Sophie Kowalevski believed it was a mistake of the uninformed to confuse mathematics with arithmetic. Arithmetic was just a pile of “dry and arid” numbers to be multiplied and divided. Mathematics was a world of elegant possibilities that “demand[ed] the utmost imagination.” To engage in mathematics fully was to elevate it to an art not unlike poetry. “The poet must see more deeply than other people, and the mathematician must do the same.”
Looking deeply into the numbers was a skill she acquired at a very young age. When Kowalevski was a child, her father, who had recently retired from Russian military service, moved the family to a rural estate near the Lithuanian border. It was a large home next to a forest and on a lake, far from any big cities. They ordered wallpaper from St. Petersburg to freshen up the home’s interior, but when the paper arrived, it became clear that there had been a miscalculation. The nursery was left bare. Instead of going through the hassle of ordering more, Kowalevski’s father fashioned an inexpensive, DIY solution. He had the room papered with the lithographed lectures on differential and integral calculus from a course he’d taken as a young officer. If there is an event that catalyzes the imagination, sending us, for the rest of our lives, restlessly after our passions, for Kowalevski, this was it. Her governess could not tear the girl away from the equation-layered room. “I would stand by the wall for hours on end, reading and rereading what was written there.” She was too young to understand its meaning, but age didn’t stop her from trying.
For the majority of her childhood, Kowalevski’s education did not keep pace with her curiosity. Her father wasn’t keen on the idea of “learned women.” Consequently, her formal instruction was spotty. “I was in a chronic state of book hunger,” she wrote in her autobiography. Kowalevski would sneak into her family’s library to consume the forbidden foreign novels and Russian periodicals heaped on the room’s tables and couches. “And here, suddenly at my fingertips—such treasure! How could anyone not be tempted.”
When her uncles visited, she probed them for stories about math and science. Through them, she learned how a coral reef was formed, how mathematical asymptotes would never kiss the curve leaning toward them, and about the Greek problem of how to square a circle. “The meaning of these concepts I naturally could not yet grasp, but they acted on my imagination, instilling in me a reverence for mathematics as an exalted and mysterious science which opens up to its initiates a new world of wonders, inaccessible to ordinary mortals.”
Kowalevski whipped through a borrowed algebra book, ducking the attention of her governess while she studied. When a neighbor, a physics professor, dropped off a textbook he’d written, as a gift for her father, the volume mysteriously ended up in his daughter’s possession. The next time the professor visited the house, Kowalevski engaged him in conversation about optics—not the simplest task. The professor was reluctant to talk to he about something that she couldn’t possibly understand. She was young—at this point in her teens—and a woman. But Kowalevski’s explanation of sine changed his mind. Because she was mostly self-taught, Kowalevski’s education had gaps. The chapter on optics, for instance, gave her trouble because she lacked a foundation in trigonometry that would have explained the function of sine. And sine was all over the place! So she began experimenting with its meaning, ferreting out an answer through trial and error. When she laid out her conclusion for the professor, his jaw hit the floor. She had pioneered her way to sine’s meaning via the same route that mathematicians had taken historically.
The professor appealed to her father, comparing Kowalevski’s considerable abilities to the famous French mathematician Pascal. She needed advanced academic training, stat.
Her father finally gave in. Kowalevski’s opportunities in Russia, however, had a well-established ceiling. Her only chances for greater professional development were abroad. But how to get there? Unmarried, she was stuck at home, subject to her father’s rules. Married, she would be forced to conform to her husband’s life in Russia. To Kowalevski and her older sister Anyuta, neither option was viable. Kowalevski opted for a third, more unconventional option. She entered into a sham marriage.
Her husband, Vladimir Kowalevski, was part of a radical political group fighting for equal education for women. When Sophie married Vladimir at age eighteen, both she and her sister were free to leave Russia thanks to their new legally bound but platonic chaperone.
Kowalevski’s first stop was Heidelberg, Germany. (Her husband went elsewhere to study geology.) But when she arrived, Kowalevski found that women were barred from university enrollment. The young mathematician, though, was practiced at using her insight as a tool to change reluctant minds. Kowalevski soon gained approval to attend lectures unofficially. One classmate, Yulya Lermontova, who became the first Russian woman to earn a doctorate in chemistry, remembered the impression Kowalevski made on the place. “Sofya immediately attracted the attention of her teachers with her uncommon mathematical ability. Professors were ecstatic over their gifted student and spoke about her as an extraordinary phenomenon. Talk of the amazing Russian woman spread through the little town, so that people would often stop in the street to stare at her.”
Next, Kowalevski traveled to Berlin, where she convinced a mathematician she greatly admired, named Karl Weierstrass, to teach her privately. (The University of Berlin, where Weierstrass taught, had an even stricter ban on women.) He was no supporter of the other sex in academics, but Kowalevski’s abilities and passion for the subject quickly earned her a place as his star student and later a trusted peer.
She wanted a doctorate in mathematics, so Weierstrass facilitated one from the University of Göttingen—a university that would grant higher degrees to women—without Kowalevski having to attend class or exams. From Berlin, Kowalevski became the first woman in Europe to earn a PhD in mathematics. Most doctoral students opted to write one dissertation; Kowalevski assembled three: two in pure mathematics and one in astronomy.
Meanwhile, Kowalevski’s sham marriage morphed into a real one. In 1875, she returned with her husband to Russia, putting mathematics aside. Weierstrass begged Kowalevski to come back to Europe and her studies. With so much distance between them, she stopped returning her advisor’s letters.
Six years after she left Berlin, having accrued several failed real estate ventures and a strained marriage, Kowalevski returned to Germany alone. Her work resumed immediately. Kowalevski published groundbreaking papers on the refraction of light in crystals and on “the reduction of a certain class of Anelian functions to elliptic functions.” In 1883, Stockholm University invited her to become a lecturer. She initially rejected the invitation, citing “deep doubts” about her ability to excel at the position until she felt ready to live up to the honor. However, within six months of her arrival, she’d been promoted to full professor and offered an editor position in the journal Acta Mathematica. Two years later she was the department chair, fluent in Swedish, and dedicated to her work with a singular passion not felt since the early days of liberation from her father’s roof.
It was then, egged on by supportive peers, that she went after what the discipline called the “mathematical mermaid,” a classical mathematical problem that had eluded many greats. For advancing the field’s understanding of this problem, which involved “the rotation of a solid body around a fixed point under the influence of gravitational force,” the Paris Academy of Sciences would issue a cash prize. Kowalevski worked furiously to complete her offering on time.
The Paris Academy of Sciences’ announcement was a shock for two reasons. First, the winner broke so much new ground on the problem that the prize’s governing body voted to increase the pot. The second was only a surprise to those who didn’t already know her. Of the fifteen entries submitted anonymously, Kowalevski’s took the prize. Her solution led the way to new areas of research in theoretical mathematics. An analysis of her work pointed out that her win had influence that was more than mathematical: “The value . . . is not only in the results themselves nor in the originality of her method, but also in the increased interest she aroused in the problem . . . on the part of researchers in many countries, in particular Russia.”
By the time of her death from pneumonia at age forty-one, Kowalevski had risen to the top of her discipline. As was custom, her brain was weighed and assessed, the size and grooves judged as an indication of ability. “[The] brain of the deceased was developed in the highest degree,” reported the Stockholm newspapers. “And was rich in convolutions, as might have been predicted, judging by her high intelligence.”
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