Ah, mathematics. That cold, unfeeling discipline which produces its results promptly and without dispute, through an armada of formulae, axioms and proofs. A machine that receives an input and spits out an answer. And mathematicians – dedicated, focused little workers, slaving away at their calculations and the minutiae of algebra and arithmetic. Or so an unfortunate number of people think.
Mathematicians exist at a very uncomfortable crossroads of academia; one that most would not chance upon at all. Math does not capture as wide a public imagination as does astrophysics or nanotechnology, yet it also lacks the immediate physicality and intuitiveness of engineering. But every once in a while, we are reminded of, and promptly humbled by the omnipresence of mathematics and the imagination that it demands. None better to exemplify this, than Amalie Emmy Noether.
Emmy Noether was born in 1882, into a Jewish German family in the Bavarian city of Erlangen. Her father, Max Noether, was a mathematician. Noether was a smart child, but had shown little inclination towards mathematics in her early years, instead learning trades that would be useful around the household like cooking and cleaning. Her fluency in English and French, which had been cultivated from an early age, actually gave her the qualifications necessary to become a language teacher. Forgoing this career option, she chose to study mathematics at the University of Erlangen, where her brother was a student, and her father had been a professor.
Right off the bat, Noether had to grapple with the sexist agendas of late 19th, early 20th century society. The university did not allow her to attend classes like its male students, but only to ‘audit’ them – essentially being taught without a graded performance. Unfettered, she waited two years, until the university allowed women to take an examination to become doctoral students – and passed. In 1907, she graduated, becoming the second woman ever to receive a degree in mathematics. The next obstacle was finding employment. Since many universities at the time, including the one she had graduated from, had policies against hiring women professors, she chose to help her father at the Mathematics Institute of Erlangen. She stayed there for several years, conducting her own research on abstract algebra and the like, while simultaneously filling in for her father when he was sick, teaching his lectures. Her work was almost completely unpaid. But over the course of those many years, Noether’s name and reputation grew along with each published work she put out.
In 1915, David Hilbert and Felix Klein, two mathematicians at the University of Göttingen, invited Noether to work with them. Hilbert and Klein were working on the mathematics behind Einstein’s theory of general relativity, which had been published relatively (no pun intended) recently. They concurred that an expert was needed to help them sort their problems out; hence the invitation to Noether. Unsurprisingly, many of Hilbert’s colleagues, the teaching staff at Göttingen, weren’t too pleased with a woman professor at their institution, some outright opposing her arrival. Noether did manage to teach at Göttingen, although her initial years came with many restrictions. She was not paid for her work and she was only able to lecture under Hilbert’s name.
When it came to proving her mettle, however, Noether did not falter. In 1918, she proved a theorem that linked symmetries in physical systems to conserved physical quantities. This would go down in history as her most famous accomplishment: the eponymous Noether’s theorem.
Noether’s theorem sounds almost childishly simple in its statement, but just as they say, ‘wisdom oft comes from the mouth of babes’, this theorem had profound and far-reaching consequences for physics. Simply put, Noether’s theorem relates the symmetry of a system, to a conserved quantity. If you’ve ever used the laws of conservation of energy, charge, or momentum to solve a problem, you’ve unknowingly been a witness to the beauty of one-half of Noether’s theorem. The full genius of it is exposed when you add in the ‘symmetry’ of the system – which, depending on what system you’re dealing with, could be translational motion, rotational motion, or a sub-atomic particle’s spin.
This monumental discovery not only filled a hole in the understanding of general relativity, but set the stage for the basis of particle physics. It's astonishing how much it affects our understanding of natural phenomena today – from the very small to the very large, it has been the gift that keeps on giving, providing us valuable insight about problems even today.
Noether’s perseverance and resolve paid off in 1919 when, after World War I, women gained many rights in Germany, and she was officially made a formal lecturer. Compounding her brilliance even more was the fact that she was both an engaging and caring teacher. There was even a small group of students known as ‘Noether’s boys’, who were extremely devoted to her and her teaching, some even travelling all the way to Russia to study with her.
She remained a noteworthy figure in Göttingen, which in itself was a hub for cutting-edge mathematics at the time, until 1933. That year saw Nazi Germany’s rise to power, and one of its edicts was to dismiss all Jewish teaching staff from their positions. Noether, being Jewish, left to the USA for a teaching position at Bryn Mawr College. She worked there until her untimely demise two years later, due to complications from an ovarian cyst surgery. She was only 53.
Noether’s contributions ranged far and wide. Her most well-known work was a universal game-changer in physics, a subject that she wasn’t even interested in. She worked extensively on ideals, rings, and non-commutative algebra, and her unique and novel approach to mathematics reformed entire fields, like topology. She had a brilliant mind for grasping at abstract concepts and navigating through them. This allowed her to see things from a non-standard angle that most mathematicians did not have the creative capabilities to. After her demise, she was praised by many as one of the greatest women mathematicians ever, including Hermann Weyl, Pavel Alexandrov, and Albert Einstein himself, who said her theorem was an example of “penetrating mathematical thinking”. Noether was a beacon of unwavering resilience all through her life, even in the face of repeated hardships.
Truly, there never was, nor ever will be, another like Emmy Noether.
An article by Dhruv Raghavan
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