Two women behind chaos science whose contributions should not be forgotten | Women in Mathematics Day

There are many examples in the history of science where the key contributions of female scientists are not adequately recorded and acknowledged. Some examples are well known to people, while others are less well known. For example, in the history of the development of chaos theory, there were two women who participated in the numerical calculation of the weather forecast model of Lorenz, the "Father of Chaos". They made important contributions to the development of chaos theory, but were ignored by historians of science. If it were not for the thanks from Lorenz, the "Father of Chaos", it would be difficult for later generations to discover this history.

Written by Ding Jiu (Professor of Mathematics at the University of Southern Mississippi)

When talking about "chaos", people always mention Edward Norton Lorenz (1917-2008), an American meteorologist known as the "Father of Chaos". But how many people know the details in the history of the development of chaos that have been intentionally or unintentionally hidden or passed over, especially when the details involve people who are considered "insignificant" by media reporters or best-selling authors?

In the fall of 2017, 54 years after the publication of Lorenz's famous scientific paper "Deterministic Nonperiodic Flow", Daniel Rothman, a geophysicist at the Massachusetts Institute of Technology and deputy director of the school's "Lorenz Center", was busy preparing for a seminar to be held early the following year to commemorate the centenary of Lorenz's birth. He once again opened Lorenz's masterpiece, which has been cited in thousands of scientific papers. Although he was very familiar with the "scientific part" of the article and had even taught relevant content in previous classes, this time, when his eyes fell on the "non-scientific part" behind the references of this article - "Acknowledgments", he read a sentence he had never noticed before, just like Columbus suddenly discovered the New World: "Special thanks are due to Miss Ellen Fetter for handling the many numerical computations and preparing the graphical presentations of the numerical material (Special thanks to Miss Ellen Fetter for handling many numerical computations and preparing the graphical presentations of the numerical material)."

Who is Fett? She is the beautiful "heroine behind Chaos" in the photo below (taken when she was 23 years old):

Ellen Fetter (born in 1940) is not the only woman whose own brilliance is overshadowed by Lorenz's shadow. There is another woman who, like Fetter, is unknown in the Hall of Chaos, but shines in another field. She worked for Lorenz before Fetter, and her name is Margaret Hamilton (born on August 17, 1936). The following is a headshot of her taken in 1995 from the Wiki page. She is nearly 60 years old, but still so intellectual and elegant:

The two of them, along with Lorenz, who eventually outclassed them, have one thing in common: all received bachelor's degrees in mathematics. They also achieved a dream that everyone strives for: longevity; Lorenz died at the age of 91, and the two women are still alive, at the ages of 83 and 86 respectively. What could be better in life than majoring in mathematics when you are young and living a long life?

May 12 is Women in Mathematics Day, an international mathematics day established in 2019 to celebrate the brilliant life of Maryam Mirzakhani (1977-2017), the first female winner of the Fields Medal. It is meaningful to review the historic contributions made by Hamilton and Fetter to the establishment of chaos theory.

Margaret Hamilton

Today, "software engineering" is a buzzword and lucrative career in computer and information science, but as a 2018 article in Software Magazine states at the outset, "Most of today's software engineers might be surprised to learn that the founder of their discipline was a woman." That woman was Margaret Hamilton.

This was one of her many contributions during her participation in the US Apollo Moon Landing Program half a century ago, and is well known in the scientific community, and she has received several government awards. However, more than 60 years ago, when she was just out of college, her numerical simulation work on weather forecasting led to the emergence of chaos theory, making her first scientific contribution in her life, but this achievement is little known to the public.

Hamilton's original name was Margaret Elain Heafield. She was born in Paoli, a small town in Indiana with only three or four thousand residents. Later, she moved to Michigan in the north with her parents. After graduating from high school there, she was admitted to the University of Michigan, the best public university in the state, in 1954. However, because her mother had studied at Earlham College, a private liberal arts college in Indiana, she also transferred to her mother's alma mater the following year, majoring in mathematics and minoring in philosophy. Three years later, she obtained a bachelor's degree in mathematics from the school and became an alumnus of the next generation with her mother. This small college, founded in 1847 by the Protestant religious group Quakers, has a total of less than 750 students. It not only imparts humanities and scientific knowledge to its students, but also focuses on influencing them with values ​​such as integrity, commitment to peace and justice, and mutual respect between people.

After graduating from college, Margaret married James Hamilton (1937-2014), who was also studying at the same school. Later, she taught briefly at a high school in Boston, a small city in the state. This is a common place for undergraduate graduates of mathematics in the United States, and some even teach for life. But they soon had new plans and moved to Boston, a city of culture and education. Mr. Hamilton went to Harvard Law School, and Mrs. Hamilton initially planned to study abstract mathematics at Brandeis University, but she soon changed her mind and went to the best engineering university in the city, Massachusetts Institute of Technology, as a computer programmer. Soon, she joined the team of meteorology professor Lorenz and did computational programming for him. She participated in the initial process of discovering the chaotic phenomenon of weather forecasting and witnessed the emergence of the concept of chaos.

When Mrs. Hamilton, a 23-year-old bachelor of mathematics, came to the research group of Professor Lorenz, who had a master's degree in mathematics and a doctorate in meteorology, in the summer of 1959, the latter had just purchased a LGP-30 desktop computer made by Royal McBee and learned how to use it by himself. The machine weighed 800 pounds and could not be operated by one person, so a skilled programmer was needed to help. This reminded me of forty years ago when my fellow graduate students and I were programming and solving problems in the computing center of Nanjing University, there was a "female programmer army" busy in the computer room. Ulam, the father of the American hydrogen bomb, also recalled in his autobiography "The Experience of a Mathematician" that in the Manhattan atomic bomb development project, "professor wives" including his wife were called together to form a "programmer army" to assist their husbands in completing the complicated calculations of the atomic bomb.

However, Hamilton was much better than ordinary programmers. Those programmers were mainly manual laborers, and manual operations such as inputting cards hardly required mathematical thinking. Hamilton, on the other hand, not only used her hands, but also her brain. Her mathematical training made her play a role in Lorenz's pioneering numerical research on weather forecasting, just like her successor Fette, which was equivalent to half of Lorenz.

The two years she spent working with Lorenz were, Hamilton says, “my formative years.” In a 2019 article in Quanta Magazine, science journalist Joshua Sokol recounted Hamilton’s recollections of her youth: “After a party ended at three or four in the morning, she realized that the LGP-30 wouldn’t produce results the next morning, so she and a few friends rushed over to start it up. Another time, bothered by all the things she had to do to get the machine running again after fixing an error, she figured out a way to bypass the computer’s clumsy debugging process. To Lorenz’s delight, Hamilton would take the paper tape that would feed the computer, roll it down the hallway, and edit the binary code with a sharpened pencil. ‘I’d poke holes in a few of the tapes and cover the others with Scotch tape,’ she said.”

Hamilton arrived at the right time, and soon Lorenz and his assistants started the journey of using 12 ordinary differential equations to numerically simulate "small-scale weather forecasts". As an energetic and thriving associate professor of meteorology, he was naturally the navigator, and Hamilton coded his calculation ideas. They set the computer to slowly print out a graph of one or several of the many variables changing over time. The printed results showed both periodic phenomena and some non-periodic phenomena, and everything seemed quite reasonable. However, one day, things changed, and this happened to be triggered by the American gentleman's common love of drinking coffee.

That day, the printout they set was not an image, but the function values ​​of several variables corresponding to the same moment of time evolution. Each line printed was the weather data for one day later. After calculating for a while, Lorenz's coffee addiction came, so he input the printed set of numbers as the initial data back into the computer for a second calculation, and then went downstairs across the hall to have coffee. When he came back an hour later, something strange happened. The new calculation showed that when the time was not long, the value of this time was basically consistent with the previous one, but as time went on, the difference between the two results showed an "exponential growth", resulting in a completely different weather situation two months later.

The following story is already a classic: Lorenz discovered that the data he entered before drinking coffee was only a printout approximation of the six decimal places involved in the calculation in the computer, rounded to three places, so the second calculation already had an "initial value error" of about 5 ten-thousandths at the beginning, and because the differential equations they numerically solved had a sensitive dependence on the initial conditions, the initial error was quickly amplified, causing the error to become unbearable after a period of time. Since actual weather forecasts rely on observational data, and observational errors are inevitable, his discovery shattered the fantasy of long-term weather forecasts. The later popular phrase "butterfly effect" vividly depicts this. (See "The Chaos Relay between Meteorologists and Mathematicians")

However, Lorenz did not write an article about his scientific discovery immediately - his famous article "Deterministic Non-Periodic Flow" was not officially published until the first half of 1963 - but he gave a report on it at a numerical weather forecasting academic conference held in Tokyo in November 1960. After the report, someone in the audience asked: "Did you change the initial conditions slightly to see how different the results are?" He replied: "In fact, we tried the same equation once to see what would happen." This shows that he and Hamilton had observed the butterfly effect of weather forecasting several times before. In the famous reportage work "Chaos: The Creation of a New Science" by American science journalist Gleick, the time of this great discovery that "sown the seeds of a new science" was moved back a year.

Lorenz became famous in the scientific community and the public media during the chaos craze that began in the mid-1970s because he was the first to discover chaos in the field of natural science in the 1960s. However, he always behaved like a gentleman and sincerely thanked anyone who helped him. Regarding programmer Hamilton, he wrote in the acknowledgments of his paper: "The writer is greatly indebted to Mrs. Margaret Hamilton for her assistance in performing the many numerical computations which were necessary in this work." This humble approach and moral sentiment are undoubtedly worthy of learning for scientific and technological personnel, especially those "star scientists."

In terms of appearance, in the writings of science journalist Greg, Lorenz's face "looks like a weather-beaten old American farmer, but his eyes are surprisingly bright, making him look as if he is always smiling. He rarely talks about himself or his work, but just listens to others."

In the summer of 1961, Hamilton left Lorenz's team to join another project, but before she left, she was very responsible for helping Lorenz interview and hire a new programmer, Fette. From then on, she embarked on a broader and more successful career path. This golden road made her body more and more "golden" because she was wearing the "Apollo suit" to fly to the moon. Her dazzling brilliance can be compared with Lorenz, her first coding mentor in her life, and may even surpass him.

She did not leave MIT, but moved to Lincoln Laboratory, where she worked until 1963, writing software programs for several important projects. Then, she joined the Instrumentation Laboratory and became the first programmer hired for the Apollo lunar exploration program. Two years later, she began to lead the airborne flight software team in the Apollo project as the director of the "Software Engineering Department". Lorenz's "Toy Weather Forecast" was just a scientific exploration by a small team of a meteorology professor, while the "Apollo Moon Landing Project" was the clarion call for mankind to enter the solar system space and an ambitious cosmic adventure by a large group of thousands of people from NASA. Leading the software development of this major project requires not only scientific knowledge, engineering background and technical experience, but also the comprehensive qualities of logical thinking and image thinking. Hamilton, who has been influenced by both mathematics and philosophy, was entrusted with the important task.

On July 20, 1969, when Apollo 11 astronaut Neil Armstrong (1930-2012) became the first person in human history to walk on the moon, people all over the world were shocked by the exciting news, and their eyes were focused on the heroic man. However, as the meaningful title of a cartoon in 2016 said: She is "The woman who put a man on the moon". Without the flight software specially developed by the software engineer team led by Hamilton, Apollo 11 would not have been able to fly to the moon, and Armstrong would not have been able to take this step in human history.

History has recorded that it was during the height of the Apollo program that Hamilton coined the now-universal term "software engineering." Years later, she recalled:

“When I first came up with the term, no one had heard of it before, at least not in our world. For a long time, it was a joke. They loved to make fun of my radical ideas. There was a memorable day when one of the most respected hardware gurus explained to everyone at a meeting that he agreed with me that the process of building software should also be considered an engineering discipline, just like hardware. Not because he accepted the new “term” itself, but because we won his approval, and the approval of everyone else in the room, because we belonged to the field of engineering in our own right.”

In an article published nine years ago in the American magazine Wired, "Software — and a Woman — at the Heart of the Moon Victory," the author wrote: "Hamilton, along with another early programming pioneer, COBOL inventor Grace Hopper, deserves great credit for helping more women enter and succeed in science, technology, engineering and mathematics fields such as software." Hopper (Grace Hopper, 1906-1992) is also a legend. She received a Ph.D. in mathematics from Yale University, was a professor of mathematics at Vassar College, one of the "Seven Sisters Schools", and in the late World War II, devoted herself to naval research and development, developed software science, and was eventually promoted to a rear admiral.

Hamilton has received numerous awards since 1986, most notably the Presidential Medal of Freedom, the nation's highest civilian honor, awarded to her in 2016 by President Barack Obama.

Alan Fetter

Ellen Fetter's name was included in Lorenz's greatest paper because she was responsible for the picture drawn by the computer-controlled printer in the article, which looked like the wings of a butterfly and was later called the "Lorenz Attractor."

In 1961, just before Hamilton was about to go to Lincoln Laboratory, Fetter was recruited by her to join the meteorological group to continue her role as a programmer. The 21-year-old girl, who had just received a bachelor's degree in mathematics from Mount Holyoke College, the first women's college in the United States and the eldest of the Seven Sisters, was recommended to Hamilton by a woman who managed the LGP-30 computer in the Department of Nuclear Engineering at MIT. After interviewing her, the latter was satisfied and asked her to use her mathematical mind to serve Lorenz's meteorological research.

On her first day on the job, as Fette entered Building 24 in the heart of the MIT campus, Lorenz handed her a computer manual and a few programming problems to practice with, and soon she was up to speed. “He had a lot of stuff in his head,” Fette recalls of her boss, “and sometimes he would come in with a yellow paper and pull a little piece of paper out of his pocket and say, ‘Let’s try this.’ ” Like her predecessor, she still speaks glowingly of Lorenz’s humility and guidance decades later. Lorenz thanked both men in his paper before later chaos chroniclers excluded them.

At this time, Lorenz found that the 12 ordinary differential equations used before seemed too many to get to the heart of the problem, so he borrowed three convection equations from the seven equation library owned by Barry Saltzman (1932-2001), a geophysicist at Yale University, to describe a simpler non-periodic system. This set of ordinary differential equations describes the state changes of water in a beaker during heating from below and cooling from above. After they were printed on a page of a paper in the Journal of Atmospheric Sciences in 1963, they became a landmark of chaos since the 1970s and had a humanized name "Lorenz equations". In the published article, Lorenz not only wrote the thank you to Fette quoted at the beginning of this article, but also thanked Saltzman, which is his usual behavior.

The three variables in these equations roughly give the relationship between the convection speed and temperature change of water in an idealized beaker, which can be represented by three Cartesian coordinates of points in three-dimensional space; mathematicians call this space phase space. Lorenz and Fetter asked the printer to draw the trajectory of the moving points as their spatial positions change over time, and found that the shape of the graph was very similar to the wings of a butterfly. According to the basic theory of differential equations, using the terminology of continuous dynamic systems, the "non-periodic flow" formed by these solution curves will not intersect with themselves, but will swim forward and rush around inside the pair of wings. A small change in the initial conditions will greatly change the trajectory of the phase flow points. When the numerical wings discovered by meteorologists flew all over the scientific community, rigorous mathematicians began to conduct theoretical research on them and found that it has the so-called "fractal structure", has chaotic dynamic system phenomena itself, and attracts nearby initial points, so they named it the "Lorenz attractor".

The "sensitive dependence of the system on initial conditions" and the butterfly wing attractor observed by Lorenz and Fetter through numerical simulation of the three convection equations formed the main topic of his landmark article "Deterministic Non-Periodic Flow" published in 1963. Ten years later, it inspired Li Tianyan and York to define the mathematical term "chaos" for the first time in the "immortal mathematical treasure" "Period Three Means Chaos", which became the cornerstone of the subsequent booming chaos theory. What made Lorenz equally happy was that in 1962, when the article was accepted for publication by the journal, he was promoted to full professor by the school.

In the same year that Lorenz published this brilliant paper, Fetter's love affair with a brilliant young man who was studying for a doctorate at MIT also matured. She married her ideal husband, which meant that she had to leave the computing work she loved and the gentleman she worked for, because her husband, John Gille (1935-), had already obtained a doctorate in geophysics and was going to teach at Florida State University in the south. After working as a computer programmer at the university for a few years, Fetter decided to return to her family and take care of her children. In the 1970s, Dr. Gille moved to the National Center for Atmospheric Research in Colorado, and the whole family naturally followed. Mrs. Gille wanted to continue working in software, so she went to the University of Colorado Boulder in the same city to recharge and take some computer science courses, but eventually she gave up such careers that were more favored by young people and turned to tax work.

Fetter's children have developed well due to their parents' careful training. Her daughter Sarah Gille also went to Yale University, her father's alma mater, to study. After graduation, she followed her father's footsteps and went to MIT to pursue a doctorate in physical oceanography. She is now a professor of physical oceanography at the University of California, San Diego. Interestingly, the department she studied for her doctorate at MIT in the 1990s was the result of the merger of the meteorology department, physical oceanography department, and geology department in 1983, so the department name is very long, called the "Department of Earth, Atmospheric and Planetary Sciences". In this way, Lorenz can be regarded as her "senior professor of the department".

Rossman, the deputy director of the Lorenz Center mentioned at the beginning of this article, is also a professor in this department. When he found the unfamiliar "Miss Ellen Fetter" in the acknowledgments of Lorenz's paper, he immediately searched for her on the Internet and finally saw her wedding announcement with the groom in the New York Times in July 1963. At this time, one of his colleagues remembered that there was a graduate student named Sarah Gill in the department 20 years ago, so Professor Rossman contacted Sarah and learned that she was the fruit of the love between Ellen Fetter and John Gill. Through Sarah's introduction, Rossman also talked to her mother on the phone and learned that before Miss Fetter, there was Mrs. Hamilton who worked with Lorenz. In this way, the two heroines hidden behind the "chaos" story that has been spread all over the world for decades finally surfaced.

Perhaps, Fette's daughter did not know the role her mother played in the history of chaos development before, but when she took a course on how to program scientifically in her undergraduate studies, one of the cases studied by the whole class was actually Lorenz's discovery on the LGP-30 computer! When she was a graduate student, one of her office partners was asked in a qualification exam: "How would you explain chaos theory to your mother?" Now, Sarah has reason to wonder, if this question was asked to her at that time, would her mother, who still remembered that period of history, tell her everything about how the printer next to the LGP-30 in the early 1960s drew the beautiful butterfly wings?

Yes, although Hamilton in his middle age may have been too busy with Apollo's grand plan to "reach the moon" to recall the past, Fette's mind has never forgotten the characteristics of numerical weather forecasting that she has personally experienced: "a slight error can lead to a huge loss." Even after leaving Boston on the East Coast, she and her husband still kept in touch with Lorenz and met him at social events, but she did not realize how famous the polite Lorenz had become. She also did not expect that at the age of 80, her name would suddenly become known to many people.

postscript

The concept of chaos began to take shape in the celestial mechanics of French mathematician Henri Poincaré (1854-1912) in the 1890s, and emerged from the weather forecast of American meteorologist Lorenz in the 1960s. After the vigorous development over the past sixty years, the land of chaos has "attracted countless heroes to compete for it." In this world, in addition to the two mentioned above, there are also American mathematician Stephen Smale (1930-), Australian scientist Robert May (1936-2020), American mathematician James Yorke (1941-), Chinese mathematician Li Tianyan (1945-2020), American mathematical physicist Michelle Feigenbaum (1944-2019), French mathematician Benoit Mandelbrot (1924-2010), etc. However, history has more or less neglected or even omitted some other people, some of whom are outstanding and some are ordinary, but their actual contributions should not be forgotten. Fortunately, some regrets have been discovered and made up for.

For example, Freeman Dyson (1923-2020), a veteran physicist at the Institute for Advanced Study at Princeton, wrote in his popular article "Birds and Frogs" published in 2009 to "speak up for" his British compatriot Mary Cartwright (1900-1998), the first female mathematician fellow of the Royal Society, because she discovered a type of "strange attractor" for second-order nonlinear ordinary differential equations twenty years earlier than Lorenz when she contributed her mathematical talents to the military's analysis of radar systems.

However, some authors who write the history of science, while affirming the contributions of some people, ignore or even disregard the considerable contributions of others. An example related to the Chinese is Gleick's book "Chaos: The Creation of a New Science", which has sold more than one million copies. In the main text of the book, he did not mention the historical role played by Tien-Yien Li in establishing the Lee-York theorem. He only squeezed out a few understated words in the 69th note at the end of the book: Written with his student Tien-Yien Li (co-written with his student Tien-Yien Li). Doesn't he know that it was Tien-Yien Li who rigorously proved this theorem? Yes, the predecessor of the theorem was a conjecture that York intuitively obtained after reading Lorenz's article, but a conjecture will always be just a conjecture if it cannot be proved. For example, French mathematician Pierre de Fermat (1607-1665) determined that his "Great Theorem" was true 350 years ago, but no one could confirm that he had actually proved it. It was not until Andrew Wiles (1953-) finally proved it in the 1990s that this conjecture became a true theorem. Did the mathematical community only agree that Fermat guessed it but not that Wiles proved it? In fact, it was mainly because of this outstanding work that Wiles won numerous awards, including the unique special award of the International Congress of Mathematicians.

There are many examples in the history of science where the key contributions of female scientists are not sufficiently recognized. Wu Jianxiong, who first experimentally confirmed that the law of conservation of parity in weak interactions does not hold, is a typical example. Franklin, who played a key role in the scientific exploration of the double helix structure of DNA, is another example. Naturally, compared with the protagonist Lorenz, whether it is Hamilton or Fette, who worked for him for two years in programming and calculation, the scientific insights they had just graduated from college in grasping the essence of the problem were naturally not as good as the host who designed research plans and had rich mathematical knowledge and scientific research experience. Fortunately, the professor they worked for was a modest gentleman with a very gentlemanly demeanor. He sincerely thanked them at the end of his written scientific work. Otherwise, on the occasion of Lorenz’s centenary, who could imagine and who would believe that more than half a century ago, two young women had helped lay the first foundation for the establishment of chaos theory?

In the early 1960s, Hamilton and Fetter were responsible for programming the huge computers for Lorenz, and the calculations revealed strange attractors and other features of chaos. Therefore, the "Quantum Magazine" article quoted above concluded: "Two female programmers played a pivotal role in the birth of chaos theory. Their previously unknown story illustrates the changing role of computing in science." Many of today's scientific achievements are no longer "single-handed" genius discoveries like in Newton's era, but the condensation of collective wisdom. Whether big or small, whether female or male, anyone's important contribution should be recorded in the archives of history and become a model for those who come after.

References

1. Joshua Sokol, “The hidden heroines of chaos,” Quanta Magazine, May 20, 2019.

This article is supported by the Science Popularization China Starry Sky Project

Produced by: China Association for Science and Technology Department of Science Popularization

Producer: China Science and Technology Press Co., Ltd., Beijing Zhongke Xinghe Culture Media Co., Ltd.

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