## From Yale University: Women in STEM: “Math — not computer science — was Grace Hopper’s first language”

March 26, 2019

Kendall Teare

kendall.teare@yale.edu

203-836-4226

For her pioneering work in computer science, Grace Murray Hopper ’30 M.A., ’34 Ph.D. has been dubbed the “queen of code” by her biographers. Yet, beneath that crown was the brain of a mathematician, according to an article in Notices of the American Mathematical Society (PDF) by Gibbs Assistant Professor of Mathematics Asher Auel that makes the details of Hopper’s doctoral training in mathematics public for the first time.

“In some sense, you could think of mathematics as the liberal arts of the sciences,” said Auel. “It is the language that you will be using in all scientific disciplines. It’s a way of knowing, a way of thinking, a way of understanding truth.” Studying math is studying problem solving — the necessary skill for anyone who wants to be able to approach a future problem that doesn’t exist now, he explained — for example, building the first computer.

“Sometimes people think Hopper was ‘only’ a foundational computer scientist or ‘only’ a naval officer,’ but that was just the beginning,” said Julia Adams, head of Grace Hopper College and professor of sociology. “It’s important that people realize that she had four or five illustrious careers, each of which would have done her great honor: mathematician, foundational computer scientist, naval officer, teacher, and public intellectual.”

Adams has invited Auel to discuss his paper on Hopper’s lesser-known “mathematical origins” at a college tea in the Hopper Head of College House (189 Elm St.) on Wed., April 3 at 4 p.m. This event is free and open to the Yale community.

In his paper, after listing the many instances in which Hopper self-described as a mathematician, Auel asserts that it was Hopper’s initial training in mathematics that gave her the tools to help build the discipline of computer science with which she’s most often associated. As Auel describes, television host David Letterman once asked Hopper about her work on the Mark I computer: “Now, how did you know so much about computers then?” To which Hopper replied, “I didn’t. It was the first one.”

Letterman’s assumption that Hopper would have needed a background in computer science in order to have succeeded in her later career has persisted in even the most authoritative biography on Hopper, which is being used as the basis for a forthcoming Google-produced biopic, said Auel. That biography and others have claimed that for her Yale doctoral degree Hopper studied everything from mathematical physics — a program Yale did not offer in the 1930s — to computer science — a program that no school offered in the 1930s.

Also, Auel says, Hopper didn’t just study “math” at Yale, she studied “pure algebra,” with a focus on algebraic number theory and geometry, highly theoretical topics with no direct practical applications at the time but which require an advanced ability to think in symbols. Auel believes that it was this ability to conceptualize abstract numerical and geometrical ideas that helped enable Hopper to invent a new language, that of computer code, which would require a deep and rigorous understanding of existing mathematical languages.

In fall 2015, Auel was teaching Math 350, Yale’s foundational abstract algebra course, for the first time. Before the term began, he went to the Mathematics Library at 12 Hillhouse Ave. to gather background on the history of abstract algebraists at Yale for his opening lecture. Paul Lukasiewicz, the now-retired, 40-year-veteran math librarian, put him on Hopper’s trail. Until then, Auel too had only associated Hopper with computer science.

On the first day of class, when Auel told his standing-room-only lecture that Hopper had taken a version of that very same course likely in the very same classroom while she was a graduate student in math at Yale, he was met with audible delight and excitement from students.

Hopper’s Yale graduate school transcript. (Image courtesy of Graduate School of Arts and Sciences, Yale University, Student Records [RU 262], Manuscripts and Archives, Yale University Library. Reproduction in consultation with family representatives Roger and Deborah Murray)

“I think that reclaiming Grace Hopper’s mathematical legacy is a great step for the math community,” said Catherine Lee ’20, a junior in Grace Hopper College who is majoring in math and is the former co-president of the Yale Undergraduate Math Society. “Studying math is isolating at times because it’s a discipline that isn’t well understood or widely appreciated in popular culture, and having strong role models can be important.”

“The Grace Hopper Conference [an annual conference for women in computer science] has been tremendously inspirational to women in computer science,” added Lee. “I hope that recognizing Hopper’s work as a mathematician will prove similarly empowering, especially at Yale, since Hopper is a very prominent figure here and there’s a serious demand right now for increased representation of female mathematicians on campus.” Lee is also a member of Dimensions, “the first organization at Yale that aims to inspire, celebrate, and empower women and gendered minorities in mathematics.”

Even within the Yale math department, Auel found that few colleagues knew about Hopper’s Yale math legacy. This is likely because, unlike the theses of her male peers, Hopper’s was never published. She presented the paper at a meeting of the American Mathematical Society, which made the March 1934 Bulletin of the American Mathematical Society as a one-paragraph-long abstract in a long list. Hopper’s thesis is now available in its entirety online at the Yale math department website.

In 1934, Hopper completed her doctoral degree with a dissertation on “new types of irreducibility criteria,” under the supervision of then Sterling Professor Øystein Ore, who is a noted figure in the history of algebra at Yale. She studied figures called Newton polygons, which are produced in an X-Y coordinate plane starting from a polynomial. These polygons are helpful in determining whether or not a polynomial can be reduced to a product of smaller degree polynomials.

Hopper hand drew the geometric illustrations for her thesis. This is her depiction of the “Minkowski sum” of the Newton polygons of two polynomials. (Image courtesy of Manuscripts and Archives, Yale University Library. Reproduction of details from Grace Murray Hopper’s Ph.D. dissertation [19] in consultation with family representatives Roger and Deborah Murray)

“While this idea of connecting Newton polygons with the irreducibility of polynomials had been floating around for about 20 years before Hopper came to it in her dissertation, the way she wrote about it was so modern,” said Auel, who will be leaving Yale at the end of spring 2019 to join the faculty at Dartmouth College.

The principle she identified is now an accepted theorem in abstract algebra, and the way she explained it is similar to how mathematicians understand it today. “The understanding she had was beyond her time,” said Auel. “Only 20 to 30 years after Hopper’s thesis did I see re-enunciations of the same philosophy in mathematics literature.”

Prior to Auel, the only scholars who’d examined the particulars of Hopper’s mathematical training were Judy Green ’66 M.A. and Jeanne LaDuke, two mathematicians who co-authored a comprehensive math history book, “Pioneering Women in Mathematics: The Pre-1940 Ph.D.’s.”, and supplementary material on the 228 mathematicians profiled. According to Green and LaDuke’s count, Hopper was actually the twelfth woman to receive a math Ph.D. from Yale. The first was Charlotte Barnum in 1895 for a dissertation on “functions having lines or surfaces of discontinuity.”

In 2019-2020, Yale will celebrate the achievements of its trailblazing female scholars like Barnum and Hopper while marking the 150th anniversary of women in Yale’s graduate and professional programs and the 50th anniversary of women in Yale College. For more information about this upcoming dual anniversary, visit the Celebrating Women at Yale website.

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Yale University comprises three major academic components: Yale College (the undergraduate program), the Graduate School of Arts and Sciences, and the professional schools. In addition, Yale encompasses a wide array of centers and programs, libraries, museums, and administrative support offices. Approximately 11,250 students attend Yale.

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