by Karen Olsson
A few years passed, and then one summer evening, as Felix and Anna were returning home from an excursion, they found their estranged friend waiting for them. “There, in front of our door, sat the pale sick man,” Anna wrote. “‘Lie!’ we cried, in joyful surprise.”
Felix Klein and Sophus Lie “shook hands, looked into one another’s eyes, all that had passed since their last meeting was forgotten.”
Lie went back to Norway in 1898, but he could only lec ture for a few months before his deteriorating health forced him to retire. He died, of pernicious anemia, the following year.
At last André goes on the offensive, that is to say, he answers Simone’s repeated requests with a long, technical description of some of his mathematical work, a treatise in the form of a letter. He knows full well that she won’t understand these “thoughts,” as he calls them: “I decided to write them down, even if for the most part they are incomprehensible to you.” He plunges into a density of terms she wouldn’t know, with only minimal efforts to say what he means by quadratic residues, nth roots of unity, extension fields, elliptic functions.
In the first half of his letter he sketches a historical context for his work, starting with the nineteenth-century watershed in algebra, that leap by which mathematicians inverted the problem of solving equations within given domains by constructing domains in which given equations had solutions. He alludes to a time when questions about numbers began to rub up against questions about equations or functions in new ways. “Around 1820, mathematicians (Gauss, Abel, Galois, Jacobi) permitted themselves, with anguish and delight, to be guided by the analogy between the division of the circle . . . and the division of elliptic functions,” he writes.
Anguish and delight! As he’s laying out his none too explanatory explanation of his research, André emphasizes the role of analogy in mathematics—which his sister might appreciate, even if the rest of it flies right over her head. Here, analogy is not merely cerebral. The hunch of a connection between two different theories is something felt, a shiver of intuition. For as long as the connection is suspected but not entirely clear, the two theories engage in a kind of passionate courtship, characterized by “their conflicts and their delicious reciprocal reflections, their furtive caresses, their inexplicable quarrels,” he writes. “Nothing is more fecund than these slightly adulterous relationships.”
Analogy becomes a version of eros, a glimpse that sparks desire. “Intuition makes much of it; I mean by this the faculty of seeing a connection between things that in appearance are completely different; it does not fail to lead us astray quite often.” This, of course, describes more than mathematics; it expresses an aspect of thinking itself—how creative thought rests on the making of unlikely connections. The flash of insight, how often it leads us off course, and still we chase after it.
Pernicious anemia, the cause of Lie’s death, is a decrease in normal red blood cells that results when the intestines cannot absorb enough vitamin B12. Symptoms include confusion, depression, loss of balance, and numbness of the hands and feet.
David Hilbert, one of the leading mathematicians of the early twentieth century, also suffered from pernicious anemia, and although he benefited from a new treatment developed in the 1920s, the disease contributed to his decline at the same time as his beloved Mathematical Institute in Göttingen, for decades a hothouse of mathematical progress, was drained of its talent by the Nazis. One evening Hilbert attended a banquet where he was seated next to the Nazi minister of education, who asked him whether it was true that the institute had suffered following the removal of its Jewish faculty and their supporters. “It hasn’t suffered, Herr Minister,” Hilbert replied. “It just doesn’t exist anymore.”
Dementia set in, and he came to believe he was living again in Königsberg, the Prussian city of his childhood. He died in 1943; hardly anyone went to his funeral.
But I digress.
Simone studies her brother’s letter closely, so closely that it hurts. She pulls some of his books down from their shelves and begins to wade through a German text on complex functions, until her head threatens to split open.
Yet might she be right, or at least not altogether wrong, to think her brother’s work should be explicable? That it should do more than just extend the work that preceded it, that it should reveal something about the world?
Meanwhile André sends her another letter, two days later, continuing their debate about ancient mathematics. In their correspondence both he and Simone propose ways in which the Greeks might’ve discovered irrational numbers, drawing geometrical diagrams in the margins to illustrate their theories. Did the discovery trouble the Greeks or inspire them? André suggests that they were disturbed by it, while Simone counters that it would’ve brought them joy.
In one, two, three drafts that she composes in reply to the second letter, again there are figures traced in the margins, as well as more geometrical ideas, discussions of Platonic dialogues, musings on the relationship between an artist’s worldview and her art, speculations about mysticism in ancient Greece.
Simone conceives of a civilization in which mathematical reasoning, mystical belief, and existential loneliness formed an energetic triangle. The Greeks, she writes, experienced intensely the feeling that the soul is in exile: exiled in time and space. Mathematics could bring some ease to the exiled soul, she says. Doing math could free you from the effects of time, and your soul could come to feel almost at home in its place of exile.
She also writes, regarding André’s explanation of his research, “I understood nothing of your sixteen-page letter (which I read several times).” She has nonetheless sent excerpts to his lawyer, thinking perhaps a description of his research might be useful for his court case. And she tells him, “Send me more things like that if the inspiration strikes. I like it very much.”
It wasn’t lost on me, as I watched math lectures on my computer, that my return to (casually) studying math, much as it was not quite intentional, much as I stumbled into it, had come to resemble one of those self-improvement projects that middle-aged people embark on, like learning a language or taking piano lessons or rereading the classics. And yet studying German or piano or Homer seemed downright practical, relative to Math E-222.
What lies behind these efforts? Nostalgia? Wanting to stave off the inevitable, already perceptible decline? I suppose both, and then of course (almost too obvious to mention, and I feel like a scold and a cliché for even bringing it up) there is simply the enterprise of attention itself, the goal of concentrating on anything at all, in this era of distraction technologies. Surely Simone Weil, as odd as some of her beliefs and proposals were, was right to emphasize the importance of sustained attention, which is something we are letting slip away, or really giving away, with little more than mild, fleeting second thoughts.
Great titles in math:
The Whetstone of Witte
What Are and What Should Be the Numbers?
New Invention in Algebra: As Much for the Solution of Equations as for Recognizing the Number of Solutions That They Have with Several Other Things Necessary for the Perfection of This Divine Science
The Sand-Reckoner
Logarithmatechnica
The Mathematical Science Reduced to Tables
Essay on Fire
Simone understands more of André’s disquisition than she lets on. She includes a detailed summary of it in her letter to his lawyer, presented in the unlikely hope that the military tribunal could be convinced of the importance of André’s work and that her brother would thus be freed.
I picture this lawyer as a harried sad sack, exhaling as he pulls from its envelope an overly long missive sent by the same pestering angular female who has been unrelenting in her visits to his office. He reads the opening lines: “Number theory (putting aside some propositions discovered by the Greeks, notably the Pythagoreans) begins in France, in the seventeenth century, with Fermat. It was then the German mathematician Gauss, in the beginning of t
he nineteenth century, who made the most decisive progress . . .”
He blinks and sets the letter down on his desk.
Sophie Germain, born in 1776 in Paris, could not, as a woman of that era, be admitted to the École Normale or the École Polytechnique, but she studied mathematics as best she was able, borrowing from other students’ lecture notes and studying the books in her father’s library. She wrote letters to the great mathematicians of her day under a pseudonym, M. Le Blanc.
“But how can I describe my astonishment and admiration on seeing my esteemed correspondent Monsieur Le Blanc metamorphosed into this celebrated person,” replied Carl Friedrich Gauss, after she was compelled to reveal her true identity. By then she had withdrawn from social life, weary of being regarded as a curiosity because of her mathematical talent. “The taste for the . . . mysteries of numbers is very rare; this is not surprising, since the charms of the sublime science, in all their beauty, reveal themselves only to those who have the courage to fathom them. But when a woman, because of her sex, our customs and prejudices, encounters infinitely more obstacles than men, in familiarizing herself with knotty problems, yet overcomes these fetters and penetrates that which is most hidden, she doubtless has the most noble courage, extraordinary talent, and superior genius.”
He challenged her to come up with her own proofs for three of his latest theorems, which she duly sent back to him. Six months later, he sent his last letter to her. He had accepted a professorship at Göttingen and, he explained, had become too busy to keep up their correspondence.
“Remain always happy, my dear friend,” Gauss wrote to Germain before terminating their exchange for good. “The rare qualities of your heart and mind deserve it, and continue from time to time to renew the gentle assurance that I may count myself among your friends, a title of which I will always be proud.”
At trial, André is defended poorly by his lawyer and sentenced to five years in prison, and so he agrees to serve in a combat squad, in return for a suspended sentence. One ordeal ends and another begins; he joins a machine-gun unit, alternately lifting chests full of ammunition and sneaking off to read math books. During a German offensive, the men are evacu ated to England, where, because he misses curfew one evening, André is put in “prison,” that is to say he’s confined to an area of tents surrounded by barbed wire. Separated from the rest of his company, he literally misses the boat when the others are sent to Morocco. He becomes, instead, an interpreter for the guards who’ve been his jailers, and as such enjoys a good deal of spare time, free to go to the library and visit colleagues at the University of Bristol. He’s managed to turn military service into a rather leisurely, math-centered existence.
The Germans, meanwhile, have reached Paris. Simone and Bernard and Selma squeeze onto the last southbound train and arrive in Nevers, in central France, barely ahead of the occupiers. They continue on foot toward Vichy, buying themselves baskets in an effort to pass as peasants, though this only makes them look like three well-off urban Jews carrying baskets.
André makes his way to London, where he flirts with English women and persuades a friend in the camp office to fill out a card stating that he has pneumonia. A hospital ship takes him to Marseilles, and he reunites with Eveline and Alain. The three of them board another ship, sailing across the Atlantic to Martinique, and from there he secures an American teaching position and a visa.
His parents and sister remain in the south of France, in Vichy and then in Marseilles, trying to obtain visas to enter Morocco or Portugal, with the idea that from there they’ll find a way to follow André to the United States.
PART THREE
7.
In a dream, André meets Jacques Hadamard and notices that his beloved old teacher is wearing an undershirt and short pants and that as a matter of fact André is wearing the same thing himself. The kindest man he has ever known, boyish even in old age. Hadamard says, I’ve been looking for you! Indeed he has made supper for them, a roast chicken so large that the table underneath has begun to sag. Hadamard smiles, not with his face but by engulfing the whole dream with his generous spirit. Sit down, sit down, he says, but there are no chairs.
The actual Hadamard would think for extended periods without words. His greatest difficulty, he said to one of his daughters, was translating his mental images into language, though over the course of a very long life he surmounted this difficulty time and time again, as is made clear by the trail of work he left behind. He published in the fields of:
function theory
calculus of variations
number theory
analytical mechanics
algebra
geometry
probability theory
elasticity
hydrodynamics
partial differential equations
theory of gasses
topology
logic
as well as education, psychology, and the history of mathematics. He was a happy generalist, known for his good nature and what André called “an extraordinary freshness of mind and character.” For twenty years, beginning in 1921, Hadamard convened a seminar on Tuesdays and Fridays at which visiting mathematicians presented their research, and often he would suggest an approach that a visitor had overlooked, or connect the topic to some far-flung province of mathematics not previously seen to be related.
But what did he think when he was thinking? What were those concepts floating free of words?
Hadamard was born in 1865 and died just shy of his ninety-eighth birthday, in 1963—a chronology I can barely wrap my head around, spanning so much of modernity—and he himself was quite interested in the question of what actually happens inside the brain of a mathematician. He was in his late seventies when he delivered a series of lectures in New York on the mental processes that underlie mathematical in vention, which became the basis for his book The Psychology of Invention in the Mathematical Field.
Mathematical invention, he writes, is an instance of invention in general, one and the same engine underlying the creation of science, art, and literature. Intelligence is perpetual and constant invention, Hadamard said. Life is perpetual invention.
By the time the book appeared, all three of his sons were dead, two of them having perished in World War I and the third in World War II. He himself had fled Europe because he was Jewish. Only his two daughters survived him, one of whom would later recall that her father wrote his papers by dictating them to her mother, Louise, indicating with a “poum” that she should leave a blank space so that he could write in a formula later. He would say something like: “we integrate—poum—we see that the equation—poum, poum, poum—equals zero, takes the form—poum, poum, poum, poum.”
And: Hadamard had a passion for mushrooms. Fungi and also ferns—once, he and Louise traveled across the Soviet Union by train, and at various stops the elderly mathematician would amble off to hunt for specimens outside the station, poor Louise fearing all the while that the train would leave without him.
Simone dreams of Crete. She travels to a village of ancient geniuses, who harvest grains and triangles, combine words and figures into a form of pure expressive speech, a giving back to the air in exchange for the gift of breath. They light fires on the beach. Fish swarm their feet in the shallows. Cherishing geometry and the ocean, they arrange their bodies in certain set configurations before they begin their open-air dances, which are cued by the rhythms of the waves. They draw pentagons in the sand, sing stories, wail for the dead, all forms of praise. The practice of an ecstatic order. They tell of the god who endowed them with speech as though he came by a few days ago. Yes, yes, they tell her, he was just here.
If only we had more access to the untranslated thoughts, to the mystery of how the mind churns. Hadamard, for his part, based his reflections partly on his own experience, albeit somewhat sheepishly. “I face an objection for which I apologize in advance,” he wrote, “that is, the writer is obliged to speak too much about himself.”
He drew from his wide reading and notably from a study undertaken by the editors of the French journal L’Enseignement Mathématique, which they published in two parts in 1904 and 1906. Called “An Inquiry into the Working Methods of Mathematicians,” the study was based on responses to a long questionnaire they’d sent out to people in the field.
Sample question: Would you say that your principal discoveries have been the result of deliberate endeavor in a definite direction, or have they arisen, so to speak, spontaneously in your mind?
Another began: It would be very helpful for the purpose of psychological investigation to know what internal or mental images, what kind of “internal world” mathematicians make use of . . .
Hadamard asked similar questions of his contemporaries in the 1940s, and as an appendix to his book he reprints a letter from Albert Einstein. Like Hadamard, Einstein characterizes his thinking as something apart from language.
“The words or the language, as they are written or spoken, do not seem to play any role in my mechanism of thought,” Einstein writes. “The psychical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be ‘voluntarily’ reproduced and combined.”
André dreams he is on a ship again, another ship like the ones that have lately carried him across the North Sea, across the Atlantic Ocean and up the eastern coast of the United States. He lies in bed yet his body remains in motion, rocking back and forth. Swelling and receding. But there is another bed in the cabin, and on it sits Daniel Bernoulli, the eighteenth-century mathematician restored to life. André gets up and follows him out onto the deck, gradually realizing that the boat is full of Bernoullis, the brothers, the sons, the nephews, also the unremembered sisters and wives of that sprawling, backstabbing mathematical dynasty. Bernoullis everywhere!
Were we privy to Hadamard’s, or Einstein’s, or anyone else’s nonverbal thoughts, would we find that their subsequent translation into language served to crystallize those mental images into a more precise and elegant form? For a long time I thought of writing this way, as the art of making thought exact, of bringing latent insights and feelings into a fuller, more realized existence. Now I think that while this may be partly true, that some cloudy thoughts may be condensed into pools of precision, the longer I go on writing, the more I sense its limitations, see the tiny word critters scuttling around an inexpressible landscape.