Mathematician Emmy Noether, Symmetries and Conservation Laws

Emmy Noether, linocut, 11" x 14", Ele Willoughby, 2018

Emmy Noether (1882-1935, pronounced NER-ter) has long been on my "to do" list of scientist portraits. Noether's Theorem is one of the most fundamental and profound theories in physics and I think it's impossible to overstate its importance. In some ways it's astonishing that Noether's Theorem wasn't discovered until one century ago in 1918 and in some ways its true import wasn't clear until much later. The theorem is so powerful that I struggled with how I could depict it visually. It can be written in many different ways. I could have reproduced her actual equations as her paper is widely available in the original German and in English translation. But, my goal with my art is to communicate science, and even writing a single equation cuts the potential audience. I hope that expressing ideas visually through geometry is more accessible to more people. So, in my portrait, I chose to depict a young Emmy in front of a blackboard with a more simple formulation of her theorem and three specific applications of it, shown schematically, using pictures and geometry. In simple terms, Noether's theorem shows us that any symmetry of a system (say, a given problem in physics, like a ball rolling or a molecule or a solar system or the universe itself) implies a conservation law.

The three examples I give are probably the best known, but just give a hint of the power of this theorem. If you do an experiment and then move three steps to the right and repeat it, you usually expect the same results. In general, a lot of things will have this translational symmetry. Noether's Theorem shows that if you get the same result in two reference frames which are shifted from one another, your system conserves momentum (p with an arrow, as a vector quantity). Thus, we have conservation of momentum in any inertial frame of reference. That means that any place where we don't have to worry about any significant differences from acceleration or gravity, we can solve physics problems by simply knowing that the total momentum never changes. In my print I show a set of x, y, z axes moved (translated) to get a new set of axes x', y' and z' and then the quantity p. Similarly, if your system doesn't care if you rotate it or how it's oriented in space, the conserved quantity is angular momentum (L with an arrow, as a vector quantity); hence in my print, I show a set of x, y, z axes rotated x', y' and z' along with conserved quantity L. Your system itself doesn't need to be symmetric. A lumpy asteroid conserves angular momentum every bit as much as a planetary system made of perfect spheres. If it's irrelevant to results whether you do your experiment at 3:00 or 6:25 then your system has a time symmetry and conserves energy (E). This method of using observed symmetries of something and then finding things which are invariant allows us to easily solve all sorts of problems in physics. Further, using observed symmetries of the Universe allows us to know which things are invariant, know more about the nature of reality and assess any new theories by checking whether they also produce the same conserved quantities.*

Here's a nice video which talks about Noether's Thereom.




Her male colleagues Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl, and Norbert Wiener described Noether as the most important woman in the history of mathematics - a compliment which betrays the biases of the times in comparing her only to those of the same sex. She was quite simply, one of the most important mathematicians period, and her impact on physics was tremendous. (My portrait betrays my own biases, focusing on the physics of Noether's Theorem, rather than her contributions to mathematics... but there you are. I'm a physicist by training, not a mathematician).

Born in Erlangen, Germany, Emmy Noether initially planned to teach girls English and French, rather than follow in her father's footsteps and become a professor of mathematics. But ultimately, she choose to study mathematics at the University of Erlangen, where he was a lecturer. Pursuing mathematics was unconventional for a woman; the university had recently declared that mixed-sex education would "overthrow all academic order" and as one of 2 female students (out of 986) she was only able to audit classes at the discretion of professors. She nonetheless managed to pass the graduation exam in 1903 and was granted a degree. She spent the winter semester at the University of Göttingen attending lectures from astronomer Karl Schwarzschild and mathematicians Hermann Minkowski, Otto Blumenthal, Felix Klein, and David Hilbert, before returning to Erlanger. She completed a dissertation supervised by Paul Gordan, On Complete Systems of Invariants for Ternary Biquadratic Forms (1907) using the "computational" approach to invariants, later superseded by Hilbert's more abstract and general approach. She later referred to this well-received thesis and the first few similar papers as "crap". She continued to work at the university for 7 years, but as a woman she was excluded from an academic position and in fact forced to worked without pay.

In 1915 she was recruited to come to the renown University of Göttingen and work with famed mathematicians David Hilbert and Felix Klein. However, some philologists and historians in the philosophical department protested that a woman must not become a Privatdozent, an additional post-doctoral rank required in Germany and certain other European nations to act as a university professor. Famously, a faculty member protested "What will our soldiers think when they return to the university and find that they are required to learn at the feet of a woman?" but Hilbert defended her indignantly, with one of my favourite lines in response to such entrenched academic sexism: "I do not see that the sex of the candidate is an argument against her admission as privatdozent. After all, we are a university, not a bath house." There she still faced hurdles and had to rely on her family to support her financially, as she was unpaid and could only lecture under Hilbert's name until 1919  despite already having published her eponymous Noether's Theorem in 1918! After Einstein published his theory of general relativity in 1915 and Noether responded by applying her invariance work to some of its complexities and this eventually lead her to prove her famous theorem. As Einstein wrote when he read her paper, "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."

The end of WWI and German Revolution of 1918-1919 lead to social change and increased rights for women. Her habilitation was approved and she obtained the rank of Privatdozent in 1919. Three years later she was promoted to an untenured professor (nicht beamteter ausserordentlicher Professor) but her work remained unpaid until the next year when she was finally granted a special position (Lehrbeauftragte für Algebra).
 
Until 1919 she focused on theories of algebraic invariants and number fields. While her incredible contribution to physics had already occurred in 1918, mathematicians remember her for her central role in the 20th century revolution in mathematics, the development of abstract algebra, and her prolific work including Ring Theory from 1920 to 1926, as well as Noetherian rings, Noether groups, Noether equations, Noether modules and more. Her revolutionary 1921 paper Theory of Ideals in Ring Domains is considered a classic and objects which satisfy the ascending chain condition are named Noetherian, in her honour. In the final stage of her career, she focused on noncommutative algebras and hypercomplex numbers and united the representation theory of groups with the theory of modules and ideals. She was a leader in the strong University of Göttingen math department until 1933. Her colleague Dutch mathematician B. L. van der Waerden made her work the foundation of the second volume of his influential 1931 textbook, Moderne Algebra; it was typical of her to allow students and colleagues to receive credit for her ideas. She supervised more than a dozen doctoral students. She was known for her patient guidance but insistence on accuracy. van der Waerden wrote that she was, "Completely unegotistical and free of vanity, she never claimed anything for herself, but promoted the works of her students above all." She learned to live frugally, having gone so long without a salary, and took no concern about her manners, housework or appearance. She used her lecturers as a time for spontaneous discussions of the latest mathematics with students and a place to explore ideas (many of which would become major publications of those students). She spent the winter of 1928–29 at Moscow State University, working with P. S. Alexandrov. She was interested in and supportive of the Russian Revolution and her political opinions got her evicted from her lodging back in Germany when students there complained of living with "a Marxist-leaning Jewess". In 1932, she won the received the Ackermann–Teubner Memorial prize for her contributions to mathematics, which came with 500 Reichsmarks and she gave the plenary address at the 1932 International Congress of Mathematicians in Zürich, a sign of her international stature in the field. Colleagues complained that she was however never elected to the Göttingen Gesellschaft der Wissenschaften (academy of sciences) or promoted to full professor. Within a year Nazi Germany moved to dismiss her and all Jewish academics from university positions. The German Student Association, aided by one of Noether's own former students, a privatdozent named Werner Weber, led the attack on Jews at the University of Göttingen. She merely laughed when students showed up dressed as Hilter's brownshirts. Dedicated to her students, she invited them to her home to discuss math and their plans for the future. Herman Weyl wrote "Emmy Noether—her courage, her frankness, her unconcern about her own fate, her conciliatory spirit—was in the midst of all the hatred and meanness, despair and sorrow surrounding us, a moral solace." Emmy Noether was able to find a position at Bryn Mawr College in Pennsylvania in 1933, where she finally gained the appreciation she deserved. In 1934 she lectured at the Institute for Advanced Study in Princeton, but remarked that she was not welcome at the "men's university, where nothing female is admitted." Tragically, she died 4 days after surgery to remove an ovarian cyst in 1935 when she was only 53.

Noether's theorem remains fundamental to physics, and has been especially vital to particle physics in the decades since her death. Her originality in mathematics was beyond compare and in Weyl's words she "changed the face of algebra by her work."

References
Emmy Noether, wikipedia article access June 6, 2018

Noether E (1918). "Invariante Variationsprobleme". Nachr. D. König. Gesellsch. D. Wiss. Zu Göttingen, Math-phys. Klasse. 1918: 235–257.

M. A. Tavel's English translation of Noether's Theorems (1918)

Matthew R. Francis, Mathematician to know: Emmy Noether, Symmetry Magazine, June 18, 2015.

Natalie Angier, The Mighty Mathematician You’ve Never Heard Of, The New York Times, March 26, 2012

*Now, if you're interested in the equation itself here's one good online explation (if say, you have most of an undergraduate degree in physics or more). A more intuitive a bit more straightforward explanation is here. The original paper is here and can be found in translation here. 

Mary Anning, Great Fossil Hunter & Paleontology Pioneer for Ada Lovelace Day

Today is the fifth annual international day of blogging to celebrate the achievements of women in technology, science and math, Ada Lovelace Day 2013 (ALD13). I'm sure you'll all recall, Ada, brilliant proto-software engineer, daughter of absentee father, the mad, bad, and dangerous to know, Lord Byron, she was able to describe and conceptualize software for Charles Babbage's computing engine, before the concepts of software, hardware, or even Babbage's own machine existed! She foresaw that computers would be useful for more than mere number-crunching. For this she is rightly recognized as visionary - at least by those of us who know who she was. She figured out how to compute Bernouilli numbers with a Babbage analytical engine. Tragically, she died at only 36. Today, in Ada's name, people around the world are blogging 

I've been thinking for a while about whom I would like to portray, in a new print, and write about for Ada Lovelace Day. Emmy Noether has been on my 'to do' list for a while.  When I was taught Noether's Theorem, which related symmetries to conservation laws, I was absolutely flabbergasted. It's one of the most profound ideas in physics (and hence a pretty fundamental aspect of the universe as we know it). It's downright astounding that she isn't a household name (and remember, we do know the names of many male theoreticians even if an understanding of their work is uncommon). I experienced one of those - perhaps transcendental - flashes of insight where I suddenly 'saw' how the theorem was true. I think those moments actual explain why many people choose to be scientists; there's nothing like it. I don't know that I can explain it fully and I'm not sure I can explain the beauty of Noether's Theorem without spontaneously trying to teach anyone willing to listen a term of classical mechanics. I definitely do not know how to convey its meaning in a portrait... though as you can see from my portrait of Lady Ada Lovelace that I am willing to carve actual equations (in reverse, no less!) to show what a person accomplished, but the fact is that even if you do not understand the computation of Bernouilli numbers the meaning of the print is conveyed; Ada showed how the calculations could be done mechanically, on Charles Babbage's wondrous computing engine. To really let anyone understand Noether's tremendous accomplishment, I would need to make Noether's Theorem transparent... though I'm sure most artists would not feel constrained by this requirement. As I've yet to figure out how to do so, I wanted to select someone's whose achievements were more accessible and obvious to understand.

Mary Anning (21 May 1799 – 9 March 1847) was the wrong class, the wrong sex and even the wrong religious denomination to gain the education, opportunity to work and communicate her results or to garner any respect as a pioneering paleontologist. Further, during her lifetime most people in Britain and elsewhere thought the Earth was a mere few thousand years old, based on a very literal interpretation of the Bible and found the idea of extinction did not fit in with the story of creation. Yet, her fossil discoveries, meticulous collection, documentation and independent work to fully understand the anatomy of the amazing Jurassic creatures she encountered in famed Blue Lias cliffs of Lyme Regis, Dorset, England, were so undeniable that she gained the recognition, admiration and respect of the paleontologists of the day. Others working in the field at the time were almost uniformly male, upper class and Anglican - most of them in fact were clergymen; while Dissenters like Anning were banned from university, the army and several professions. It's hard to fathom today the way that the contributions of working class people could be blatantly ignored or appropriated by their so-called 'betters'. During her lifetime she was barred from even attending meetings of the Geological Society of London (let alone applying for membership) because of her sex and many wealthy "paleontologists" published claims of discovery based on having purchased fossils from Anning and other working class fossil hunters (neglecting to mention Anning's name or contribution). Nonetheless, many contemporary geologists and paleontologists became so impressed with her knowledge that they would visit her to consult on the anatomy and classification of fossil finds as well as collaborate on what we would today call field studies, including her friend Henry De la Beche, William Buckland (who acknowledged her role in recognizing the significance of coprolite to the Geological Society), Richard Owen,  Thomas Hawkins (whom she chastised for 'enhancing' his fossil skeletons by mixing finds before this unscientific habit resulted in scandal), Roderick Murchison, the famed Swiss palaeontologist Louis Agassiz (who thanked her in his book Studies of Fossil Fish, and named two species in her honour). She corresponded with the foremost geologists of her day, Charles Lyell, and Adam Sedgwick who taught Darwin. 

Her family was poor and her parents struggled with adequately support their 10 children (only two of whom survived infancy and childhood). Mary's formal education only included reading and writing which she was taught by the local Congregationalist school. Her father was a cabinet maker (who tragically died young) who like other locals supplemented his income by selling 'curios' to tourists, which were mainly ammonite fossils called 'snake stones' at the time. I've shown two of these spiral, fossil shells of these extinct cephalopods in my print. Mary made her living by selling fossils from a small roadside stand and ultimately a small store she set up in her home. She made her first significant find, the first ichthyosaur skeleton to be correctly identified, with her brother Joseph when she was only 12 years old. I've shown Mary Anning standing in front of the Blue Lias cliffs in front of an ichthyosaur skeleton, like the one she painstakingly excavated. She discovered and excavated the the first two plesiosaur skeletons ever found and the first pterosaur skeleton located outside Germany as well as many important fossil fish. I've also taken the artistic license to show a plesiosaurus fossil above her, based on one she herself found. The cliffs I've illustrated are so packed with fossils, not to underplay the work of finding these fossils, but to try and illustrate the some of the huge scope of her finds. She also worked on invertebrates. Her research showed that belemnite fossils contained fossilised ink sacs like those of modern cephalopods. She was also the first to recognize that coprolites, known as bezoar stones at the time, were fossilised... well, animal droppings (feces). While this sounds distinctly unglamorous, the study of coprolites pioneered by Anning and Buckland were vital to understanding ancient ecosystems. Her friend Henry De la Beche painted the first widely circulated representation of a prehistoric (deep time) scene, based on her finds, and he sold prints to benefit her financially.

Despite her limited formal education she read the scientific literature of the day, copying technical anatomical diagrams and made her own dissections of animals to make comparative studies with her fossil finds. Despite the many barriers to her, she did succeed in having her 1839 letter to the editor Magazine of Natural History published, which corrected an erroneous claim about the classification of extinct sharks.

Gathering fossils along the cliffs, especially following storms when new fossils could be found due to the recent erosion and during the winter months was dangerous work. The tides could come in quickly and the cliffs were not always stable. She narrowly escaped being crushed in a landslide which claimed the life of her faithful dog Tray.

Though she resented the way her own work was ignored or appropriated by others, she did become increasingly well-known, as her life advanced and posthumously. She struggled financially for most of her life, and she did receive help from colleagues who held her in high esteem. Nine years before her death she was given an annuity, raised by members of the British Association for the Advancement of Science and the Geological Society of London.  After she was diagnosed with cancer, in 1846 the Geological Society raised money from its members to help with her expenses and the council of the newly created Dorset County Museum made her an honorary member. When she died, Henry De la Beche wrote and read her eulogy at a Geological Society meeting beginning,

"I cannot close this notice of our losses by death without advertising to that of one, who though not placed among even the easier classes of society, but one who had to earn her daily bread by her labour, yet contributed by her talents and untiring researches in no small degree to our knowledge of the great Enalio-Saurians, and other forms of organic life entombed in the vicinity of Lyme Regis ..."

Members of the Society commissioned stained glass windows in her parish church in her memory. Charles Dickens wrote of her in 1865 that "[t]he carpenter's daughter has won a name for herself, and has deserved to win it." 


She has since been recognized for her scientific contributions by the Royal Society, paleontologists, geologists, historians of science and even in some popular culture. It is often repeated that she is the subject of the well-known tongue-twister

She sells seashells on the seashore
The shells she sells are seashells, I'm sure
So if she sells seashells on the seashore
Then I'm sure she sells seashore shells.

but there is no solid evidence to support this; in fact, this is likely metafolklore, a folktale about a tongue twister. She appears in in The French Lieutenant's Woman (1969) by John Fowles, and she is the subject of the historical novels Remarkable Creatures (2009) by Tracy Chevalier and Curiosity (2010) by Joan Thomas.

This post contains much information from the remarkably thorough wikipedia entry on Mary Anning, the Natural History Museaum biography of Mary Anning (which calls her "the greatest fossil hunter ever known") and though it was a novel and takes some license with personal relationships for the sake of an engaging story, the very enjoyable 'Curiosity' (2010) by Joan Thomas (which I reviewed here). Edited August 26, 2023 to correct myth about tongue twister.