In Joyce’s Ulysses, the centenary of which was duly marked by Professor Sam Slote in the February post of this blog, the protagonist, Leopold Bloom, is plagued by a very ancient mathematical conundrum: constructing with ruler and compass a square from a given circle such that the area is preserved, which we often refer to in common parlance as “squaring the circle”. Hoping to collect the £1,000,000 prize, the unfortunate Bloom is unaware that the impossibility of the problem was proved by German mathematician Carl Louis Ferdinand von Lindemann in 1882 in a result that formed one of many stepping stones towards a mathematical “modernism” after the turn of the century. It is therefore curiously fitting that 1922 is also an important centenary year for modern mathematics: this April marks 100 years from the first time the trailblazing modern mathematician, Emmy Noether, was finally promoted to “außerordentliche Professorin” at Germany’s University of Göttingen — the beacon of cutting-edge mathematics in the 19th and 20th centuries — and began to receive a wage shortly after. This had hitherto been denied to her due to her gender and, in part, her Jewish heritage.
Noether, with a pedagogical farsightedness, taught mathematics as if it were a Humanities subject, and it is as a teacher and collaborator who was remarkably unselfish with her ideas that she is ultimately remembered.
Tom Hedley
Emmy Noether was born in 1882 in Bavaria into a middle-class Jewish family with a clear affinity for mathematics: her father, Max Noether, was well established professor of mathematics at the University of Erlangen. Her secondary school grades paint a picture of someone who was an exceptional mathematician and linguist, but also who failed most spectacularly at home economics. Noether was advised to train as an English and French teacher, more customary for educated females in the German Reich, but she opted to study mathematics at the University of Erlangen instead. As one of only two females in the entire establishment, she completed her doctorate in 1907. Unable, as a woman, at this stage to progress to any higher academic grade, she lectured unpaid in place of her father to cover his periods of illness.
During this time, her research garnered enough of a reputation that she was called upon by David Hilbert and Felix Klein, the discipline’s eminent leaders in Göttingen, to help them resolve lingering uncertainties in Einstein’s theories of relativity. Her insights here led to the discovery of a theorem that has been lauded as a ground-breaking feat of creative and abstract thought, published in 1918 and now known as “Noether’s theorem”. While Klein and Hilbert fought hard to secure for Noether a lecturing role, the faculty board protested at the thought of a woman in their ranks, provoking Hilbert’s famously ill-tempered retort at the board meeting: “Gentleman, we are in a university, not a bathhouse!” After some negotiation, Noether was permitted only to lecture unpaid under Hilbert’s name in Göttingen, and she survived frugally from a small inheritance left by a relative.
In the aftermath of the First World War, the easing of restrictions for women in the Weimar Republic allowed for Noether’s return to Göttingen in April 1922, for which she initially received a small stipend before securing a proper (but nonetheless moderate) teaching salary. Turning her mind to the abstract study of algebraic structures, now known as “modern” algebra, Noether carved out a reputation as a pioneering figure in the evolving discipline of mathematics, becoming the figurehead of an informal school of devout followers — the so-called “Noether Boys”. In terms of method, Noether’s skill lay in sidestepping the specific natures of the usual mathematical objects (numbers, spaces, functions etc.) and uncovering the abstract structural relationships behind them all. This highly productive school has since become emblematic of the modernist wave that reshaped the discipline of mathematics in early 1900s, and its creative zeal was not lost on Noether, who stridently held the view that mathematics is, above all else, an art form, not a science. If we can speak of a modern mathematics that sits under the modernist banner alongside art and culture, then Noether’s fingerprints are to be detected on an astonishing portion of it. In her own words: “My methods are really methods of working and thinking; this is why they have crept in everywhere anonymously.”
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Her unconventional teaching methods dissuaded the more traditionally minded, for Noether’s classes tended to involve collaborative attempts to fashion new proofs and fast-paced debates on mathematical ideas — a far cry from the authoritative lecturing style we associate with tertiary mathematics education to this day. In accordance with her personal philosophy, Noether, with a pedagogical farsightedness, taught mathematics as if it were a Humanities subject, and it is as a teacher and collaborator who was remarkably unselfish with her ideas that she is ultimately remembered. In this most productive and illustrious period, Noether’s career was brought to an abrupt halt. Alongside all Jewish academics in the wake of 1933 and the rise of German Nazism, she was removed from her post, which she reportedly accepted with a characteristic level-headedness and selflessness, devoting most of her time and energy to supporting other colleagues in the same predicament. Undeterred from her mathematical work, Noether led a covert study group on algebraic field theory in her living room with her students before securing passage to the USA, where she was granted a professorship in Bryn Mawr College, a women’s liberal arts university in Pennsylvania.
Finding at last financial security in the well-supported academic community in the US, Noether gave the algebraic treatment at Bryn Mawr and Princeton until her untimely death at the age of 53 in 1935 — just eighteen months after her appointment — due to unexpected complications from routine surgery on an ovarian cyst. On 1st May 1935, an obituary for Noether appeared in The New York Times, and it was written by none other than fellow intellectual refugee Albert Einstein, who was, of course, uniquely familiar with the fruits of Noether’s distinctive way of thinking. He writes:
In the judgment of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since higher education of women began. […] Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in simple, logical and unified form the largest possible circle of formal relationships. In this effort toward logical beauty spiritual formulae are discovered necessary for the deeper penetration into the laws of nature.
With Einstein’s suggestion that the modern mathematics associated with Noether must be viewed as the “poetry of logical ideas” we have perhaps come full circle: in the 1920s, mathematics began to behave like an art form, not a science, which makes the addition of modern mathematics to the modernist era all the more compelling. Curiously, it is an acclaimed scientist like Einstein who most sharply observes this nature of mathematics, with its creative credentials most clearly on display in Noether’s avantgarde school of abstract algebra.
About Tom Hedley:
Tom Hedley is a PhD candidate in the Department of Germanic Studies at Trinity College Dublin and early career researcher at the Trinity Long Room Hub. A graduate of TCD (BA German and Mathematics) and the Friedrich-Schiller-Universität Jena (MA Literature, Art & Culture), Tom’s current research explores the changing understanding of space and spatiality in modern mathematics and German modernism (ca. 1890–1930), which is funded by the Irish Research Council. As of September 2021, he is the guest Junior Fellow at the Descartes Centre at Utrecht University, Netherlands, where he will work with other researchers and MA students in the history of mathematics research area. Tom is also a producer and editor on the Hublic Sphere Podcast, a podcast created by Early Career Researchers at the Trinity Long Room Hub.