Emmy Noether: Mathematical Symmetry and the Laws of Conservation

By Jaime Seltzer

“My methods are really methods of working and thinking; this is why they have crept in everywhere anonymously.” – Emmy Noether

Emmy Noether was born in 1882 in Bavaria to a very clever family: her father was a professor at the University of Erlangen, and two of her three younger brothers became scientists. Emmy went to finishing school to study to become a teacher of English and French, graduating in 1900. However, on facing the completion of her coursework and passing her exams, Emmy suddenly realized that teaching language was not what she wanted to do at all.

In the early 1900s, Emmy went back to school to audit classes, as German universities still did not allow women to matriculate: first to Erlangen, then to Göttingen, and finally back to Erlangen when, in 1904, they first allowed women to attend on the same footing as men. The entire time Emmy matriculated, she attended school with one other girl in a sea of young men, and she had to personally ask permission from every professor whether she was allowed to sit in their lectures or not. Her particular mentors were Paul Gordon and his successor Ernst Otto Fischer, well-respected mathematicians of the day, who were very supportive in her endeavors. She earned her doctorate in mathematics from the University of Erlangen in 1907, graduating summa cum laude.

Emmy’s work attracted the attention of David Hilbert back at Göttingen, who had been asked to look at some of the mathematical proofs for Einstein’s Theory of Relativity. Hilbert needed an expert in a particular branch of math, and Emmy was the best young Doctor of Mathematics for the job.

Once again, Emmy faced a dilemma. Of course she wanted to work in mathematics, but who would have her? She was allowed to teach at Erlangen, but the university had only just accepted female students and still did not accept female professors. She ended up working there for eight years without pay, lecturing only under her mentor’s name. The papers she began publishing gained her some acclaim, however, and a year after she was awarded her doctorate, she was invited to join a professional mathematics society.

Emmy’s work attracted the attention of David Hilbert back at Göttingen, who had been asked to look at some of the mathematical proofs for Einstein’s Theory of Relativity. Hilbert needed an expert in a particular branch of math, and Emmy was the best young Doctor of Mathematics for the job. He wooed Emmy from Erlangen back to Göttingen, but the university did not approve; she worked there for seven years without pay, while Hilbert practically tore out his hair convincing her to stay and trying to convince the university that they ought to perhaps pay her for the important work she was doing. Hilbert, upon hearing the phrase but she is of the lady persuasion or something silly like that, famously replied, “I do not see how being a woman is an argument against her admittance. Gentlemen, this is a university, not a bathhouse!” This is only one of the many reasons that Hilbert was awesome; he was a genius mathematician in his own right, who would mentor many incredible people alongside Emmy.

Even Einstein actively campaigned to provide Emmy with an official position, which she finally earned in 1919 – but she was only paid beginning in 1922, and it was a pittance in comparison to what the other professors were making.

Hilbert was right to support Noether, not just because it was morally right, but because Emmy was a font of brilliance who would bring prestige to the university. Noether had developed a theorem just as she was moving her mathematical work and her incredible, incredible brain to Göttingen. It was known as Noether’s theorem, and it that would change the world of science and mathematics forever. Its implications, while vast, are within the understanding of the average person, and the basics of the math are within the grasp of a high school student studying calculus. That is indubitably what makes it so brilliant and so intriguing: the formula relies not on advanced mathematics, but the sort of non-linear, creative, big-picture mathematical thinking for which Emmy Noether is still known.

Emmy Noether

Emmy Noether and her card in the Women in Science game.

The idea goes something like this. Wherever you find symmetry, there is conservation. In this case, ‘symmetry’ could refer to the artistic sense of symmetry with which we are all familiar, in the form of, say, a wheel – or, it could refer to any quantity that does not affect the outcome of a formula or an experiment any way you ‘turn’ it – like time. Noether’s Theorem states that any such quantity has a corresponding Law of Conservation.

For example, an experiment that would occur the same way no matter when you performed it could be used to prove the Law of Conservation of Energy. An experiment that would occur the same way no matter where you performed it could be used to prove the Law of Conservation of Momentum. An experiment that would occur the same way no matter how you spun a wheel could be used to demonstrate the conservation of angular momentum. And Noether could demonstrate that this was the case, mathematically, for dozens of different quantities by using the same, basic, deceptively simple math.

One of the best parts of Noether’s Theorem was its flexibility and applicability; it’s considered a “workhorse” of theoretical physics because it underpins such a huge body of research that it’s hard to imagine the world of physics as it is today without it.

Her brilliance did not stop there. Once Emmy was able to use mathematics to predict Laws of Conservation, the next question is which kinds of actions still preserve the law: in other words, what actions are symmetrical? Emmy called this set of actions a group. Group theory predicted the existence of certain particles before they were ever discovered, and is used in chemistry to describe crystalline and molecular structures. Felix Klein gave a famous lecture that used Noether’s group theory to define geometry as a whole!

(It’s also the underpinning of how to solve a Rubik’s cube, though of course you don’t have to understand group theory to be using it subconsciously to solve the puzzle.)

Emmy was working on ring theory when the Nazis eliminated her position, and the position of all those of Jewish ancestry in 1933. The faculty fought for her, signing a testimonial on her fantastic work and incredible character; one compared her brilliance to Meitner’s; her students led an active protest. However, the Nazis followed that up a year later by dismissing all women from university positions, so Emmy was doubly out of luck. Despite her circumstances, she continued teaching from her home for some time, even while some of her students wore the uniform of the Nazi party. A colleague of hers said that she never once doubted the sincerity of any of her students: if they were there to learn, she was happy to teach them.

When the situation became untenable, Einstein intervened on Emmy’s behalf again, this time to secure her a position at Bryn Mawr College in the United States. Unfortunately, a year and a half later it was discovered that she had cancer. The operation went badly, leading to a post-operative infection that killed her within twenty-four hours.

Emmy was not only a brilliant mathematician. She possessed the talent of taking a vast array of information and tying it neatly together with a simple and elegant mathematics that was uniquely her own. She was a determined and vehement pacifist, who taught members of the Nazi party in her own home. Because of Noether, our understanding of fields like quantum physics would never be the same.

References
Angier, N. (2012, March 26). The Mighty Mathematician You’ve Never Heard Of. The New York Times. Retrieved from http://www.nytimes.com/2012/03/27/science/emmy-noether-the-most-significant-mathematician-youve-never-heard-of.html?_r=0
Conrad, K. (2005). Why is group theory important? In University of Connecticut: Math 216 – Abstract Algebra I. Retrieved from http://www.math.uconn.edu/~kconrad/math216/whygroups.html
Francis, M. R. (2015, June). Mathematician to know: Emmy Noether. Symmetry. Retrieved from http://www.symmetrymagazine.org/article/june-2015/mathematician-to-know-emmy-noether
O’Connor, J. J., & Robertson, E. F. (2014, November). Emmy Amalie Noether. In School of Mathematics and Statistics: University of St. Andrew’s. Retrieved from http://www-history.mcs.st-andrews.ac.uk/history/Biographies/Noether_Emmy.html
Weyl, H. (2014, November). Hermann Weyl’s speech at Emmy Noether’s funeral. In School of Mathematics and Statistics: University of St. Andrew’s. Retrieved from http://www-history.mcs.st-andrews.ac.uk/history/Extras/Weyl_Noether.html

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