematical Society in 1951. Cartwright was awarded several medals honoring her mathematical achievements and became Dame Cartwright in 1969.

A major force of modern computer technology was Grace Murray Hopper (1906-1992). After earning her 1934 Ph.D. in mathematics from Yale University, Hopper taught at Vassar until World War II, when she joined the U.S. Navy Reserve. She pioneered computer software and coined the term "computer bug" after she discovered that an insect had shorted out two tubes. Rear Admiral Dr. Grace Hopper also invented the modern subroutine, the computer language APT, and played a significant role in verifying the computer language COBOL.

Olga Taussky-Todd (1906-1995) has the distinction of being the first woman to hold a formal appointment at the California Institute of Technology; in 1971 she became the first female full professor there and 10 years later, professor emeritus. Her doctorate in mathematics was from the University of Vienna. Taussky-Todd worked with Emmy Noether at Göttingen and then taught in England. Before moving to California, Taussky-Todd worked again with Noether at Bryn Mawr; the two often traveled to Princeton to lecture, where they frequently met Albert Einstein (1879-1955). Taussky-Todd's research focus was matrix theory and its application to emerging computer technology. She authored more than 200 papers and received innumerable honors, including the Ford Prize, the Austrian Cross of Honour for Science and Art (Austria's highest scientific honor), an honorary D.Sc. from the University of Southern California, and Fellow of the American Association for the Advancement of Science; she also served as Vice-President of the American Mathematical Society.

Nazi politics also forced mathematician Hanna Neumann (1914-1971) to flee Germany. Neumann was a doctoral student at Göttingen in 1938 when she realized that she would have to leave the country to marry her Jewish fiancé Bernhard Neumann, a fellow mathematician who had immigrated to Britain. Hanna Neumann continued her studies at Oxford University, where Olga Taussky-Todd supervised her research. Neumann's thesis studied the problem of determining the subgroup structure of free products of groups with an amalgamated subgroup. After her degree from Oxford in 1944, she taught and continued her research on Hopf's problem or, as it is now known, Hopf's property, questioning whether the free product of finitely many Hopf groups is again a Hopf group; she also worked on near-rings. Neumann and her husband accepted offers from the Australian National University, where in 1963 she became the chair of Pure Mathematics, a new position created for her. In recognition of her great contribution to the field of mathematics, Hanna Neumann was elected in March 1969 to the Fellowship of the Australian Academy of Science.

*Impact*

These brilliant, pioneering women forged the way for female mathematicians in the latter half of the twentieth century. Some examples: Evelyn Boyd Granville (1924- ) obtained her doctorate from Yale in 1949, one of the first two African-American women to receive a Ph.D. in mathematics. Louise Hay (1935-1989), after her doctorate from Cornell in 1965, was named head of the department of mathematics at the University of Illinois at Chicago; at that time she was the only woman to head a major research-oriented university mathematics department in America. In 1971 Hay, along with Alice T. Schafer (1915- ), were co-founders of the Association of Women in Mathematics.

The first award for Distinguished Service to Mathematics from the Mathematical Association of America went to Mina Rees (1902-1997) in 1962; eight years later she was elected the first woman president of the American Association for the Advancement of Science. Twenty years later, Berkeley mathematics professor Julia Robinson (1919-1985) became the first woman president of the American Mathematical Society.

By the close of the twentieth century, women had made a definite and permanent contribution to the field of mathematics. They had followed their predecessors and refused to allow prejudice or bias to thwart their ambitions. Many female mathematicians successfully combined career and marriage, raising children who often grew up themselves to be scientists and mathematicians. These extraordinary women were an inspiration to their sons and daughters, to their students, and to their colleagues.

**ELLEN ELGHOBASHI**

*Further Reading*

### Books

McGrayne, Sharon Bertsch. *Nobel Prize Women in Science.* New York: Carol Publishing Group, 1993.

Morrow, Charlene and Teri Perl, eds. *Notable Women in**Mathematics. *Westport, CN: Greenwood Press, 1998.

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

### Internet Sites

Agnes Scott College. Biographies of Women Mathematicians. "Other Resources on Women Mathematicians and Scientists." http://www.agnesscott.edu/Lriddle/women/resource.htm

Association for Women in Mathematics (AWM). "Emmy Noether Lectures. http://www.math.neu.edu/awm/noetherbrochure/Introduction.html

Four Thousand Years of Women in Science. http://crux.astr.ua.edu/4000WS/4000WS.html

History of Mathematics. http://aleph0.clarku.edu/~djoyce/mathhist/mathhist.html

University of St. Andrews Scotland. School of Mathematics and Statistics. "Emmy Amalie Noether." http://www-groups.dcs.stand.ac.uk/~history/Mathematicians/Noether_Emmy.html

University of St. Andrews Scotland. School of Mathematics and Statistics. The MacTutor History of Mathematics Archive. http://www-groups.dcs.st-and.ac.uk/history/index.html

University of St. Andrews Scotland. School of Mathematics and Statistics. "An Overview of the History of Mathematics." http://www-history.mcs.st-and.ac.uk/history/HistTopics/History_overview.html

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# Emergence of Women at the Highest Levels of Mathematics

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**Emergence of Women at the Highest Levels of Mathematics**