When I began, I didn’t know how I would decide when I was done. I thought I might take the year-end, statewide exams for high school students, but the first question I read on the algebra exam asked me to solve a problem by using a formula that I had never heard of, and I decided, No good. I wasn’t studying math as a competitive task, and I didn’t want a grade as a measure of whether my work had been worthwhile.
Amie could tell where I was from the questions I asked. She would occasionally say, “You can move on. I think you’ve done enough.” When I seemed stuck on the idea of a too-exquisite grasp of a concept or a procedure, she would say, sometimes with a mild exasperation, “Can’t you move on now?”
I was cut from the herd once more by a process called implicit differentiation. It wasn’t clear to me if I didn’t understand it, or whether I had failed to understand the steps that lead to it—that is, if I had fallen behind earlier and had, in effect, faked my way to it.
When I wrote Amie, she answered, “I’m sorry about ID. It’s a multistep procedure. You have to keep track of what you’re doing. For me it’s very visual. I have to say, ‘What’s going on here? What does this mean?’ There has to be a story that explains it.” Her remarks reminded me of what it is to have an intelligence that is welcoming to such challenging material. I see only steps and abstruse rules, and she sees a story.
“I really think you should move on,” she added. “You really are not going to finish in the foreseeable future at this pace.”
By the age of sixty time no longer seems unlimited. At least that is when I first felt, obscurely, or now and then, that I am running out of time. A few years ago, the writer Roger Angell, who lately turned one hundred, told me, “When I was in my sixties, I thought about death all the time. Now I never think of it.” I don’t think about my end so much as I think that it is unreasonable to assume that the party will last forever. I know three landscapes intimately, two from my childhood and New York City, where I have lived for forty years. I feel close to no longer having sufficient time to learn another one. “I can only buy old guitars now,” Ry Cooder, who is roughly my age, told me. “I haven’t got the time anymore to break new ones in.”
A STUDENT’S RELATION to the world is indirect, and only more so when the student is older. The things that occupy one’s thinking do not bear on ordinary life. If they apply, it is abstractly, in that they might make possible a different future.
I review formulas and equations like an actor reviewing lines. They occupy the forefront of my thinking. It is strange to be older and about as skilled at something as a child is likely to be.
Mathematicians do not make irrational assertions about mathematics, the way the rest of us do about things that we believe are true. We are broadcasters and proclaimers of self-sustaining remarks. It appears we never run out of them. We deliver them even when we think we aren’t. The mathematics that is acknowledged to be true is aloof from human bias and shortcomings. In mathematics no one has to believe anything. There is no question of faith. There are no math opinions about whether something is true or not true.
Human beings deal mostly with incomplete or unproven truths. It would be very difficult, perhaps impossible, to construct an axiomatic catalog of truths that was anything like comprehensive, because ideas of truth don’t agree. The standard of truth would seem to be absolute, but in many cases it’s only partial; what appears true to me is not necessarily true to you. Struggling to understand a math problem is different from struggling to understand an argument. In math, when a voice within me says, Why can’t you understand this? I can’t answer, Because I don’t agree with it.
As I got older, I got better at reading, having read for many more years, of course, but also from maturing, of having developed the capacity to respond more deeply and in more complex ways, and of having worked to make more of myself than what I seemed able to be or fated to be. To grow beyond the restrictions of my raising. I had thought that some part of this greater capacity would lend itself to learning math. Learning math isn’t necessarily congenial to outside influences, though, another way in which math is cold.
Thinking is a strange pursuit. You are making an approach on an idea. It doesn’t want to stay where you can see it clearly or reveal all of itself to you at once. It doesn’t hold a shape, it dissolves, it shows you different sides, it slips away. As hard as I try, before long I realize I am thinking of something else, and I have to start over and reconsider and reevaluate, only sometimes advancing. With calculus I am not infrequently confronted with ideas that I can’t efficiently grasp. They seem just beyond my reach. There was a period near the end of my studying when Amie wasn’t always taking my calls. I imagined her looking at her phone and thinking, Not now. The patience on the part of learned people for half-bright beginners is not inexhaustible.
Day in and out I deal with things that no one I know deals with. It limits my conversation. I can’t in the evening say to a dinner companion, “We have a g(x), what is the function that defines f(g(x))?” although it is a perfectly simple question.
Ever tried. Ever failed. No matter. Try again. Fail again. Fail better.
—Samuel Beckett, Worstward Ho!
Failure has become not a way station, as graduation speakers sometimes describe it, or an obstacle to confront and subdue, but a shadow. It never falls on me entirely, but it never leaves me, either. Perhaps unconsciously and self-destructively, I have chosen a task designed to show that I am less than who I had believed myself to be. Perhaps I flattered myself about my abilities and have taken a serious task lightly and now, having committed myself to putting aside my regular life for more than a year, I am discovering that I might fail for real. I might have enacted something I believe, which is that most difficulties in life are the result of bad judgment. During these periods, I see nothing romantic about failure. It feels like a condition that has separated me from the people around me; I look at others the way a sick person looks at the well.
By Kathryn Schulz in her book Being Wrong, I was made aware of Anne Carson’s poem “Essay on What I Think About Most,” which begins,
Error.
And its emotions.
On the brink of error is a condition of fear
In the midst of error is a state of folly and defeat.
Realizing you’ve made an error brings shame and remorse
Or does it?
Failure as a companion makes me feel a little crazy, deluded in my purposes, like a monk who sees lower angels from the ones that everyone else in the monastery sees. For solace I looked for figures in myths and folk tales who fail repeatedly, but I couldn’t find any. Sisyphus fails of course, but he is fulfilling a punishment, not attempting a task. Parsifal fails to ask the right question, but only once. Repeated failure doesn’t appear to be a mythological subject.
Something I said to a friend, a philosopher, led him to write me:
There are indeed some questions which are unanswerable, and some tasks which are unfulfillable, but many questions and tasks humans propose to themselves are answerable and fulfillable after some period of failure. Nevertheless, it is striking that many humans regard any failure as shameful. The mechanization of much of our lived world has led us not only to have unrealistic expectations of success in all endeavors but also to be both impatient with and frightened by failure. But regarding failure as shameful long precedes the industrial revolution; one of the reasons Socrates struck so many people as odd and even frightening was his lack of this attitude toward failure.
This made me feel encouraged to go on, in my own halting way.
IN FAILING, THOUGH, I had company. After I was nearly done with calculus and was downcast about my performance I heard Amie’s husband, Benson, say one night at dinner, “I’m failing all the time. One time I was describing to someone how hard math is, because you’re failing ninety percent of the time, and a mathematician happened to hear me and he said, ‘Benson, you succeed ten percent of the time, you’re amazing.’ I said, ‘No, I was exaggerating.’”
He shook his head. “Doing math is like I walk you to a wall, and it’s a hundred feet high, and it’s a sheer, straight wall, and it looks provably impossible to climb, but you have to climb it,” he went on. “Everyone says it’s not possible, and I say, ‘Not with your current abilities. Your learning rate has to keep doubling. Levitate is the only strategy. You have to get to the point where you fly, and that’s how you get over the wall.’”
Imagination is the biggest nation in the world, and in deep mathematics human beings can fly.
At times, it is like walking into a room where the parts of a machine are on the floor, some of which you know how to put together, but you don’t know how to make the whole machine, even though you think you understand what each part does, and meanwhile you’re not sure what the capabilities of the completed machine are, either. Some you grasp, and some are to be revealed by its operation. The machine was put together for you the day before, or maybe only moments before, as a demonstration, but there are too many procedures to be consecutive in your mind, or some of them were skipped. They are discrete and until you find an order among them you will not know how to manage the difficulty.
In its higher ranges, mathematics involves a combination of literalness and abstraction. A formula might be abstract in its reasoning and claims, but it is intended to be used literally. “A good mathematician balances between rigorous logic and intuition,” Deane told me. By the time a mathematician reaches the positions that he and Amie and Benson occupy, the path from one intuitive judgment to another is not always clear. You are now and then in the dark parts of the map, where the territory might be infinite. Leaps of imagination are required. You can leap to somewhere you don’t recognize, though, while also aware that every step you took was correct. It is furthermore possible that others might have taken the same steps and arrived somewhere different from where you did. It is a little like giving directions to someone who tells you later that the directions were wrong, but you had told them the way to a place that you’d been to many times.
I had imagined mathematics as a landscape and my studies as a series of tours and encounters, but travel books tend to be cheerful, and when I am despondent about math, I reflect, What the hell am I thinking? This has been so difficult, and I have grown so isolated that I no longer am confident that I understand the parameters of a social exchange—how much to reveal, how deeply to go into a subject, when to withhold a remark, the whole calculus of conversation. I’ve become a kind of mathematical shut-in. I don’t go anywhere except in my mind. My favorite travel books—In Patagonia, Great Plains, The Sudden View, Life on the Mississippi—have weather and anecdotes and changes of scene, not to mention characters, the whole assembly moving the effort along as if in a procession. I keep repeating myself—math is hard, I thought I’d be better at it; math isn’t flawed, I am—but it is because in an abstract realm where one has only oneself for company the scenery doesn’t change appreciably very often, never mind the difficulty of having only a barely workable sense of the language and the landscape’s being obscured from view in many places and in others being harsh and unwelcoming. On such terrain one can’t help but be thrown back on oneself. For long periods, I have occupied the same ground day after day. I feel less the winsome and ingratiating traveler and more like the people I have read about who by a shipwreck or some other disaster were stranded in the Arctic. Each day repeated the day before, and the grueling task conducted amid a spiky landscape of ice that went all the way to the horizon in all directions. Of course I mean only metaphorically and hyperbolically. In those cases, though, each day the landscape looked unaltered, but it had been, subtly, because the ice had moved, because it was sitting on the sea.
Only later is it borne in on me that I am as if sitting on the sea, with the terrain shifting beneath me. The unconscious changes that are subject to different stimuli and a different timeline are not always immediately apparent.
I feel like I’ve been through a romance that has hit the rocks, though. Calculus is walking away, and I refuse to give up on it. I know that breakthroughs arrive after periods of confusion, and that I should welcome the uncertainty; surely there is enough rhetoric in the culture supporting the point, it’s practically a T-shirt sentiment—“Embrace the Confusion”—but I’m not always able to sit easily with the anxiety. I remind myself that the ability to bear anxiety is a trait of emotional well-being. I remind myself of Keats’s concept of negative capability—the artist’s capacity, that is, for accepting “uncertainties, mysteries, doubts, without any irritable reaching after fact and reason,” which he wrote in a letter to his brothers in 1817. The energy that ought to be available to me for learning, though, is still often diverted into the service of an indignant resistance. I feel I ought to be able to manage it and am annoyed that I can’t. In such periods I think, What am I, helpless? Then I have a response and a response to the response and a response to that response until after a while I have forgot where I was in my textbook, and I’m just a disputatious cacophony of voices.
One morning, though, briefly, I have a respite of a kind, maybe a fleeting insight. The frustration of doing calculations is replaced by my intuitively apprehending the position the calculations occupy in a larger design, their ability to make hidden things visible. Even though I often can’t perform them correctly, I see what they are meant to do. I can see the outline of the plan, even if only shakily.
It got me so excited that I had to go outside and walk around until I calmed down.
When I came back in u substitution returned and my poetic intuitions were demised. Once again calculus enumerated all of my faults and the reasons we were ill-suited and insisted that we see other subjects.
“U substitution is a tool,” Amie said. “It’s important, it’s used a lot, but it isn’t fundamental to the discipline of calculus.”
She sensed I wasn’t cheered. “Listen,” she said, “you’re not going to learn calculus at a deep theoretical level. It isn’t possible in a year. You need to be familiar with its workings, but that’s all.”
Then, “You should take a break. You’ve done really, really, really a lot.”
I wasn’t sure if she thought that I had or was only trying to make me feel better, but I said, “If I do, here’s what will happen—”
“You’ll forget the math.”
“Yes.”
“You won’t forget.”
“I suppose it’s there, isn’t it,” I said, “but it doesn’t really feel like it.”
It’s a strange thing I have done, locking myself away. I didn’t think it was strange when I began, because I didn’t foresee how long it would take or how relentless it would be, or, I guess, how obsessively I would pursue it. I have lain awake at night thinking about a lot of things, but I never expected that among them would be mathematics. No one I know has embarked on an undertaking that is entirely interior and reclusive, so there is no one I can ask about how it’s supposed to go. It was meant to be a lark, and it has become a reckoning. And lest I feel prideful, I shouldn’t forget that I’m secluded with disciplines that are handled by adolescents.
“You have my permission to stop,” Amie said.
I considered it, in order to return to a life where I am something like competent. Where my mistakes aren’t a referendum on my abilities. Where they are matters of inattention and haste, missed opportunities, and sometimes poor judgment, which is all that my mistakes in calculus are, but the imperious requirement of a correct answer is a remorseless imperative. In ordinary life, by means of circumstance or luck, I not infrequently escape the consequences. There isn’t a daily record of my shortcomings. I think of a poem by William Meredith that I read when I was young, “Hazard Faces a Sunday in the Decline,” in which Hazard is a painter in a difficult period. “The cat is taking notes against / his own household,” Meredith writes. For me, this reckoning figure resonates.
Wherever I leave off will mean that there is far more that I don’t understand than there is that I do understand. I had thought that there would be a clean break, that having studied junior varsity calculus I would have learned everything about mathematics that someone should be expected to know and could collect my gold star.
A parting look at infinite things: to certain spiritual-thinking people in Russia whose beliefs were not orthodox, Cantor introduced a justification for their practices. By proposing novel infinities and naming previously incomprehensible ones, Cantor had brought these collections into being. In Naming Infinity, Loren Graham and Jean-Michel Kantor discuss Name Worshippers, who believed that repeating the name of God in the Jesus prayer, “Lord Jesus Christ, Son of God, have mercy on me, a sinner,” brought them into a divine presence. In the same way, Michael Harris writes that Russian mathematicians believed that “mathematical objects were brought into being in the course of giving them names.”
Naming makes an ineffable thing real. “Mathematicians sometimes bring into being objects that no one has ever thought of before,” Loren Graham told me. “How do you know it exists? That it isn’t something that you think exists, but you can’t convince other people, like Cantor with set theory? The first step is giving it a name. In the Bible God says, ‘Let there be light,’ and, having named it, there is light, as if the naming had been the creating. How do I know there’s a God? I can name him.” Until Cantor named specific infinities, no one had thought that there might be more than one.
When Cantor was seventy-two, he went back into the hospital. Several times he wrote to his wife, asking to be allowed to come home. Instead, he died where he was, from a heart attack, in 1918.
A quasi-mathematical means of considering God’s existence is Pascal’s wager, which is sometimes said to have introduced probability theory. A person making the wager has to choose a position among four that are offered. Either God exists, or He or She doesn’t (or They don’t, either), and one must decide whether or not to believe. If God exists, and you believe, you are saved. If God exists, and you don’t believe, you lose. If God doesn’t exist, and you believe, you denied yourself pleasures. If God doesn’t exist, and you didn’t believe, you didn’t waste your time. Pascal concludes that the risks of not believing are greater than those of believing, so one might as well believe.
Arguments for the existence of God are called ontological arguments, concerned, that is, with the nature of being. The first ontological argument I know of was made in 1078 by St. Anselm, who describes God as “something than which nothing greater can be conceived.” (“The true infinite or Absolute, which is in God, permits no determination,” Cantor wrote in 1883.) Even a fool would agree that something that is greater than anything else that can be thought of must exist in the mind, since the fool understood the concept, Anselm writes. If it exists in the mind, it exists outside the mind, since such a circumstance would qualify as a greater state of existence than one embodied merely in thoughts. If it exists only in thoughts, then it is both something than which nothing greater can be conceived and something than which something greater can be conceived, a contradiction. Therefore, something greater than which nothing can be conceived exists both in the mind and in the world.
Gödel had an ontological proof, which he kept secret out of a concern that if people believed that he was a deist they might think less of him. In 1970, when he thought he was dying, he told a few people. He died in 1978, and his proof was published among a volume of his Collected Papers, in 1987.
In 2017, some German computer scientists said that they had run trials of Gödel’s proof and that his assertions had held. Gödel’s ontological proof is difficult and not understood entirely even by many philosophers, so I approach it cautiously. All I feel safe saying is that the computer program did not prove the existence of God; it proved that the form of logic that the proof involves, called modal logic, which is essentially an if/then logic, was consistent more or less with itself. To be persuaded of the proof’s outcome, you would have to believe that all the propositions in the proof were true, and not everyone does.
Gödel’s proof, which is an exemplification of St. Anselm’s proof, using different methods, is usually stated in the following way:
Definition 1: x is godlike if and only if x has as essential properties those and only those properties which are positive
Definition 2: A is an essence of x if and only if for every property B, x has B necessarily if and only if A entails B
Definition 3: x necessarily exists if and only if every essence of x is necessarily exemplified
Axiom 1: If a property is positive, then its negation is not positive
Axiom 2: Any property entailed by—i.e., strictly implied by—a positive property is positive
Axiom 3: The property of being godlike is positive
Axiom 4: If a property is positive, then it is necessarily positive
Axiom 5: Necessary existence is positive
Axiom 6: For any property P, if P is positive, then being necessarily P is positive
Theorem 1: If a property is positive, then it is consistent, i.e., possibly exemplified
Corollary 1: The property of being godlike is consistent
Theorem 2: If something is godlike, then the property of being godlike is an essence of that thing
Theorem 3: Necessarily, the property of being godlike is exemplified
Extended to mathematics, this argument suggests an axiom that would answer all mathematical questions. Apprehending this axiom is likely outside of our abilities, the reasoning goes, but that does not mean that it doesn’t exist. The only mind capable of creating such an axiom is God’s. This agrees with Gödel’s incompleteness theory in suggesting that for any system in which at least some arithmetic can be done there are statements that cannot be proved within the system. Two circumstances arise: the axioms will be inconsistent—that is, they might prove, for example, that 0 = 1—or there will be certain true statements, such as 1 + 1 = 2, that they won’t be able to prove. It is also a Platonic notion, since it suggests that mathematics is the result of apprehending truths outside ourselves and that mathematics is not merely a complex structure built by human beings from arithmetic.
In suggesting an absolute explanation for mathematics Gödel arrived where Cantor had—there is an ultimate region in which mathematics resides, and it is located in the mind of God.
MATHEMATICIANS HAVE OBSERVED that unlike other arts, mathematics is the result of centuries of shared effort and in that way resembles a church as much as a creative discipline. I find these thoughts expressed by Robert Langlands in “Is There Beauty in Mathematical Theories?” a talk given at the University of Notre Dame in 2010.
Although I am not equipped to join the church of mathematics, I see that it could be a church. Per Langlands, like a church it represents the work of many people over centuries. There is a primary religious experience in the form of the transcendental engagement of doing mathematics; there are priests in the form of mathematicians who understand the mysteries, and sacred texts in the form of theorems, which instead of being inscrutable like proverbs are unambiguous, although requiring training and special knowledge to comprehend. To the initiates, though, their import is clear: God had a plan and He (or She or They) revealed it in the form of mathematics, and, so as to understand something of the plan’s design and magnificence, the divinity gave us the intelligence to apprehend it, an idea that first appears in the Middle Ages.
You could also have a companion church that believes in the hegemony of beauty and the tenet that math is discovered not created and therefore enigmatic and suggestive of forces impossible to know, and you could have a schismatic branch that believes that mathematics is created by mathematicians and that beauty has nothing to do with it; nevertheless, mathematics is worth worshipping.
MY OWN EXPERIENCE with religion is limited. The only sacred text I am familiar with is the Bible, and I cannot separate it from my father. In the middle of his life, he became a Christian Scientist; I don’t know why exactly. The questions one might ask of the dead pile up, and it is only one question I might ask him. He worked in an office on a high floor of a building in New York City, and from things he said later I pieced together the impression that he had begun to feel deeply anxious when he had to pass an open window. In addition, he had painful headaches.
Our family might have been better off if he had taken a more worldly approach, if he had occupied a couch in an analyst’s office, say, but maybe not. Some people can’t take self-examination and collapse instead of getting better. Or shed their old lives for new ones, the way some people survive car wrecks that kill everyone else. The practice of Christian Science was a lean one, at least for a child. You were made to feel, in a plainspoken way, that if you fell short of the necessary faith, you would pay for it with your well-being. You would get sick and become crippled from polio and God would abandon you.
Part of my father’s spiritual practice was to read from the Bible and from Science and Health early in the morning, before leaving for the train that took him to the city. One morning when I was four or five, I woke early and, wandering downstairs, saw him on the couch at the far end of the living room. A book was open on his lap, and the light from the lamp beside him hid his face. A friend of mine once said, “Your children never want to see you scared.” I wasn’t aware that my father was scared, but I had the feeling that he was doing something, not illicit, perhaps, but solitary and obscurely desperate, and I was unsettled. He kept the books in a drawer in a table beside his bed, and I used to open the drawer and look at them, especially the Bible, with its softbound leather cover and marbled end pages, and wonder what he used it for, what it did for him—that is, what secret did it contain?
The Bible is, among many other things of course, a catalog of antique miracles and a plan for moral behavior. Nearly every page contains a story about how to behave or how not to. Only Dante, perhaps, spent as much time thinking of consequences as the writers of the Bible did. It’s an enlarging book, of course, the way Shakespeare is enlarging, and Dante and Tolstoy. I can read the Bible partly as history or as a key to the states of mind of ancient people or as a guide to the complexities of belief. A good portion of it involves the attempt to find form for the magical. It can make you feel in the presence of holy events and holy people. It can reduce you to a state of wonder. A not inconsiderable number of people believe that the key to it is mathematics. I regard it as a work of art, a repository of human knowledge. A description of the terrible struggle to remain completely alive in the face of harsh circumstances.
TOO MUCH FAITH makes a person liable to being credulous, whereas too much intelligence might restrict faith. Faith that is humility, a belief that we don’t know and might never know the larger design, the intention, the plan if there is one, seems sensible to me, but so does the belief that there is no plan.
It does not ask much of the imagination to regard mathematics, especially as it grows more abstruse, as a trail that followed to its end might bring one into the presence of God in the form of a super-axiom. From the figure described in the Old Testament it isn’t difficult to picture a deity sufficiently pleased with His creation, even sufficiently vain, that He would like its design to be appreciated and so has left traces of its plan in the form of schematics. Or that there is a kind of endgame for humans in finally apprehending the design of the universe, a moment when paradise would arrive on earth. I’m not saying that I believe these things, only that a person can’t say that they couldn’t possibly be true.
In the eighteenth century, the belief that studying mathematics was a form of worship was exemplified by Maria Gaetana Agnesi, an Italian woman who wrote one of the first significant textbooks on calculus. According to a piece about Agnesi by Evelyn Lamb, published on the website of the Smithsonian, Agnesi was drawn to the thinking of the Jesuit philosopher Nicolas Malebranche, who wrote that “attention is the natural prayer of the soul.” A Christian life involved strengthening one’s intellect, and Agnesi considered the study of calculus to be a form of prayer.
The mathematician Leonhard Euler, in the eighteenth century, believed that imaginary numbers, such as the square root of –1, “are neither nothing, nor less than nothing, which necessarily constitutes them imaginary, or impossible.” To Leibniz they were “a fine and wonderful refuge of the Holy Spirit, a sort of amphibian between being and not being.” Leibniz invented the dyadic or binary system, in which all numbers are represented by 0 and 1 and combinations of them. Leibniz equated 0 with nothing, and 1 with God, who had created all things out of nothing. Leibniz was so excited about his system that he asked the president of the mathematical tribunal to China, a Jesuit named Grimaldi, to describe it to the emperor of China. This is reported in the paper “On the Representation of Large Numbers and Infinite Processes,” by Arnold Emch, which was published in 1916. “Leibniz hoped that in this manner the Chinese Emperor might be won over to Christianity,” Emch writes.
A modern example of a distinguished mathematician who was also a God-seeker is Alexander Grothendieck, who died in 2014. I know about him from Deane, who revered him. Grothendieck was an algebraic geometer, meaning that he studied equations that describe physical spaces. Michael Harris describes Grothendieck’s work as “part of a search for total purity.” Another writer said of Grothendieck that following his work one has the “impression of rising step-by-step towards perfection. The face of Buddha is at the top, a human, not a symbolic face, a true portrait and not a traditional representation.” In 1988, Grothendieck fasted for forty-five days, intending to force God to show himself.
Grothendieck was born in Berlin in 1928 and died in a small town in France called Ariège, in the Pyrenees. According to Winfried Scharlau, a German mathematician who has written a biography of him, Grothendieck lived a “very unusual life on the fringes of human society.” François Hollande, the president of France, described Grothendieck when he died as “an out-of-the-ordinary personality in the philosophy of life.”
In the late 1950s and early ’60s, Grothendieck was married and had three children, as well as a son from a woman he wasn’t married to. He sometimes let people who were down on their luck stay at his house for weeks at a time. He taught in Brazil, and at the University of Kansas, and he lectured at Harvard. In 1966, Grothendieck received the Fields Medal, the most prestigious award in mathematics, but to protest the Soviet Union’s having imprisoned two writers, he refused to go to Moscow to accept it. In 1970, when he was forty-two, he quit mathematics altogether.
He became a Buddhist, but in 1980 he took up a type of Christian mysticism, and for many nights in a row he “played chorales on the piano and sang,” Scharlau writes. In 1991, Grothendieck secluded himself and worked on his mathematical meditations. According to Scharlau, these “cover biographical, religious, esoteric, and philosophical themes,” but they haven’t been translated. By the time Grothendieck died he had written thousands of pages. His obituary in The New York Times, written by Bruce Weber and Julie Rehmeyer, quotes the following remarks from his writings: “Among the thousand-and-one faces whereby form chooses to reveal itself to us, the one that fascinates me more than any other and continues to fascinate me, is the structure hidden in mathematical things.”
Plato’s conception of mathematics was metaphysical more than theological. He believed that mathematical truths made the mind receptive to higher truths and an awareness of the Good, an entity different from the Greek gods in that it didn’t look like a person or have supernatural attributes. The Good was the divine force that organized ideas. For later thinkers, understanding that terms of infinity applied in mathematics enabled a person to understand the expansive quality of truth and its place as a concept central to a higher form of existence. The sixteenth-century mathematician Giordano Bruno saw mathematics as a link between the terrestrial and celestial worlds.
One night at dinner, toward the end of my studying, Amie said that, so far as she could tell from the discussions we had had, my book might turn out to be interesting, but she hoped it wouldn’t have too much about Platonism, which startled me sufficiently that I was unable to frame a reply. She is, and I guess I was surprised to learn, a fairly vehement anti-Platonist. The home ground of her discoveries is not a non-spatiotemporal realm, she said, a trifle emphatically for my comfort, but an abstract one created by mathematicians.
“Human beings have shone a light on numbers, and we’ve picked out a logical system,” she said. “You can’t bring God into this. It’s unnecessary.”
After that I shut up around her about Plato.
She also said, “I have to admit I was kind of alarmed when I realized how bad your arithmetic skills were.”
“How did you know that?”
“From the things you would ask.”
Mathematics did not embrace me. It tolerated me. With practice, I developed a low-grade proficiency at finding derivatives and antiderivatives and factoring polynomials, but those are only methods. I was pleased with being able most of the time to describe how fast the train from Omaha was traveling as it passed through Kansas, but that simple competence didn’t resemble the capacity to have thoughts in another language. There is always a gap between mere ability and the invocation of insight or imagination. Of course, I didn’t really expect to get there. I didn’t even expect to get to where I might understand one of Amie’s papers. Although I had viewed calculus as an end in itself, it isn’t; it’s simply an acquisition of practices, a station on the way to an unspecified front.
“Innocently to amuse the imagination in this dream of life is wisdom,” Oliver Goldsmith writes in The Vicar of Wakefield, published in 1766. If one is fortunate, and sometimes I was with mathematics, more by reading about its ideas and its reach than by doing it, but if one is fortunate, one has something that engages one’s attention, a diversion that makes one lose the feeling of time’s passing, as in childhood. Time passes as it does in childhood, because one is adrift among the senses. There is only narrative—this happened, then this happened, and after that … Adults contribute the notion that there isn’t enough time.
I’m not sure what I thought I would accomplish, although I hoped to achieve something more than a fair to middling facility at math. My ambition had been to expose myself to something challenging and to become different from who I was. I didn’t know if things might change gradually or suddenly, or how they might change, or if they would change at all. Just because I wanted them to didn’t mean that they would.
The things that were happening to me, though, were happening somewhere else than in the forefront of my thinking and differently from the way that I had imagined they would happen. What I had thought would be a process of enlarging was instead often a process of concentrating. As a practical matter, I had learned to solve problems tactically, by viewing them as combinations of parts, which was broadly valuable. Meanwhile, though, certain habitual manners and patterns of undermining myself, certain hesitancies, even a kind of self-supporting narcissism, necessary to sustain myself against threats of inconsequence, appeared to have relaxed their hold. Having been shown day in and out not to be useful, they seemed to have withdrawn, quietly, without announcement, and without my having noticed their leaving. Apparently I had enacted Judith Rodin’s belief that mastering a task later in life makes a person more confident. I hadn’t mastered my task, but I had at least kept at it.
I began then to regard failure differently. I had thought of my failures as setbacks amounting to a nearly perpetual defeat, but perhaps they also had other meanings and consequences. Failure as an emotional event was disruptive and in the day-after-day sense was static and oppressive, an affront to my vanity, but failure as a psychic event appeared to be an incitement, a provocation of the what-are-you-going-to-do-about-it kind. In Hinduism there is the story of the churning of the ocean of milk in order to obtain amrita, the nectar of eternal life. The ocean delivers several valuable gifts, including amrita, but the first thing it delivers is a poison strong enough to demolish the world. Shiva, the god who destroys and creates the world, drinks the poison, which turns his neck blue. Shiva destroys things so that they can be re-created in a better form, illusions as well as material things. In challenging myself, perhaps I had been about demising illusions that I clung to. Or perhaps in failing I had simply been discovering what it means to allow things to fall apart.
On the night at the dinner table where Amie shut me up about Platonism, Benson said that doing mathematics was for him a “quasi-religious experience.” He has been working for several years on a problem called Hilbert’s thirteenth problem, which has been open at least since Hilbert proposed it in 1900 but also has elements of problems first studied by the Babylonians in 3000 BCE. (It involves seventh-degree polynomials—that is, equations of the form X7 + aX6 + bX5 + cX4 + dX3 + eX2 + fX + g = 0.)
Everything else fell aside when he worked, he said, and he found himself feeling connected to people who were thinking of the same things hundreds of years ago. “It’s not like you’re talking to God, it’s like you are a god,” he said. As a boy he had been a tennis player with a regional ranking, but doing anything but math had come to seem trivial and silly, although he read at night because mathematics often got him too wound up to sleep.
I asked when he decided to be a mathematician, and he said, “There’s a great mathematician named Dennis Sullivan, and one time we’re with a bunch of people, at a conference in Helsinki, we’re on a boat, and he asked us, ‘When did you know you wanted to be a mathematician?’ Some people said, ‘When I was in grad school.’ I said, ‘I can name the day when I knew I wanted to become a mathematician.’ So Dennis Sullivan looked me up and down, and he said, ‘You were fourteen.’ I said, ‘How in the world could you possibly know this?’ and he said that either he has a theory, or he heard of a theory, that the moment that one wants to become a mathematician, all emotional maturity ends. And he said, ‘Don’t feel bad, Benson, I was thirteen.’”
I wish that I were able to invent such an exchange, because it would suggest that while I hadn’t been any good at mathematics, I had nevertheless absorbed something of its essence.
The incremental way that calculus reckons change suggests the patterns by which our lives advance, either years at a time or slight change upon slight change, much of it either so slight or so grand as to elude ordinary notice. The movement from one second to another. The revolutions of night and day. The years that pass more quickly than we can account for. The inherent motion of life, its rushing river quality. Our lives progress, infinitesimally, but also majestically, one breath, one heartbeat after another, the increases accumulating while the range grows smaller, with perfection as one limit and death as another. If our existences are described by words, they are also described by numbers in their pure states. It is a further application of the unreasonable effectiveness of mathematics, one that is a little chastening to acknowledge. This led me to wonder if what Maxwell loved about calculus was its majestic precision, its beauty, its God’s eye view.
I WROTE AMIE with a calculus question, and she said that what I was trying to learn was considerably beyond introductory calculus and useful only if I were trying to visit the discipline’s higher ranges. “You have my permission to stop,” she wrote, so I did.
I did not foresee that learning adolescent math would lead me to notions of divinity. In my defense I will point out that I did not blaze a trail; I followed footprints worn into history. As a child I sometimes had the sense of an accompanying presence, of something immaterial behind everything. It wasn’t a thought so much as an intuition, a sense of being in the company of, a proximity. This manner of thinking is called Immanence, in which the divine is believed to be among us, as it were, sensed but not seen. So many people have experienced this—after all, it has a name—that it seems quaint to regard it as original or even unusual. I simply add myself to the company of those who have felt it. There is a period of childhood when the balance between conscious awareness and the Unconscious is not weighted so much toward consciousness, when one receives sensations differently and is less inclined to examine them. Everything is too immediate and too novel for examination, and anyway one hasn’t yet developed any examining faculties.
Why have I found this subtext of God knowing so interesting? Partly because it was unexpected, the notion that numbers are hidden in the world and the divine might be hidden in numbers, but also because, like many people, I want to believe in something more than the ordinary terms of life. It’s an ancient human longing, as anyone knows, and there is solace in being a member of a benign and well-wishing human community. You’ll die alone is an insult and a threat. I have found it inspiring to share the company of large thinkers. The non-sneerers. The gropers toward knowledge. The knowledge fluent. The failers and perseverers, and the substantial and vulnerable women and men in all cultures and places and of all races who laid themselves open to inspiration, mystery, and joy. They have raised my spirits. They have extended the boundaries of the lived world, and although I can’t be one of them, I can cheer them on and exult in their accomplishments.
Even if mathematics didn’t seem to want to have much to do with me, I see no reason to nurture a grievance. I can still celebrate it as an idea carried down through history like a sacred knowledge by so many different minds, durable and adaptable and only partly explored and some of it, perhaps even much of it, presently out of reach entirely. It wasn’t my intention to become good at it, I wanted to submit myself to it and allow myself to respond to what it acquainted me with. Surely I wish I had done better at it, but it taught me not to be complacent in my thinking. An experience that only flattered my vanity would have taught me nothing at all.
Studying mathematics made me aware of a natural structure, elusively apparent and perhaps ultimately impenetrable. An implicit orderliness. An unfolding, moment by moment, on an apparently spectacular scale of something that no force can interrupt, something that is perhaps force itself. A trembling quality to life, both fearsome and fragile, a pattern that even to a novice like me is as clear as the grain in a piece of wood. I am aware that this way of regarding the world might be seen as unoriginal and romantic, but I find it consoling nevertheless, and mathematics, by my slight acquaintance, delivered me to it, which is more than I had asked that it do.
I am pleased for what learning of a pure type has done for me. I say pure meaning that I have no means of using algebra, geometry, or calculus; I’m not trying to get into college or preparing for a new career. Despite my resistance and my incapacity, mathematics broadened me. By the close company of a discipline that insisted that I think and reason, I was enlarged. Like the philosopher at the dinner table, I understand the value of inquiry now and am more inclined to listen and less inclined to resist or pronounce. I assume that a problem has dimensions that I haven’t yet grasped or am even unaware of and that only a receptive examination can advance my understanding, meanwhile knowing—what I didn’t before—that all thought, all knowledge, all opinions and beliefs are everlastingly subject to revision.
I am grateful to have learned this. I wish I knew who to express this gratitude to.