Book Review Beyond the Limit: The Dream of Sofya Kovalevskaya Reviewed by Ann Hibner Koblitz 1 Beyond the Limit: The Dream of Sofya husband’s, not hers. And several people Kovalevskaya took great glee in re- Joan Spicci counting to me the Forge Books, 2002 aphorism often attrib- $26.95, 490 pp. uted to Hermann Weyl ISBN 0-765-30233-0 that there have been Twenty-five years ago when I began research only two women in the on my doctoral dissertation on the Russian history of mathemat- ics, and one of them mathematician, writer, and social activist Sofia wasn’t a mathemati- Kovalevskaia (1850–1891), I was surprised to learn cian [Kovalevskaia], that most mathematicians I encountered had while the other wasn’t some smattering of information (often false) about a woman [Emmy Noe- her. The more diffident and cautious among them ther]. would recount their “knowledge” in the form of The plethora of mis- questions, such as “Is it true that she slept with her information about Kovalevskaia made me realize advisor and that he did all of her best work?” or that I had a larger task on my hands than I had ini- “Did she really abandon math for literature?” But tially supposed. Naturally, I needed to comb the I also met a few mathematicians whose thought- archives and libraries of Russia, Sweden, and else- lessness or arrogance led them to state confidently where to put together the pieces of her life and set that all of Kovalevskaia’s papers were erroneous or her within the intellectual and cultural milieu of her that she was awarded the Prix Bordin of the French time. But equally importantly, I wanted to trace the Academy of Sciences out of gallantry. One Swedish various legends and insinuations to their sources mathematician (who will remain nameless) conde- and uncover the roots of commonly held misper- scendingly chided me for being interested in ceptions. This historiographical study proved Kovalevskaia as a mathematician at all, saying that although she might have been something of an 1The most charitable interpretation of this bit of non- amateur mathematician, the Cauchy-Kovalevskaia sense might be that since the theorem is often called Theorem (one of her best-known results) was her Cauchy-Kovalevskii in Western European languages (“ii” is a masculine ending for names in Russian, while “aia” is feminine), he thought that the reference must be to a Ann Hibner Koblitz is professor of women’s studies man. Also, Kovalevskaia’s husband, Vladimir Kovalevskii, at Arizona State University. Her email address is was indeed a scientist in his own right, but he was a pa- [email protected]. leontologist, not a mathematician. JANUARY 2004 NOTICES OF THE AMS 39 almost as engrossing as did my research on the cir- Kovalevskaia’s life was eventful. Besides her cumstances of Kovalevskaia’s life and work and in- pioneering efforts to open up higher educational volved questions of changing political and cultural opportunities for women in Russia and continen- atmosphere as well as the idiosyncrasies of indi- tal Europe and her literary endeavors, she partici- vidual historians, mathematicians, and other com- pated in the Paris Commune, aided Polish and mentators. Russian revolutionaries whenever possible, and For those who know little about Kovalevskaia, was on cordial terms with many of the most promi- let me digress for a moment to give a short sum- nent mathematical and cultural figures of her day. mary of her life. Sofia Kovalevskaia (née Korvin- She numbered among her colleagues and friends Krukovskaia) was born into a noble household in the eminent mathematicians Charles Hermite, Henri Russia in 1850. Like many young gentlewomen of Poincaré, Hermann Schwarz, Carl Runge, Pafnutii her generation, she was caught up in the social and Chebyshev, and Emile Picard. She was acquainted political movements of Russia in the 1860s, among with such scientific and literary luminaries as which was nihilism. Adherents of this philosophy Charles Darwin, George Eliot, Dmitri Mendeleev, Ilia denied the value of everything in traditional tsarist Mechnikov, Ivan Sechenov, Herbert Spencer, Ivan society, had great faith in the power of education, Turgenev, Fedor Dostoevskii, Henrik Ibsen, and believed that a transformative social revolution August Strindberg, and her salons in St. Peters- was imminent, and maintained that the best way burg, Moscow, and Stockholm were attended by to help the revolution along was through the study these and other leading intellectual figures. of the natural sciences and medicine. Kovalevskaia In fact, what emerged most prominently from embraced nihilism and decided to pursue an my research in Russian and Swedish archives and advanced degree in mathematics. She married libraries was the extent to which Sofia Kovalevskaia nihilist book publisher and sometime paleontolo- was not, as implied in the Weyl remark cited above gist Vladimir Kovalevskii in order to further her or in Bell’s Men of Mathematics,2 some sort of efforts to get a university-level education. Their barely tolerated amateur or outsider. On the con- stormy, unconventional marriage resulted in a trary, she was a participating member of the Eu- daughter, born in 1878, and ended with Vladimir’s ropean mathematical elite: she was consulted on suicide in 1883. hiring decisions, was asked for letters of recom- Kovalevskaia studied in Heidelberg and Berlin mendation, organized conferences, and so on. Ko- with Karl Weierstrass, Leo Königsberger, Paul valevskaia actively contributed to the “mathemat- DuBois-Reymond, and others and with their help ical culture” of late nineteenth-century Europe, and in 1874 persuaded Göttingen University to recog- during her lifetime she was accorded the respect nize her three research works as sufficient for a she deserved. It was not until her contemporaries degree. She was the first woman in the world to get retired or died that the legends and distortions a doctoral degree in mathematics in the modern began to surface in the European mathematical sense of the term, and she was among the first community. women to earn advanced degrees in any subject. In the 1980s several serious works on Ko- Because of the prejudices of the time, she had dif- valevskaia and her mathematics appeared in ficulty establishing herself in a professional posi- tion, and it was not until late 1883 that her colleague English. My doctoral dissertation was published in Gösta Mittag-Leffler was able to arrange a post for 1983 and was soon followed by a comprehensive her at the newly established university in Stock- book on Kovalevskaia’s mathematics by Roger holm. Kovalevskaia became an Extraordinary Cooke, the English translation of a careful biogra- Professor (in modern American parlance, assistant phy by the Soviet academician Pelageia Polubari- professor) at Stockholm University in 1884 and nova-Kochina, and a volume of Contemporary joined the editorial board of the new journal Acta Mathematics devoted to Kovalevskaia’s life and 3 Mathematica at the same time. mathematical legacy. Since the 1980s there have In the last three years of her life, Kovalevskaia certainly been enough reliable accounts of received the Prix Bordin of the French Academy of 2 Sciences for her work on the revolution of a solid Eric Temple Bell, Men of Mathematics (New York: Simon body around a fixed point, a full professorship in and Schuster, 1937), 423–429. 3 Stockholm, and corresponding membership in the Ann Hibner Koblitz, A Convergence of Lives. Sofia Ko- Russian Imperial Academy of Sciences (the rules valevskaia: Scientist, Writer, Revolutionary (Boston and had to be changed to permit her membership). At Basel: Birkhauser, 1983; revised 2nd ed. New Brunswick, NJ: Rutgers University Press, 1993); Roger Cooke, The the same time she also distinguished herself as a Mathematics of Sonya Kovalevskaya (New York: Springer- writer; she published a much-acclaimed memoir Verlag, 1984); Pelageya Kochina, Love and Mathematics: of her childhood, several essays on political and Sofya Kovalevskaya (Moscow: Mir, 1985); Linda Keen, ed., social topics, and (in conjunction with Mittag- The Legacy of Sonya Kovalevskaya, Contemp. Math. (Prov- Leffler’s sister, Anne Charlotte Leffler) two plays. idence: American Mathematical Society, 1987). 40 NOTICES OF THE AMS VOLUME 51, NUMBER 1 Kovalevskaia’s life and research that one could ex- her characters in a manner one usually associates pect the disparagement and innuendo to cease. with the genre of romantic fiction. The book there- Unfortunately, it seems as if some mathemati- fore has the feel of a novel, though the author cians are more comfortable with the old rumors and refers to her study as a “historical work” (p. 7). tales about their predecessors than they are with Spicci limits herself to a relatively short but the reasoned results of careful scholarship. Eric exciting period in Kovalevskaia’s life. The book Temple Bell’s Men of Mathematics, for example, begins in the winter of 1865 with a description of still appears to be popular despite the fact that it the adolescent Sofia’s infatuation with her older is riddled with errors and his section on Ko- sister Aniuta’s beau, the famous author Fedor valevskaia is inaccurate and coyly sexualized.4 And Dostoevskii. It ends with Vladimir and Sofia Felix Klein’s idiosyncratic two-volume history of happily going home together to Russia in 1874 nineteenth-century mathematics (which casts doubt upon the successful completion of their doctoral on Kovalevskaia’s originality) is still commonly studies. The long narrative (close to 500 pages) regarded by mathematicians as authoritative, even stresses Kovalevskaia’s determined efforts to attain though, as Jeremy Gray noted, historians have a university education and doctorate in mathe- learned “to handle [it] with care.”5 matics, her nihilist political beliefs and those of Even at the present time one occasionally comes her sister and friends, and the on-again, off-again across sexist treatment of Kovalevskaia.