by Karen Olsson
Simone dreams her brother is a tooth—her own tooth, but not her own. Stuck inside her mouth and schooling her as always. She pushes at him with her tongue to wiggle him loose; although she doesn’t want to be separated she still has that compulsion to dislodge him, to feel the bloody gap where he used to be.
Moreover, after many years of writing I find I have traveled down so many forking paths, motivated by a sensation that something like the truth awaits at the end of a path, only to wind up verbalizing the various ways that I have taken too many forks, been eluded by truth, realizations, epiphanies, et cetera, only to learn that the path either doesn’t end or that it leads, as in a formal garden, into a cul-de-sac enclosed by hedges, where I come upon a moss-streaked stone pedestal that once supported a statue, the statue having been for some reason taken away.
Have you ever worked in your sleep or have you found in dreams the answers to problems? Or, when you waken in the morning, do solutions which you had vainly sought the night before, or even days before, or quite unexpected discoveries, present themselves ready-made to your mind?
André dreams of a bucket of candied fruit, one that he bought in actual waking life from a factory that was selling off its seconds, just before his family crossed the Atlantic. On the voyage they devoured those sticky scraps of pear and citron in the evenings, before all the passengers crowded into single-sex rooms to sleep in bunk beds, lumped together like litters of kittens. In the dream he puts his hand straight into the syrup, as others press around him, holding out plates or palms, touching his jacket, and though he would like to save all the sweets for Eveline and Alain, he can’t turn these people down, no, in spite of himself he distributes shiny gobs of second-rate fruit to these mewling strangers.
Mathematical discoveries do not occur in dreams, Hadamard claims, or if they do, they are probably absurd. Yet Hadamard can’t resist including a strange exception to the rule, reported by an American mathematician named Leonard Eugene Dickson, who had heard the story from his mother. When she was a girl, she and her sister had been keen on geometry, both competing against and collaborating with each other. They once “spent a long and futile evening over a certain problem,” only to give up and go to bed, but during the night, in their shared bedroom, Dickson’s mother dreamed of the problem and stated the solution, while asleep, in a loud and clear voice. Her sister heard her, got out of bed, and took notes. At school the next day, the sister gave this (correct) solution to the problem, which Dickson’s mother, though she had dictated it in a dream, had no recollection of knowing.
In general, though, new ideas are far more likely to present themselves to a person who is just waking up, Hadamard notes, adding that he was once jolted out of sleep by a loud noise, and “a solution long searched for appeared to me at once without the slightest instant of reflection.”
During the lull between waking and willing, the haphazard miracles of the liminal mind.
I can remember a dream I had in college—oddly enough, since I don’t usually remember dreams the next day, much less years later—which was about matrices, rectangular arrays of numbers. The matrices of my dream were life-size, with detachable rows and columns that would hover over a person and act upon him or her in some inscrutable way.
In the introduction to an article published in 1990, the mathematician Robert Thomason explains that a dream ushered him toward the work he was presenting, and because of it he chose to include as his coauthor a friend and colleague who’d committed suicide the year before. “The first author must state that his coauthor and close friend, Tom Trobaugh, quite intelligent, singularly original, and inordinately generous, killed himself consequent to endogenous depression,” wrote Thomason. But Trobaugh had returned to him in a mathematical dream: “Ninety-four days later, in my dream, Tom’s simulacrum remarked, ‘The direct limit characterization of perfect complexes shows that they extend, just as one extends a coherent sheaf.’”
Although Thomason, “awaking with a start,” knew the idea was wrong, he pursued the argument, at dream-Tom’s insistence, and discovered the path to the right idea in the shoals of the wrong one, landing quickly upon the results of the article.
Thomason himself died suddenly a few years later, of diabetic shock.
As far as method is concerned, do you make any distinction between invention and redacting?
The Latin cogito, meaning “to think,” derives from a prior meaning that is “to shake together,” notes Hadamard in a footnote to his book. Augustine had observed as much, he writes, and also that intelligo means “to select among.”
Which is to say that cogitation is, at its verbal core, recombination and selection. Hadamard was a friend and admirer of the mathematician Henri Poincaré, and here he follows in the older man’s footsteps. In a 1910 essay titled “Mathematical Creation,” Poincaré characterizes math as practically a form of spontaneous combustion, “the activity in which the human mind seems to take least from the outside world.” As such, he says, it ought to tell us something about the essence of thinking.
What does mathematical creation consist of? asks Poincaré, who blazed his way through a large territory of mathematics and physics by relying on his remarkable geometric intuition. It requires not only the combining of existing facts but the avoiding of useless combinations: making the right choices. The facts worthy of study are those that reveal unsuspected relationships between other facts. Moreover, much of this combining and discarding and retrieving goes on without the mathematician’s full awareness, occurring instead behind the scrim of consciousness.
Case in point: Poincaré’s own struggle to prove the nonexistence of a certain kind of function. He recalls how every day he sat at his worktable for an hour or two, trying different things, with no luck. Then one evening he drank a cup of black coffee and couldn’t sleep. “Ideas rose in crowds,” he writes. “I felt them collide until pairs interlocked, so to speak, making a stable combination.” By the next morning he had the outline of his results, establishing that a class of such functions did in fact exist, and he was able to promptly write up his work.
The prior banging of head against wall is necessary to the revelation, Poincaré insists: “These sudden inspirations . . . never happen except after some days of voluntary effort which has appeared absolutely fruitless and whence nothing good seems to have come, where the way taken seems totally astray.”
The cruelty in all this is that the head-banging hardly guarantees the revelation, that to be an ambitious mathematician is to spend much if not most of one’s time being stuck. Though maybe instead of saying being stuck I should instead say chasing after tops.
André Weil, in his description of the role of analogy in mathematics, those “slightly adulterous relationships” he conjured in one of his letters to Simone, may have valued the chase over the capture, writing that “the pleasure comes from the illusion and the far from clear meaning; once the illusion is dissipated, and knowledge obtained, one becomes indifferent at the same time.”
The flicker of a parallel, the suspicion of a connection, excited him, more so than nailing it down, working out the details. As though knowledge itself were a bit of a letdown: it’s being on the cusp that brings the greater thrill.
André dreams himself back to the École Normale Supérieure. What ease he feels in Paris, home at last, making his way along the riverbank, then down an alleyway and finally to a long, narrow set of stairs, which zigzags up the side of a building made of large white bricks. There, on a high balcony, friends he doesn’t recognize are waiting for him because he is supposed to lead a seminar on the topic of “Bethlehem fields.” When he wakes up he still has the term in his head, reaches for the definition before he remembers that there is no such concept and that he is living in an American town called Bethlehem, Pennsylvania.
Simone dreams she has developed a new physical theory, of continuous atomic states, freedom of the electrons to assume any energy level whatsoever. Gradually she becomes aware that she has
n’t simply invented a theory but has imposed it on her surroundings, which turn more and more diaphanous, which begin to dissolve before her dissolving eyes.
Would you say that a mathematician’s work should be interrupted by other occupations or by physical exercises which are suited to the individual’s age and strength?
One night after having watched an algebra lecture, I dream that I am standing in front of a blackboard, next to a woman I’ve played pickup basketball with (that is, a woman I identify as the basketball player, even though my dream version doesn’t look much like her, not least because her dream hair is bright orange), and I point to a symbol on the board and remind her that the subgroup under discussion is a normal subgroup because it’s the kernel of a homomorphism.
Poincaré speculated that the elements of thought were something like the hooked atoms imagined by the philosopher Epicurus. When the mind is at rest, our thought atoms remain as if hooked to a wall, stationary, and so never meet, but by thinking we agitate them, cogito. As a result they collide, interlock, and surprise us.
Several of his discoveries leaped to mind while he was walking someplace, or, in one case, while stepping onto an omnibus.
Said Georg Cantor of one of his own results: “I see it, but I do not believe it.”
Another night I dream that I’m back at college, only it’s not 1990s college but rather Radcliffe College circa the 1950s, a distinctly black-and-white world as in an old movie, and I’m at a tea where the special guest is André Weil. He’s there to speak about his work, but at the moment he’s drawing an analogy between something in math and a woman’s breasts, he’s smirking and positioning his hands in front of his ribs like he’s supporting the weight of some serious mammaries. The few men in the audience chortle, while the others, the women, are unimpressed. I can tell I’m the only one who wants to cut André some slack—that is, I want the other women to see past the off-putting delivery and appreciate the math.
There’s a question-and-answer period following the talk, and I think of a question that I believe will rescue him, a means of illuminating something about his work, but I’m not called upon; instead the woman sitting next to me is, and in response to her question Weil leans toward her and tells her she has beautiful eyes. After that, as people begin to cluster around him, he starts for the door, pushing through the crowd, and like a reporter trailing an accused man away from the courthouse I catch up to him and yell out a question.
It’s not the one I came up with earlier, though. “How do you pronounce your name?” I shout. Confused, he mutters something, and I say, “Your name, your last name!”
“Vay,” he says dismissively, and marches off.
Naturally the mode of mathematical thinking varies by thinker. According to Hadamard’s informal survey of colleagues in the United States, George Birkhoff would visualize algebraic symbols. Norbert Wiener would think either with or without words. For George Pólya, one word might appear in his head, and ideas would precipitate around it. Hadamard goes so far as to specify the “strange and cloudy imagery” that arises in his own mind as he follows a simple proof about prime numbers, listing the steps of the proof on the left-hand side of the page, his mental images on the right. The images are, for example, “a confused mass,” or “a point rather remote from the confused mass.” In fact, he writes, every time he undertakes mathematical research he develops a set of such images, which helps hold everything together.
Poincaré believed he’d inherited his mathematical talent from his grandmother.
Do you experience definite periods of inspiration and enthusiasm succeeded by periods of depression and incapacity for work?
For much of his life Cantor suffered from depression and was repeatedly hospitalized. In the late nineteenth century he came up with revolutionary ideas about the concept of infinity, which his contemporaries viciously cut down. Beginning in middle age he also dedicated himself to trying to prove that the true author of Shakespeare’s works was Francis Bacon.
Simone dreams she is in a refugee camp, relieved and excited to have reached a place where she might at last suffer boundlessly. She throws away her shoes, then goes looking for a friend she’s never met. What does he look like? How will she know him? By his purity, she thinks. One half of the camp is forest, the other desert. She heads straight for the desert in bare feet that are already starting to burn. Finally! A world cleansed of luxury and split open by such fierce light.
André dreams he can no longer dream. A doctor—his father?—a doctor who doesn’t look like his father but is nonetheless, André understands, Dr. Bernard Weil, gives him the diagnosis. It is just the two of them in a small, square box of a room with metal walls, a metal ceiling. But, but: the objection hovers just out of his reach, he almost knows and is almost capable of presenting the obvious counterexample to this theory that he can’t dream, but he doesn’t quite have possession of it, and rage rises up in him, he wants to strangle the doctor who is and is not his father.
If any persons who have been well acquainted with defunct mathematicians are able to furnish answers to any of the preceding questions, we ask them instantly to be kind enough to do so. In this way they will make an important contribution to the history and development of mathematical science.
“I am trying to answer in brief your questions as well as I am able,” Albert Einstein wrote to Hadamard. “I am not satisfied myself with those answers and I am willing to answer more questions if you believe this could be of any advantage for the very interesting and difficult work you have undertaken.”
8.
“My dear brother,” Simone in France writes to André in America. “My attitude has not changed. I don’t wish to live in America for a whole pile of reasons.” She and their parents are sharing an apartment in Marseilles, and while Bernard and Selma intend to emigrate as soon as possible, Simone wants to remain in France, to suffer whatever the French are made to suffer—“even if a real famine occurred, I would undergo it like the others . . .” André’s case, on the other hand, is different; she thinks he’s where he ought to be, for the sake of his work. “A mathematician is so rare an animal that he deserves to be preserved, be it only on the score of curiosity.”
She would go to America only on one condition, and that’s if she could be sure that it would help her realize her ambition to organize a cadre of front-line nurses. It’s a fixation of hers, a proposal that nurses be dropped by parachute onto the battlefield to treat the injured and dying, though many would surely become casualties themselves.
A flock of kamikaze benevolents, floating above the front, landing among the bleeding, the legless, men drawing their last breaths. They would bandage the wounded and offer comfort, at least until they themselves had their heads blown off. Simone would be among them, naturally. She is desperate to see the plan carried out, petitions officials, brings it up over and over, her fantasy of sacrificing herself converted into a patriotic mission—though not so much patriotic, in her mind, as humane. To plant nurses right in the middle of the slaughter would be an act of defiance, a strike against inhumanity itself.
Reading Tacitus, she learns that it was customary among ancient Germanic peoples, the barbarians and seminomadic tribes of the first century, to bring a young girl, surrounded by elite warriors, to the front line of battle.
It’s hard to see how Simone could’ve been so serious about such an untenable idea. All I can hazard is that she had hit upon a notion so perfectly in tune with her psychology, one that so encapsulated her search for meaning in affliction—her desire to transmute her own alienation into difficult charity and at the same time to summon her own annihilation—that she refused to consider its flaws.
As time goes by she gravitates more and more toward Catholicism, though it wouldn’t be quite right to call her a convert, since (as she explains to the priests whom she ropes into scholastic debates) she’s unsure whether she could truly belong to the church, whether she even wants to belong. Her stated preference
is to suffer alongside the sinners in the wilderness, though in practice she’s more like someone standing just outside the church door, hectoring the priest with questions.
In Marseilles, the terrible housing conditions of Indochinese laborers rile her; she also agitates for refugees from neighboring countries and corresponds with a Spanish anarchist interned in a French camp. More often than not, it seems, the people who inspire her moral outrage are a step removed, different from her. Publicly, at least, she doesn’t express the same kind of alarm about, say, injustices endured by women. She hardly mentions the plight of Jewish people; later she’ll write that the best course for Jews would be to intermarry and assimilate.
In high school I indulged in a more conventional dream of an airborne vocation, thinking I might become a foreign correspondent. This idea may have been shaped in part by Margot Kidder as Lois Lane, climbing up the elevator shaft of the Eiffel Tower in Superman II, and even more so by a certain credit card commercial that used to play on television: a man and a woman living in different places fly to meet each other in some European capital for the weekend (a trip made possible by the credit card, I guess), and although I don’t know whether the pretty actress in the ad was supposed to represent a newspaper reporter, that’s what I thought she was, because, as I remember it, she was wearing a trench coat.
So unusual in Pennsylvania is the aurora borealis that when the sky becomes streaked one summer night with plumes of light, like smoke from some extraterrestrial volcano, and the horizon below turns bright pink, people suspect that the Germans are behind it. André knows better, he’s seen a show of lights like this before, in Finland, but it seems impossible that they would be visible so far south, and so he is invaded by the sense, however absurd, that what they’re witnessing in the sky must be a portent. Portent of what? He sits in silence with Eveline and Alain on the porch of their rented house, all three of them dumbfounded by the spectacle. He pictures his parents, his sister, asleep in Marseilles; the separation, the guilty realization that it’s probably been a day or two since he thought about them; the smallness of his family, the shrinking of possible outcomes and at the same time the tiny being now growing in his wife’s belly, the child of them all.