On any list of history’s great mathematicians who were ignored or underappreciated simply because they were women, you’ll find the name of Emmy Noether. Despite the barricades erected by 19th century antediluvian attitudes, she managed to establish herself as one of Germany’s premier mathematicians. She made significant contributions to various math specialties, including advanced forms of algebra. And in 1918, she published a theorem that provided the foundation for 20th century physicists’ understanding of reality. She showed that symmetries in nature implied the conservation laws that physicists had discovered without really understanding.
Joule’s conservation of energy, it turns out, is a requirement of time symmetry — the fact that no point in time differs from any other. Similarly, conservation of momentum is required if space is symmetric, that is, moving to a different point in space changes nothing about anything else. And if all directions in space are similarly equivalent — rotational symmetry — then the law of conservation of angular momentum is assured and figure skating remains a legitimate Olympic sport. Decades after she died in 1935, physicists are still attempting to exploit Noether’s insight to gain a deeper understanding of the symmetries underlying the laws of the cosmos.
Yay, it’s story time!
Albert Einstein was in over his head. He had worked out his general theory of relativity, but he was having problems with the mathematics that would have to correspond. So Einstein pulled in a team of experts from the University of Göttingen to help him formulate the concepts. The team was led by David Hilbert and Felix Klein, who were held in extremely high regard for their contributions to mathematical invariants. But their legacy, in part, is the community of scholars they fostered at Göttingen, who helped the university grow into one of the world’s most respected mathematics institutions. They scouted talent. For the Einstein project, Emmy Noether was their draft pick.
Noether had been making a name for herself steadily. In the eight years prior, she worked at the University of Erlangen without a salary or a job title. By the time she left for Göttingen, she had published half a dozen or so papers, lectured abroad, taken on PhD students, and filled in as a lecturer for her father, Max Noether, who was an Erlangen mathematics professor suffering from deteriorating health.
At the time, Noether’s specialty was invariants, or the unchangeable elements that remain constant throughout transformations like rotation or reflection. For the general theory of relativity, her knowledge base was crucial. Those interlinked equations that Einstein needed? Noether helped create them. Her formulas were elegant, and her thought process and imagination enlightening. Einstein thought highly of her work, writing, “Frl. Noether is continually advising me in my projects and…it is really through her that I have become competent in the subject.”
It didn’t take long for Noether’s closest colleagues to realize that she was a mathematical force, someone of extraordinary value who should be kept around with a faculty position. However, Noether faced sharp opposition. Many of the people who supported the push to make her a lecturer also believed that she was a special case and that, in general, women shouldn’t be allowed to teach in universities. The Prussian ministry of religion and education, whose approval the university needed, shut down her appointment: “She won’t be allowed to become a lecturer at Göttingen, Frankfurt, or anywhere else.”
The shifting political landscape finally cracked open the stubborn set of regulations governing women in academia. When Germany was defeated in World War I, socialists took over and gave women the right to vote. There was still a movement internally to get Noether on staff, and Einstein offered to advocate for her. “On receiving the new work from Fräulein Noether, I again find it a great injustice that she cannot lecture officially,” he wrote. Though Noether had been teaching, on paper her classes were David Hilbert’s. Finally, Noether was allowed a real position at the university with a title that sounded like fiction. As the “unofficial, extraordinary professor,” Emmy Noether would receive no pay. (Her colleagues joked about the title, saying “an extraordinary professor knows nothing ordinary, and an ordinary professor knows nothing extraordinary.”) When she finally did receive a salary, she was Göttingen’s lowest-paid faculty member.
Pay or no pay, at Göttingen she thrived. Here’s how deeply one line of study, now called Noether’s theory, influenced physics, according to a physicist quoted in the New York Times: “You can make a strong case that her theorem is the backbone on which all of modern physics is built.” And the dent she made in mathematics? She was a founder of abstract algebra. In one paper, published in 1921 and titled “Theory of Ideals in Rings,” Noether dusted her work free of numbers, formulas, and concrete examples. Instead she compared concepts, which, the science writer Sharon Bertsch McGrayne, explains, “is as if she were describing and comparing the characteristics of buildings—tallness, solidarity, usefulness, size—without ever mentioning buildings themselves.” By zooming way, way out, Noether noticed connections between concepts that scientists and mathematicians hadn’t previously realized were related, like time and conservation of energy.
Noether would get so excited discussing math that neither a dropped piece of food at lunch nor a tress of hair sprung from her bun would slow her down for a second. She spoke loudly and exuberantly, and like Einstein was interested in appearance only as it related to comfort. Einstein loved his gray cotton sweatshirts when wool ones were the fashion; Noether wore long, loose dresses, and cut her hair short before it was in style. For Einstein, we call these the traits of an absentminded genius. For Noether, there was a double standard—her weight and appearance became the subject of persistent teasing and chatter behind her back. Like the trivial annoyances of title, pay, and politics, the comments didn’t bother Noether. When students tried to replace hairpins that had come loose and to straighten her blouse during a break in a particularly passionate lecture, she shooed them away. Hairstyles and clothes would change, but for Noether, math was her invariant.
With a mind working as rapidly as hers, it was a challenge for even Noether to keep up with her own thoughts. As she worked out an idea in front of the class, the blackboard would be filled up and cleared and filled up and cleared in rapid succession. When she got stuck on a new idea, students recalled her hurling the chalk to the floor and stomping on it, particles rising around her like dust at a demolition. Effortlessly, she could redo the problem in a more traditional way.
Both social and generous with sharing ideas, many, many important papers were sparked by Noether’s brainpower and published without her byline but with her blessing. In fact, whole chunks of the second edition of the textbook Modern Algebra can be traced back to her influence.
Politics in Germany affected her career again. Though Noether had established herself as one of the greatest mathematical minds of the twentieth century, the Nazis judged only her left political leanings and her Jewish ancestry. In May 1933, Noether was one of the first Jewish professors fired at Göttingen. Even in the face of blatant discrimination, perhaps naively, the math came first. When she could no longer teach at the university, Noether tutored students illegally from her modest apartment, including Nazis who showed up in full military gear. It wasn’t that she agreed with what was happening, but she brushed it aside for the dedicated student. “Her heart knew no malice,” remembered a friend and colleague. “She did not believe in evil — indeed it never entered her mind that it could play a role among men.”
For her generosity, Noether’s friends were wholly dedicated to her. Understanding that staying in Germany would put her in serious danger, in 1933 her friends arranged for Noether to take a position at Bryn Mawr College in the United States. It was meant to be a temporary post until she could land somewhere more prestigious. But just two years after she arrived, Noether died while recovering from a surgery on an ovarian cyst. She was fifty-three. Following her death, Einstein wrote a letter to the New York Times. “Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began.” Today, some scientists believe her contributions, long hidden beneath the bylines and titles of others, outshine even the accomplishments of the ode’s writer.
Physicists tend to know Noether’s work primarily through her 1918 theorem. Because their work relies on symmetry and conservation laws, nearly every modern physicist uses Noether’s theorem. It’s a thread woven into the fabric of the science, part of the whole cloth. Every time scientists use a symmetry or a conservation law, from the quantum physics of atoms to the flow of matter on the scale of the cosmos, Noether’s theorem is present. Noetherian symmetries answer questions like these: If you perform an experiment at different times or in different places, what changes and what stays the same? Can you rotate your experimental setup? Which properties of particles can change, and which are inviolable?
Conservation of energy comes from time-shift symmetry: You can repeat an experiment at different times, and the result is the same. Conservation of momentum comes from space-shift symmetry: You can perform the same experiment in different places, and it comes out with the same results. Conservation of angular momentum, which when combined with the conservation of energy under the force of gravity explains the Earth’s motion around the sun, comes from symmetry under rotations. And the list goes on.
The greatest success of Noether’s theorem came with quantum physics, and especially the particle physics revolution that rose after Noether’s death. Many physicists, inspired by Noether’s theorem and the success of Einstein’s general theory of relativity, looked at geometrical descriptions and mathematical symmetries to describe the new types of particles they were discovering.
Emmy Noether’s theorem is so vital to physics that she deserves to be as well known as Einstein. – Brian Greene
Noether’s theorem to me is as important a theorem in our understanding of the world as the Pythagorean theorem. – Christopher Hill
Mathematicians are familiar with a variety of Noether theorems, Noetherian rings, Noether groups, Noether equations, Noether modules and many more. Over the course of her career, Noether developed much of modern abstract algebra: the grammar and the syntax of math, letting us say what we need to in math and science. She also contributed to the theory of groups, which is another way to treat symmetries; this work has influenced mathematical side of quantum mechanics and superstring theory.
Story from Headstrong – 52 women who changed science and the world, by Rachel Swaby, and Fermilab/SLAC National Accelerator Laboratory Symmetry Magazine.