# Emmy’s beautiful mathematics

Noether’s theorem, which links together symmetry and physics, is one of the most beautiful and elegant concepts I have ever learnt. I didn’t expect to find out that Noether was really Emmy, a woman. And yet, what took me aback was my own surprise. I had always believed that women were just as capable as men of doing anything, however, when I learnt this beautiful piece of mathematics, I believed unquestionably that it had been written by a man, that the author could be a woman had not even crossed my mind. Of course, at the time I knew about Marie Curie, a female scientist so famous that could be called a legend among the hallways of the physics department. But I had never heard of Noether, nor any other woman on my physics course, and I was in the second year! This realisation opened my eyes to my unconscious bias and made me want to learn more.

Noether was really a mathematician, one of the leading founders of abstract algebra, who approached problems in a completely novel way. Her work in physics, which had so caught my attention, was just something she did on the side, to help a poor physicist who couldn’t figure out the maths of his new theory. As a result came Noether’s theorem, described technically by the equation above, which says that for every continuous symmetry of a physical system, there exists an associated conservation law. Conservation laws are fundamental in physics, as they allow us to determine phenomena that can or cannot happen in physics. Noether theorem links them to symmetries of the systems and allows us to determine which physical quantities are conserved uniquely from the properties of the Lagrangian, a function of the energy of the system. Many conservation laws had been known, such as energy and momentum conservation of a closed system, but Noether’s theorem resolved paradoxes in those conservation laws arising in new theories of physics, such as General Relativity. To say this theorem is an important result in physics is an understatement.

Emmy grew up in a family of mathematicians, who somehow failed to notice her aptitudes and didn’t encourage her to pursue mathematics. She started training as a language teacher when she became fascinated by mathematics and started attending lectures at the university of Erlangen. As a woman, she could not officially enrol, so she would simply audit the lectures. Some years later women started to be officially allowed to take classes, but the policy on women’s rights would always have a hard time catching up with her. For some time, once she had passed her doctoral thesis, she was only allowed to teach classes at the University of Gottingen that were advertised as Hilbert’s. Years later, she was able to gain a position at the university although badly paid. She would never get to be a full professor in Germany, or even gain the wages due for her work. Being forbidden from teaching at the university of Gottingen due to her Jewish heritage in 1933, she moved to Bryn Mawr, a single-sex school in the USA. Her time at the college was accompanied by difficult circumstances, as she couldn’t teach graduate courses, find a permanent position, had health problems and the political situation of Germany was increasingly bad. However, she saw things differently and said that “the last year and a half had been the happiest in her whole life, for she was appreciated in Bryn Mawr and Princeton, as she had had never been appreciated in her own country”. Unfortunately, she was to die soon after, from complications of a surgery to remove a tumour.

When reading about Noether’s life, it was her personality that struck me the most. She was a fantastic woman and an incredible mathematician, whose informal lifestyle caused many jokes that she would simply ignore. Her appearance, dress and weight were usually commented upon, so was her voice, deemed “loud and disagreeable” because it was not soft and refined as other women’s. She cared enormously for her students, with whom she shared her ideas and whom she taught with passion and enthusiasm, regardless of their political position (to the point that one of her students used to come to her house to take class wearing a nazi brown SA shirt). Her students held her in high esteem as she made them feel like she was one of them, “almost as if she too were thinking about the theorems for the first time.” She applied to both mathematics and life a general principle of simplification and removal of the unnecessary. She wore comfortable men’s shoes and coats, and during a time, she would go six days a week to eat the same dinner at the same time, at the same table of the same restaurant. According to Noether’s only American graduate student, “her methods of thinking and working were simply a reflection of her way of life: that is, to recognise the unnecessary, brush it aside and enter wholeheartedly into the present”.

She was also held in high esteem by her colleagues, and it was thanks to their continuous campaigning that she was able to get her teaching positions, first at the university of Gottingen and later on in Bryn Mawr College and the Institute of Advance Studies in Princeton (although sadly she died before she could join the latter). Hermann Weyl, a professor in Gottingen before the Second World War, said that he was “ashamed to occupy a preferred position beside her, whom I knew to be my superior as a mathematician in many aspects”. After she made important contributions to Einstein’s theory of General Relativity, Einstein wrote to Hilbert: “Yesterday I received from Miss Noether a very interesting paper on invariants. I’m impressed that such things can be understood in such a general way. The old guard at Göttingen should take some lessons from Miss Noether! She seems to know her stuff.” He was later to write, in her obituary for the New York Times, “In the judgment of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began.”
Noether, unlike some of her male colleagues, did not receive much recognition during her life and was instead criticised for many unimportant things. As historians, Crease and Mann said: “Had Noether been a man, her appearance, demeanour, and classroom behaviour would have been readily recognised as one of the forms that absent-minded brilliance frequently assumes in the males of the species”. I find Noether inspiring. Inspiring because of her achievements, which are made particularly striking given the tidal forces she had to fight against to pursue her lifelong passion for mathematics. Inspiring for her drive, her attitude towards students and colleagues, her dismissal of criticism. Inspiring because of her beautiful mathematics.

Today’s post is a celebration of Ada Lovelace’s Day, international celebration day of the achievements of women in science, technology, engineering and maths (STEM)!

Most of the biographical anecdotes of this post have been obtained from the book:  Nobel Prize Women in Science: Their Lives, Struggles, and Momentous Discoveries by Sharon Bertsch McGrayne

## 11 thoughts on “Emmy’s beautiful mathematics”

1. bf says:

Emmy Noether’s most important work isn’t of course this beautiful theorem (which I never even heard of in college, possibly because I studied mathematics not physics) but the creation of a number of key ideas in modern algebra. Her ideas are pervasive in a vast area of mathematics (encompassing commutative algebra, algebraic and arithmetic geometry) that has seen impressive progress since her work gave us a language to work with. Her work is so important that it has spawned an adjective (Noetherian) and a noun (noetherianity).

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1. Noether’s theorem is well known among physicists, and given its importance for Analytical Mechanics and General Relativity, one can only wonder what more could she have contributed if only she spent her time working on physics rather than algebra! It’s a pity we don’t have the chance to study more of her work while doing a physics undergrad.

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2. One nitpick: Emmy Noether is probably better described as a founder of commutative algebra, ring and module theory, or invariant theory. Abstract algebra includes a lot of work due to earlier mathematicians (Gauss, Galois, Cauchy, etc etc etc). Nice piece!

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1. It says a lot about the contribution of a scientist when members of different disciplines can argue about what field benefited most from their contributions! 🙂

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1. There is no argument, although Noether’s theorem is a highly significant contribution to modern physics, her contributions to the development of modern algebra and structural mathematics are far more important.

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