Emmy Noether – Her rocky road to the top of world mathematics

(Preface and afterword to Lars Jaeger’s new book about Emmy Noether; https://www.lovelybooks.de/autor/Lars-Jaeger/Emmy-Noether-Ihr-steiniger-Weg-an-die-Weltspitze-der-Mathematik-4828335597-w/)

Why is there no Nobel Prize for mathematics? There are several answers to this question. A widespread but unconfirmed anecdote tells that the only reason there are no mathematicians on stage when the prizes are awarded in Stockholm is that once Alfred Nobel’s lady of the heart had given preference to a Swedish mathematician. However, it is more likely that Nobel simply underestimated the importance of mathematics. According to his will, every year those scientists are honored (for the first time in 1901) who have brought a particularly great benefit to mankind. Mathematics probably seemed to Nobel to be of little use in direct applications.

Twenty or thirty years later, he would probably have thought very differently. For mathematics had established itself as the foundation of all sciences. It paved the way for a completely new physics, provided the statistics for the new gene theory in biology, and dominated the work processes in chemical laboratories. But before it could unleash this power, it had to overcome the deepest crisis in living memory. The scholars of the 19th century encountered internal contradictions that called into question the entire basic framework of mathematics, which was believed to be absolutely certain. Mathematics shared this fate with physics, chemistry and biology, because in the decades around 1900, in a process unique in world history, all the natural sciences without exception lost their footing and had to reinvent themselves – each for itself – from scratch.

Emmy Noether is one of the central figures in this complete reorientation of mathematics. Her achievements stand at least on an equal footing with those of the most famous mathematicians of the 20th century: David Hilbert and John von Neumann. Since she decisively advanced the introduction of higher abstraction, Emmy Noether is even one of the most influential persons of all time in mathematics. Almost incidentally, she also solved a central problem of modern physics and thus cleared the way for today’s understanding of quantum field theory: the “Noether theorem” is today one of the most important, if not the leading principle of theoretical physics.

The fact that her name is still practically unknown today, despite her outstanding importance in mathematics, is mainly due to one circumstance: Emmy Noether was a woman. With great difficulty she had to fight for a place at university, first as a student, then as a research associate and associate professor in what was then the world center of mathematics: Göttingen. Because it was inconceivable to her male colleagues that a woman could penetrate mathematics to its depths, a curious discrepancy arose between the admiration for Emmy Noether’s achievements and the inability to grant a woman the same opportunities as anyone else. For Emmy Noether’s achievements were undeniable and were not doubted even by those who would have preferred to see the university enterprise remain purely in male hands. From the end of the 1920s, she was world-famous among experts and received the highest honors. But on the university career ladder, Emmy Noether had hit the famous glass ceiling early on: Men with lesser mathematical abilities were rewarded with attractive positions and earned enough money to support a family. Emmy Noether was denied this kind of recognition in Germany until the end. It was only in the last two years of her short life, while emigrating to the United States, that the now world-famous mathematician was awarded a significant salary.

After her early death in 1935, Emmy Noether’s mathematics lived on, her findings revolutionized mathematics and are now among the foundations of all scientific fields. But her person fell into oblivion. Only a few biographers took up her story, including Auguste Dick, Cordula Tollmien, Mechthild Koreuber and Peter Roquette.

It is only in recent years that Emmy Noether’s life, marked by hardships and setbacks, has been remembered in wider circles. A number of fellowships and other funding measures have been launched in her name to support women’s scientific careers. Emmy Noether would certainly have liked this.

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Her profound mathematical insights, central today, and her no less profound physical Noether theorem made Emmy Noether one of the historically rare people whose work was so significant for both (theoretical) physics and (pure) mathematics that she must be considered to be at the forefront of both disciplines. Emmy Noether would today without question deserve the two highest honors of these disciplines: the Nobel Prize in Physics and the Fields Medal in mathematics.

Up to the present there is no one who has received both prizes, only one physicist has been awarded the Fields Medal, but not the Nobel Prize: Edward Witten in 1990. The Fields Medal has been awarded since 1936 and has an age limit of forty years. The latter would have been just enough for Emmy Noether, who was thirty-nine years old when her most famous (mathematical) paper was published in 1921. The Nobel Prize in Physics was awarded in 1901 and Emmy Noether more than deserved it, but the value of her work in 1918 was not recognized as fundamentally important until well after her death.

Even beyond her mathematical brilliance, Emmy Noether was a great idol in her personality, her attitude and her humanly fascinating, warm-hearted behavior toward others. Above all, she did not have the ambition of always wanting to be in the foreground, as is often the case with men, even in mathematics.

Even today, Emmy Noether can be considered a great role model for many girls and women: Because of her strength to do the opposite of what society expected of her as a woman at that time; because of her inner strength to follow her goal against many a male resistance, and this at a time when women were still considered intellectually inferior to men; also because of her energy and assertiveness; because of her self-assurance and belief in her destiny; because of her determination to courageously go her own way; last but not least, because of her warm-heartedness towards all people, even potential enemies.

It then took another twenty-two years after Emmy Noether’s death, as we have seen, for the first German woman to become a full professor of mathematics at a university: Ruth Moufang was only the third German woman after Emmy Noether to achieve a habilitation. Yet, even in the 1960s, women were not yet properly recognized as professors in general. A shameful example of this was that during the annual meeting of Nobel laureates in Lindau, Ruth Moufang, as an official delegate of the rector of Frankfurt University, was not allowed to participate in the scientific program, but only in the so-called “ladies’ program.”

Today, there are numerous female professors of mathematics worldwide. Emmy Noether, however, probably still towers above the vast majority of them: in a display at the 1964 World’s Fair that depicted and described the important mathematicians of the modern world, she was the only woman. Would a woman join them in a world’s fair today? Hardly – except probably Maryam Mirzakhani, who died in 2017 at the age of forty, one of only two female winners of the Fields Medal to date (the second very recently, in 2022).