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
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.
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 hasn’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.”