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  For

  James Wilkinson,

  Sara Barrett,

  and

  Sam Wilkinson

  As I made my way home, I thought Jem and I would get grown but there wasn’t much else left for us to learn, except possibly algebra.

  —HARPER LEE, To Kill a Mockingbird

  Fall

  1.

  I don’t see how it can harm me now to reveal that I only passed math in high school because I cheated. I could add and subtract and multiply and divide, but I entered the wilderness when words became equations and x’s and y’s. On test days I sat beside smart boys and girls whose handwriting I could read and divided my attention between his or her desk and the teacher’s eyes. To pass Algebra II I copied a term paper and nearly got caught. By then I was going to a boys’ school, and it gives me pause to think that I might have been kicked out and had to begin a different life, knowing different people, having different experiences, and eventually erasing the person I am now. When I read Memories, Dreams, Reflections, I felt a kinship with Carl Jung, who described math class as “sheer terror and torture,” since he was “amathematikos,” which means something like nonmathematical.

  I am by nature a self-improver. I have read Gibbon, I have read Proust. I read the Old and New Testaments and most of Shakespeare. I studied French. I have meditated. I jogged. I learned to draw, using the right side of my brain. A few years ago, I decided to see if I could learn simple math, adolescent math, what the eighteenth century called pure mathematics—algebra, geometry, and calculus. I didn’t understand why it had been so hard. Had I just fallen behind and never caught up? Was I not smart enough? Was I somehow unfitted to learn a logical, complex, and systematized discipline? Or was the capacity to learn math like any other attribute, talent for music, say? Instead of tone deaf, was I math deaf? And if I wasn’t and could correct this deficiency, what might I be capable of that I hadn’t been capable of before? I pictured mathematics as a landscape and myself as if contemplating a journey from which I might return like Marco Polo, having seen strange sights and with undreamt-of memories.

  We reflect our limitations as much as our strengths. I meant to submit to a discipline that would require me to think in a way that I had never felt capable of and wanted to be. I took heart from a letter that the French philosopher Simone Weil wrote to a pupil in 1934. One ought to try to learn complicated things by finding their relations in “commonest knowledge,” Weil writes. “It is for this reason that you ought to study, and mathematics above all. Indeed, unless one has exercised one’s mind seriously at the gymnasium of mathematics one is incapable of precise thought, which amounts to saying that one is good for nothing. Don’t tell me you lack this gift; that is no obstacle, and I would almost say that it is an advantage.”

  I could have taken a class, but I had already failed math in a class. Also, I didn’t want to be subject to the anxiety of keeping up with a class or slowing one down because I had my hand in the air all the time. I didn’t want a class for older people, because I didn’t want to be talked down to and more cheerfully than in usual life, the way nurses and flight attendants talk to you. I could have sat in a class of low achievers, a remedial class, but they aren’t easy to find. I arranged to occupy a chair one afternoon in an algebra class at my old school, where twelve-year-olds ran rings around me. The teacher assigned problems in groups of five and by the time I had finished the first problem they had finished all of them correctly. They were polite about it, and winning in the pleasure they took in competing with one another, but it was startling to note how much faster they moved than I did. I felt as if we were two different species.

  Having skipped me, the talent for math concentrated extravagantly in one of my nieces, Amie Wilkinson, a professor at the University of Chicago, and I figured she could teach me. There were additional reasons that I wanted to learn. The challenge, of course, especially in light of the collapsing horizon, since I was sixty-five when I started. Also, I wanted especially to study calculus because I never had. I didn’t even know what it was—I quit math after feeling that with Algebra II I had pressed my luck as far as I dared. Moreover, I wanted to study calculus because Amie told me that when she was a girl William Maxwell had asked her what she was studying, and when she said calculus he said, “I loved calculus.” Maxwell would have been about the age I am now. He would have recently retired after forty years as an editor of fiction at The New Yorker, where he had handled such writers as Vladimir Nabokov, Eudora Welty, John Cheever, John Updike, Shirley Hazzard, and J. D. Salinger. When Salinger finished Catcher in the Rye, he drove to the Maxwells’ country house and read it to them on their porch. I grew up in a house on the same country road that Maxwell and his wife, Emily, lived on, and Maxwell was my father’s closest friend. In the late 1970s, as a favor to my father, Maxwell agreed to read something I was writing, a book about my having been for a year a policeman in Wellfleet, Massachusetts, on Cape Cod, and this exchange turned into an apprenticeship. Maxwell was also a writer. Around the time he spoke to Amie, he was writing So Long, See You Tomorrow, which is the book I give to people who don’t know his work, because it is regarded as one of the great short novels of the American twentieth century, and I know that if they like it they will probably like the rest of his writing. I loved him, and I wanted to know what he had seen in calculus to delight him. He died, at ninety-one, in 2000, so I couldn’t ask him. I would have to look for it myself.

  The following account and its many digressions is about what happens when an untrained mind tries to train itself, perhaps belatedly. It is the description of a late-stage willful change, within the context of an extended and disciplined engagement, not a hobby engagement. For more than a year I spent my days studying things that children study. I was returning to childhood not to recover something, but to try to do things differently from the way I had done them, to try to do better and see where that led. When I would hit the shoals, I would hear a voice saying, “There is no point to this. You failed the first time, and you will fail this time, too. Trust me. I know you.”

  After a time my studies began to occupy two channels. One channel involved trying to learn algebra, geometry, and calculus, and the other channel involved the things they introduced me to and led me to think about. While it was humbling to be made aware that what I know is nothing compared with what I don’t know, this was also enlivening for me. I am done doing mathematics, so far as I was able to, but the thinking about it and the questions it raises is ongoing. The structure of my narrative reflects this dual engagement. It is organized more or less according to the order in which I learned new things, much as in a travel book one visits places along the writer’s path.

  What did I learn? Among other things, that while mathematics is the most explicit artifact that civilization has produced, it has also provoked many speculations that do not appear capable of being settled. Even those figur
es occupying the most exalted positions in regard to these speculations can’t settle them. A lifetime doesn’t seem sufficient to the task.

  Some things I had to learn were so challenging for me that I felt lost, bewildered, and stupid. I couldn’t walk away from these feelings, because they walked with me in the guise of a gloomy companion, an apparition I could shake only by working harder and even then often only temporarily. There were times when I felt I had declared an ambition I wasn’t equipped to achieve, but I kept going.

  Finally and furthermore and likewise and not least, I had it in for mathematics, for what I recalled of its self-satisfaction, its smugness, and its imperiousness. It had abused me, and I felt aggrieved. I was returning, with a half century’s wisdom, to knock the smile off math’s face.

  * * *

  “HOW DO YOU think this will go?” I asked Amie.

  “If I had to guess, I would say you will probably overthink.”

  “How so?”

  “X is a useful thing. I can solve for it—I can manipulate it—and I can hear you say, ‘What does that mean?’”

  Do I whine like that, I thought, then I said, “What does it mean?”

  “It’s a symbol that stands for what you want it to stand for.”

  “What if I don’t know what I want it to stand for?”

  “See, this is what I’m talking about.”

  “Well, wait, that’s—”

  “Here is some advice,” she said firmly. “I get it that you try to put things into a framework that you can understand. That’s fine, but at first, until you become comfortable with the formal manipulation, you have to be like a child.”

  She must have seen something in my expression, because she added, “To be a good mathematician you have to be very skeptical, so you have the right temperament.” Then, “It’s possible I can explain algebra and geometry to you in a way that you’ll grasp, but we might have trouble with calculus.”

  2.

  So far as I can tell, mathematicians welcome novices but provisionally. They know if an amateur has trespassed the boundaries of his or her understanding, and they are prone to classifying. This tendency is displayed in the essay “Mathematical Creation” by Henri Poincaré, which appears in the issue of the philosophical journal The Monist for July of 1910. It begins, “A first fact should surprise us, or rather would surprise us if we were not so used to it. How does it happen that there are people who do not understand mathematics?”

  To Poincaré, mathematics is a matter of reasoning. People can reason through ordinary circumstances, why can’t they reason through slightly demanding chains of mathematical symbols when those chains are only smaller and simpler chains connected to one another?

  Because people remember the rules only partly, he says, and what they remember they use wrong. More than rules they only half recollect, they should follow a problem’s logic.

  This had not been not my experience as a boy or now, either. My experience has been that I might understand what rule applies, but I don’t necessarily understand how to employ it or why it applies in one case and not in another which seems the same case or very closely related. Or I don’t know which in a series of rules to use first and in what order the rest should follow. It has been as if I were trying to read but because of some deficiency or inhibition saw only single words without understanding that they formed sentences.

  According to Poincaré, most people have ordinary memories and spans of attention. Such people are “absolutely incapable of understanding higher mathematics.” Others have a little of the “delicate feeling” necessary to go with powerful memories and spans of attention, and so can master details and understand principles and sometimes apply them, but these people will never create mathematics. A final group, an elite, has the delicate feeling in various degrees and so can understand mathematics and even if their memories are nothing exceptional can create mathematics to the extent that their intuitions have been developed. I am a hybrid of the first and second class, but mainly of the first class, the ordinary one.

  In the essay “A Mathematician’s Apology,” published in 1940, the British mathematician G. H. Hardy writes, “Most people are so frightened of the name of mathematics that they are ready, quite unaffectedly, to exaggerate their own mathematical stupidity,” but mathematicians don’t usually like it when people say that they can’t do math and especially when they say that they don’t see the point of trying to. Mathematicians tend to regard such an attitude as taking pride in being ignorant. I don’t think that mathematicians realize, though, that what is opaque to many, maybe most, of us is clear to them, and that they have as if been granted special circumstances.

  What went on in the minds of the boys and girls whose work I copied, I had no idea until I came across the following sentences in The Weil Conjectures, a kind of memoir and imaginative biography of André and Simone Weil by Karen Olsson, who got her degree in math at Harvard. Obscurely echoing Poincaré, Olsson describes feeling as an adolescent that the principles of algebra and geometry “didn’t need to be taken in and memorized, the way you had to take in and retain other things. They were, it seemed, already at hand. As though there were simply some latent machine I could turn on with logic, and then! An entire world I never suspected.”

  I studied music in college and sometimes when math was trampling on me now I would tell myself that it doesn’t make sense that someone can learn a complicated practice of one kind and not another, music or languages or philosophy, say, but not math. If the capacity of our brains were the explanation for abilities with numbers, then it would seem strange as a matter of adaptation that some of us had a region of the brain developed for mathematics and some didn’t, since there would be a striking and almost disabling deficiency in terms of numbers among those of us who were without the attribute, and in terms of its evolutionary advantage the attribute would appear to be all but inconsequential and maybe even largely useless.

  * * *

  TO PREPARE FOR our meeting, Amie suggested that I read Algebra for Dummies, which I had hardly begun when it was borne in on me that it didn’t matter who it was for, it was still algebra. I read with a companion self, a twelve-year-old boy who had no desire to sit in algebra class again. Algebra, he reminded me, if you really thought about it, was impossible. We had already proved that. Why prove it twice?

  I was surprised to find it difficult. I assumed that in growing older I had also grown smarter. High school math I expected to be totally within my capabilities. I expected to find myself thinking, How could I have found this so challenging?

  As a boy I didn’t know how to learn anything. So far as I can tell, learning involves an ability to see things consecutively and according to a set of relations. As a child I felt overwhelmed by all that went on around me. I don’t think that I was more sensitive than children usually are, but I grew up in a turbulent household. Rather than organize my thoughts, I looked for reasons to avoid having them. By myself in the woods, turning over stones to find salamanders and catching pond turtles, is how I passed much of my childhood. I knew that it was strange to be so solitary, but I didn’t know how to be different. I have my own theories about why this is so, but I don’t think they are sufficiently diverting to justify lingering any longer in Confession Gulch. As a boy I turned the pages of my algebra textbook and did what part of my homework I could while I waited for a breakthrough, a grace, an illumination that seemed to have arrived for the bulk of my classmates, and I wondered why it hadn’t for me.

  Because I had been good at arithmetic, I was placed in the advanced math class, which meant that I would take algebra in eighth grade. A few days before the end of summer vacation, I realized that I had forgot how to divide. I thought that if I told anyone, I would have to go to seventh grade again. At the bus stop, on the first day of school, I challenged another boy to divide two big numbers and closely observed how he did it.

  I remembered this when I read a mathematician’s remark that algebra is a fo
rm of arithmetic. My impression was that algebra was less a subject than a practice into which one was inducted by the algebra priests after a series of mortifications. The letters and equations that the teacher drew on the board did not seem related to the numbers I had handled in other classrooms. For one thing, a problem in arithmetic was vertical, one number beneath another, and a problem in algebra, an equation, was horizontal. I felt as if in a permanent present, unable to see how the past and the future were joined. In Ulysses James Joyce writes that the present is the drain that the future goes down on its way to becoming the past.

  3.

  When I read the observation made in 2007 by the Russian mathematician Yuri Manin that algebra was once connected to language, I see some of why I am again having difficulty. The algebra that the ancients knew in Egypt, India, Babylonia, Greece, and Persia is essentially the algebra that is taught in high school. The antique version is called rhetorical algebra, because the problems were described in prose and not in symbols or letters. The version that followed, called syncopated algebra, used abbreviations for common operations. Letters as symbols arrived in the sixteenth and seventeenth centuries, when mathematics began to figure more in commercial and scientific life, and needed to be easier to use and more accurate. The modern version is called symbolic algebra. The symbols are arbitrary. A simple equation tends to be solved for x but it could be some other letter, too. In Geometria, published in 1637, Descartes suggested that the earlier letters of the alphabet represent known quantities and the ones after p represent unknown ones, which remains the general practice in teaching. In higher mathematics, though, the conventions are not universal. Amie’s field is dynamical systems, which studies the behaviors of a structure such as a solar system that is constrained by particular rules. “In my work,” she wrote me, “certain letters are used to symbolize certain things—p, q points; t time; x a numerical unknown; r a positive number; ε, epsilon a small number; μ, mu a measure; z, w complex numbers; A a matrix; M a manifold; n, m integers; X a set, and so on.” I appreciated her adding, “and so on,” as if she imagined that I were reading along and thinking, That’s probably what I’d do, too.