Considering the overwhelming evidence of the shape of space and the deep revelations of the exploration reachable by imagination and logic, the history of Mathematics has always revolved around masculinity. For centuries, women in the field of mathematics have been underappreciated, undervalued, judged and ungratefully overlooked because of gender differences. Indeed in many cultures across the globe, mathematics is seen as a male-dominated field which most women have tried to break into in vain. Since the dawn of humanity, many names such as Riemann, Gauss, Tao, Zhang and many more have been associated with some of the most beautiful mathematical discoveries, but sadly, all are men. Consequently, this misleading fact has significantly evolved in the public consciousness and reinforced with some scholars and historians, with books such as Men of Mathematics acting as an example of this misguided course. Although there is no slight doubt that men often excel in this field, the history and the achievement of women in mathematics is significant, as is made clear through the struggles of Sophie Germain, Emmy Noether, and Sofia Kovalevskaya.
At the time when women had limited access to mathematics as a field of science or philosophy and were considered intellectually inferior to men, Emmy Noether won the admiration of her masculine colleagues. As a matter of fact, Noether who was born in 1882 in Erlangen, Germany to a mathematician father always had an interest in mathematics but was only allowed to enroll for a certificate in teaching foreign languages because women were not allowed to pursue a career in the field of mathematics. According to Cavna (n.p), it was only until Noether completed her certificate course was she able to pursue her passion in mathematics, but not officially. Fortunately, at this time, the conditions for women in mathematics as a field of science in Germany were improving and women were now allowed to acquire basic knowledge of mathematical operations and shapes. When she was 18, the young German scholar was already a certified English and French teacher who taught in schools for girls in 1900, however, she decided to quit and chose to pursue mathematics at the University of Erlangen (Morrow, Charlene, and Teri Perl). Though she was not technically allowed to enroll in official classes because at the time women were only allowed to audit classes with the permission of a certified instructor. Eventually, after auditing for a while at the University of Gottingen between the year 1903 and 1904, she returned to Erlangen where women were now allowed to become full-time students. Luckily, she sat for a doctorate exam after attending classes for a period of two years at the University of Erlangen, where she received her Ph.D. degree in 1907 (Huff). Sadly, the university could not hire her as a junior professor even though she had an outstanding dissertation on algebraic invariants because of her gender, leaving her with an option of assisting her father with his work at the same university. Even so, she began conducting her own separate researches and eventually, she started publishing papers based on her own work while still helping her father with some of his classes.
Soon after, particularly after the end of World War I, Emmy Noether was invited to her former University, Gottingen, by David Hilbert and Felix Klein to work on one of Einstein's theories. According to Morrow, Charlene, and Teri Perl (n.p), Hilbert and Klein invited Noether because of her knowledge of algebraic invariants which they used to explore the mathematics behind Einstein's theory of general relativity. Regardless of her gender, they felt that her expertise could help them with their progress, however, there were vehement objections from some faculty members against a woman teaching in the department. Consequently, she was only allowed to teach some classes under Hilbert's name.
Despite the frustrations, Emmy Noether faced during her studies at Gottingen, and later as a researcher, she made significant discoveries that completely changed the course of mathematical discoveries. For instance, according to Cavna (n.p), in 1918, Noether discovered that if a quantity that characterizes a physical system (Lagrangian) does not change when the coordinate system changes, then some quantity is conserved. Ideally, this means that when a Lagrangian does not depend on the changes in time, then the conserved quantity is the energy. In fact, this relation between symmetries of a physical system and its conservation laws was later named after Emmy as Noether's theorem earning her admission as an academic lecturer in the process. The theorem has currently proven to be a key contributor to theoretical physics. Similarly, Noether was also involved with the investigations under general theories of ideals which helped draw together different important mathematical developments. But perhaps, her true contribution as a shrewd mathematician was the appearance of 1920, "Concerning Moduli in Noncommutative Fields such as Differential and Different Terms" which she wrote in collaboration with Werner Schmeidler, one of her colleagues at Gottingen (Kosmann-Schwarzbach 57). By 1927 however, all her researchers were based on noncommutative algebras, their linear transformations, and their applications. In her collaboration with Helmut Hasse and Richard Brauer, she was able to build up a theory of noncommutative algebras concentrated on their application to commutative fields by means of vector cross product. Later, when Nazis took control of Germany in 1933, Noether was one of the few professors who got dismissed prompting her to take refuge in the United States. Bryn Mawr College had offered her a teaching position, where she went as a visiting professor of mathematics. Even more, she lectured and conducted research at the Institute of Advanced Study in Princeton, New Jersey until her death in 1935 as a highly respected mathematician having made a tremendous contribution in the field of mathematics.
Even though she had many personal tragedies during her short life between 1850 and 1891, Sofia Kovalevskaya had a remarkable career in mathematics. She was born in Moscow, Russia to Vasily Korvin-Krukovsky, a general in the Russian army and a member of the nobility. Having loved mathematics at a young age, Sofia started to receive education from tutors and governesses initially at their family's estate in Palabino and later at St Petersburg. According to Kovalevskaya, Vorontsova, and Vasilyevna Korvin-Krukovskaya (n.p), Sofia had a unique childhood having been raised by a strict governess which often made her nervous and withdrawn for most of her life. However, this did not stop her from papering her room with her father's old calculus notes on differential and integral analysis which provided the young Russian with an introduction to calculus. In fact, she managed to study these notes and also discussed abstract and mathematics concepts with her uncle Pyotr Vasilievich Krokovsky, who had an interest in mathematics as well. Her father, nonetheless, did not appreciate her excellence in mathematics because like everyone else in those days, he did not believe that women could be highly educated, prompting her to study in secrecy away from both her father and governess. Polyakhova (n.p) states that, by the age of fourteen, Sofia had already taught herself trigonometry to be able to understand a book authored by a neighbor to her family. As a matter of fact, when the neighbor realized how determined she was, he convinced her father to let her take mathematics classes.
Upon finishing her secondary schooling, Sofia's ambition was to study mathematics at university, sadly, the nearest university that accepted women at the time was in Switzerland. Having known that it was impossible for unmarried women particularly her age to travel alone without permission from their father, she entered a marriage of convenience with Vladimir Kovalevskaya in September 1868. And so, one year later, Sofia and her newly found husband left Russia and traveled to Heidelberg, Germany. In Heidelberg, Sofia hoped to major in mathematics and natural sciences, but unfortunately, just like in Russia, she was informed that women were not allowed to take courses on arrival, although she was permitted to attend lectures and seminars in both mathematics and physics. So, two years later, the young mathematician moved to Berlin where she began to study under one of the greatest calculus experts, Karl Weierstrass. She was still not allowed to register officially at the University of Berlin. Morrow, Charlene, and Teri Perl (n.p) indicate that after three years under Weierstrass, Sofia had already completed three papers, all of which were considered worthy of a doctorate. Luckily by the end of 1874, on Weierstrass' initiative, she was awarded a doctorate by the University of Gottingen for her work on partial differential equations.
Work was still hard to find even after receiving her doctoral degree, prompting her to return back to her family in Moscow. According to Polyakhova (n.p), her marriage of convenience turned into love and as a result, in 1878 she and Vladimir had a daughter. Consequently, she abandoned everything she learned in Germany and started developing her literary skills instead while writing fictional stories, theater reviews and as well as scientific articles. But even so, due to frustrations, Mrs. Kovalevskaya moved back to Berlin, Germany in 1880 in pursuit of work in the mathematics field, leaving her husband back in Moscow. Fortunately, by 1883, she was invited to lecture at the University of Stockholm having spent two years moving between Berlin and Paris working on a project. Her career immediately changed for the better when she arrived at Stockholm since, after only six months at the University, she was not only offered a five-year contract as a professor of mathematics but also appointed as an editor of new journal Acta Mathematica (Kovalevskaya, Vorontsova, and Vasilyevna Korvin-Krukovskaya). Even more, she was honored by being made Chair of Mechanics and gained tenure at the same university. Her greatest triumph, however, came in 1888 when she won the prestigious Prix Bordin award organized by the French Academy of Sciences when she entered her paper, "On the Rotation of a Solid Body about a Fixed Point". So impressed was the Academy by her profound theory and her highly regarded paper that they adjusted the prize money from 3000 to 5000 francs. As a result of her success in developing a theory for an unsymmetrical body where the center of mass is not on an axis in the body, she was elected as an associate of the Imperial Academy of Sciences a position which had never been given to a woman before. Even though she died shortly after her election on February 10, 1891, Sofia had a remarkable career in her short life. She is known for the theory of differential equations, Cauchy-Kovalevsky theorem for analytic partial differential equations and the rotation of an unsymmetrical solid body around a fixed point.
Marie-Sophie Germain, born in Paris on April 1, 1776, to a French merchant Ambroise-Francois and his wife Marie Germain, was a French mathematician who contributed significantly to the study of elasticity, the theory of numbers and acoustics (Bucciarelli, Louis, and Dworsky). She was born in an era of revolution. In her year of birth, for instance, the American Revolution began, followed by a revolution in her own backyard which began on her thirteenth birthday. Even though revolution was full of perseverance and hard work, Sophie, in many ways embodied the spirit of revolution into which she was born and raised.
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