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This article comes from the official account of Wechat: back to Park (ID:fanpu2019), author: William Dunham (Professor of Mathematics, Mullenburg College, USA), translator: Feng Shu

In 1891, the great mathematician Sophia Kovalevskaya died suddenly, leaving the world with endless sighs. The math genius had a chance to study math with her cousin. Without the support of her family, she had a chance to go to college through "fake marriage", all because she was a woman.

After Sofia, it has become more and more common for women to enter the field of mathematics, but the inequality they face needs to be further eradicated. It is hoped that one day, women will no longer be marginalized in the field of mathematics.

If readers have been doing statistics, it is clear that men appear more often than women in this book. This imbalance reflects the historical advantage of men in mathematical science. But does this mean that women did not contribute to the subject in the past, do not contribute now, and will not contribute in the future?

The answer to the above question is "No", "of course not" and "Please be serious". The figure of women in the history of mathematics can be traced back to classical times, and today women are more active than ever before. If women want to survive in the field of mathematics, they have to face obstacles that male mathematicians can hardly imagine, not only because of their lack of encouragement, but also because of their strong resistance to women joining the field of mathematics.

First of all, we admit that Archimedes, Newton, Euler, Gauss and others are all men in the short list of the most influential mathematicians in history. Before 1900, the number of women in mathematics was very small. One of the frequently mentioned is Hypatia of Alexandria, who lived around 400 AD. The Marquise Chartley (Emilie du Chatelet,1706-1749) and Maria Agnesi (Maria Agnesi,1718-1799) were active in the 18th century, while Sophie Germain (Marie-Sophie Germain), Mary Somerville (Mary Somerville,1780-1872) and Ida Lovelace (Ada Lovelace,1815-1852) were active in the early 19th century. In the middle and latter part of the 19th century, Sophia Kovalevskaya (Sofia Kovalevskaia, also known as Sonia Kovalevsky in literature) was also on the list.

Among these women, Hipatia is an influential geologist, teacher and writer. The Marquis Chartley is famous for translating Newton's works to the French. Somerville is famous for translating Laplace's works to the British. In 1748, Agnesi published a mathematics textbook, which received due recognition. Lovelace worked with Charles Babbage when he built his first "analytical engine".

Germain and Kovalevskaya are the most versatile mathematicians on this list. The former studies both pure mathematics and applied mathematics. We mentioned her study of Fermat's Great Theorem in the chapter Fermat / Fermat. In 1816, Germain won a grand prize from the French Academy of Sciences for his work on mathematical analysis of elasticity. Kovalevskaya earned her doctorate and held positions at the university, making the pioneering achievements of the women of her time. In the process, she has won the respect of her male colleagues who were once skeptical of her in all respects.

So, before the 20th century, there must have been female mathematicians. What surprises us is not the small number of them, but their real existence. Because women not only have to overcome the usual obstacles faced by people who are eager for mathematics, that is, the real difficulties of advanced mathematics, but also have to overcome the obstacles brought by various cultural levels. Let's discuss the three biggest obstacles that stand in their way.

The first obstacle is the general negative view of women in this discipline, which is deeply ingrained in the minds of many men and women. Its core is to believe that women do not have the ability to do pure mathematics. This idea has been deeply embedded in the minds of many people, including very influential people. Emanuel Kant is said to have said that women grow beards "when they use their beautiful heads to think about geometry". It is discouraging that such a comment comes from such an important philosopher. Unfortunately, such a view was by no means an isolated case in the past. At that time, many high school girls who wanted to study trigonometry or calculus were persuaded by mentors, parents or friends to pursue subjects such as home economics or English, which are more suitable for women's way of thinking. Believe it or not, this situation continues.

One of the many evidences that women cannot engage in mathematical research is that very few women are engaged in this research. In other words, the lack of women in mathematics is used to prove that they are not capable of doing the subject. Of course, the reasons for these arguments are absurd. This is the same view that boils down to the lack of African-Americans in Major League Baseball before World War II because they do not have the quality to play the game. As Jackie Robinson, Henry Allen and many others have amply demonstrated, the lack of black players in Major League Baseball does not prove their lack of ability, but rather their lack of opportunity.

The specific characters mentioned above fully show that women can also study mathematics. We can prove this with female mathematicians who have been very active recently. Grace Young played a very important role in the improvement of higher integral theory at the beginning of the 20th century. Julia Robinson (Julia Robinson) was the solver of Hilbert's tenth problem, and Emmy Noether was one of the most accomplished algebraists in the 20th century. The view that women cannot study mathematics is unfounded.

However, there is a related view that women should not study mathematics. At best, it is a waste of time; at worst, it is harmful. Just as children should not go near highways, women should not go near math.

Let's take Florence Nightingale as an example, she later won a reputation in the field of medical art. When she was young, her mother was surprised that she showed great enthusiasm for math, so she asked, "what is math for married women?" As we mentioned in the chapter "Utility / practicality", there is nothing more useful in the human cause than mathematics. But Nightingale was told it was useless. Given the traditional roles imposed on 19th-century women, mathematics will be seen as useless to them anyway.

Moreover, the woman was told that studying math would undermine her social attractiveness. What's more, it is said that there is any medical evidence that women who think too much will transfer their blood from the reproductive organs to the brain, with dire consequences. What makes us curious is that men don't seem to have to worry about similar blood flow.

Such views quickly become a stumbling block to action or, more accurately, to action. Germain had to publish her math paper under a masculine pseudonym; Kovalevskaya, despite her incontrovertible ability, did not gain academic status at first. Even the great Eminot was given the cold shoulder when she sought a junior position at the University of Gottingen in Germany. Her detractors are staunchly opposed, and some worry that once a woman walks through this door, it will bring an unstoppable retrogression. To this end, David Hilbert responded with the following ingenious satire: "I don't understand why the gender of this candidate has become the basis for opposing her inauguration." after all, we are a university, not a bathing place. " In the end, Nott got the job, and the math group (the University of Gottingen) lived quite well.

The second obstacle is the lack of formal education. The subject of mathematics needs training, high intensity training. In order to get to the forefront, you have to start with the basics, which takes years of effort for a subject as old and complex as mathematics. In the past, few women have begun such a hard journey. Therefore, it is almost impossible for them to succeed in advanced mathematics.

How do men learn this subject? They usually receive tutoring or one-on-one lectures. We have seen Leibniz consult Christian Huygens, while Euler studies with John Bernoulli. This is the process of training masters to carry the torch to the future. Few women have such a chance.

Men enter the university after proper training, where their talents and abilities will be further developed. Gauss studied at Helmstadt University, Wenzel at the Paris Polytechnic Institute in France, and Russell at Cambridge University.

Germain, by contrast, is a very promising man, but he was even refused to enter the university lecture hall because of his gender. She can only listen to lectures at the door of the classroom, or borrow notes from compassionate male students to copy them, so she secretly keeps up with the progress. In Gauss's words, her success proved that she was the "most courageous" woman.

As a result, too many women have never actually come into contact with the world of advanced mathematics. It is worth mentioning that many of the female families mentioned above are relatively well-off and have the advantages of the corresponding class. Germain is free to use her father's library. Somerville eavesdropped on her brother's tutoring lessons. The daughters of these wealthy families clearly have the right to choose not to conform to more timely traditions. As Michael Deacon commented on the future of math research for poor women: "Poverty and female identity are too heavy stumbling blocks."

It would be interesting to compare this situation with the situation of female writers of roughly the same period. Reading and writing are part of a lady's training, although it is seen as a necessary social skill rather than a means to an artistic career. However, many women are still qualified for writing. If they have enough time, training and ability, they may make use of these conditions to create poetry or literature. Jane Austen is an example. Her works are carefully observed and refined by her extraordinary talents. Austin can read and write. She is an artist. Her works have made her one of the great men of English literature.

It is true that many girls have learned some elementary calculations. But unlike literature training, math learning ends here. The progress of advanced mathematics requires an understanding of geometry, integrals and differential equations, each of which is based on the former. If there is no corresponding training, few people can master them. When women's training needs are rejected, they don't have the math tools. Their door to the future of science was slammed shut. We will never know who Jane Austen of mathematics is, who was abandoned by mathematics because she lacked the necessary formal education.

All this is a thing of the past. What's the situation now? The apparent obstacles have disappeared and universities are no longer enforcing the ban on women imposed on Germain. On the contrary, judging from the enrollment data of mathematics majors at American universities, we have reason to be optimistic. In the academic year from 1990 to 1991, American research institutions awarded 14661 undergraduate diplomas in mathematics, of which 6917 were girls, accounting for about 47 per cent. Nearly half of this was unthinkable in the field of mathematics, which was dominated by men a century ago.

But when we look at advanced degrees again, the numbers are disappointing. In the same academic year, women accounted for only 2 percent of those who received master's degrees in mathematics and only 1 percent of those with doctorates in mathematics. This situation shows that despite the rapid growth in the number of women receiving undergraduate education, they are rarely able to continue their training and enter the graduate stage, and from here on will produce tomorrow's research mathematicians and university professors. so the situation is still unbalanced between men and women.

Why do women rarely go on to graduate school? Historically, many women have aspired to be a preparatory teacher at the university level, so there is no need for a research degree. In some cases, because women are under the above concepts, lower self-evaluation does have a negative impact on the pursuit of a higher level of success. Courage and finding mentors who can inspire you and help you remove obstacles to learning advanced math are the key to success. Men have too many peers and role models, while women always feel lonely in the highly competitive academic field. Their path to formal education is different from their male peers in many ways.

Even when women overcome all kinds of negative perceptions and get a solid education, they still face many obstacles: women lack the support to fully engage in their work to meet the needs of daily life.

Mathematical research requires a large chunk of time free from all kinds of interference. Research mathematicians spend a lot of time sitting there thinking. This is true in the past and today, but not everyone has such a large chunk of time. As mentioned above, the easiest way is to be very rich. According to legend, Archimedes is part of the Syracuse lineage. The Marquis of Marquis de l'H ô pital,1661-1704 was so rich that he hired John Bernoulli to coach him in the burgeoning field of calculus and became famous in Europe. Among the women we mentioned above, the Marquis Chartley is a marquise, Lovelace is a countess, and Agnesi is the child of a rich family. None of these people live by washing clothes.

On the other hand, support came from European societies, which were the think tanks of that era. Sponsorship from various societies in Berlin, Paris and St. Petersburg has supported countless scholars. Euler, who took positions in Berlin and St. Petersburg, is a mathematician who took advantage of such opportunities to succeed.

Or you have an undemanding job that allows you to do research and meditation in your spare time. It was during his diplomatic work in Paris that Leibniz, as we have already mentioned, found time to study mathematics and eventually created calculus. District Judge Fermat never seemed to try his best to do the work of the court, but devoted himself to mathematical research.

In short, it is harmless for potential mathematicians to have money, to be a member of an academic community, or to spend only part of their time working. Of course, today's main sponsorship of mathematicians comes from research universities, which provide offices, libraries, travel expenses, like-minded colleagues and moderate teaching tasks. In return, the school wants mathematicians to think deeply about the frontiers of the subject.

Compare the historical roles of women: staying at home while their husbands or brothers are working outside, raising children, cooking, mending, and taking care of household chores. Even if they are trained in mathematics, how can they have time to think about differential equations or projective geometry? The environment expects them to be completely different.

In fact, women rarely even have their own space. As Virginia Woolf reminded us in her essay on such topics, women have little room for solitude, thinking, writing (or doing mathematical research). Woolf tells a story about Shakespeare's imaginative sister Judith, who has the same talents as her brother. When her brother William devoted himself to her career as a writer, her life was responsible for the daily needs of the family. According to Woolf, Shakespeare's sister is as aggressive, imaginative and eager to learn about the world as he is. But she wasn't sent to school. She didn't have a chance to learn grammar and logic, so she could only read a little about Horace and Virgil. She occasionally picks up the book. Read a few pages. Then her parents would come in and remind her to mend her stockings or don't forget to cook instead of indulging in books and paper and ink.

Brother and sister, one is the provider of support, and the other is the recipient. The difference is too big.

Again, Leonhard Ora, the father of 13 children. Someone has to raise the children, change their diapers and wash their clothes. But this man is not Leonhard. Take another look at Srinivasser Ramanujin (1887-1920), a very talented mathematician at the beginning of the 20th century. But in daily life, he is as helpless as a child, and his wife takes care of everything in his life. Then look at Paul Eldesh, a man we met in the chapter Arithmetic / arithmetic, who only learned how to butter bread at the age of 21. Apparently, he received unusual support from his mother in the early stages of his mathematical discovery.

What would happen if we swapped it? If Mrs. Euler, Mrs. Ramanujin and Mrs. Eldesh succeed in math, will their partner meet their daily needs? If these women have become famous, can they devote a lot of time to math? No one knows the answer. But if women can get the same support as these men, then more of them will appear in the chronicle of math. There is no doubt about it.

In the life of Sophia Kovalevskaya, "the greatest female mathematician before the 20th century", all the above-mentioned obstacles, such as negative ideas and difficulties in mathematics education and lack of systematic support, have appeared.

Kovalevskaya was born in Moscow in early 1850 and grew up in a relatively well-off scholarly home. She had an English tutor and had the opportunity to study math. There is an interesting story that her bedroom wall is covered with old handout notes from her father's calculus course. The young girl was fascinated by these strange formulas, which surrounded her quietly like friends. She vowed to know the secret one day.

Of course, it takes training. At first, she learned arithmetic. She was allowed to attend her cousin's tutoring course, which the family did basically to persuade her cousin to study harder. In this way, she acquired the knowledge of algebra, but her cousin still couldn't learn it. Next, Kovalevskaya borrowed a book he wrote from a physicist who lived nearby. While reading the book, she encountered the difficulty of trigonometry, a subject she knew next to nothing. Unwilling to give up but without proper guidance, Kovalevskaya started her research from scratch. When her physicist neighbor realized what she was doing, he was surprised to find that "she had created the whole subject of trigonometry for the second time."

Such achievements show extraordinary mathematical creativity. When she was 17, she and her family came to St. Petersburg. There, Kovalevskaya persuaded her father, who was opposed to her math, to take a tutoring course in calculus. Although she is a woman, with such talent, she should have entered the university immediately. Unfortunately, for a 19th-century Russian woman, she did not have such a choice.

From a modern point of view, her reaction to these disappointments is a little extreme. At the age of 18, she decided to "fake" a marriage to a young scholar who was going to Germany, through which she hoped to get further higher education. The man is Vladimir Kovalevsky, a paleontologist who volunteered to participate in the "fake marriage", which he believes is good for women's liberation. The two of them set off for the University of Heidelberg, ostensibly maintaining a marriage, but in fact engaged in the research they were interested in.

Kovalevskaya did as well as ever in Heidelberg, so in 1897 she set her sights on a higher goal: the University of Berlin and its venerable senior professor of mathematics, Carl Verstras (Karl Weierstrass,1815-1871). The determined Kovalevskaya arranged a meeting with the world-famous scholar and begged for his guidance. Wilstras sent her away after asking some very challenging questions, and he didn't want to see her again.

But he saw her again. A week later, Kovalevskaya returned with the answer in his hand. According to Vilstras, her work shows "genius intuition about dimensions." this is rare even among students in the past or higher-level students. She transformed one of the world's most influential mathematicians from a skeptic to an admirer.

As a result, the old Vilstras and the young Kovalevskaya began a long-term cooperation. Her energy and insight won his respect, and he arranged for her to come into contact with many math groups in Europe. Under the guidance of Verstras, Kovalevskaya began to study partial differential equations, Abel integrals, and the dynamics of Saturn's rings. As a result of these achievements, she received a doctorate in mathematics from the University of Gottingen in 1874. She is the first woman to receive a doctorate from a modern university.

Throughout her life, Kovalevskaya was interested not only in mathematics, but also in issues such as social and political equity. As a supporter of liberal activities, she supports feminism and Polish independence. She was writing for a radical newspaper. With the help of her husband, she secretly entered Paris during the commune in 1871, when the city was surrounded by Bismarck's army. In this adventure, she was hit by a bullet from a German soldier. In Paris, she fell ill, was injured, and got in touch with the radical leaders of the besieged city. This is a figure who is eager to realize his social beliefs.

In addition to being a scientist and revolutionary, she is also a writer. Kovalevskaya wrote novels, poems, plays and memories of Childhood, an autobiographical record of childhood. She spent her youth in Russia, so she met Dostoevsky and met Turgenev, Chekov and George Eliot later in life. The socially responsible mathematician entered the famous art circle.

In short, Sophia Kovalevskaya has all kinds of amazing talents. Smart, decisive and sharp-tongued, she was described as "simply dazzling" by her contemporaries. The picture below shows this woman with an extraordinary personality, and people have created a lot of best-selling books and TV series about her.

Sophia Kovalevskaya on the stamp, like all TV series, her story begins with a comedy and ends in tragedy. Despite her special marital background, she fell in love with her husband, and the couple gave birth to a daughter in 1878. But five years later, after a business failure cost him a lot of money, a frustrated Vladimir Kovalevsky killed himself on chloroform. Sophia became a widow and single mother.

Fortunately, she is also a world-class mathematician. With the enthusiastic help of another disciple of Verstras, Mitge Raffler, she was appointed to teach at Stockholm University in Sweden. In 1889, she became a tenured professor at the school for the first time for a woman in the field of mathematics.

Those days in Stockholm were not without difficulties. Her inherent prejudice against women hinders her open and firm support for the cause of progress. Conservative scholars, who were impeccable about her math, turned to accuse her of coming into contact with a famous German socialist. Mr Vilstras and Mr Mitger-Raffler also politely suggested that Mr Kovalevskaya adopt a more cautious political attitude. But she didn't do it.

On the math side, she was named editor of the Journal of Mathematics, and she was the first woman to hold the position. She contacted mathematicians such as Hermit and Chebyshev (whom we met in the Arithmetic / arithmetic chapter) and became an important link between the Russian mathematical community and the Western European mathematical community. In 1888, Kovalevskaya won the Balting Prize of the French Academy of Sciences for her paper "the rotation of a rigid body around a fixed point", which led to international fame, media reports and congratulatory letters. Such cheers are enough to earn her membership of the Royal Russian Academy of Sciences (as a woman, such an academic position is not enough to support her in her home country).

In 1891, a promising future seemed to be in front of the famous figure, but what he did not expect was the sudden arrival of disaster. On the way to France, Kovalevskaya began to cough as if she had a common cold. But when she returned to Stockholm, her health got worse in the rainy and cold climate. When she got home, she became too weak to work. After a coma, Kovalevskaya died on February 10, 1891 at the age of 41.

As always, when such a genius left forever, she left the world with wonder, endless doubts and unfulfilled dreams. People's praise came from all over Europe, and the sadness that followed was sincere. We cannot estimate what contribution Kovalevskaya could have made to mathematics, nor do we know how much such a contribution would improve the status of women in the discipline.

Such a genius as Kovalevskaya is rare, but since her death, it has become more and more common for women to enter the field of mathematics in the 20th century. But then there was a troublesome problem. When we dedicate this chapter to female mathematicians, does it make them more marginalized and regarded as outliers? Should we feel guilty? As many women enter professional fields such as medicine and law, few people talk about "female doctors" or "female lawyers". In this chapter, we are not saying that the mathematics profession should be divided into two groups: mathematicians and female mathematicians. This is certainly not our intention, and it is not the real status quo. But there is such a danger.

This is Julia Robinson's point of view. As her popularity grew, when she entered the National Academy of Sciences and won the MacArthur Award, she was regarded as the winning woman in the male domain. In a very important essay, she wrote: "all these concerns are pleasant but also confusing. I am a mathematician. I prefer to be remembered just because I have proved some theorems or solved some problems, not because I am the first woman of one kind or another."

Although there is a need to further eradicate the inequalities faced by women, we have reason to be confident that Robinson's wishes will be realized. Many prejudices and obstacles are disappearing, and the number of women who devote themselves to mathematics has begun to increase. Even if this problem has not been completely solved, it is undeniable that progress has become a fact. We hope to ask "where are the women in the near future?" Such a chapter would be considered totally unnecessary.

This article is an excerpt from "those things in Mathematics" (people's posts and Telecommunications Publishing House, March 2022), with the title added by the editor.

A brief introduction to the author

William Dunham (William Dunham) is a professor of mathematics at Mullenburg College in the United States. His masterpiece is the Journey of Calculus: from Newton to Lebesgue, the Journey of Genius guidance. Dr. Dunham has won the George Polya Award, the Trevor Evans Award and the Lester R. Ford Award from the American Mathematical Association. "those things in Mathematics" won the Mathematical Excellence Award of the American Publishers Association.

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