1. Overview
Julia Hall Bowman Robinson (1919-1985) was a pioneering American mathematician renowned for her groundbreaking contributions to computability theory, particularly in the realm of decision problems. Her most significant work was instrumental in the ultimate resolution of Hilbert's tenth problem, a monumental achievement in the field of Diophantine equations. Beyond her profound mathematical research, Robinson also made notable contributions to game theory, including early work on the travelling salesman problem and a key proof regarding Nash equilibrium. Her career was marked by overcoming institutional barriers, as she became the first female mathematician elected to the National Academy of Sciences and the first female president of the American Mathematical Society. Throughout her life, she was also actively engaged in political activism, advocating for democratic causes. Robinson's legacy extends beyond her mathematical theorems, inspiring future generations and expanding the role of women in mathematics.
2. Life
Julia Robinson's life was characterized by intellectual brilliance, resilience in the face of personal and professional challenges, and a deep commitment to both mathematics and societal engagement.
2.1. Early Childhood and Education
Julia Hall Bowman Robinson was born on December 8, 1919, in St. Louis, Missouri, to Ralph Bowers Bowman, who owned a machine equipment company, and Helen (Hall) Bowman, a former school teacher. Her mother passed away when Robinson was two years old, and her father later remarried. She had an older sister, Constance Reid, who became a notable mathematical popularizer and biographer, and a younger sister, Billie Comstock.
At the age of nine, Robinson was diagnosed with scarlet fever, which was soon followed by rheumatic fever. This period of illness caused her to miss two years of formal schooling. Upon her recovery, she received private tutoring from a retired primary school teacher, enabling her to complete the fifth, sixth, seventh, and eighth-year curricula in just one year. Despite an IQ test in junior high school where she scored 98, which she attributed to being "unaccustomed to taking tests," Julia excelled academically. At San Diego High School, she distinguished herself as the sole female student enrolled in advanced mathematics and physics classes. She graduated with a Bausch-Lomb award, recognizing her overall outstanding performance in science.
In 1936, at the age of 16, Robinson enrolled at San Diego State University. However, she found the mathematics curriculum there to be unsatisfactory. In 1937, her father committed suicide due to financial insecurities, a tragic event that deeply impacted her. In 1939, she transferred to the University of California, Berkeley for her senior year, seeking a more challenging academic environment. During her first year at Berkeley, she took five mathematics courses, including a number theory course taught by Raphael M. Robinson. She earned her Bachelor of Arts degree in 1940 and married Raphael M. Robinson in 1941.
2.2. Early Career and Research
Following her graduation, Robinson continued her academic pursuits as a graduate student at Berkeley. She served as a teaching assistant within the Department of Mathematics and later as a statistics lab assistant under Jerzy Neyman at the Berkeley Statistical Laboratory. Her work in the latter role led to her first published paper, titled "A Note on Exact Sequential Analysis."
In 1948, she earned her PhD degree under the supervision of Alfred Tarski. Her doctoral dissertation, "Definability and Decision Problems in Arithmetic," marked a significant early contribution to her field. In this work, she demonstrated that the theory of the rational numbers was an undecidable problem. She achieved this by showing that elementary number theory, which was already known to be undecidable due to Gödel's first incompleteness theorem, could be defined in terms of the rationals.
Her thesis highlighted the depth of her findings, noting that the variety of relations between integers (and operations on integers) that are arithmetically definable in terms of addition and multiplication of integers is very great. For instance, from Theorem 3.2 and Gödel's result, she concluded that the relation which holds between three rationals A, B, and N if and only if N is a positive integer and A=BN is definable in the arithmetic of rationals.
3. Major Mathematical Achievements
Julia Robinson's mathematical career was distinguished by several pivotal contributions that profoundly impacted the fields of computability theory, number theory, and game theory.
3.1. Computability Theory and Decision Problems in Arithmetic
Robinson's doctoral research laid foundational groundwork in computability theory, specifically addressing decision problems in arithmetic. Her dissertation, "Definability and Decision Problems in Arithmetic," published in 1948, established that the theory of rational numbers is an undecidable problem. This was a crucial finding, as she demonstrated that elementary number theory could be defined within the framework of rational numbers. Since elementary number theory had already been proven undecidable by Gödel's incompleteness theorem, Robinson's work extended this understanding to the arithmetic of rationals, highlighting the inherent limitations of algorithms in certain mathematical systems.
3.2. Hilbert's Tenth Problem
Robinson's most celebrated contribution was her pivotal role in the resolution of Hilbert's tenth problem. This problem, posed by David Hilbert in 1900, asks for an algorithm to determine whether any given Diophantine equation (a polynomial equation with integer coefficients) has integer solutions. Robinson began investigating methods for this problem in 1948 while working at the RAND Corporation.
Her research focused on the Diophantine representation for exponentiation and her innovative use of Pell's equation. This work led her to formulate the J.R. hypothesis (named after Robinson) in 1950. Proving this hypothesis became central to the eventual solution of Hilbert's tenth problem. Her published research attracted the attention of other mathematicians, leading to significant collaborations. She first met Martin Davis in 1950, who was working on showing that all listable sets were Diophantine, a complementary approach to Robinson's focus on proving specific sets (like prime numbers and powers of two) were Diophantine. Their collaboration began in 1959, and they were later joined by Hilary Putnam. Together, they demonstrated that the solutions to a specific "Goldilocks" equation were key to unlocking Hilbert's tenth problem.
In 1970, the problem was finally resolved in the negative, meaning it was proven that no such algorithm can exist. This monumental achievement was the result of the combined efforts of Robinson, Davis, Putnam, and the young Soviet mathematician Yuri Matiyasevich, who provided the final piece of the puzzle by proving the J.R. hypothesis. Through the 1970s, Robinson continued to collaborate with Matiyasevich on corollaries of their solution. She once stated that there exists a constant N such that, given a Diophantine equation with any number of parameters and unknowns, it can be effectively transformed into another equation with the same parameters but in only N unknowns, such that both equations are solvable or unsolvable for the same parameter values. At the time of the solution's initial publication, N was established as 200. Subsequent joint work by Robinson and Matiyasevich further reduced this number to 9 unknowns.
3.3. Game Theory
During the late 1940s, Julia Robinson spent approximately a year at the RAND Corporation in Santa Monica, California, where she conducted research in game theory. Her 1949 technical report, "On the Hamiltonian game (a traveling salesman problem)," is recognized as the first publication to use the phrase "travelling salesman problem", a foundational problem in combinatorial optimization.
Shortly thereafter, in 1951, she published a paper titled "An Iterative Method of Solving a Game." In this paper, she provided a proof that the fictitious play dynamics converges to the mixed strategy Nash equilibrium in two-player zero-sum games. This problem had been posed by George W. Brown as a prize problem at the RAND Corporation, and Robinson's solution was a significant early result in the development of game theory.
4. Academic Career and Activities
Julia Robinson's academic career was marked by her exceptional mathematical talent and her perseverance in navigating the academic landscape, which presented unique challenges for women in her era.
4.1. Professorship at UC Berkeley
After marrying Raphael M. Robinson in 1941, Julia Robinson faced an institutional barrier at the University of California, Berkeley. A university policy at the time prevented family members from working together in the same department. As a result, she was not permitted to teach in the Mathematics Department, despite her desire to teach calculus. Instead, she remained affiliated with the statistics department.
Even after Raphael retired in 1973, it was not until 1976 that Julia Robinson was offered a full-time professorship position at Berkeley. This offer came after the department learned of her nomination to the prestigious National Academy of Sciences, indicating a recognition of her immense contributions to mathematics that transcended the previous departmental restrictions.
4.2. American Mathematical Society Activities
Her autobiography reveals her initial reluctance to accept the nomination for the presidency of the American Mathematical Society (AMS) in 1982. She recognized that her selection was partly due to her being a woman and having the "seal of approval" from the National Academy. After discussions with her husband, Raphael, who thought she should decline to conserve energy for mathematics, and other family members who disagreed, she ultimately decided to accept. Robinson felt that as a woman and a mathematician, she had no alternative but to take on the role, stating, "I have always tried to do everything I could to encourage talented women to become research mathematicians. I found my service as president of the Society taxing but very, very satisfying." This statement underscores her commitment to advancing the role of women in mathematics, viewing her acceptance as a duty to inspire and support future generations of female mathematicians.
5. Awards and Honors
Julia Robinson received numerous awards and honors throughout her career, recognizing her profound impact on mathematics and her pioneering role as a woman in the field.
5.1. Member of the National Academy of Sciences
In 1975, Julia Robinson made history by becoming the first female mathematician to be elected to the National Academy of Sciences. This recognition came after Yuri Matiyasevich successfully solved Hilbert's tenth problem, a solution that heavily relied on Robinson's J.R. hypothesis and insights related to the Fibonacci number sequence. Saunders Mac Lane nominated Robinson for this esteemed membership. Furthermore, her former mentors and colleagues, Alfred Tarski and Jerzy Neyman, traveled to Washington, D.C., to personally advocate for her, emphasizing the critical importance of her work and its tremendous contributions to mathematics.
5.2. Other Awards and Honors
In 1982, Robinson was selected as the Noether Lecturer by the Association for Women in Mathematics. Her lecture was titled "Functional Equations in Arithmetic," further showcasing her expertise in number theory. Around the same time, she was awarded the prestigious MacArthur Fellowship, which included a prize of 60.00 K USD. In 1985, the year of her passing, she also became a member of the American Academy of Arts and Sciences, adding another significant accolade to her distinguished career.
6. Political Activities
Beyond her academic and mathematical pursuits, Julia Robinson was deeply involved in political activities, demonstrating a commitment to civic engagement and democratic principles. In the 1950s, she was an active participant in local Democratic Party activities. Her involvement was hands-on and grassroots, as she dedicated her time to registering voters, stuffing envelopes for campaigns, and even ringing doorbells in neighborhoods where such efforts were crucial.
Her sister, Constance Reid, recounted Julia's deep involvement during those years in the "nitty-gritty" of Democratic Party politics. She registered voters, stuffed envelopes, and rang doorbells in neighborhoods where people expected to be paid for their vote. Robinson even served as Alan Cranston's campaign manager for Contra Costa County when he successfully ran for state controller, his first political office. Additionally, Robinson volunteered for Adlai Stevenson's presidential campaigns, further illustrating her dedication to political causes she believed in.
7. Death and Legacy
Julia Robinson's passing marked the end of a remarkable life dedicated to mathematics and public service, but her legacy continues to influence and inspire.
7.1. Death
In 1984, Julia Robinson was diagnosed with leukemia. She bravely battled the illness but ultimately succumbed to it, passing away in Oakland, California, on July 30, 1985, at the age of 65. In her final wishes, Robinson requested that there be no funeral service. Instead, she asked that those wishing to make a gift in her memory contribute to the Alfred Tarski Fund, a fund she had been instrumental in establishing in honor of her late teacher, friend, and colleague, Alfred Tarski. This request reflected her characteristic modesty and her enduring commitment to the academic community.
7.2. Legacy and Commemoration
Julia Robinson's influence continues to be felt in the mathematical community and beyond. Her older sister, Constance Reid, a renowned mathematical biographer, played a significant role in preserving and promoting Julia's legacy. Constance Reid won the Mathematical Association of America's George Pólya Award in 1987 for her article "The Autobiography of Julia Robinson," which offered a personal and insightful account of her sister's life and work.
In recognition of her contributions and to inspire young minds, the Julia Robinson Mathematics Festival was named in her honor. This festival was sponsored by the Mathematical Sciences Research Institute from 2007 to 2013 and has been supported by the American Institute of Mathematics since 2013, providing engaging mathematical experiences for students.
Her life and pivotal work on Hilbert's tenth problem were also chronicled in a one-hour documentary titled Julia Robinson and Hilbert's Tenth Problem. Produced and directed by George Csicsery, the film premiered at the Joint Mathematics Meeting in San Diego on January 7, 2008. The documentary received positive reviews in prominent publications such as Notices of the American Mathematical Society and College Mathematics Journal, further cementing her place in the history of mathematics and bringing her story to a wider audience.
