Sofia Kovalevskaia - a biographical sketch

A Symposium, The Legacy of Sonya Kovalevskaya, sponsored by the Association for Women in Mathematics and the Mary Ingraham Bunting Institute was held at Radcliffe College on 27-28 October 1985. There were also three special sessions at the meeting of the American Mathematical Society held on the 25-27 October 1985 which were organised to fit in with the Symposium. A selection of the papers presented to the Symposium and the special sessions of the American Mathematical Society were published by the American Mathematical Society in Contemporary Mathematics 64 (1987), 1-298. This is an excellent collection of papers divided into four sections: Kovalevskaya - her life and work; Geometry and dynamical systems; Analysis; and Applied Mathematics. We present below a version of the talk Sofia Kovalevskaia - a biographical sketch given by Ann Hibner Koblitz.

Sofia Kovalevskaia - a biographical sketch

Sofia Korvin-Krukovskaia was born in Moscow on 15 January 1850. Her father was a general in the Russian artillery and a large landowner, and her mother was from a family of German scholars who had settled in Russia during the time of Catherine the Great. In most respects, Sofia had a typical upbringing for a girl of her class and time. She was left largely in the care of nurses and governesses, spoke English and French almost as well as she did Russian, and was reared in the belief that her future would be settled by her marriage with a young man of suitable wealth and family position.

There were two circumstances which served in the long run to set her apart from the majority of young gentry girls, however. One was her father's love of science and mathematics, and his preference for Sofia over her brother and sister. He permitted her to be tutored in mathematics to a level of preparedness achieved by few men of her time.

The other circumstance which proved to be crucially important in Sofia's future development was the socio-political milieu in Russian educated circles in the 1860s, when Sofia was growing to adulthood. The nihilist philosophy, which came into currency at this time, taught that the traditional structure of tsarist society had to be changed. The nihilists believed in the power of education to improve humanity's lot, and felt that the most useful branches of knowledge were the natural sciences. They taught that science would hasten the (peaceful) social revolution that they all considered inevitable, and they viewed a scientific career as an active blow against backwardness and autocracy. Moreover, the nihilists joined this faith in science and social revolution to a firm conviction about the equality of women, and a resolve to help women attain professional status and independence.

Through her older sister Aniuta, Sofia encountered the nihilist philosophy at an early age. She found the philosophy congenial, with its emphasis on the natural sciences, revolutionary social change, and women's equality. Nihilism provided for Kovalevskaia a framework within which her interests in mathematics, politics, and women's emancipation were seen as harmonious with one another. She remained true to nihilist precepts throughout her life, and always identified herself as a member of the movement. It is impossible to understand her without putting her in the nihilist context.

At the age of eighteen, Sofia married fellow nihilist Vladimir Kovalevskii, who later became an eminent palaeontologist. The marriage was at first what the nihilists called "fictitious." That is, it was contracted to allow Kovalevskaia the possibility of studying in a university abroad, and was not intended to become a real union. But eventually Sofia and Vladimir consummated their marriage and had a daughter. The story of the Kovalevskii relationship is long and complicated and does not really concern us here. Suffice it to say that the marriage was not happy.

After her marriage in 1868, Kovalevskaia went to St Petersburg in the hope that she would be allowed to enter an institution of higher education there. Universities in Russia, like those of all Europe at the time, were closed to women. But Russian women, fired by the nihilist philosophy, expected the universities to open to them soon. When this optimism proved unfounded, the women turned their attention to the universities of Western Europe.

To a large extent, Russian nihilist women of Kovalevskaia's generation opened higher educational institutions in continental Europe to women. Russian women were among the first officially enrolled female students in Zurich, Bern, Geneva, Heidelberg, and elsewhere. Kovalevskaia herself was the first woman student at Heidelberg University. She enrolled as an auditor for the spring semester of 1869.

At Heidelberg, Kovalevskaia soon gained a reputation as a talented student of mathematics and the sciences. She worked with the famous researchers Gustav Kirchhoff, Hermann Helmholtz, Leo Königsberger, and Paul DuBois-Reymond, all of whom praised her and spread her fame among their colleagues. Konigsberger thought so highly of her capabilities that he persuaded her to move to Berlin. He felt that she could derive the most benefit from working with the great analyst Karl Weierstrass, and so in 1871 she became his pupil.

Weierstrass and Kovalevskaia developed a warm personal and professional relationship. He called her the most talented of his pupils (who included such eminent mathematicians as Gcorg Frobenius, Hermann Schwarz, Leo Königsberger and Carl Runge) and watched out for her interests as carefully as he did for his own.

Kovalevskaia, in her turn, accorded him the affection of a daughter, propagandised his methods in her works, and always credited him as the source of her ideas. She often exaggerated her intellectual debt to him, thereby damaging her reputation for originality. The friendship (also extended to Kovalevskaia by Weierstrass's two spinster sisters) lasted throughout Kovalevskaia's life.

The relationship of Kovalevskaia and Weierstrass, by the way, caused much gossip in the European intellectual community at the time, especially after Kovalevskaia was appointed to a position at Stockholm University in 1884. For many conservative European academics, the only way to explain a woman's scientific competence was to tell themselves and others that she had beguiled some man into being her ghost-writer. I should point out that as far as I know, during Kovalevskaia's lifetime this gossip was confined to non-mathematical circles. Mathematicians gossiped about her politics and her bohemian lifestyle, but none of them defamed her friendship with her former adviser.

Within three years of moving to Berlin to work with Weierstrass, Kovalevskaia produced three doctoral dissertations. According to Weierstrass, each of them was deserving of the doctoral degree. But neither Weierstrass nor Kovalevskaia wanted to take any chances. They realised that since she was the first woman to apply for her degree in mathematics, her case would have to be especially strong.

One of Kovalevskaia's dissertations was a commentary on a classic paper of Laplace, in which she improved Laplace's calculation of the shape of Saturn's rings. The second paper used Weierstrass's function theory to reduce a class of abelian integrals to simpler elliptic integrals. The third work, the best of the three papers and one of Kovalevskaia's two most famous researches, was called "Toward a Theory of Partial Differential Equations." This paper contains what is now known as the Cauchy-Kovalevskaia Theorem.

Unknown to Weierstrass and Kovalevskaia, Cauchy had included a proof of this theorem in an 1842 paper; an 1875 priority dispute between Gaston Darboux and Kovalevskaia (waged on their behalf by Hermite and Weierstrass) revealed Cauchy's prior work. But according to Poincaré, Hermite, and others, Kovalevskaia's solution was the most elegant, and she had made several important observations that the others had not. She noted, for example, that certain differential equations have no solutions even when they have "formal power series" solutions. Also, she commented on the conditions under which certain sets of partial differential equations are integrable.

Thanks to the representations of Weierstrass, DuBois-Reymond, Lazarus Fuchs, and others, Kovalevskaia was granted her doctoral degree, summa cum laude, from Göttingen University (in absentia) in 1874. She was the first woman to be granted a doctoral degree in mathematics in the modern sense of the word and one of the first in any field.

But when she returned to Russia, she discovered that the combination of her sex and her politics and her German degree made her unacceptable as a job candidate. Chebyshev and other Russian mathematicians were devotes of the French rather than the German school of analysis. They were in the process of throwing off the domination of German scientists in their Academy of Sciences, were evolving their own approach to mathematical problems, and consequently looked with suspicion on those who had done their work entirely abroad.

In Western Europe, Kovalevskaia's status as Weierstrass's student would have been to her advantage. But there were other difficulties as well. Her reputation as a nihilist was shocking to conservative European academics, and her sex was also an obstacle. The most immediate barrier to her mathematical career, however, seems to have been the social convention of the time. Kovalevskaia was married, and married women did not live apart from their husbands, nor did they support themselves with teaching positions.

Even Weierstrass, although he had broken with tradition enough to accept and even do battle for a woman student, was conventional enough to be influenced by such considerations. For a long time, he had not felt that it would be necessary for Kovalevskaia to obtain an official degree. Since she was married, Weierstrass reasoned that Kovalevskaia was only studying mathematics for the intellectual satisfaction it gave her. Her husband would support her, Weierstrass thought; she would therefore have no need of official recognition of her scientific competence.

Only after Kovalevskaia told Weierstrass the true story of her marriage and something of her political beliefs, did he agree that a degree might be useful to her sometime in the future and persuade Göttingen to grant her one. And only after Vladimir Kovalevskii conveniently removed himself from the scene in 1883 did Weierstrass feel completely happy about actively seeking a university position for Kovalevskaia.

But in the mean time, there was a hiatus in Kovalevskaia's scientific activity. From 1874, when she returned to Russia, to 1878, while she was pregnant with her only daughter, Kovalevskaia more or less abandoned any attempt at original scientific research. Partly, there were practical reasons for this. It was impossible for a woman to get a teaching job at any level higher than the equivalent of the fourth grade. This was the result of a common "Catch-22" situation encountered by many women attempting to gain access to previously male spheres of activity. Namely, in order to teach at the higher levels, one needed a Russian master's degree. But women were forbidden to take the qualifying exam to obtain that degree.

There were also emotional reasons for Kovalevskaia's temporary decision to stop trying to obtain some form of scientific employment. As she wrote in her semi-autobiographical novella A Nihilist Woman, she found herself enchanted with Petersburg life, especially after the exhausting years of study in Heidelberg and Berlin. She plunged into literary circles, tried her hand at writing, had a salon of sorts, was active in the movement to establish a women's university in St Petersburg, and so on.

Moreover, sometime in 1874 or 1875 Vladimir and Sofia decided to consummate their marriage. Vladimir's increasing inability to have a scholarly life of his own, and his unwillingness to support Kovalevskaia's scientific interests, seem to have induced her to try to devote herself to her husband's interests and submerge her scholarly self. But after years of a sort of on-again, off-again marriage, with Kovalevskaia becoming increasingly bored and Vladimir increasingly unstable (he abandoned palaeontology for various business speculations and wound up bankrupting the family), Kovalevskaia found that she could no longer live without her mathematics.

The return to mathematics was not easy for Kovalevskaia. It was another three years before she abandoned Vladimir for good and set off on her own. In the meantime, she began to talk to Russian mathematicians again, gave talks at Russian Congresses of Natural Scientists and the Moscow Mathematical Society, entered into correspondence with Weierstrass and Weierstrass's Swedish student Gösta Mittag-Leffler, and began to hope that eventually she could become a professional mathematician.

Kovalevskaia separated from Vladimir in 1881, and went to live in Paris and work on her mathematics. She became a member of the Paris Mathematical Society, and came to the attention of the great French mathematicians of the time - Charles Hermite, Emile Picard, Henri Poincaré, Joseph Bertrand, etc. But although she was an active participant in the mathematical life of both Berlin and Paris, she was in a kind of holding pattern - she had no official position, and no hope of one while her marital status was so ambiguous.

Mittag-Leffler had been trying to get her a position since she separated from Vladimir in 1881, first in Helsinki and then in Stockholm, but he had run into opposition. In Helsinki, the opposition centred a round the fact of Kovalevskaia's nationality and political convictions, rather than the fact that she was a woman. Mittag-Leffler found that Finnish university administrators worried because Kovalevskaia was a known nihilist. They feared her appointment would draw the anger of the tsarist government, which occupied Finland at the time. In Stockholm, protest was more diffuse. But Mittag-Leffler and Weierstrass were sure that Kovalevskaia's ambiguous marital status would be used against her.

In April 1883, Vladimir committed suicide. He was involved in a stock swindle for which he faced prosecution, he was on the verge of bankruptcy again, and he had been denied his Russian doctoral degree by an examination committee jealous of his reputation among his Western European colleagues.

Kovalevskii was mourned by palaeontologists all over the world. But mathematicians seem to have been secretly pleased. In the case of Hermite, Weierstrass, and Mittag-Leffler, they did not try very hard to hide their pleasure, at least not among themselves. For them, the suicide seemed like the answer to a prayer, a graceful way out of what had seemed earlier to be an insoluble dilemma. For purposes of obtaining a university position, nothing could have been better for Kovalevskaia than recent widowhood. Widowhood was eminently respectable. A widow was her own boss, in control of her own fate.

Within an almost indecently short time after Vladimir's suicide, Kovalevskaia had been offered a position as privatdocent at Stockholm University. She accepted the post immediately, wound up her affairs in Russia, and in November 1883 sailed off to Sweden to begin her new life as a professional mathematician.

In the scant seven years which elapsed between her first appointment and her death in 1891, Kovalevskaia chalked up an impressive array of firsts and achievements! In 1884, Kovalevskaia became an editor of the Scandinavian mathematical journal Acta Mathematica, and one of the chief propagandisers and fund-raisers for the journal. To my knowledge, she was probably the first woman on the editorial board of a major scientific journal. She was acknowledged on the masthead, too, as Emmy Noether was not in the case of Mathematische Annalen over thirty years later.

In 1884 Kovalevskaia was appointed extraordinary professor for five years. This was also most likely a first in modern times, and understandably enough, Mittag-Leffler had to fight to achieve Kovalevskaia's appointment. Eventually, he and several other research scientists on the faculty had to agree to a "trade." They withdrew their vociferous objections to two male non-entities who had been nominated for tenure on the condition that the latters' supporters vote for Kovalevskaia.

In 1888, Kovalevskaia was awarded the Prix Bordin of the French Academy of Sciences for her work on the rotation or a solid body about a fixed point - the famous Kovalevskaia top. This is Kovalevskaia's best known work, and the Prix Bordin was in some sense the triumph of her career, making her famous all over Europe. One of the techniques used in this paper, Kovalevskaia's so-called asymptotic method for determining conditions of algebraic integrability, has returned to the limelight in the past ten years or so.

The Prix Bordin episode is an interesting one. The contest was, on the face of things at least, anonymous. The Academy announced a public competition for the prize, and entries were submitted with quotations heading them for identification instead of names. But there is overwhelming evidence that the French academicians decided to make the motion of a rigid body the topic of the 1888 Prix Bordin contest precisely because they knew that Kovalevskaia was working on the problem. Hermite was writing to Mittag-Leffler about the possibility of increasing the prize money for Kovalevskaia months before the contest was officially closed. He even went so far as to assure Mittag-Leffler that the deadline could be extended if Kovalevskaia could not finish in time, or alternately, they could accept her partial solution.

According to a letter of Charles Hermite to Thomas Stieltjes, the rigging of an Academy prize for an eminent younger mathematician was a custom of the Academy. The same honour was accorded to Picard and Stieltjes as well as Kovalevskaia. The question of the anonymity (or lack thereof) of the Prix Bordin is an important point because it illustrates how Kovalevskaia was regarded in the mathematical world of her time. She was considered important enough to have a prize tailored to her area of expertise. Not coincidentally, the prize was timed to give her maximum visibility in the months before she would come up for tenure in Stockholm.

The question of the anonymity of the Prix Bordin is also important because the anonymity of the prize is sometimes made much of in recent feminist accounts, which maintain that Kovalevskaia would never have been awarded the prize if the French academicians had known the winning entry belonged to a woman. This, as we have seen, is exactly the opposite of the true situation.

In 1889 Kovalevskaia was appointed to the chair of analysis at Stockholm University, and the Prix Bordin certainly helped smooth her way. But even then, at the last minute her colleagues had to deal with objections on the grounds of her political views. Kovalevskaia never thought it necessary to hide her radical political sympathies, and freely aired her socialist opinions in Stockholm society. Fortunately, opposition to her was overcome, partly because of a petition initiated by Hermite and Mittag-Leffler, and signed by Beltrami, Picard, Poincaré, and other eminent mathematicians. Kovalevskaia was given a lifetime professorship in June 1889, and became the first woman in modern times to be so honoured. (For comparison, let me note that Maria Sklodowska Curie was appointed to her late husband's chair at the Sorbonne in 1906, and Emmy Noether, as we know, did not achieve a professorship until Bryn Mawr.)

This was a banner year for Kovalevskaia. Besides her full professorship, she received the Oscar Prize of the Swedish Academy of Sciences for further work on the rotation problem. Moreover, she was made the first woman corresponding member of the Russian Imperial Academy of Sciences; the rules had to be changed so that she could become a member. In fact, at the time of her premature death, there was some indication that Kovalevskaia was being considered for full membership in the Academy. This was an extremely prestigious position, the post that Kovalevskaia coveted above all others.

It should be emphasised that during these seven extremely productive years of her life, Kovalevskaia by no means devoted herself exclusively to mathematics. She tried her hand at writing, and in the five years before her death published her childhood memoirs, a play, several essays on political and social themes, some poetry. Her works were well-received, especially in Russia and Scandinavia. One enthusiastic British critic even compared her to Turgenev - slightly overblown praise, in my opinion.

In addition, Kovalevskaia remained a political activist and champion of women's rights. In her youth, she had participated in a minor way in the Paris Commune of 1871. Later, she was on the fringe of various revolutionary organizations and was watched by the tsarist police. Also, judging by fragmentary archival evidence (fragmentary because Mittag-Leffler apparently burned most of her political correspondence on Kovalevskaia's death), she handled funds for revolutionary groups and smuggled female political refugees out of Russia on her passport.

Moreover, Kovalevskaia was an ardent advocate of higher education for women, and a vocal feminist. As I mentioned, she helped organise the women's university in St Petersburg. She also sponsored the careers of young women in the sciences, and recommended women faculty to infant women's colleges in Europe and America. She even organised several prominent women to attend a ceremony honouring the virulently anti-feminist writer August Strindberg (who had once called her as a female professor of mathematics a "monstrosity"), on the grounds that women should be above such viciousness and illogic as Strindberg had displayed.

Sofia Kovalevskaia died soon after her forty-first birthday, in 1891, of pneumonia. She was at the height of her mathematical career, and at the beginning of a promising career in literature as well. And, lest these listings of her literary, political, and scientific achievements cause us to forget, Kovalevskaia was a single mother during the period of her greatest professional activity.

On Kovalevskaia's mathematics: I do not want to exaggerate her accomplishments. She was not Gauss. She was not a path breaker, she founded no school of mathematics, nor did she "revolutionise" her field in any way. But at the time of her death, she was the equal of any male mathematician of her generation, and was so considered by her contemporaries. She has earned a respectable place in the history of nineteenth century mathematics. Unfortunately, sometimes she does not get the respect she deserves.

As to Kovalevskaia's long-range impact on mathematics - well, in a sense, very few people have a smashing impact on mathematics. Maybe Euclid, Archimedes, Al Khorezm [better known as Al-Khwarizimi], Gauss, Poincaré, Hilbert - mathematicians would dispute even those, want to add and subtract, depending on their field of expertise. Obviously Kovalevskaia does not belong in those ranks.

But the Cauchy-Kovalevskaia Theorem, to which she gave most complete final form, is basic to the theory of partial differential equations. And her asymptotic method for determining conditions of algebraic integrability is finding increasingly widespread applications today.

This rejuvenation of a mathematician so long dead is unusual, and is a tribute to Kovalevskaia's depth of vision. She had once suggested to Picard that her method could be used for integrating equations of the type that later become known as Korteweg-DeVries equations, but Picard had dismissed the idea. Now, ninety years later, Kovalevskaia has been proved right.

I think, though, that we have to look deeper for Kovalevskaia's impact on mathematics. Kovalevskaia interacted with mathematics, to the credit of both. She was an important conduit of ideas between East and West, as I mentioned earlier. Also, in some sense, Kovalevskaia is a symbol representing certain historical traditions in the mathematical community. They do not start with her, but they achieve an important boost with her. Traditions of openness, liberality. I think it is no coincidence, in other words, that the first woman academic to be fully integrated into the professional life of her time was a mathematician.

In conclusion. I would like to summarise what I consider the salient points to be drawn from this sketch of Kovalevskaia's life, and how I think my work on her differs from that of previous commentators.

First, Kovalevskaia (and the whole generation of Russian women scientists of which she was a part) cannot be separated from the socio-political milieu, a milieu in which she was encouraged to defy the traditions of patriarchy and seek higher education in the sciences. It is, after all, in some sense astounding that Russia produced not only the first professional female mathematician, but also the first professional woman chemist, physiologist, zoologist, and ecologist as well. The philosophy of nihilism goes a long way toward explaining this flowering.

Second, Kovalevskaia was not Weierstrass's mascot or old man's darling, not some amateur clinging to the outskirts of the scientific community, not some barely tolerated eccentric. Rather, she was a fully accepted member of the mathematical world: consulted on hiring decisions, solicited as a journal editor, chosen as the speaker at prestigious conferences and societies, and routinely ranked by contemporaries with Hermite, Weierstrass. Mittag-Leffler, Picard, and Poincaré, as one of the best mathematical analysts of her age.

Kovalevskaia was not a mathematical revolutionary, as I have said above. But she was considered an eminent mathematician in her lifetime, and the increasing applications of her asymptotic method to work in mathematical physics today are an indication of the strength of her mathematical intuition and skill. She deserves a respectable place in the history of nineteenth century mathematics. Hopefully, with the publication of this volume and the most recent biographies of her, she finally has it.

Last Updated January 2021