Sofia Kovalevskaya: ILE

 

"It is impossible to be a mathematician without being a poet in soul".

Sofia Kovalevskaya (1850-1891)'s childhood at Palibino contained an accident of wallpaper that became one of the most typologically telling origin stories in the history of mathematics. When the family redecorated the manor house in 1858, they ran one roll of wallpaper short for the nursery. Rather than order another from St. Petersburg, they papered the wall with what they had in the attic — pages from Ostrogradsky's lectures on differential and integral calculus, which Sofia's father had attended during his military training. She was eight years old. "I remember how I spent whole hours of my childhood in front of that mysterious wall," she wrote later, "trying to make out a single sentence and find the order in which the pages ought to have followed one another. From long daily contemplation of them, the appearance of many of the formulas burned itself into my memory, and the text itself left a deep imprint in my brain, although at the time I was studying it I could not understand it at all." When she later took her first formal calculus lessons as a teenager, her teacher was astonished at how quickly she advanced — as if she already knew the material. She did, in some pre-rational sense that had nothing to do with having been taught it.

All this is quite typical of ILE's relationship with knowledge: the concepts absorbed before they can be formally understood, the patterns that burn themselves into memory through intuitive contemplation rather than systematic instruction. Ne perceives structure before Ti can articulate it — the mathematical wallpaper gave her Ne the raw material it needed years before the formal Ti apparatus arrived to make sense of it.

Her uncle Pyotr Vasilyevich Korvin-Krukovsky, likely the same ILE type as she, was the other early formative presence, and his description in her memoirs is almost a portrait of what ILE finds most exciting in a human interlocutor. He was "an eccentric and a dreamer" who had given away his estate and lived on a small pension, who visited Palibino for weeks at a time and made the household livelier by his presence, who "had a great respect for mathematics and often spoke about the subject." He talked about the quadrature of the circle, about asymptotes — lines that approach a curve forever without ever reaching it — about concepts she couldn't yet grasp but that "acted on my imagination, instilling in me a reverence for mathematics as an exalted and mysterious science." And crucially, he read scientific journals and caught fire: "But more than anything my uncle was carried away when he came across in some journal the description of a new important discovery in science. On such days heated arguments were carried on at the table" — and he would retell the article, "involuntarily embellishing and supplementing it, and drawing bold conclusions." 

At fourteen she encountered trig formulas in a neighbor's physics textbook before having studied trigonometry. Rather than stopping at the obstacle, she attempted to derive the formulas herself from first principles. By a "strange coincidence," she described, she hit upon the same approach that had been historically used — substituting chord for sine for small angles. When she told the neighbor, Professor Tyrtov, how she had reasoned her way to the formulas, he was sufficiently impressed to persuade her father to allow serious mathematical study. Ne's characteristic route to knowledge is often the independent rediscovery — finding the non-obvious approach that happens to converge with what the field already knows, arriving at the right place by a road no one had told her about. Of course, it also has a reverse side — many Ne types were quite disappointed to know that their auto-didact musings are old hat to specialists in the field.

Her fictitious marriage with Vladimir Kovalevsky, a paleontology student and radical willing to cooperate, further demonstrates the independent will of the type and its dislike of social prohibitions. While not a political theorist, Kovalevskaya, like many ILE's, was interested in radical politics, and even wrote a novel about a nihilist young woman who follows her new revolutionary husband to penal servitude.

At Berlin, having been refused admission to the university, she got herself in front of Karl Weierstrass — one of the greatest mathematicians of the century — by solving the test problems he laid in front of her at their first meeting. Weierstrass had intended the problems to send her away; instead he was surprised by her exceptional mind. He went on to tutor her privately for four years, consider her the most gifted of all his many students, maintain a lifelong friendship with her, and describe the world after her death in terms that suggest how completely she had engaged both his mathematical judgment and his personal affection: "People die, ideas endure" — not a bad slogan for ILE, in general.

Her most significant paper was the one on partial differential equations now known as the Cauchy-Kovalevskaya theorem — established conditions under which these equations have unique solutions, generalizing Cauchy's earlier work. The mathematical style is consistent with ILE: reaching the general result by a route that organizes the prior landscape rather than simply adding to it, seeing the structure that makes the specific cases instances of something larger.

In her quote about mathematics and poetry, she is saying they share a perceptual mode — the capacity to see what others miss, to look through the surface appearance to the structure underneath. This is Ne describing itself. The leading function in ILE operates precisely this way: it perceives patterns and connections that are hard to immediately see otherwise, and this perception precedes and enables the Ti systematization.

Her emotional self-management was a consistent difficulty for her — unlike chiller ILE's like Raymond Smullyan, she definitely was closer to the "constructivist", ethically rigid pole of this secondary dichotomy. "My fame has deprived me of ordinary feminine happiness... Why can no one love me? I could give a loved person more than many women — so why do people love the most insignificant, and only me does no one love?" Her professional bonds consistently worked, her private ones consistently failed to resolve in the way she needed. Her initially fictive husband fell in love with her for real, but eventually killed himself, and she rejected another proposal from sociologist Maxim Kovalevsky (no relation) due to not wishing to commit herself to marriage — evidence for ILE's weak and rejected Fi. 

Her motto, adopted from the French: "Dis ce que tu sais, fais ce que tu dois, adviendra que pourra" — "Say what you know, do what you must, come what may." The third clause is the key one: come what may. The ILE's characteristic relationship with consequences — produce the work, release it, accept that you cannot (and shouldn't want to) control what happens next.