Lady of the Rings

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The aftershocks of the Lawrence Summers brouhaha ripple on. Summers, you may recall (well, it was several news cycles ago) scandalized the academic establishment by suggesting that the scarcity of female scientists and mathematicians might have its origins in the different biology of men and women. Our own Stanley Kurtz has a nice follow-up piece on the Summers flap in the current City Journal.

Reading Stanley's piece, I got to thinking of Emmy Noether, who died just 70 years ago last week. I am going to leave you to deduce what, if anything, you can from Emmy Noether's story. It's a story worth telling, in any case, so here it is.

Emmy Noether was the greatest female mathematician of the 20th century, and quite possibly of all time. She was the daughter of a mathematician, too, so there is plenty of scope for nature-nurture speculation here. Emmy's father, Max Noether, was a professor of mathematics in the south-German town of Erlangen, just north of Nuremberg. Emmy was born there in 1882. Her career has to be seen in the context of the German Empire in which she grew up, the empire of Bismarck (Prime Minister and Chancellor to 1890) and Wilhelm II (German Emperor — "Kaiser" — from 1888 to 1918). Of the place of women in that society, the historian Gordon Craig has this to say:

"The woman's predicament," Ernest Rhys has written, "is the test of the moral and human worth of any given state of society." Judged by this standard, Germany's condition in the nineteenth century was truly deplorable. Its ruling class was as intent upon keeping the female population in a state of dependence as it was upon combating socialism. All the expedients of the law, every form of financial and moral pressure, were employed to maintain male dominance in state, society, and the home. Women were denied basic civil rights … and were excluded from both any share in the governance of the country and employment in any of its administrative agencies. Moreover, unless they defied convention and won an independent position in the arts, their share in their country's cultural life was minimal, in comparison with that of their contemporaries in France.

— Germany 1866-1945, by Gordon Craig

Wilhelmine Germany was an exceptionally misogynist society, even by late 19th-century standards. The German expression Kinder, Kirche, Küche (children, church, kitchen), supposedly identifying a woman's proper place in society, is known even to people who don't speak German. It is popularly thought to have originated with the Nazi Party. In fact it goes back much further than that, and was used approvingly of the attitude displayed by Wilhelm II's lumpish consort, the Empress Augusta Victoria, except that on her lips it was supposed to have been uttered as Kaiser, Kinder, Kirche, Küche. Similarly, every literate person is familiar with the great French and Russian novels of anguished, transgressing 19th-century womanhood, Flaubert's Madame Bovary (1856) and Tolstoy's Anna Karenina (1877). Very few are acquainted with the German equivalent, Theodore Fontane's Effi Briest (1895). (There is a Penguin Classics translation by Douglas Parmée.)

So when Emmy Noether decided, around age 18, to take up pure mathematics as a career, she had set herself to climb a steep mountain. She did what she could, auditing her father's classes, even managing to get a doctorate in 1907, only the second doctorate in mathematics given to a woman by a German university. She still wasn't allowed to teach at university level, though. For eight years she worked at Erlangen as an unpaid supervisor of doctoral students, and occasional lecturer. There was, however, nothing to stop her publishing, and she steadily became known for brilliant work in mathematics.

These were the years following Albert Einstein's unveiling of his Special Theory of Relativity in 1905. Einstein was absorbed in trying to work out the General Theory, in which he aimed to bring gravitation under the scope of his arguments. There were, though, some difficult problems to be overcome, mainly involving the conservation of energy. In June and July of 1915 Einstein presented his General Theory, unresolved problems and all, in some lectures at Göttingen University. Einstein noted of this event: "To my great joy I succeeded in convincing Hilbert and Klein."

This was an occasion for joy indeed. David Hilbert and Felix Klein were, even at this fairly late point in their respective careers (Hilbert was 53, Klein 66), two giants of mathematics, while Einstein — he was 36 — was still not far beyond the wunderkind stage. Hilbert and Klein had, of course, followed the development of Einstein's ideas with interest before he came to lecture in 1915. Now "convinced" (convinced, presumably, that Einstein was on the right lines), they gave their attention to the outstanding problems in the General Theory. They knew of some work Emmy Noether had done in the relevant areas, and invited her to Göttingen.

Noether duly came to Göttingen, and within a matter of months produced a brilliant paper resolving one of the knottier issues in General Relativity. Einstein himself praised the paper.

Emmy Noether had arrived. Her professional troubles were not yet over, though. World War One was into its second year — Emmy's younger brother Fritz (another mathematician) was in the army. Göttingen, though liberal by the standards of Wilhelmine universities, still balked at putting a woman on the faculty. David Hilbert, a broad-minded man who judged mathematicians by nothing but their talent, fought valiantly for Noether, but without success.

Some of the arguments on both sides have become legendary among mathematicians. The faculty: "What will our soldiers think when they return to the University and find that they are expected to learn at the feet of a woman?" Hilbert: "I do not see that the sex of a candidate is an argument against her admission as a Privatdozent [that is, a lecturer supported from fees paid to him by students]. After all, we are a university, not a bathing establishment." (Aber meine Herren, wir sind doch in einer Universität und nicht in einer Badeanstalt. You can't help but like Hilbert.)

Hilbert's solution to the Noether problem was characteristic: He announced lecture courses in his own name, then allowed Noether to give them.

In the general liberalization of German society following defeat in WWI, however, it at last became possible for a woman to get a university teaching position, if only of the Privatdozent variety. Noether started teaching in 1919. In 1922 she actually got a salaried position at Göttingen, though she had no tenure, and the meager salary was soon obliterated by hyper-inflation.

It was during these early postwar years that Noether did the work she is mainly remembered for. It's not easy to explain that work to non-mathematicians. Suffice to say the following: Modern mathematicians don't just restrict their attention to numbers and geometric figures. They study many weird and wonderful mathematical objects, all discovered (or invented — there is a philosophical point here you can discuss among yourselves) in the 19th and 20th centuries. These objects are purely abstract creations. They have names like "group," "field," "module," "lattice," "manifold," "scheme." Well, one key object has the name "ring." This object, this aery abstraction — it has nothing, or very nearly nothing, to do with circles or annuli — turns out to have a complicated and fascinating inner structure. It was Emmy Noether who first defined this object in all its generality, and discovered the rules governing its inner structure. She was the Lady of the Rings.

By the early 1930s, Emmy Noether was at the center of a vigorous group of researchers at Göttingen. She still held a low-level position, ill-paid and without tenure, but her power as a mathematician was not in doubt. Her colleagues regarded her with awe and affection, though since they were all male, and Kaiser Wilhelm's Germany was only a dozen or so years in the past, the affection expressed itself in ways that would not be accepted nowadays. Noether did not at all conform to the standards of femininity current in that time and place — nor, it has to be said in fairness to her colleagues, any other time and place. She was stocky and plain, with thick glasses and a deep, harsh voice. She wore shapeless clothes and cropped her hair. She had a rough temper, and her lecturing style was generally described as impenetrable.

Hence all the disparaging quips, not meant unkindly at the time, that have become part of mathematical folklore. Best known is the reply by her colleague Edmund Landau, when asked if he did not agree that Noether was an instance of a great woman mathematician: "Emmy is certainly a great mathematician; but that she is a woman, I cannot swear." Norbert Wiener described her somewhat more generously as "an energetic and very nearsighted washerwoman whose many students flocked around her like a clutch of ducklings around a kind, motherly hen." Hermann Weyl expressed the common opinion most gently: "The graces did not preside at her cradle." Weyl also tried to take the edge off the appellation Der Noether (der being the masculine form of the definite article in German): "If we at Göttingen … often referred to her as Der Noether, it was … done with a respectful recognition of her power as a creative thinker who seemed to have broken through the barrier of sex … She was a great mathematician, the greatest."

Ill-paid and un-tenured as her position at Göttingen was, Noether lost it when the Nazis came to power in the spring of 1933. Having been once barred from university teaching for being a woman, she was now more decisively barred for being a Jew. The appeals of her Gentile colleagues and ex-colleagues — led, of course, by Hilbert — counted for nothing.

There were two common avenues of escape for Jewish or anti-Nazi intellectual talents: to the U.S.S.R., or to the U.S.A. Emmy's brother Fritz chose the former, taking a job at an institute in Siberia. Emmy went the other way, to a position at Bryn Mawr College in Pennsylvania. Her English was passable, she was only fifty-one, and the college was glad to acquire such a major mathematical talent. Alas, after only two years Emmy Noether died of an embolism following surgery for removal of a uterine tumor. Albert Einstein wrote her obituary for the New York Times, though for reasons I don't know it was printed as a Letter to the Editor. From which the following:

Beneath the effort directed towards the accumulation of worldly goods lies all too frequently the illusion that this is the most substantial and desirable end to be achieved; but there is, fortunately, a minority composed of those who recognize early in their lives that the most beautiful and satisfying experiences open to humankind are not derived from the outside, but are bound up with the development of the individual's own feeling. thinking and acting. The genuine artists, investigators and thinkers have always been persons of this kind. However inconspicuously the life of these individuals runs its course, none the less the fruits of their endeavors are the most valuable contributions which one generation can make to its successors.

— "The Late Emmy Noether," New York Times, 5/5/35