In the early 20th century, mathematics was a world largely reserved for men. Those who sought to break these barriers confronted centuries of systemic inertia, both in institutions and in collective consciousness.
In Göttingen, David Hilbert had just begun popularising his reformist vision: a radical re-examination of the foundations of mathematics as people understood them. But amid this great optimism surrounding mathematics, women were still largely excluded from academic life.
Invited to Göttingen by Hilbert, Emmy Noether was denied a teaching post simply because she was a woman. For four years she lectured under Hilbert’s name rather than her own, as she was not permitted to teach officially. When the faculty objected, Hilbert famously responded: “I do not see that the gender of the candidate is an argument against her admission. We are a university, not a bathing establishment.”
Noether’s work extended deeply into both mathematics and physics. Speaking at a memorial address, her contemporary Hermann Weyl argued that her strength was to operate with abstract concepts, to strip away the inessential, and to rephrase problems so that the whole structure could be surveyed. It’s only natural that to understand Noether’s stubborn mind, we need to look at both her life and her mathematics.
Early life and challenges
Amalie Emmy Noether was born on 23 March 1882 in the small town of Erlangen, the first child of Max Noether and Ida Kaufmann, both from families of Jewish merchants. Max deserves a biography of his own: a leading mathematician in algebraic geometry and professor at the University of Erlangen. But in this story, he is the backdrop. Emmy is the main character.
Her path was anything but conventional. As a young woman in Bavaria, the most respectable career open to her was that of a teacher of modern languages. She excelled at it too: she earned top marks in the state examinations for French and English, qualifying her to teach in girls’ schools. She could have followed that socially acceptable path. Instead, in a move that feels characteristically stubborn in hindsight, she chose mathematics.
At the time, Bavarian universities did not allow women to enrol as full students; officials claimed that mixed education would “overthrow academic order”. It was also basically impossible for women to sit the Realgymnasium examination required to enter university. So, in the winter semester of 1900, Emmy attended lectures at the University of Erlangen as an auditor. She could be present and take notes, but only with the permission of each professor. She was one of only two women among roughly a thousand students.
The historical central building of the University of Erlangen. Image: Akriesch, CC BY 2.5
Despite the barriers, in 1903 she passed the Realgymnasium examination as an external candidate, becoming formally eligible for university study, even though the law still hadn’t properly caught up with her. Around this time, she also spent a semester in Göttingen as a visiting student, attending lectures by figures such as Felix Klein and David Hilbert; this is likely when Hilbert first noticed her.
Soon after, restrictions in Bavaria were lifted and in October 1904 Emmy finally enrolled as a full student at Erlangen. She was the only woman studying mathematics there. She completed her doctorate under Paul Gordan, the “king of invariant theory”, and a close friend of her father, marking the beginning of her way of thinking that would allow her to redefine algebra itself.
Her doctoral thesis and much of her early work dealt with some objects called invariants. Invariants are things that remain unchanged after you apply a transformation. For instance, in a coordinate plane, if you rotate a segment or change the coordinates, the length remains the same. Unfortunately, a lot of the study of invariants involved hands-on, heavy computations—there was no non-constructive way yet to find what remains unchanged. However, mathematicians then cared about these unchanging things because they reveal what is truly essential in a system. Emmy Noether had a talent for this abstract thinking, and although her early work would not cultivate it, it helped in fine-tuning her later methods.
Einstein’s co-author
Meanwhile, on the northern side of the country, physics was undergoing its own revolution with Albert Einstein in the front line. In 1915, Einstein’s general relativity replaced Newtonian dynamics as such: gravity is not just a regular force, but it is spacetime curvature, and motion under gravity is simply motion along the paths in that curved geometry. Thus, matter and energy curve spacetime themselves.
Central Göttingen. If Hilbert called you here, would you show up? Image: Daniel Schwen, CC BY 2.5
While the physics was elegant, the machinery required mathematical tools outside the standard toolkit. Hilbert felt that something was wrong with this behaviour that was only empirically described. Energy had to go somewhere, and in Einstein’s universe, spacetime itself was curving, so what does “conservation of energy” even mean? It meant that matter wasn’t the only thing bending spacetime; it was the initial curvature that had to bend spacetime and that bending added new energy, which bent it further, until the curvature became both the cause and the consequence of the curve under the mass. Had the physical universe broken its promise?
Hilbert, therefore, did the most sensible thing he could. He called Emmy Noether to Göttingen.
Conservation of energy is itself an invariant; it is the thing that everyone knew for certain that it would not change, no matter how much spacetime bends. However, Hilbert had discovered a paradox. Noether seemed to be the best person to tackle this. And she did. In her 1918 paper Invariante Variationsprobleme, she rigorously proved something more general. Whenever nature has symmetry, something is conserved. Time symmetry gives energy. Space symmetry gives momentum. Rotational symmetry gives angular momentum. In her abstract approach, she noticed that an object could be asymmetric, but if the laws of physics themselves are symmetric, then energy is conserved. And in Einstein’s universe of a curved spacetime, you can achieve this conservation locally. This was proved by Noether.
She was not allowed to present her own findings publicly because she was not a member of the Royal Academy (in fact, there were no women members). Felix Klein presented it in her name. Einstein himself wrote to Hilbert about Noether’s work that he was “impressed that one can comprehend these matters from so general a viewpoint. It would not have done the old guard at Göttingen any harm had they picked up a thing or two from her”. And even still, her framework was so general that the result we know as Noether’s theorem appears only as a lemma in her paper.
An ascending chain of abstraction
While Noether was rigorously rescuing Einstein’s great new theory of relativity, Göttingen was still denying her habilitation. Habilitation was the institutional test that certified that someone could be trusted to lecture and supervise students. So she was working unpaid, lecturing under Hilbert’s name and supervising doctoral students under her father’s name. Despite massive support from titans like Hilbert and Klein (prompting Hilbert’s famous line), she was denied recognition and pay. Only in 1919 would she finally be habilitated and allowed to conduct lectures.
Noether’s famous 1918 Paper
Inspired by her work in invariant theory, Noether became fascinated with Hilbert’s 1890 breakthrough. Up to that point, invariant theory meant endless calculation. Hilbert proved that there must always be a finite generating set of invariants without ever writing them down. In a strange coincidence, the man who would become Noether’s supervisor called this paper “theology”: believing in the existence of something unseen.
Noether herself moved from studying particular objects to understanding the underlying structure behind them. She became familiar with Dedekind’s theory of ideals. An ideal is a special subset of a ring that absorbs multiplication, allowing one to treat divisibility and factorisation in a structural way rather than element-wise.
In this period, she began developing what would become her signature way of doing mathematics. Her proofs introduced, for the first time, arguments that if you have two ways of building the same algebraic structure, you can swap pieces and substitute in variables without gaining a different structure. This rigidity forces the structure to be unique. Another novel approach was understanding modules and rings by writing it as an intersection of simpler pieces. To prevent this representation from becoming chaotic, she imposes a finiteness condition.
This will culminate in her 1921 paper, Idealtheory in Ringbereichen (ideal theory in rings). Building on earlier work by Emanuel Lasker, she proved that every ring could be better understood by breaking it into a finite intersection of ideals. Crucially, she showed that this only works when the ring satisfies a finiteness condition: the ascending chain condition. If you build larger and larger ideals, the process will eventually stop. In a well-behaved ring, chains of ideals cannot climb forever. This guarantees a kind of mathematical “niceness” of such rings, a coherence and a quiet disciplined structure among objects that otherwise feel wild. Rings that satisfy this condition are now called Noetherian. On the footnote of her paper, she noted that this theorem had first been observed by Dedekind for number fields and by Lasker for ideals of polynomials. Reflecting Weyl’s description of her ability to strip away everything irrelevant, she wrote: “In both cases the theorem finds only specific applications, however. Our applications depend without exception on the axiom of choice.”
Chalkdust’s idea of an ideal ring. Image: Mattbuck, CC BY-SA 2.0/Adobe Firefly
The human cost
Emmy Noether was described as warm. She would be unusually spirited when her dedicated students would ask her questions. Her colleagues remember that they used to go on walks together talking nothing but mathematics. She eventually got a more secure position and her salary increased, and she put money aside for her nephew’s university tuition. She loved sharing mathematics. She was making mathematics a more habitable place, especially for those that used to be in her place decades prior.
However, soon politics would touch Göttingen again. In 1933, with Nazi racial laws closing in, Noether’s right to teach would be revoked. According to her biographer Auguste Dick, she tried her best to encourage colleagues facing the same fate with “her own optimism and showed how deeply grateful she was for every bit of help she received from non-Jewish friends”. When her right to teach was revoked, she hosted mathematical lectures in her small apartment. When one of her students showed up in a Nazi uniform, she showed no concern and even laughed about it later. Later, she was evicted from her apartment, when her lodgers complained of living with a “Marxist-leaning Jewess”.
Germany was no longer a safe place to be for a lot of the Göttingen elite. Albert Einstein and Hermann Weyl took refuge at Princeton, and Emmy Noether ended at Bryn Mawr, also in the United States. Her time at Bryn Mawr was pleasant, unlike her brief visits to Princeton, where, according to her, “nothing female was admitted”. She formed a reading group of young women mathematicians, often called the “Noether girls”. She mentored her doctoral student Ruth Stauffer with a warmth her colleagues often described as almost maternal. There, she was finally somewhere she could freely teach and think about the future.
That future ended abruptly. In April 1935, after what was expected to be a routine operation, Noether died from post-surgical complications. She was 53. Ruth would defend her thesis without her, and the mathematical world lost a mind that was very much in motion.
Noether’s ashes were buried in the cloisters ouside the Bry Mawr old library. Bryn Mawr’s mascot is the owl. Go owls. Image: Jeffrey M Vinocur, CC BY 2.5/Adobe Firefly
Legacy
So much of Noether’s legacy lives on in the language of mathematics. In this article I have tried to outline two such pillars: Noether’s theorem and Noetherian rings. But the real purpose of this piece is to remind the reader that behind those names stood a real person, a mathematician who had to fight for the right to exist inside the very institutions that needed her most. Emmy Noether’s story reminds us that brilliance should never have to struggle so hard to exist.
Today, the doors are more open than they were in Göttingen, but they are still not wide enough. Pay attention to whose voices fill seminar rooms. Notice whose names are attached to the theorems we study. Ask who feels permitted to imagine themselves at the centre of this discipline. Today, the doors are open. The room is not yet shared.
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