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# On mathematics and gender politics - Emmy Noether: Pioneer of abstract algebra and modern theoretical physics

Anyone who takes a closer look at the history of mathematics quickly realizes that only male geniuses seem to ever have worked in this field. Whether the “Princeps mathematicorum” Carl Friedrich Gauss or the Swiss Leonhard Euler, whether Isaac Newton, Gottfried Wilhelm Leibniz, or David Hilbert, whether Pythagoras, Archimedes, Euclid, Fibonacci, or al-Chwarizmi, in the […]

Anyone who takes a closer look at the history of mathematics quickly realizes that only male geniuses seem to ever have worked in this field. Whether the “Princeps mathematicorum” Carl Friedrich Gauss or the Swiss Leonhard Euler, whether Isaac Newton, Gottfried Wilhelm Leibniz, or David Hilbert, whether Pythagoras, Archimedes, Euclid, Fibonacci, or al-Chwarizmi, in the group of great historical mathematical figures we find not a single woman. Until the twentieth century, the very list of known mathematicians is quite short. The sole known female mathematician in ancient times was Hypatia of Alexandria, of which, however, no works has survived the times and who has been remembered mainly for the spectacular circumstances of her assassination. The early Enlightenment period recognizes Émilie du Châtelet, who was a mathematical expert on Newton’s physics, but achieved much more fame as a partner of Voltaire. In the nineteenth century, Sophie Germain earned honors for proofing Fermat’s conjecture (now theorem) for a certain group of primes (the so-called ” Sophie-Germain prime numbers “), and in 1889 Sofya Kovalevskaya became the world’s first female professor of mathematics. Today, however, the latter is still more known for the stubbornly persisting – and likely false – rumor that the reason there is no Nobel Prize for mathematics is because Kovalevskaya had had a liaison with Alfred Nobel and then left him for another man, Gösta Mittag-Leffler. It is only in the second half of the twentieth century that the list of mathematicians became longer, but there is still no particular name sticking out, with one single exception which, however, outside of mathematics (and theoretical physics), hardly anyone knows: Emmy Noether. And that it took until the year 2014, until a woman was awarded the highest award for mathematics, the “Fields Medal” (it is the Iranian Maryam Mirzakhani), is equally worth a deeper reflection.

The search for reasons for this almost complete absence of women on the list of great mathematicians is like poking into a wasp nest. And one can all too quickly get into big trouble, as Larry Summer, after all former US Treasury Secretary in the second cabinet of President Bill Clinton, knows all too well. After his foolish testimony in January 2005 of females missing gene for scientific talent and on the (when it some to math and science) “innate differences between men and women”, he was forced to resign from his position as president of the prestigious Harvard University. In fact, there is no serious academic reference supporting a statement that females have a lower average mathematical intelligence than males, although there are both cognitive tasks where men perform better on average, as well as those where women do. There is evidence that the variance (variability) of the intelligence distribution in men is broader than in women, in other words, there are more very stupid and very intelligent men than women. However, this statement is not uncontroversial, as differences in variance between men and women do not appear in all countries and appear to depend on the degree of social equality between women and men. However, a simple look at the history of academic institutions gives a much clearer indication for the reasons of the striking under-representation of women in mathematics: Far into the 20th century women were simply forbidden to teach mathematics at universities. To come to mathematical fame and honor under these circumstances was extremely difficult, if not impossible.

Nevertheless one single female figure did stand out in the male-dominated guild of mathematicians in the early twentieth century, Emmy Noether. But she, too, was suffering from the limitations of her time. For one can easily assume that if Noether had been a man he would be considered one of the greatest mathematicians (and theoretical physicists) of the 20th century. Her name should thus be on a level with those of David Hilbert, Albert Einstein, Erwin Schrödinger, or Werner Heisenberg. But Emmy Noether did not ever receive even the privilege of a full tenure professorship.

Emmy (Amalie) Noether was born on March, 23 1882 in Erlangen, Germany, as the daughter of the mathematician Max Noether. As of 1909, the great David Hilbert regularly invited her to Göttingen, the leading mathematical center in the world at that time, and in 1915 encouraged her to apply for a habilitation (postdoctoral qualification for teaching at a university). This was followed by a controversial discussion in the faculty, in which many argued in principle against a habilitation of women. Hilbert’s authority and his statement that “a faculty is not a bathing establishment” brought the desired success among the Göttingen mathematicians. However, the corresponding request for a special permit (the habilitation of women at Prussian universities was prohibited until 1919) was rejected by the respective minister on the grounds that “the admission of women as a private lecturer continues to pose considerable reservations in academic circles.” Emmy Noether had no choice but to announce her lectures under the name of Hilbert, as his assistant – without being paid. Only when in the early Weimar Republic the byelaws for habilitations changed, so that also women were allowed to acquire the right to teach mathematics, Emmy Noether in 1919 became the first woman in Germany to receive such a habilitation – with still very low pay, though (the impetus for this did, however, not come from the Göttingen mathematicians, but from Albert Einstein, who had learned to appreciate her through her work on questions on the theory of relativity). She was, however, never to get a permanent academic position, and even in 1930, she now being a well-known and internationally recognized mathematics scholar, her promotion to the *Göttingen Gesellschaft der Wissenschaften* (academy of sciences of Göttingen) was refused.

Emmy Noether is regarded as the founder of modern abstract algebra (“mother of modern algebra”), one of the most important innovations of 20th century’s mathematics. In the mid-1920s, she accumulated a small following of highly talented students from around the world, the so-called “Noether boys” (even if two women were among her PhD students). Many of her students, for their part, became important mathematicians (including Grete Hermann, whose role in the mathematical formulation of quantum mechanics has just recently been recognized by historians) and played a significant role in the implementation of abstract algebraic methods in different fields of mathematics. Numerous mathematical structures and theorems are today named after Emmy Noether (Noetherian rings, Noetherian spacer, Noetherian scheme, Noetherian modules, and others).

Her most famous contribution, however, Emmy Noether made in theoretical physics in 1918, with her research paper *Invariante Variationsprobleme* (invariant variation problems). The so-called “Noether Theorem” presented therein developed century into one of the most important foundations of modern physics in the second half of the 20th. The motivation and initiation of her work came from a problem Hilbert thought Einstein’s Theory of General Relativity was facing: The principle of (local) energy conservation seemed to be infringed in it, he concluded. Noether solved the apparent incompatibility of Einstein’s theory with this basic principle of physics in an impressively elegant way and was thereby able to develop a much more general relationship. Einstein was so impressed by her work that he wrote: “*Yesterday I received from Miss Noether a very interesting paper on invariants. I am impressed that the things can be understood in such a general way. The old grey guards at Göttingen should have taken some lessons from Miss Noether!*”

The Noether Theorem states that each continuous (mathematically accurate, differentiable) symmetry of a physical system corresponds to a conserved quantity, and vice versa. What does this mean? By “symmetry” is meant a transformation which does not alter the behavior of a physical system. For example, it makes no difference whether an experiment is carried out today, tomorrow or in a year (or billions of years). Physicists also speak of the “homogeneity in time”. Analogously, there is the “homogeneity in space”: Where I perform an experiment does not matter, either. The result should be the same (in the case of identical boundary conditions, i.e. no differently acting external forces). This may be obvious, but the consequences of these invariants are anything but trivial. As from these arise directly conserved quantities in physics. The homogeneity in time corresponds to the important physical principle of energy conservation: in every physical system, the total energy remains unchanged. Analogously, homogeneity in space corresponds to the conservation of momentum. In fact, it is generally true that whenever a symmetry can be found in a mathematical law of nature, there must be a corresponding conserved quantity. An astonishing relationship, which required the ingenious insight of Emmy Noether (the most prominent mathematician and the most prominent physicist of the 20th century had both failed to solve that problem). Particularly in modern quantum field theories, where much more abstract symmetries than spatial or temporal homogeneity play very important roles (so called “gauge symmetries” which e.g. relate to the conservation of electrical charge in electrodynamics), the Noether theorem provides important indications on the properties of the fundamental physical structures in our world, as well as useful calculation methods and indications for possible experimental validations of physical theories.

Her mathematical insights make Emmy Noether one of the few people in history (probably the first since Isaac Newton), whose work was so important both for physics and pure mathematics that it would have deserved both, the Nobel Prize in Physics *and *the Fields Medal of Mathematics (the Fields Medal has only existed since 1936 and is limited to mathematicians under the age of 40 (which excludes David Hilbert, Noether was 36 years old at the time she wrote her most famous paper), the Nobel Prize since 1901).

Emmy Noether is at last to be given credit for the fact that today women have much better access to mathematical education and academic positions. In the US today, for example, the “Noether Lecture”, are held annually by the *Association for Woman in Mathematics* honoring women who have made fundamental and sustainable contributions to mathematics. The list of reputable female mathematicians has displayed rapid growth since the second half of the twentieth century, unmasking the statements of Larry Summers as an unpleasant mixture of arrogance and ignorance. One does not have to be a feminist to presume that our descendants in 100 years may recognize the second half of the twentieth century as the time when humanity has finally unlocked a long-unused part of its mathematical and intellectual capital. Emmy Noether could thus ultimately also prove to be a pioneer in this respect.

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