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 figures 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.”
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 form 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.
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.
The movement from rhetorical algebra to symbolic algebra resembles the passage from arithmetic to algebra. If no one tells you that you are leaving one field for another and instead behaves as if the fields are the same, even though they appear to be different, it is easy to become confused, at least it confused me. I think this confusion is peculiar to mathematics, which has a quality of otherness. It seems to be both literal and abstruse. Extremely complicated chains of symbols can express a single, unambiguous thought, whereas language can be made literal only by reducing it to simplest terms, often prohibitory: “Thou shalt not kill.” “No smoking.” “Keep Off.” Computer translations of literary texts rarely satisfy because too many choices are involved, having to do with not only which words to use but also in what arrangement so as to serve the most explicit meaning, let alone the writer’s intentions, let alone art. Mathematics is severe and faultless. It became the language of science because of its precision. The theory of relativity can be written in prose, but e = mc² is more succinct.
ARITHMETIC BECAME ALGEBRA because the ancients found themselves doing repeated calculations to compute, say, the area of a piece of farmland and, while some of the calculations could be done in one’s head, if a person thought carefully enough, it was easier to automate them. Eventually a symbol came to stand for the quantity that one was trying to solve for, which is where my difficulties began. Some people are sufficiently comfortable manipulating numbers that manipulating symbols comes naturally to them. They might find the solving of an algebra problem to be something like a game—apply some rules and receive an answer. I simply found it mystifying.
“If you’re the kind of person who has trouble keeping track of things, whose mind likes to wander down certain paths and doesn’t want to be put on task, like maybe you are, I think solving an algebra problem can be very difficult,” Amie said. “The presence of symbols makes it seem that there are too many possibilities.”
I wish someone had said on my first day in algebra class, “To start, all you need to know is that you are answering a problem whose solution, instead of involving a single unknown, as it does in arithmetic, involves a second unknown, which we call x.” Or, “Algebra is arithmetic, but you just don’t know at the beginning what all the numbers are.” The arrival of x inserts an abstraction into what in arithmetic had been a literal exchange. The simplest algebra problems (I know now) are only one degree removed from arithmetic. Instead of 2 + 5 = x, there is, say, 2x + 5 = 9, which is easy enough to be solved visually, but algebra insists on procedures—subtract 5 from 9 and divide 4 by 2, still arithmetic, but the mind has to accommodate deferring an answer in order to assess what procedures apply and in what order; dividing both sides of the equation by 2 to begin arrives at the same answer but less straightforwardly.
In the first weeks of algebra class, I felt confused and then I went sort of numb. Adolescents order the world from fragments of information. In its way adolescence is a kind of algebra. Some of the unknowns can be determined, but doing so requires a special aptitude, not to mention a comfort with having things withheld. Furthermore, straightforward, logical thinking is needed, and a willingness to follow rules, which aren’t evenly distributed adolescent capabilities.
In Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers, by Joseph Mazur, I read the following: “At its surface, algebra seems to be the art of manipulating symbols according to some rules for doing so. But then, the modern student knows that all that has to be done is to translate the problem into symbolic notation, and let the rules of symbolic manipulation take it from there.” I can appreciate such remarks now, but if someone had said that to me when I was twelve years old, I wouldn’t have known if he was messing with me or not.
I found a description of my circumstances in “Mysticism and Logic,” by Bertrand Russell. “In the beginning of algebra, even the most intelligent child finds, as a rule, very great difficulty,” Russell writes. “The use of letters is a mystery, which seems to have no purpose except mystification. It is almost impossible, at first, not to think that every letter stands for some particular number, if only the teacher would reveal what number it stands for.”
Arithmetic is unequivocal. Algebra is a means for making statements that apply widely.
I SAT IN algebra class afraid that I would be called on and handed in homework that I believed was evidence of my dullness of mind. I can understand Russell now when he writes:
It is not easy for the lay mind to realise the importance of symbolism in discussing the foundations of mathematics, and the explanation may perhaps seem strangely paradoxical. The fact is that symbolism is useful because it makes things difficult.
But how little, as a rule, is the teacher of algebra able to explain the chasm which divides it from arithmetic, and how little is the learner assisted in his groping efforts at comprehension! Usually the method that has been adopted in arithmetic is continued: rules are set forth, with no adequate explanation of their grounds; the pupil learns to use the rules blindly, and presently, when he is able to obtain the answer that the teacher desires, he feels that he has mastered the difficulties of the subject. But of inner comprehension of the processes employed he has probably acquired almost nothing.
On my second engagement I tell myself that in a problem there is something that I need to find. Looking for it requires a detachment I can’t always enact. I wrote Amie, “Today I went through several pages of problems and got every one of them wrong. Very discouraging.”
To make me feel better, she wrote, “I am trying to imagine a similar task I could set for myself. Something like learning Japanese?”
I told a friend, Deane Yang, who is a professor of mathematics at NYU, how often I was wrong, and he said, “Getting things wrong is the trick of our trade.”
I still don’t know what he meant.
It requires unusual abilities to become a mathematician, that and years of painful training in which the intellect is forced to bend upon itself.
—David Berlinski, A Tour of the Calculus
Reading Algebra for Dummies, I am surprised to find that I recall almost nothing of algebra. I had got lost so quickly that very little had made an impression. I can still recite the “Prologue to the Canterbury Tales” in Middle English, which I was required to learn as a senior in high school. I remember, “Kingdom, phylum, class, order, family, genus, species.” And that in 585 BCE, Thales predicted an eclipse of the sun. With algebra, I come up empty.
When I thought I had read a sufficient amount, I went to Chicago to see Amie. I sat beside her on a couch in her living room. I held my pencil and notebook ready. My manner was like that of the novice on his first day in the monastery poised to have the head monk reveal how to find God. She said, “I’m not sure where to start.” I had been expecting her to say something like, “There’s a train in Omaha heading for Dallas and leaving at three in the afternoon.” Instead, we sat silently. A dog barked. I smiled weakly.
There is a belief among certain academics that a subject is less efficiently learned from an adept than from someone who is studying it or has just finished studying it. The adept’s long acquaintance makes it difficult for him or her to see the subject in its simpler terms or to appreciate what it is like to approach the subject as a greenhorn. As I sat uneasily beside Amie, it was borne in on me that I was asking a mathematician with a trophy case whose standing is international to teach me math that she had learned nearly half a century earlier as a precocious child and hadn’t used since. Furthermore, she had for the most part embraced it intuitively and then layered upon it many other practices, explorations, and diversions. Her learning had a kind of family tree of associations, and all I had was what I had picked up piecemeal in a few weeks of study.
What I might have said to her of the difficulty I was having was, “Pretend you were a child receiving this information for the first time. Can you remember how you heard it so that it was sensible to you?” A further complication developed, which is that what is difficult for me had not been difficult for her, and I don’t think she could see why I had such trouble learning what she had found simple. “How do you think you would have thought about this if you hadn’t been able to think of it as you had,” is the kind of question I would have had to ask, and being philosophical more than practical, it isn’t a discussion that would have solved my difficulties. I might have learned something about her, but not likely anything about math.
In On Proof and Progress in Mathematics, William Thurston writes, “The transfer of understanding from one person to another is not automatic. It is hard and tricky.” We had been working together in a halting way for several weeks when I realized that I was going to have to learn a lot of this on my own.
As a child, I don’t think I grasped the concepts of the general and the specific. It seems simple now. Another algebra day-one statement, ideally: in arithmetic the terms are particular; in algebra we are going to make generalizations, meaning we are going to do math without knowing all the terms. In An Introduction to Mathematics, by Alfred North Whitehead, I come across simple information that I might have found helpful.
“The ideas of any and of some are introduced into algebra by the use of letters, instead of the definite numbers of arithmetic. Thus, instead of saying that 2 + 3 = 3 + 2, in algebra we generalize and say that, if x and y stand for any two numbers, then x + y = y + x. Again, in the place of saying that 3 > 2, we generalize and say that if x be any number there exists some number (or numbers) y such that y > x.”
Whitehead gives five examples of the fundamental laws of algebra:
x + y = y + x,
(x + y) + z = x + (y + z),
x × y = y × x,
(x × y) × z = x × (y × z),
x × (y + z) = (x × y) + (x × z).
The first is the commutative law of addition; the second is the associative law of addition; the third and fourth are the commutative and associative laws of multiplication; and the fifth is the distributive law of addition and multiplication. As for the concision that symbols provide, Whitehead writes that in prose the first rule, instead of being four letters, would be, “If a second number be added to any given number the result is the same as if the first given number had been added to the second number.”
I forced myself to advance. I say advance, but sometimes only time was advancing. My progress might be sideways and sometimes backward. On the other hand, I had no standard to compare myself to. I had never known an older person who was trying to learn math. Older self-improvers usually memorize poetry or study a language, which they can practice with other people. I was able only to sit by myself in a room and review mathematical rules and terms in the hope of making them familiar. I won’t say that it was like learning prayers, but it had an in-the-service-of feeling, as if I were secluded. It was similar in that prayers, like mathematical procedures and principles, have specific applications.
Meanwhile, a part of me was resisting the effort, perversely, as if there were a pleasure in failing, or at least obstructing, even if a sour one. The energy being claimed by the resistance I might have used for the task, and until I had it, I wouldn’t be firing on all cylinders. Other days the problems tipped over like targets in a shooting gallery, and I went ahead intrepidly.
Possibly not everyone knows that algebra is thought to be the contribution, although maybe not entirely the invention, of a Persian mathematician and librarian named Muhammad ibn Musa al-Khwarizmi, who lived in Baghdad in the ninth century. Al-Khwarizmi wrote a book called Al-Jabr W’al Muqabalah, which translates to “Calculation by Restoration and Reduction.” Al-Jabr has been translated to algebra and is the first time the word appears. In Imagining Numbers, the mathematician Barry Mazur says that al-jabr and al-muqabalah also refer to processes. “Al-jabr is the operation of moving quantities from one side of an equation to the other,” he writes, and “al-muqabalah is the operation of collecting ‘like’ terms.”
Algebra perhaps had antecedents, according to a scholar named Peter Ramus, writing in the sixteenth century. In the entry for algebra in the eleventh edition of the Encyclopedia Britannica, Ramus is the source for the assertion that “there was a certain learned mathematician who sent his algebra, written in the Syriac language, to Alexander the Great, and he named it almucabala, that is, the book of dark or mysterious things,” which is a pretty good title for a book about algebra.
For a reason I don’t know, perhaps from taking things too literally, historical accounts of algebra often mention that Jabr is from the verb jabara and means “to join” and that an algebrista in Spain was a “bone-setter.” Al-Khwarizmi wrote that his book concerned “what is easiest and most useful in Arithmetic, such as men constantly require in cases of inheritance, legacies, partition, lawsuits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computations, and other objects of various sorts and kinds are concerned.”
I am not alone in finding algebra largely incomprehensible. Darwin, another self-improver, studied mathematics with a tutor during the summer of 1828, when he was nineteen. “The work was repugnant to me, chiefly from my not being able to see any meaning in the early steps in algebra,” he writes in his autobiography.
In the theater of my mind, my adult self was prepared to step in with an I’ll-handle-this attitude to defend the boy I had been against mathematics. As an older, somewhat educated person, I could see that mathematics was wrong, because based on illogical and inconsistent propositions. It was taught to children because they are impressionable. Adults would see through it. The rules and procedures pressed on children amounted to an indoctrination and were not rules so much as articles of faith.
I carried this attitude into my renewed engagements. I practically enameled myself with it. When algebra wouldn’t yield, I adopted, as Amie had predicted, a position of over-literalness, which viewed math skeptically, as a con, really. The more overwhelmed I became, the more I insisted that math submit to being interrogated. I believed that I could refuse to accommodate math’s self-serving willfulness. Why anyone had tolerated it was a question I couldn’t answer. It seemed like being a mathematician was like being in a cult. In exchange for accepting a wagonload of irrational claims, you lived in a perfectly ordered world.
Amie I figured had agreed to these arrangements before she was old enough to see that they were unfounded. Because they were told to her by adults whom she trusted, she accepted them, and had lived for years under principles that she never had realized were unsound. I looked forward to disabusing her, which I think was unworthy of me, but I also felt sympathy for her situation. To have one’s lifelong assumptions overthrown in middle age is not a simple matter. It needed to be done with care and consideration. Tread carefully, I thought.
Misreading a symbol or failing to register the meaning of one, I am sometimes lost for days, left in the dust of the algebra train as it heads for the horizon without me. It is a commonplace that to the degree that mathematics is an imaginative pursuit it is also an art, but such a thing does not happen to me in the other arts. I can find pleasures in a book or an artwork or a piece of music that I don’t completely understand. In any other serious field of imaginative work there is no necessarily correct interpretation, but in mathematics you must be certain. Eventually, what you don’t know will stop you, ask for your papers, and detain you for questioning.
Practicing other arts, one can proceed, at least a certain distance, as an innocent and even blindly. A mathematician can also proceed blindly, but not as a novice. The mists and darknesses are only for adepts. As a writer, a painter, or a musician your limitations assert themselves sooner or later, but you might go a ways before they do. Occasionally, they become part of your style. I read once of David Hockney’s answering the question, Why do your shoeless figures always have socks on, by saying, I can’t draw feet. You can’t do math without an awareness of what is behind you, the stately progressions, the panorama of understandings, findings, and breakthroughs. Mathematics is rigid, but for those who comprehend it, the rigidity becomes liberating, a kind of touchstone from which you can launch journeys and to which you can confidently return. Math is modern and historical at the same time. Nearly all beginner math—that is, algebra, Euclidean geometry, and calculus—was known in the eighteenth century and in the case of algebra and Euclidean geometry is ancient.
As for inconsistencies, I collided early with pi, which has multiple and unconnected uses. In fact it seems nearly ubiquitous, an apparition hovering between the background and foreground in a multiplicity of mathematical statements, which I regard as suspicious. Its ubiquity makes it appear to have no real identity; it seems more like a placeholder than a real thing. Perhaps like me not everyone recalls that pi, an endless number beginning 3.141, is the ratio of a circle’s circumference to its diameter. It is also the equivalent of 180°, because π = c/d, circumference over diameter. A diameter is equal to twice the radius, the radius being the line from the center of a circle to its edge, so diameter equals 2r. By convention, a circle’s circumference is 360 degrees when the radius is equal to 1. Pi then equals 360°/2r or 180°. Pi makes so many other appearances in mathematics, though, and is useful in so many situations that it seems absurd. It seems like a mathematician at a loss just writes something like, “Thus we see that,” and adds pi, and it’s quitting time.
Amie told me that pi appears so often because many mathematical formulas derive their features from repeating patterns. They exemplify pi, but not in ways that are always immediately apparent.
“Have you ever heard of Buffon’s Needle?” she asked. “It’s another way of computing pi. You take parallel lines on a piece of paper. You space them an inch apart, because that’s exactly the length of your needle. Then you toss the needle into the air like you would a coin, and you count the number of times it hits one of the lines. It will never hit both, because it’s exactly the length of their separation. That’s a probability zero event. Anyway, you divide the number of hits by the number of tosses. That answer’s going to be some proportion less than one, I believe it’s 2/pi. There is a lot of symmetry in this problem. Pi has to do with anything that involves periodicity or cyclical behavior.”
This only deepened my reservations.
PROBABLY HUNDREDS AND maybe thousands of mathematicians have written books about mathematics. I wasn’t equipped to write from a scholarly perspective or an insider’s one, either. As an older novice, but one with a certain acuity, I hoped I might see things that mathematicians had overlooked as being too familiar or perhaps had never even noticed.
It wasn’t long before I had such an insight. I was staring at a page of equations, and it was borne in on me that mathematics is a language and an equation is a sentence. The subject appears on the left-hand side of the equal sign, along with the verb in the form of the transaction being conducted, and the figure on the right-hand side is the object: two old friends, 4x and 5, go walking and meet 5’s uncle, 13.
I was pleased with this insight, which struck me as deep and even lyrical. It cheered me in light of the discouragements I had encountered and reinforced my sense of purpose.
Mathematicians know what mathematics is, sort of. I have heard: mathematics is the craft of creating new knowledge from old using deductive logic and abstraction. The theory of formal patterns. Mathematics is the study of quantity. A discipline that includes the natural numbers and plane and solid geometry. The science that draws necessary conclusions. Symbolic logic. The study of structures. The account we give of the timeless architecture of the cosmos. The poetry of logical ideas. Statements related by very strict rules of deduction. A means of seeking a deductive pathway from a set of axioms to a set of propositions or their denials. A science involving things you can’t see whose presence is confined to the imagination. A proto-text whose existence is only postulated. A precise conceptual apparatus. The study of ideas that can be handled as if they were real things. The manipulation of the meaningless symbols of a first-order language according to explicit, syntactical rules. A field in which the properties and interactions of idealized objects are examined. The science of skillful operations with concepts and rules invented for the purpose. Conjectures, questions, intelligent guesses, and heuristic arguments about what is probably true. The longest continuous human thought. Laboriously constructed intuition. The thing that all science, as it grows toward perfection, becomes. An ideal reality. A story that has been being written for thousands of years, is always being added to, and might never be finished. The largest coherent artifact that’s been built by civilization. Only a formal game. What mathematicians do, the way musicians do music.
Bertrand Russell said that mathematics, by its nature as an explorative art, is “the subject in which we never know what we are talking about, nor whether what we are saying is true.” Darwin said, “A mathematician is a blind man in a dark room looking for a black cat which isn’t there.” In Alice’s Adventures in Wonderland, Lewis Carroll has the Mock Turtle say that the four operations of arithmetic (addition, subtraction, multiplication, and division) are ambition, distraction, uglification, and derision. A complicating circumstance is that mathematics, especially in its higher ranges, is hard to understand. It begins as simple, shared speech (everyone can count) and becomes specialized into dialects so arcane that some of them are spoken by only a few hundred people in the world.
No scripture is as old as mathematics is. All the other sciences are younger, most by thousands of years. More than history, mathematics is the record that humanity is keeping of itself. History is subjective and can be revised or manipulated or erased or lost. Mathematics is objective and permanent. A² + B² = C² was true before Pythagoras had his name attached to it, and will be true when the sun goes out and no one is left to think of it. It is true for any alien life that might think of it, and true whether they think of it or not. It cannot be changed. So long as there is a world with a horizontal and a vertical, a sky and a horizon, it is inviolable and as true as anything that can be thought.
Mathematicians live within a world that is invincibly certain. The rest of us, even other scientists, live within one where what represents certainty is So-far-as-we-can-tell, this-result-occurs-almost-all-of-the-time. Because of mathematics’ insistence on proof, it can tell us, within the range of what it knows, what happens time after time.
As precise as mathematics is, it is also the most explicit language we have for the description of mysteries. Being the language of physics, it describes actual mysteries—things we can’t see clearly in the natural world and suspect are true and later confirm—and imaginary mysteries, things that exist only in the minds of mathematicians. A question is where these abstract mysteries reside, what is their home range. Some people would say that they dwell in the human mind, that only the human mind has the capacity to conceive of what are called mathematical objects, meaning numbers and equations and formulas and so on, the whole glossary and apparatus of mathematics, and to bring these into being, and that such things arrive as they do because of the way that our minds are structured. We are led to examine the world in a way that agrees with the tools that we have for examining it. (We see colors as we do, for example, because of how our brains are structured to receive the reflection of light from surfaces.) This is a minority view, held mainly by neuroscientists and a certain number of mathematicians disinclined toward speculation. The more widely held view is that no one knows where mathematics resides. There is no mathematician who can point somewhere and say, “That is where math comes from,” or “Mathematics lives over there,” while maybe gesturing toward magnetic north and the Arctic, for me a suitable habitat for such a coldly specifying discipline.
The belief that mathematics exists somewhere else than within us, that it is discovered more than created, is called Platonism, after Plato’s belief in a non-spatiotemporal realm that was the region of the perfect forms of which the objects on earth were imperfect reproductions. By definition, the non-spatiotemporal realm is outside time and space. It is not the creation of any deity, it simply is. To say that it is eternal or that it has always existed is to make a temporal remark, which does not apply. It is the timeless nowhere which never has and never will exist anywhere but which nevertheless is. The physical world is temporal and declines, the non-spatiotemporal one is ideal and doesn’t.
A third point of view, historically and presently, for a small but not inconsequential number of mathematicians, many of them exalted ones, is that the home of mathematics is in the mind of a higher being and that mathematicians are somehow engaged with Their thoughts. Georg Cantor, the creator of set theory, which in my childhood was taught as part of the “new math,” said, “The highest perfection of God lies in the ability to create an infinite set, and its immense goodness leads Him to create it.” And the wildly inventive and self-taught mathematician Srinivasa Ramanujan, about whom the movie The Man Who Knew Infinity was made in 2015, said, “An equation for me has no meaning unless it expresses a thought of God.”
In Book 7 of Republic, Plato has Socrates say that mathematicians are people who dream they are awake. I partly understand this, and I partly don’t.
IN THE ISSUE of the Bulletin of the American Mathematical Society for October 2007, Terence Tao, whom many of his contemporaries regard as the greatest living mathematician, whatever that means, published a paper called “What Is Good Mathematics?” It might be mathematics that solved a complex problem, Tao says. Or used a technique handsomely or invented a new one. Or combined areas that hadn’t been combined before. Or one that by means of insight simplified an important concept or found a unifying one. Or discovered something not known before (and it interests me that he uses discovered instead of created). One with an important application to fields other than math, such as statistics or computer science. One that illuminated an argument. Or was visionary and had the potential to provoke new work. Or was rigorous. Or beautiful, or elegant, or creative, or useful, or strong. Tao regards all of these circumstances as consequential and does not arrange them in a hierarchy.
Fields are not inherently robust, Tao says. He imagines a field might become “increasingly ornate and baroque.” Or have brilliant conjectures but few prospects for proofs. Or consist of a collection of problems without a theme that unifies them. Or one that has lapsed into pure technique and grown lifeless. Or another grown lifeless because it restates only classical truths, even if more simply, elegantly, or succinctly.
WITHIN A MATTER of weeks, I had read that equations were sentences and mathematics was a language so often that I realized they were clichés. This was deflating.
The mathematician Alonzo Church, who taught Alan Turing, told another of his students, David Berlinski, “Any idiot can learn anything in mathematics. It requires only patience.” I would sit with a pencil and paper trying to solve an algebra problem and sometimes I could go only so far before my mind would halt, because I had used up what little I knew that might apply. It hadn’t occurred to me to think of algebra as the bright boys and girls I had been among had thought of it, as a series of related procedures. They were constructing a map. I was collecting postcards from places where anxiety or incuriousness had kept me from leaving my hotel.
In my second encounter I was subject to the same limitations with respect to equipment and attitude that I had been subject to as a child. No matter how we change or what happens to us, something fundamental seems to insist on staying the same or at least coming along for the ride.
Solving algebra problems, I knew what was to happen— I was to find the number represented by x—but I couldn’t without understanding the rules to reveal what x stood for. This is a consequence of there being only one answer to a student math problem, which is designed less to make a person think than to use theorems and properties learned lately. It occurs to me only now that when I was a boy I might have looked at the lesson that the problems were illustrating and understood that the homework was to be solved by those methods. The textbook’s structure was deliberate and not arbitrary, as it seemed to be. The problems on page 164 were asking me to use the information from pages 162 and 163, not from page 34 or information I would not receive until page 310. I saw my math textbook less as a practical manual than as an inscrutable text in which the answer I needed might be anywhere and might even be concealed.
Amie’s daughter, Beatrice, studying math at Harvard, told me, “Math is about learning patterns, so that you don’t have to relearn them. You have to understand that everything in math is connected.” The question of whether one sees patterns in a discipline’s design has partly to do with whether one’s neurology equips one to see them and partly with whether one is made aware from an early age that there are patterns to be seen.
Sometimes now, as Russell described, I had an experience that I didn’t have as a boy, that of solving a problem in a procedural fashion by having found the pattern that the problem was asking me to find, even though I didn’t understand the reason for there being a pattern or why the pattern applied to the case. This was a form of cheating, of acquiring a serviceable method without a foundation, but it typically followed such periods of frustration that I felt entitled to it.
After a while I began to see that the limitations I thought I was uncovering in mathematics were not flaws, they were examples of the limited range of my thinking, and perhaps of my ability to think at all. I decided I might benefit from being more receptive. One morning I sat trying to believe that the means for solving a problem might appear from anywhere within the circle of my awareness and not only from where I had already insisted that it couldn’t, as if I might see it from the corner of my eye, which for a while made doing algebra a little like attending a séance.
I came to realize what perhaps should have been obvious, that I was creating difficulty by behaving perversely. I had been personifying math as a spiteful thing, and as a result it was demonstrating the capacity to behave spitefully. In a kind of half-smart, half-cunning way I believed that math had tripped me up before, and I wasn’t going to allow it to again. I was permanently on alert for deceitful propositions. I was so occupied with this ridiculous position that I didn’t see that there were more sensible means of engagement.
I am aware that this is not a profound realization, but it was important to me, and I see no reason to pretend to be smarter than I am. It led me to realize that while I had learned a lot of things in my grown-up life, I had spent very little time trying to learn something that was difficult for me to learn.
Studying algebra required remembering the math that I had been taught before algebra, meaning the properties of fractions, negative numbers, exponents, and decimals. If I ever knew, I had forgot how to divide fractions or what happens when a fraction is raised to a power or what it means when there is an exponent in the denominator or what to do when the exponent is negative. So for a while I had to put algebra aside and sit in the remedial room.
To review percentages I was in sixth grade, according to the texts, which, especially as I found percentages difficult, was lowering to my self-esteem. I was trying to find where I had lost my way, and I hoped not to get lost again. I was trying to assemble a set of reliable assumptions with which I could safely advance.
Now I think being wrong almost all the time for weeks is sort of funny, but I didn’t then. It hadn’t occurred to me that not only might I find algebra difficult, but maybe algebra actually was difficult. Doing algebra again was like meeting someone you hadn’t seen in years and being reminded why you really never liked him or her. To learn, I had to begin to take my chances, to think of myself as good at some things, maybe, not good at others, surely, and to apply myself, with a degree of humility, to being edified by something that was more demanding, more severe, deeper, and less tolerant than I had expected. I knew that by accepting my shortcomings I would be able to proceed without pretensions or pressures, although I risked finding out that I had no capacity to learn math and that the shortcoming was proof of my limited intellectual powers. I would rather have had algebra on the ropes.
JUST AS THERE are impairments in reading such as dyslexia, there are impairments in mathematics. The mathematical version of dyslexia is called dyscalculia. Someone with dyscalculia cannot tell from a glance how many objects are in a small group. They also tend to have difficulty reading clocks and sheet music and in estimating how far away an object in the distance is.
I don’t have dyscalculia, but doing math as a boy made me anxious, and it still does, a little. Nobody called it math anxiety then, but they do now. It’s a syndrome. Math makes some people apprehensive to the point of dread. A severe aversion to mathematics is called high math anxiety, HMA, and also math trauma, which sounds overdramatic to me, but not everyone thinks so. HMA apparently makes a person’s heart beat faster. The amygdala is part of the brain’s limbic system, where, among other things, emotional responses and instincts for survival reside. From the paper “Avoiding Math on a Rapid Timescale: Emotional Responsivity and Anxious Attention in Math Anxiety,” by Rachel Pizzie and David Kraemer, I learned that someone with high math anxiety responds to math with “aversive, distancing behavior and increased threat-related amygdala reactivity,” so it wasn’t all in my head, although it was.
HMA, it turns out, is widespread. It afflicts about 10 to 15 percent of college students, and about 20 percent of the rest of us. (By comparison, social anxiety afflicts about 6.8 percent of us, and 3.1 percent of us have generalized anxiety disorder.) Grade school boys tend to have more math anxiety than grade school girls, but in college, women tend to have slightly more than men, although this may have to do with the obstacles that woman math students have traditionally faced from male professors. Until 1992, when a sufficient number of people complained, Barbie dolls said, “Math class is tough,” but Ken dolls didn’t.
You can have math anxiety because you don’t do well at math, but you can also not do well at math because you have math anxiety. This circumstance is called bidirectional. Moreover, math anxiety is contagious. In India, when parents with high math anxiety helped their children do math homework, some of the children got math anxiety. Through gestures and attitudes conveying unease, a teacher with math anxiety can afflict students.
A test called the Abbreviated Math Anxiety Scale identifies HMA. A person ranks experiences such as watching a math teacher diagram a problem on a blackboard, starting a new chapter in a math textbook, and being given a math quiz unexpectedly. There are also statements such as “Working on mathematics homework is stressful for me,” and “I get nervous when taking a mathematics test,” and “I believe I can think like a mathematician,” to which one answers never, seldom, sometimes, often, or usually. Lesser diagnoses of high math anxiety include low, some, moderate, or quite a bit.
I answered a sufficient number of questions on an HMA test I found online to qualify as having some math anxiety, but I don’t know how much, because I didn’t take the whole test. I am comfortable regarding myself as math averse or maybe math resistant.
Anxiety in relation to curriculum and subject isn’t common. With the exception of physics, in which mathematics figures, there isn’t science anxiety generally, at least I have never heard anyone say so, and there isn’t literature anxiety or history anxiety, either. Remembering that the Battle of Hastings was fought in 1066 or that Crime and Punishment is about a murderer is a simple task of recall. Doing mathematics is a serial task, involving several components of memory and various rather than single areas of the brain. Reading “Math Anxiety and Its Cognitive Consequences,” whose lead author is Mark Ashcraft, I find that the first component activated by a math problem is the search for a means of finding an answer. A second component seeks the tools, and a third employs them. The more steps to a mathematics problem, the more someone with HMA might think, Quit.
Someone with HMA does mathematics in a state of vigilance. He or she has an habitual response that occurs so quickly and offhandedly that it hardly registers in consciousness. According to Pizzie and Kraemer, this response is similar to “the response of phobic individuals to phobic stimulus.” The abrasion of the engagement interferes with absorbing what is meant to be learned. Since mathematics advances by a progression of methods and just gets harder, math anxiety, like seasickness, doesn’t go away, and not infrequently it worsens. For mathematically disabled children, math anxiety may reach a climax around ninth grade, with algebra, to which I can only say, I knew it!
I remember feeling nervous in algebra class and then, as I fell behind, embarrassed and determined not to let anyone know. Children compare themselves with other children, and how they feel about themselves affects who they think they can be friends with. Also, what they think they can do, and who they imagine they are. In “The Classroom Environment and Students’ Reports of Avoidance Strategies in Mathematics,” the lead author of which is Julianne C. Turner, I find remarks by a University of California professor named Martin Covington saying that for many students, “to be able is to be worthy, but to do poorly is evidence of inability and is reason to despair.”
The attempts I made to keep people from knowing that I didn’t understand math are common strategies that Covington calls “ruses and artful dodges,” the purpose of which is “to escape being labeled as stupid.” Adolescence is a period of secret-keeping, and it didn’t occur to me to wonder whether anyone else was having as difficult a time as I was. The boys and girls whose ruses had failed obviously, the dunces, I pretended I had nothing in common with.
A grievance now, a pedagogical one, maybe more than one: writers of algebra textbooks appear to take their readers to be more accomplished than I am, and so they skip steps in a collegial kind of you-and-I-know-that-these-steps-are-too-simple-to-include spirit, wink-wink, but it isn’t too simple for me, who is following line by line and doesn’t expect skips. A failure to follow can cost me hours, because I am dogged and also because I don’t always realize that a step is being skipped, I just know that the reasoning I think I am pursuing no longer makes sense. By the time I hear from Amie what the writer has left out I am good and worked up, I am hot to trot.
The omissions make me suspect that a math joke is being played on me, the plodding specifist. I end up with three or four textbooks open and writing Amie as if sending dispatches from some beleaguered outpost and needing an answer right away, because everything has broken down. Since I was typically so annoyed and frustrated, my approach usually took the form of an assault, with algebra in the crosshairs—goddam you, I’ll force you to obey straightforward terms. I was feeling an adolescent’s retributive anger, fed by slights and resentments, vehement and close to irrational, nurturing bruised feelings, and instigated by the world’s dishonesty. I understand how I had found it easy to walk away from math. I had given it a chance. It was a brute, malign, and mechanical thing, hostile to innocence and hope. I did not start out being mathematically averse, I remind myself. I did well for a time, with arithmetic. I was assaulted and weeded out, and how was that fair?
No one likes getting things wrong in public. It was not fun to be called on and to demonstrate not only that I did not have the right answer, I didn’t even know how to arrive at it. On the worst occasions, I didn’t even know what question I was being asked.
One day I sat in a class that Amie was teaching to undergraduates on linear algebra, a subject taught after calculus. It took place in a room like a small theater with rows of seats rising toward the back wall. For an hour and a half I had no idea what she was talking about. Occasionally she wrote equations on the blackboard and then she would turn and ask the class, “Do you understand?” I asked her afterward how she could tell if they were being truthful. “I can see it in their faces,” she said.
As a boy in math class I would take a seat toward the back of the room. It hadn’t occurred to me that my face might reveal something I was trying to keep secret. I just didn’t want to be asked for an answer.
THERE IS A degree of uneasiness in learning a subject that taxes one’s capacities. There is additional uneasiness if one lost to it on the first encounter. I worry not only, Am I up to it, but, I wasn’t up to it before, will I be up to it now? And how much of having not been up to it was from how I was taught and how much rests with me and my ableness to learn?
I thought I should use the algebra textbook being used at the school I went to. This book has at the beginning the sentence “A variable is a letter used to represent one or more numbers.” I would find it easier if the sentence were written, “In algebra, when you don’t know a number, you use a letter, called a variable.” In the phrasing of the textbook explanation, questions arise for me: How do I use a variable? In what circumstances? Does one letter represent more than one number, or do I need additional letters? In my version I am presented with a rule and am prepared to understand a following rule: when more than one number is unknown, you will use more than one letter, one for each unknown.
These textbooks are enormous. They are written by many people. There is a general boosterish quality to the prose, as if learning math is not only fun! but also obscurely patriotic, the duty of an adolescent citizen-in-waiting. I acknowledge that there is a great deal of material to consider in the writing of a textbook. To do it succinctly and well would be an achievement of good thinking and good writing. It seems an irony that the most precise of sciences is often presented imprecisely. Whether the textbooks intend it or not, by being unrigorous, they make math more complicated and obscure than is necessary. A rigorous program would allow a plodder such as me to follow each step and build confidence. When I had trouble with one textbook, I found another, but nearly all of the books were poorly written. In addition to leaving things out, they were careless about language, their sentences were disorderly, their thinking was frequently slipshod, and their tone was often cheerfully and irrationally impatient.
I felt asked by these books to solve puzzles that appeared to depend on a cliquish, arcane knowledge, one that was, in fact, apprehensible if explained in an uncomplicated way, using clear thinking and plain speech. It was partly opaque because their ham-handed writing and slovenly thinking made it so. Furthermore, it seemed to flatter the writers’ vanity to regard themselves as keepers of rituals and secrets. Occasionally I could find a principle illustrated simply in another book, and then I could go back and write, “Obscenity you, jerkoff,” in the margin of the first book, since I owned it, and in eighth grade it was the property of the school district.
I planned to allow six weeks to learn algebra. Studying six hours a day, six or seven days a week, I would spend about as much time as a student spends in a classroom in a year. The weeks began to pile up, though, and, while I noted the pages left in the textbook, I began to think, How can I have difficulty understanding what twelve-year-olds can? An exchange of emails I had with Amie (“This example is wrong!” “It’s not wrong, it’s trying to show you a different way of handling the problem”) concluded with her writing, “It’s been really instructive conversing with you about your readings. It seems maybe that you assign too much authority to the author, that you expect to be told the stone-cold truth and what to do at every step, whereas the authors are dwelling on possible exceptions and uncertainties before making the truth and the main point clear.”
If I wasn’t to grant authority to the author, though, whom was I to grant it to? On the other hand, I saw, once again, that the more obstinate I grew, the more I missed the point. I was continuing, rather flamboyantly, to enact Amie’s belief that I would overthink. Actually, I was brushing up against paranoia. I felt I had to follow each phrase carefully, lest the meaning of a sentence be lost, and I lead myself into overthinking. It was like going into a skirmish with all your soldiers arguing with one another.
I might have felt better if I had shed this stance, or exchanged it for another, but I didn’t know what to exchange it for. My sense of myself was caught up in the effort. I was trying to learn something that I worried my intelligence did not equip me for, and I was really, even if I’m a little embarrassed to say it, afraid that I would fail totally unless I defended myself.
I had begun with a sort of math-works-for-me-now attitude, and math had answered, Nuh-uh. I wasn’t going to have to work for algebra, but I was going to have to get along with it. That threw me against all the uncertainties that had so plagued and harassed my adolescent self, and I wasn’t eager to subject myself to them again.
At a party I ran into a colleague from The New Yorker, Calvin Trillin, who asked me what I was working on. I said a book about mathematics. He looked at me closely and said, “For or against?”
Years ago I listened to a philosopher engage in an argument at a friend’s apartment. Instead of defending a position, he cared only to know more and to understand the other person’s point of view. The way he was arguing may be common among philosophers, but I hadn’t heard anyone take part in an exchange while evincing such humility and receptivity. His example became an aspiration for me, although I have never been able to enact it, too hotheaded and too easily baited. Nevertheless, it was borne in on me that I was going to have to personify some element of it if I was going to learn algebra. Unless I could come up with some proof, I knew I couldn’t continue to call Amie and complain about algebra’s seeming to be irrational.
One day I called Amie with yet another inquiry and realized that I had framed it not as a demand or an accusation but in such a way that it allowed an explanation. For perhaps the first time, I hadn’t insisted that Amie abandon mathematics and confirm my objections, but had allowed myself to think that in order to understand the matter, I would have to follow a line of reasoning different from a defense of my position. I had to disengage myself from believing that my identity was attached to my being right. That I was able to, barely, was a surprise to me.
ONE OF THE first procedures to defeat me involved multiplying polynomials, expressions that contain more than two terms. The solution to (1 + x – y)(12 – zx – y), a problem I was given in Algebra, by I. M. Gelfand and A. Shen, involves multiplying each term in the first parentheses by each term in the second: 12 times 1 = 12; 12 times x = 12x; 12 times –y = –12y; –zx times 1 = –zx; –zx times x = –zx2; and so on. Then you combine those terms that can be combined. Each problem seemed to rely on a different assumption. I would think that I had learned a rule for combining terms, for example, but when I applied it my answer was wrong. In this case, Amie pointed out that I was trying to combine terms that were incompatible. zx + zx² combined, I thought, into 2zx³, although if I had thought harder I would have realized that I had added z + z and multiplied x² by x, meaning I had been inconsistent. One term is (zx)(1) and the other is (zx)(x), so they were not like terms at all.
“They have to be exactly the same to combine?”
“That’s right.”
“So, you mean it’s actually literal?”
“That’s the beauty of it,” she said. “It’s not mysterious. It’s not magic at all.”
This was frustrating for me, even unsettling, even anger-making, because the procedures had seemed not literal, not reliable, and actually magical. The rules I thought I could depend on had failed me, because I had understood them imperfectly and applied them inconsistently, just as Poincaré had predicted that saps like me would do.
“For the moment, think of it as a monastic discipline,” Amie said. “You have to take on faith what I tell you.”
AMIE AND I have always been close and I could see that she wanted to help me, but she couldn’t always see how to. I think she might agree with Poincaré that mathematics was a question of logical remarks following one another in an orderly way, and that this was especially true of mathematics as rudimentary as the version I was struggling with. I was compromised, clearly, by my own shortcomings, but not in a way that her experience might help her understand. My habits of mind were simply different from hers. She didn’t have the same shortcomings, and so far as amateur mathematics was concerned, she probably had no shortcomings at all.
As the math grew more difficult, Amie’s explanations tended to become either too complex for me or too opaque. Sometimes what she told me made sense when I was talking to her, but I couldn’t repeat it on my own. I might recall most of what she had said, often I had written it down, but the explanation relied on procedures I hadn’t understood sufficiently to reenact. Now and then I could hear in her voice an exasperation from her not understanding why I couldn’t seem to grasp the simplest concepts. It is one thing to teach someone what appears to be a straightforward discipline and another to understand why it doesn’t seem straightforward to him or her. One process is procedural, and the other requires a sympathy of imagination that has little to do with being a mathematician. For Amie mathematics was logical, and for me it wasn’t. She saw patterns where I saw chaos, incoherence, obfuscation, and conspiracy.
When I told Amie how much trouble I had with math as a boy she said, “You were probably taught badly.” I have no idea if the men and women who taught me and my friends were good at it or had any enthusiasm for it, either. They had turned the pages of the same textbooks year after year, and, so far as they knew the material, could probably have taught math in their sleep. They drove old cars and took second jobs in the summer, and some of them were likely in unhappy marriages or drank or were lonely and felt that everything that was ever going to happen to them had already happened and who knows what was on their minds as they looked out at a classroom of adolescents, only a few of whom appeared to be paying attention, and recited problems they’d been reciting for years. I’m not comfortable finding fault with them. If I were to return to algebra class, I would approach Mr. Carmine Biazzo’s desk on the first occasion where I had difficulty and ask to have the matter explained again. He might just want to get on with his day and tell me that he was busy. Or he might put everything else aside and try to address my confusion. There is also the chance that I might still not get it, and he might conclude that in trying to help me he was spending his time unwisely.
Learning is a form of adaptation and of receptivity. Learning math more complicated than arithmetic means absorbing and remembering a wagonload of information and then using it to reason. Powerful impressions are more likely to last. As people get older, though, especially older than sixty-five, their ability to collect new memories diminishes. Learning also takes longer. Moreover, creative thinking is compromised, since the mind is inclined to follow patterns it knows. This may be a reason why pure mathematics is regarded traditionally as a young person’s pursuit.
The ability to learn mathematics is thought to decline around forty, when the brain begins slowing its handling of procedural operations such as calculating. Older people learn and forget at roughly the same pace that younger people do, but calculating takes an older person twice as long. In the paper “Acquiring Skill at Mental Calculation in Adulthood,” Neil Charness and Jamie Campbell say that middle-aged people perform as older ones do, but if they practice, they perform more as younger people do. If speed is valued more than accuracy, the decline in ability is obvious. If accuracy is valued more than speed, the decline is less obvious and maybe not even very pronounced. Younger people tend to read faster than older people. Older people tend to remember more of what they’ve read.
From brain scans it appears that older people engage more of their faculties in solving a problem than young people do. Older brains might be less robust, but they may also have become more efficient. The Scaffolding Theory of Aging and Cognition says that brains respond to declines by recruiting assistance—that is, by replacing a response typically dedicated to a single area with a pattern of layered responses involving several areas. “HAROLD” is an acronym for “hemispheric asymmetry reduction in old adults,” a form of brain plasticity. I know about it from the research article “Creativity and Aging,” by Gene Cohen. It means that brain activity in older people tends to be “less lateralized” than in younger people, Cohen says, meaning that the brains of older people might enlist areas that usually have one function to collaborate with another function, which is called bilateralization. Cohen likens it to the brain’s moving, perhaps in a compensatory way, “to an all-wheel drive.” A study at the University of Toronto found that older people did as well as younger ones on visual tests relying on short-term memory, but the areas of the brain that the younger people used were weaker in the older people, so the older people engaged other areas. One of the other areas was the hippocampus, which would more usually be invoked for a task such as learning a long speech.
Dr. Carol D. Ryff, at the University of Wisconsin’s Institute of Aging, told me about stereotype embodiment theory, which was proposed by the Yale psychologist Becca Levy. It says that the culture presents older people as moving slowly, being hard of hearing, talking too loud, and unable to read small print. These depictions are funny when we’re young; then we grow old and enact them, and they undermine a person’s sense of well-being. “There are certain fields where you get better with age, though,” Dr. Ryff told me. “You’re not going to have a twenty-two-year-old wunderkind psychotherapist. Most of Freud’s brilliant theories didn’t arrive until his fifties.”
I told Dr. Ryff that I was trying to learn math, and that I had a math allergy. “Someone with math anxiety, later in life, with a different perspective can really shine and discover something new,” Dr. Ryff said. “It’s incredibly healthy for the brain as well.” I wasn’t completely confident that I had yet developed a new perspective, so I was less cheered than I might have been.
I do not think as fast as I used to, and I do not think by means of the same associations and patterns. I do not mind this. Thinking quickly often meant that I responded impulsively and made difficulty for myself. I used to believe that such behavior was evidence of my having a passionate nature. Also, that it was winning. Now I just think I was an idiot. Anyway, I am no longer so prone to irresponsible remarks, and I am grateful for having fewer rough edges.
Sometimes I have to wait for words to arrive, but I seem also to have handled this in a manner that is suggested by some of the brain studies. Not long ago I found myself trying to recall the name of a friend’s cat. What I came up with, Leonard, was not close. My mind supplied a generic image of a dog, then a German shepherd, then a German shepherd that belonged to an older couple I knew who lived in my friend’s town. I heard the dog’s owner, a cultivated old European man named Serge Chermayeff, calling the dog: “Myyyyyyyyllllllowwww.” The cat’s name was Milo. This took about three seconds.
When I turned sixty-five, I thought, I am as far from fifty as I am from eighty, which sobered me right up.
My capacity for slower thinking, for holding an idea longer in mind, and treating it more deeply, my ability to do this has grown, as I might have hoped when I was younger, if I had known that such a thing was possible. A lot of things about getting older are beyond our imagining when we are younger, or understanding, even if they were made clear. I had no idea when I was young of the wealth and depth of experience and learning and sometimes wisdom that an older person brings to a conversation. I wince when I think of how little of other people’s lives I understood when I was young, especially the lives of older people. I tended to think of older people as having always been who they were at the moment I was talking to them. I lived almost entirely within myself, which I think is a defective way of being a person, considering all the freedom, the enlargement of oneself, that a penetrating sympathy makes possible.
Early in my studying algebra for the second time, I found I could learn things sufficiently to employ them in an exercise, but not sufficiently to remember them when they reappeared in a different context weeks later, say. It occurred to me that I wasn’t really doing math; I was doing what in my childhood was called learning by rote. In the journal Science, I read of a study by researchers in France and at MIT saying that learning multiplication tables is more like memorizing a laundry list than it is doing math.
Learning a language, one might forget a word and do without it, there are other words, but the vocabulary of mathematics has very few synonyms. The difference in being older was that unlike in high school, when I saw no relations among fields or methods, when the numbers and equations almost seemed to blur on the page, now I (slightly) more often understood what is connected to what and how, and usually I can figure out where to find a solution. I wish I had been able to do this when I was young, but I hadn’t.
I have deeply regretted that I did not proceed far enough at least to understand something of the great leading principles of mathematics, for men thus endowed seem to have an extra sense.
—Charles Darwin, The Autobiography of Charles Darwin
My suspicion that by not learning adolescent mathematics, my ability to think expansively might have been impeded seems to be supported by the study “Origins of the Brain Networks for Advanced Mathematics in Expert Mathematicians,” by Marie Amalric and Stanislas Dehaene. Some scientists think that the areas of the brain that handle mathematics are ones involved principally in language, and some think they are ones that handle number thinking and spatial reasoning. The language endorsers believe that language developed first and that numbers and mathematics developed as a consequence.
Scanning the brains of fifteen mathematicians and fifteen non-mathematicians while they considered complicated mathematics and also questions mostly involving history, Amalric and Dehaene found that in the mathematicians and not in the others the regions involved in considering math problems were separate from those that make sense of language. Language operations take place mainly in the left hemisphere. The problems that the mathematicians considered, which were in algebra, geometry, analysis, and topology, used areas in the front and middle of the brain that are engaged in thinking about space and number.
The part of the brain that performs precise calculations is different from the one that estimates them. An earlier study that Dehaene was involved in found that the calculating part used an area of the brain typically invoked when remembering. The approximating, though, took place among a network of areas participating in “visual, spatial and analogical mental transformations.” By “transformations” I assume they meant stages of reasoning.
They concluded that the mathematical areas were associated with “an evolutionary knowledge of number and space.” This might be because mathematical insights sometimes are worked out as forms of approximation, which is sympathetic to intuition and involves spatial concepts such as the number line, a thing a person tends to see in the mind’s eye. Their conclusion might have been endorsed by Einstein, who said, “Words and language, whether written or spoken, do not seem to play any part in my thought process.”
Amalric and Dehaene believe that ease with space and number in childhood might be a reliable predictor of how well someone will do with math. I sometimes walked into walls as a child, and one of my adolescent friends told me, “You have no spatial relationships,” which I took as a rebuke.
It turns out that if we believe we can learn, we do better than if we don’t. After my first encounter with algebra, it had never occurred to me again that I could learn math.
Amie thought I should read How to Solve It, by George Pólya, a handbook for mathematical problem-solving which was published in 1945 and was part of her reading as a freshman at Harvard. Mathematics, Pólya writes, has two identities. As a field it is a “systematic deductive science.” As an endeavor, it is “an experimental inductive science.” Classically, it involves two types of reasoning, analytic and synthetic. One moves forward and one backward. To prove A equal to E, synthetic reasoning establishes that A is equal to B, B to C and C to D and D to E. Seeking to equate A to E, analytic reasoning likens E to D and D to C and so on. Analysis forms a plan that synthesis enacts.
In Discourse on the Method of Rightly Conducting the Reason, and Seeking Truth in the Sciences, which was published in 1637, Descartes describes practices he established for himself for finding solutions. In old-fashioned prose so full of life that he seems nearly present, as if testifying, he writes, “I was then in Germany attracted thither by the wars in that country, which have not yet been brought to a termination; and as I was returning to the army from the coronation of the Emperor, the setting in of winter arrested me in a locality where, as I found no society to interest me, and was besides fortunately undisturbed by any cares or passions, I remained the whole day in seclusion, with full opportunity to occupy my attention with my own thoughts.”
Descartes arrives at four precepts that “would prove perfectly sufficient for me, provided I took the firm and unwavering resolution never in a single instance to fail in observing them.” They amount to a kind of diagram for how to think. He writes:
The first was never to accept anything for true which I did not clearly know to be such … to comprise nothing more in my judgment than what was presented to my mind so clearly and distinctly as to exclude all ground of doubt.
The second, to divide each of the difficulties under examination into as many parts as possible, and as might be necessary for its adequate solution.
The third, to conduct my thoughts in such order that, by commencing with objects the simplest and easiest to know, I might ascend by little and little, and, as it were, step by step, to the knowledge of the more complex; assigning in thought a certain order even to those objects which in their own nature do not stand in a relation of antecedence and sequence.
And the last, in every case to make enumerations so complete, and reviews so general, that I might be assured that nothing was omitted.
After the manner of Descartes, Pólya organizes problem-solving according to four phases. The first requires understanding a problem, so as not to be undermined by misguided action. The second involves planning an approach that one carries out in phase three. In phase four a person examines the problem for any lessons it provides.
Pólya amends each phase. In assessing a problem it is important to consider whether one knows similar problems with similar unknowns and can borrow tactics or find a simpler version of the problem to solve. Straying too far from a problem risks losing sight of it altogether, though. Borrowing from Descartes, Pólya says that the best practice is to state the problem and then separate it into parts. A way out of being stalled is to identify the parts that are difficult and then look for similar examples that have solutions. One could also try putting the parts in a different order. “Difficult problems demand hidden, exceptional, original combinations, and the ingenuity of the problem-solver shows itself in the originality of the combination,” Pólya writes.
No idea is bad unless a person is uncritical. Accepting a guess as a truth, as superstitious people do, is misguided, but so is ignoring a guess, as pedantic people do. As regards ideas, it is only bad not to have any.
Resolve matters, too. “It would be a mistake to think that solving problems is a purely ‘intellectual affair,’” Pólya writes. “Determination and emotions play an important role. Lukewarm determination and sleepy consent to do a little something may be enough for a routine problem in the classroom. But, to solve a serious scientific problem”—or a pressing one or to accomplish a serious task, I can’t help thinking—“will power is needed that can outlast years of toil and bitter disappointments.”
Figures farther back than Plato stand behind Pólya’s assertion that “teaching to solve problems is an education of the will,” and that a rigorous education imposes character. I read Pólya closely, not only because Amie had recommended it, but also because, while I wasn’t moving toward a future in which I was necessarily going to solve complicated math problems, I was surely, in growing older, moving toward one with complicated problems.
Something in me gave way, slowly slowly, against the conclusion that when I got a wrong answer, I was more likely wrong than math was. I had railed against poor writing, withheld information, apparently illogical procedures, the clannish perversity of mathematical reasoning, basically anything I’d encountered that hadn’t seemed clear as water, and I hadn’t won a single round, except maybe against poor writing. The wear and tear was wearying, and I knew Amie didn’t want to hear me rant any longer. I decided to try to find what pleasures math had if I didn’t always fight it.
I tried to study things objectively rather than examine them for hidden information intended to mock and deceive me. I am aware that many people would not feel aggrieved the way that I did and would begin their studies less combatively, but those weren’t my circumstances.
Now and then I backslid, usually on occasions when the rules weren’t obvious or easily intuited or seemed arbitrary, and then I’d get angry and have to have Amie calm me down. Studying factorization, for example, I learned that 2–2 is not 2 × –2, which is –4. By the rules of exponents, Amie said, 2–2 is 1/22, which is 1/4. The discussion over why this was so was long and contentious, and, although my objections were cunning, I lost.
“All the rules are absolutely logical. Please don’t say anymore that there are contradictions in math,” Amie told me. “There aren’t.”
About contradictions: In the sciences other than mathematics, the most authoritative finding is the result of observation and deduction and the acceptance of things that appear to be true pretty much all of the time. In mathematics, the paramount authority is proof, and a single exception among infinite cases disqualifies a result. “I’m almost certainly right” is for disciplines where the result can be argued.
A mathematician’s engagement with a problem resembles an artist’s, but is different in that an artist is not confined to a single outcome. Hemingway wrote forty-seven endings to A Farewell to Arms. A math problem typically has a single solution no matter how many ways a mathematician tries to approach it. In Love and Math, the mathematician Edward Frenkel writes, “With math you know what you’re trying to solve and either you do or you don’t.” Unlike an artist, a mathematician has the satisfaction of knowing that no revision will change an outcome and that he or she is right for all time and in all circumstances. According to Reuben Hersh in The Mathematical Experience, a mathematical proof aspires to be “so bound up with what is right in the universe, that God almighty could not set it aside.” This makes mathematics different from other arts in that the best mathematics is permanent. Even though W. H. Auden, in “The Dyer’s Hand,” writes, “The whole aim of a poet, or any other kind of artist, is to produce something which is complete and will endure without change,” decisions about permanence in the other arts are only agreements and revocable. Cultural assertions and their revisions, no matter how emphatic, reflect only who we believe ourselves to be at any moment. Cultures have conscious and unconscious lives, too.
An artist in another field must also always wonder what other form a work might have taken and whether that form might have succeeded better than the one that he or she chose. A mathematical solution is absolute, although in How Not to Be Wrong, Jordan Ellenberg points out that a lifetime might not be sufficient to achieve it. “Fermat’s problem took 350 years,” he writes. (Fermat’s problem, usually called Fermat’s last theorem: no three positive integers A, B, and C satisfy the equation An + Bn = Cn if the value for n is a whole number greater than 2.)
SOME PEOPLE HAVE a talent for obliquely penetrating the concealed design of complex math problems. Where the approaches of most mathematicians appear straightforward, these others digress, sometimes eccentrically. The mathematician Goro Shimura often collaborated in the 1950s with Yutaka Taniyama. About Taniyama, in The Map of My Life, Shimura writes that “though he was by no means a sloppy type, he was gifted with the special capability of making many mistakes, mostly in the right direction. I envied him for this, and tried in vain to imitate him, but found it quite difficult to make good mistakes.”
Models for proofs descend from Euclid. In its earliest history mathematics involved observations, inferences, and superstitions about numbers, but after Euclid it was deductive. In The Music of the Primes, Marcus du Sautoy likens this change to the one when alchemy became chemistry in the seventeenth century.
Mathematical proofs typically use the law of the excluded middle, meaning that they are either/or. Most proofs are straightforwardly deductive, but mathematics also uses an indirect method, borrowed from logic, called proof by contradiction, which is sufficiently well known that I probably don’t need to add that it is also called “reductio ad absurdum.” This method proves a statement true by assuming it is false and then showing that such a conclusion produces a contradiction. Euclid’s proof about prime numbers, those numbers such as 2, 3, 5, 7, 11, 13, and so on that can be divided only by themselves and 1 without leaving a remainder, is the first proof of there being an endless number of primes. It is also a proof by contradiction.
Euclid supposes that there is a largest prime, P. Then he defines a number Q by multiplying all the primes and adding one, in other words that Q = (2 × 3 × 5 × 7 … × P) + 1. Q is not prime, because, by the terms of Euclid’s argument, P is the largest prime. Divided by any of these prime numbers, though, Q will leave a remainder of 1. If Q is not a prime, then there is a prime that can divide it, which would be a prime greater than any on the list (and might be Q itself). This contradicts the notion that there is no greater prime than P, so the assumption of there being a final prime is false. This is fairly clear-cut, but I had to go through it with Amie so many times that I no longer remember why I found it so hard.
Euclid didn’t invent the concept of proof so much as systematize it. The concept was introduced in the sixth century BCE by Thales and by Pythagoras as a means of establishing assertions that weren’t obviously true. According to Ian Hacking, in Why Is There Philosophy of Mathematics At All?, this appears to be the result of the Greek culture’s being essentially argumentative, whereas cultures in China and Mesopotamia, which also had sophisticated mathematics, were authoritarian. Hacking refers to an observation made by Geoffrey Lloyd, the British historian of ancient science and medicine at Cambridge, that “the hierarchical structure of a powerful education system, with the Emperor’s civil service as the ultimate court of appeal, had no need of proofs to settle anything.”
A MATHEMATICAL PROOF is a faultless argument establishing a statement from a preceding one or from an axiom, which is a remark so obvious that it doesn’t need a proof: “between any two points a single straight line can be drawn” is an axiom. A successful proof is a theorem. A statement in mathematics that has no proof is a conjecture, as in the Goldbach conjecture, the most famous unsolved problem in mathematics. The Goldbach conjecture says that every even whole number greater than 2 is the sum of two primes. It has been found to hold as far as 400 trillion, but a number might exist that refutes it, so it’s a conjecture.
Propositions, lemmas, and corollaries also have proofs. Theorems may be considered to be very important propositions. Lemmas are minor propositions that lead toward a theorem, and corollaries are propositions that follow from a theorem. These terms are subjective, though, and deciding when to call a result a lemma or a corollary rather than a theorem or even a proposition is a matter of a mathematician’s judgment.
Descartes believed that the best proofs could be understood at one reading. Leibniz thought that the best proofs advanced toward a result that couldn’t be grasped all at once. For Bertrand Russell the appeal of a proof is not only its result, as people tended to think, but also the elegance of its structure. “An argument which serves only to prove a conclusion is like a story subordinated to some moral which it is meant to teach,” he writes. “For aesthetic perfection no part of the whole should be merely a means.” Impatience, he thought, led people to overvalue a proof’s result at the expense of its claims.
AMONG THE MOST useful methods for arriving at a proof, applicable to problem-solving in general, is to work fluidly, that is, forward and backward. In “A Brief Introduction to Proofs,” William Turner writes, “Start with your hypothesis and ask yourself what does this imply, but also look at your conclusion and ask what you need to prove to get to it.” Sometimes a proof by contradiction is easier, but “if you have trouble proving it false, try to use the reason you are running into difficulties to prove it true.” Turner likens a proof to a map leading from a hypothesis to a result. Whether it was built consecutively or from both ends to the middle matters less than whether it is correct.
Mathematics has two pursuits: to find patterns and to prove that the patterns are lasting ones. Before the seventeenth century mathematics was more a component in other sciences than a science itself. If a piece of mathematics was useful, it didn’t necessarily require a validation for why it worked. The modern figure influential in changing this was the German mathematician Carl Friedrich Gauss, who believed that mathematics must be justified and that proving things is what a mathematician should do.
A work of art rewards repeated engagements. For this reason, mathematicians like to return to certain proofs and mathematical documents. Gauss said, “You have no idea how much poetry there is in a table of logarithms.”
I passed the first weeks of my sojourn in the algebra highlands learning rules such as the commutative law (a + b = b + a; ab = ba) and the distributive law (c × (a + b) = c × a + c × b)). I had also to learn the properties of fractions; how to add and multiply negative numbers; the terms governing powers and negative powers and the multiplication of powers; and how to manage polynomials, the expressions that have letters and numbers, such as (a + b) (a + 2b), and so on. As a boy my eyes glazed and they still do when I see formulas and equations, especially ones I haven’t seen before and especially ones that I don’t understand. They provoke a kind of weariness of the soul. Even so, formulas and equations were not the reason I had failed to learn algebra. The reason I had failed to learn algebra was word problems. From the first day of this endeavor, opening the first textbook, I worried about word problems. I recalled them as devious and irrational, the evil end of the language ladder whose higher end was poetry.
Gustave Flaubert sent his sister Caroline, who was studying mathematics, a word problem in 1841. “A ship sails the ocean,” he wrote. “It left Boston with a cargo of wool. It grosses 200 tons. It is bound for Le Havre. The mainmast is broken, the cabin boy is on deck, there are 12 passengers aboard, the wind is blowing East-North-East, the clock points to a quarter past three in the afternoon. It is the month of May. How old is the captain?”
I found this problem in Mathematics and the Imagination, by Edward Kasner and James Newman, which was published in 1940, and if they hadn’t written that it “cannot be answered even though there seems to be plenty of information supplied,” I would have tried. I regarded all word problems as illogical, so I wouldn’t have seen Flaubert’s as being any more illogical than this one, from the Khan Academy website: André, a collector, bought 4 baseball cards. The next day, a Tuesday in March, on which it was raining, his dog ate 50 percent of his collection, leaving André 20 cards. André’s father is a doctor, and his mother is a biology professor. They gave André the cards to start his collection. How many cards did André’s mother and father give him? Which does have an answer: 36. (½ (x + 4) = 20).
Another numbing word problem, a relic one, from A History of Pi, by Petr Beckmann, provided by “The able Chinese mathematician Sun-Tsu (probably 1st century A.D.): A pregnant woman, who is 29 years of age, is expected to give birth to a child in the 9th month of the year. Which shall be her child, a son or a daughter?”
Sun-Tsu’s solution: “Take 49; add the month of her child-bearing; subtract her age. From what remains, subtract the heaven 1, subtract the earth 2, subtract the man 3, subtract the four seasons 4, subtract the five elements 5, subtract the six laws 6, subtract the seven stars 7, subtract the eight winds 8, subtract the nine provinces 9. If the remainder be odd, the child shall be a son; and if even, a daughter.” Occasions such as these reinforce my sense of mathematics, in its nature, as a series of questionable propositions.
Amie did not endorse my studying word problems. She said that they taught nothing of any lasting use and, once disposed of, never appeared in mathematics again, an argument I wish I had been equipped to present to Mr. Carmine Biazzo. I understood that they were a byway to her, but in the math curriculum, beginner division, they are prominent, at least as I remember it. Perhaps this is because ancient algebra begins with word problems. The earliest document with algebra on it is the Ahmes Papyrus, an Egyptian scroll in the British Museum that was copied by a scribe named Ahmes “in the year 33, in the 4th month of the inundation season.” Scholars believe that this might be around 1650 BCE. The scroll describes itself as “the entrance into the knowledge of all existing things and all obscure secrets,” and it is thought to be a math textbook. In addition to tables concerning addition and multiplication, instruction on how to handle fractions, geometric definitions, and methods for finding areas and volume, it has eighty-four problems, one of which, the twenty-fourth, an example of rhetorical algebra, begins, “A heap and its 1/7 part become 19. What is the heap?” According to the translation made by the Mathematical Association of America in 1927, “The author assumes 7, which with its 1/7 makes 8, and then, to find the answer, he multiplies 7 by the number that multiplying 8 will give 19.” In Algebra for Dummies I find that x + x/7 = 19 works, too.
When I began this project, I knew like I knew my name that I couldn’t do word problems. Reading one, I would hear a companion voice say, “No way you’re solving this. Not you, not now. Flintstones, meet the Flintstones, they’re … Fine, ignore me. I’ll just tag along, shall I?” Reading a word problem, I would have the sensation that I was sliding over the surface of the text, as if in a car that was skidding on black ice. I could find no place to stand from which I could say confidently, “This is the information I need to solve this problem, this is how it goes together, and I can ignore the rest.”
The other children seemed already to know what to do. So who had taught them? Why had I been left out? I say this to demonstrate how far I had wandered from the simple and straightforward. Word problems were both a practical and a metaphysical problem for me. They were a species of taunt and the sneering face of trouble in the mirror.
“Word problems aren’t the same as math problems,” Amie said sympathetically. “They lie in some space between reading and math, and they can really fell people who have issues with reading and attention. So much intervenes in the head of a twelve-year-old who isn’t, or is not yet able, to grasp complicated sentences. If you can’t keep all the terms in a paragraph in mind until you reach the end, you’re in trouble.” I decided not to say that I wasn’t twelve.
Words in word problems don’t behave as they do in conversation or as words generally do on the page. Instead, they have a background meaning; they conceal the means of solving the problem, as if they were encrypted. On the surface they are a statement of circumstances—a traveler, a train, Topeka and Dallas, say—but they are also vessels of meaning for numbers and relations of consequence. Prose writing typically has an explicit meaning. It might suggest other meanings, but it intends to convey a thought or a series of thoughts. Word problems use language subversively. The car, if there is one, isn’t actually a car. It is irrelevant to ask if the car is a sedan or a wagon, a Ford or a Chevy, new or secondhand—information that might help you picture it—because the car stands for velocity. It isn’t a metaphor, it is a car being used as a proxy for an object moving in a specified direction with reference to time.
I was an imaginative child, that wasn’t my problem. My problem was that no one had explained to me the steps of inquiry involved or even that there were steps of inquiry. Word problems embody a type of dubious abstraction that algebra thrives on. I am not the only one who has noticed this. In Ulysses, a remark by Buck Mulligan leads the Englishman Haines to ask Stephen Dedalus his idea of Hamlet. Haines and Buck Mulligan have the following exchange:
You pique my curiosity, Haines said amiably. Is it some paradox?
Pooh! Buck Mulligan said. We have grown out of Wilde and paradoxes. It’s quite simple. He proves by algebra that Hamlet’s grandson is Shakespeare’s grandfather and that he himself is the ghost of his own father.
I read Amie a simple problem, “Cletus is driving from Austin to Houston, the total distance is 162 miles. There are 5,280 feet in a mile. How many hours will it take Cletus to get to Houston if he drives at an average rate of 88 feet per second?”
“This is a good one,” she said, “because it’s not written like a real series of sentences. In real life you expect each sentence to comment on the previous sentences, but here you’re handed three unconnected ones.”
“What would you do?”
“I’d think, Where do I start? I would always draw a picture.”
Drawing a line on a piece of paper, she said, “I’d put Cletus at one end. First I multiply by 60 because there’s sixty minutes in an hour, then I multiply by 60 again, because 60 seconds in a minute. So 3600x is the number of seconds. 3600 × 88 × x divided by 5280 is the number of miles he would travel, which is 162. We want him to travel that distance in x hours…”
In school every word problem became an equation of the 5/12 = x/20 type. Eventually Amie arrived at 2.7 hours, but since she didn’t write such an equation, and I couldn’t adapt her words to the form, I was still as lost as I could be.
If I couldn’t solve word problems, I didn’t feel I could say I had mastered, or at least passed, algebra. Doing them was like visiting a place where something unhappy had occurred and hoping that I might be able to shed my disabling attachment to it. Also, if I couldn’t get past them, I didn’t see how I could go on to geometry and calculus with any confidence.
As a boy, I tended to read the problems too quickly, feeling that the sooner I got to the end the smarter I was, and if I didn’t have an equation in mind, I sulked. I had a magic-thinking notion that if I knew what I was doing the equation would reveal itself, as if written in invisible ink.
Reading a problem a second or third time wasn’t going to tell me anything I didn’t already know. And anyway the words would blur again. A car was traveling between two places I’d never been. How the car related to the time, or the distance to the place, I couldn’t tell, even though I had sat in a classroom where the reasoning had been demonstrated plenty of times. I don’t know why I hadn’t been able to absorb it. I just hadn’t. In a way that no other subject is, except perhaps physics, math is indifferent to the well-being of those who fall behind. It throws off stragglers remorselessly.
In my second engagement, I read problems carefully, methodically even, like a particularly dim detective, doubting everything. I should be able to do this, I thought. It’s not like it’s magic. I continued to believe, though, that they were deceptive. Here is how I would go wrong: Jane is 20 years younger than Rebecca. Rebecca and Jane first met 2 years ago. 14 years ago, Rebecca was 3 times as old as Jane. How old is Rebecca now?
Begin with Rebecca, I thought, who is 20 years older than Jane, so, to give her current age a variable, she is R = J(ane) + 20. Then I thought, I also have the matter of 14 years, so I wrote another equation saying 14Y = 3R = J + 20. They don’t go together, because they can’t. They are useless. And I wince at having thought of them, let alone for congratulating myself at dismissing the superfluous mention of their meeting two years earlier.
The solution perhaps is obvious; at least it was for me after Amie explained it. Jane, being twenty years younger than Rebecca, is J = R – 20. Fourteen years ago, R – 14, Rebecca was three times as old as Jane, R – 14 = 3(J – 14). Since R – 20 = J, it can be substituted into the equation in place of J, in order to eliminate the second variable, therefore R – 14 = 3(R – 20 – 14).
R – 14 = 3(R – 34). For math greenhorns like me, I will explain, so as not to skip steps, that negative numbers added to each other produce smaller negative numbers; smaller being a matter of greater distance on the number line from zero. If I begin at –20 and take 14 steps in the negative direction on the number line, I arrive at –34, a smaller number than either –20 or –14.
3 × R becomes 3R and 3 × –34 becomes –102, thus,
R – 14 = 3R – 102
Anything done to one side of an equation must be done to the other, so to consolidate the terms I add 102 to both sides, 102 + (–102) = 0 and 102 + (–14) = 88.
R + 88 = 3R
Subtracting R from the left-hand side of the equation leaves 88, and subtracting it from 3R leaves 2R.
2R = 88
R = 44
Fourteen years ago, Rebecca was 30 and Jane was 10.
I didn’t expect to become accomplished at word problems, but I didn’t expect to have them trounce me again, either. I refused to give up on them until I felt at least modestly competent. If I could get four out of five correct, I felt that would be sufficient. Instead, I often went something like one or two for five. After I finished the problems in the textbook, I did them on a website that had an inventory of them, but I worked so doggedly that I used up their archive and eventually I was offered problems I recognized. Occasionally I would have forgotten the solution and could work them again, but after a while it was like seeing a movie so many times that you can repeat the dialogue.
Learning algebra, a person is crossing territory on which footprints have been left since antiquity. Some of the trails have been made by distinguished figures, but the bulk of them have been left by ordinary people such as me. As a boy, trying to follow a path in a failing light, I never saw the mysteries I was moving among, but on my second pass I began to. Nothing had changed about algebra, but I had changed. The person I had become was someone whom I couldn’t have imagined as an adolescent. Math was different, because I was different.
The first mystery, waiting like a welcoming party in the vestibule of mathematics, concerned the origin of numbers. It’s a simple speculation, where do numbers come from? They don’t typically appear in creation stories. In Chinese Myths, Anne Birrell writes that the Coiled Antiquity myth, which belongs to “a minority ethnic group of south-western China,” describes “how numbers were created” and “provides the etiological myth of the science of mathematics,” but she does not give a source for this assertion, and I have not been able to find one, so I have to accept her word for it. So far as I am aware otherwise, no culture has a story where a creature or a spirit gives numbers to humans. No one in any scripture I know of climbs a mountain and gets numbers or finds numbers while wandering in the desert or has a number dream or a number vision, and I don’t know of a figure in a myth or legend who does, either. No country has a holiday for the day they got numbers. Gods and protectors of numbers are also rare. Plato says in Phaedrus that he had heard that in Egypt there was a god named Theuth, “who invented number and calculation, geometry and astronomy, not to speak of draughts and dice,” but that is the only other ancient reference to gods and the creation of numbers that I have been able to find, and it isn’t clear that Plato didn’t make this up. Babylonian, Indian, African, Norse, and Native American myths and traditions, so far as I can determine, are about other things than numbers. The people who wrote creation stories probably thought of numbers as practical objects, like the axe and the wheel, and didn’t feel that they required a mythic explanation.
I don’t remember learning numbers any more than I remember learning to walk or to speak. I think this is common. As very small children, most human beings can look at a small collection and say how many objects are in it. This is called subitizing. Pigeons, crows, monkeys, and dolphins can do it, too. In humans this capacity seems to fail above seven objects. In Numbers and the Making of Us, though, the anthropologist Caleb Everett describes an Amazonian tribe who have no words for numbers and can reckon correctly only with groups of three or fewer, which suggests that recognizing numbers is at least partly something we learn and not an inherent human trait.
Arithmetic begins with combining and condensing. Mathematics begins with measuring, when whole numbers show the ability to become fractions and decimals. Arithmetic is visible. Whole numbers describe objects—three horses, six birds, two bushels of wheat. You can add one collection to another or remove one and compute the result. Mathematics, on the other hand, is largely invisible. You can represent it with symbols, you can discuss it, but you can’t necessarily see it. I can imagine an endless series of numbers, but I can’t see it. I can see only a metaphor. Arithmetic is a part of actual experience. Mathematics, as James Robert Brown identifies it in Philosophy of Mathematics, is a cerebral task. “Deriving the √2 has nothing to do with sense experience,” he says.
COUNTING LEADS TO adding and subtracting and the fancier operations of multiplying and dividing. Invoking them at first, are human beings inventing mechanical procedures or observing properties inherent in numbers? As soon as numbers appear are they already intractable? They weren’t intended to be, any more than they were intended not to be. At first, they assess obvious and near-at-hand qualities of the natural world, but thousands of years later they demonstrate the ability to describe grander aspects of nature intimately, the orbits of the planets, for example. People used numbers for centuries before they even knew there were planets.
Someone who says that human beings created the operations of arithmetic cannot say that we created the results. 2 objects and 2 objects are always 4 objects. We did not say, We are adding 2 objects to 2 objects and for the sake of clarity deciding that the sum will usually be 4 objects. On all occasions, in all universes, the sum will be 4, even if the term that denotes them is not 4. It is a property of their being, an inflexible trait. “Prayer,” a poem by Ivan Turgenev, written in 1883, begins, “Whatever a man pray for, he prays for a miracle. Every prayer reduces to this: ‘Great God, grant that twice two be not four.’”
NUMBER SCHOLARS TEND to think that counting was discovered by different cultures at different times, but Abraham Seidenberg, writing in “The Ritual of Counting,” published in the Archive for History of Exact Sciences in 1962, thought that “various peculiar features of counting practices” shared among many cultures meant that counting began in one place and spread. Seidenberg thought that counting arose from creation rituals in which the number of people taking part was specified by the ritual. It might have happened that a man entered the ritual first, an explanation for the near ubiquity in ancient cultures of odd numbers being thought of as male and even numbers as female.
Seidenberg says that older cultures sometimes had counting prohibitions and specialty practices for reckoning objects. In parts of Africa, counting a person was believed to bring the person to the notice of the gods, who might decide that the person needed more suffering. Among certain African people who disliked using words for numbers, a speaker would say a number’s first syllable and make a gesture and the person he or she was speaking to would say the rest of the word, to spread the risk. In Oran, in Algeria, counting grain required a counter in a state of “ceremonial purity.” The counter would say “‘In the name of God’ for ‘one’; ‘two blessings’ for ‘two’; ‘hospitality of the Prophet’ for ‘three’; ‘we shall gain, please God’ for ‘four’; ‘in the eye of the Devil’ for ‘five’; ‘in the eye of his son’ for ‘six’; ‘it is God who gives us our fill’ for ‘seven’; and so on up to ‘twelve,’ for which the expression is ‘the perfection of God.’”
I find this mixture of ceremony and superstition very moving and sensible, acknowledging, as it does, the fragility of human circumstances, the wish for protection, and our innate belief in magic and the supernatural—in a shared human heritage, that is, and all that lies past what we can be sure of.
It seems not unreasonable to speculate that numbers are latent in the design of the world, but latent where? Think of three animals in an ancient forest. Two humans are observing them. One of them identifies the group, for the first time, as three. The one who identified them as three did not invent the amount. The concept of three is inherent and independent of its first being named. It was three the day before being identified and the day before that and so on. The observer invented a term for describing a quality, but the quality that made the collection three existed whether it was described or not. The person who had no word for its magnitude would still recognize that the three objects had in common a trait with three other objects.
Is this simpleminded? I hope not. It is the kind of thing that fascinates me.
NUMBERS ARE SIMILAR to language in that no one would say that there weren’t thoughts before language, although language appears to make possible a means of naming and ordering thoughts and of having complex thoughts that wasn’t possible before language. Likewise, beginning with arithmetic, numbers make more complicated mathematics possible. Mathematics leaves the village of counting like a pilgrim headed for the wilderness, a large part of which consists of imaginary trees.
NUMBERS APPEAR TO have been found more than created. The decision whether to call such a magnitude four or quarto or quatre is a matter of language. 4 is a symbol outside of language and not subject to it. Counting invoked numbers, but then it is as if a conjuring gesture occurs or an alchemical property is revealed. An accompanying world comes into being that has its own logic, procedures, rules, and circumstances that owe nothing to a human presence. The trick might be said to have been performed by the first person to realize that one more thing could always be added to any collection of things.
Mathematics might partly be defined as the intent to uncover the properties of numbers. It is a complex pursuit that progresses toward no apparent end and that is complicated in ways that we appear to be incapable of imagining entirely. Mathematics seems to have a vitality that we can’t account for, either, and to be indifferent to whether we try. It exists whether we think about it or not. What distinguishes mathematics among the imaginative arts is its precision. A mathematician might think of any possibility, but what he or she thinks of has to be proved to exist. In mathematics some possibilities exist, others don’t. There is no such thing as conceptual mathematics like there is conceptual art. You can say, as Douglas Heubler did in 1970, that you intend to take a photograph of every person on earth to document his or her existence, and you can set about trying and run out of time, but you can’t invoke an actual world where the square root of 4 is 0. You can only pretend there is such a world.
MATHEMATICS IMMEDIATELY DEVELOPS two identities, two sides, two aspects, two natures, divides into yin and yang, night and day, the practical and the mystical, because as soon as there are numbers there is the question of where they came from. Many mathematicians regard thinking about where numbers and mathematics come from as philosophy and not mathematics, but I don’t see how they are so separate. Mathematics is an imaginative practice, and mathematical philosophy is an inquiry, but the objects at the beginnings of both fields, numbers, contain both use and speculation equally. Numbers name things, but they also represent metaphysical concerns. The first aspect is precise, and the second is speculative, and both are ingrained. It is a light/shadow feature of their being. As with a stranger, where are you from doesn’t have to be asked, but it’s present in the encounter. I can care or not care, have the habit of mind interested in thinking about the mystery or not, but I can’t claim that it isn’t there.
Numbers have a kind of sway over us, but we have no sway over them. Nothing we do can change what they are or how they behave. They were here before us and will remain if we disappear. They are always past, present, and future, and we cannot be sure that we are. I can say that the Statue of Liberty exists whether I think about it or not, because you and I know where it is and what it looks like. I can also say that the number 4 exists whether I think about it or not, but neither you nor I know where. If I say that it exists only in my mind, or in yours, then where would it be if not a single human mind were thinking of it? It isn’t lights out then for numbers.
All objects other than numbers live either in the world of space and time or in the realm of the imagination, which is a precinct of space and time. By the nature of their dual identity, numbers can travel between these places and connect them. In a letter to the physicist Wolfgang Pauli on October 24, 1953, Carl Jung writes of numbers that they “possess that characteristic of the psychoid archetype in classical form—namely, that they are as much inside as outside. Thus, one can never make out whether they have been devised or discovered; as numbers they are inside and as quantity they are outside.”
A number would appear to be as simple as a letter—both are serial implements—but letters are literal and numbers have esoteric attributes. If I write the letter A, it is the letter A; it doesn’t represent something, it is something. If I write 4, though, it isn’t 4 in the sense that A is A. A is concrete, the manifestation of a sound, and 4 is a symbol, a term denoting a collection. It has no scale or identity. It can be four cats or four galaxies. I can write 4, but I can’t say that it is 4, at least not all the possible embodiments of 4. I can demonstrate 4 only obliquely, by gathering 4 things, AAAA, for example.
Numbers were invoked by counting, a form of organization. Letters changed speech from something ephemeral into something capable of being preserved, another form of organization. By means of addition or subtraction or some other mathematical operation, one number can deliver us to another, something letters can’t do, though, unless you think that adding letters to one another to spell a word is similar, which it isn’t; you can’t divide a word by a word or a letter by a letter. You can’t have half a letter. Or the square root of a letter. Or 3.65 percent of a letter. (Only in mathematics is A/B a sensible remark.) Numbers have two primary incarnations, positive and negative, but they also have hidden attributes, such as being prime. By agreement, we can change how words are spelled, but we can’t change arithmetic. We can allow theater or theatre, but 5 + 7 we can’t do anything to at all.
Numbers are a mystery enfolded into ordinary life. They surround us the way radio waves and dark matter do, is how I think of it, and, like hurricanes and white sharks and big cats, they suggest the edges of the inapprehensible. Numbers appear to be unambiguous, in that when I write a number, I can identify properties attached to it, but except within the context of the logic that numbers embody I can’t say for sure why there are properties.
Numbers did not initially provoke wonder or reverence. They did that later, partly because they invoked notions of infinity and therefore of God, and, after that, because they appeared to be a language in which nature could be expressed. And because on examination they showed themselves to be complex in ways that had nothing to do with what we thought about them. Not being able to settle on an origin story for numbers means that there is no origin story for mathematics, either. It is a question that is always modern.
Even a slight acquaintance with numbers equips a person to consider large mysteries in mathematics. The simplest unsolved problem in mathematics is called the Collatz conjecture. It is named for the German mathematician Lothar Collatz, who introduced it in 1937, but it is likely older than that and is maybe even ancient, meaning that it might have been known to mathematicians in Babylon, China, and India.
The Collatz conjecture says that every number will return to 1 after a simple process: If it is even, divide it by 2. If it is odd, multiply it by 3 and add 1. 5 becomes 16 becomes 8 becomes 4 becomes 2 becomes 1. 9 reaches 1 in 19 steps. The conjecture is known to be true for all numbers with fewer than 19 digits, numbers less, that is, than a quintillion, but no one knows if it holds for an infinite quantity of them.
“Hilbertian optimism,” named for David Hilbert, the German mathematician of the late nineteenth and early twentieth centuries, is the belief that every mathematical problem is in principle solvable. In 2010, Jeffrey Lagarias, an American mathematician who is an authority on the Collatz conjecture, described solving it as “completely out of reach of present-day mathematics.” The Hungarian mathematician Paul Erdős said a proof was “hopeless. Absolutely hopeless.” It is thought possibly to be undecidable.
NUMBERS ARE LIKE wolves. Only with difficulty can they be made to cooperate past any simple and self-interested exchange. A person sharing their company has to discover what they are inclined by their temperaments to do. The best way to get along with them is to let them do what they want. Essentially they’re unbending and can’t be made to act in a way that violates their natures. The study of their behavior, to extend the analogy, is number theory. Practical math, called applied math, useful in commerce and engineering, is a different creature. Applied math is the dog, compliant and eager to be helpful.
Antiquity: the first figure known to speculate about the origin of numbers was Pythagoras, a pre-Socratic philosopher of the sixth century BCE. Pythagoras is one of the great seer-like/quasi-holy men, at least he is alleged to be. He appears to have been the founder of a manner of living that had strict dietary practices and required five years of silence as a means of learning self-control. I say appears, because hardly anything reliable is known about him. Pythagoreans tended to attach his name to their writings, so knowing what he wrote, or if he wrote anything at all, is complicated. Some scholars think that the absence in the two hundred years following his death of anyone quoting from any of his books suggests that he didn’t write any, perhaps from insisting on silence. It is generally accepted that he left no trustworthy record of himself. These days he would be a celebrated mystic/thinker with no internet footprint.
In Life of Pythagoras, Iamblichus, an Arabian philosopher who lived about eight hundred years after Pythagoras, says that Pythagoreans viewed Pythagoras as a type of indefinite being. “Of rational beings, one sort is divine, one is human, and another such as Pythagoras,” Iamblichus writes. Pythagoreans believed that Pythagoras knew that the soul was immortal and was reincarnated, sometimes into animals, which he might have learned by visiting the underworld. The philosopher and mathematician Dicaearchus, a student of Aristotle, said that Pythagoras knew that after enough time events recur and that nothing is happening for the first time. Other Pythagorean highlights: he killed a poisonous snake by biting him; once while he was crossing a river, the river spoke to him; on the same day, at the same time, he was seen in two places; he had a thigh made of gold, a mark of divinity; he performed ten thousand miracles. Herodotus says that Pythagoras knew a great deal about sacrifices and rituals and that he agreed with the Egyptians that a body should not be buried in wool, which is impure.
In “Pythagoras as a Mathematician,” an essay by Leonid Zhmud published in Historia Mathematica, Zhmud says that Pythagoras seems to have been responsible for the theory of even and odd numbers, which Zhmud says is “the first example in the theory of numbers.” It begins, “The sum of even numbers is even; the sum of an even number of odd numbers is even; the sum of an odd number of odd numbers is odd.”
The most famous object with Pythagoras’s name on it is the Pythagorean theorem, a² + b² = c², where a and b are sides of a right triangle and c is the hypotenuse. Numbers such as 3, 4, and 5 that satisfy this equation are called Pythagorean triples and were known in Babylon, which Pythagoras visited. His contribution may have been to see their application to geometry.
Pythagoreans believed that the world’s design depended on numbers and that nature had a mathematical structure. In The Theoretic Arithmetic of the Pythagoreans, published in 1816, Thomas Taylor writes that Pythagoras defined number as “that which prior to all things subsists in a divine intellect, by which and from which all things are coordinated.” Iamblichus attributes to Pythagoras the observation that “number is the ruler of forms and ideas, and is the cause of gods and daemons.”
The Pythagoreans gave certain numbers qualities. According to Taylor each of these numbers “was philosophically adorned with various attributes ranging from physical to supernatural to mythic to aesthetic to moral.” Pythagoreans believed that One, the unity, was divine, and they called it Apollo. Two was audacity, because before all other numbers it separated itself from the One. Furthermore, two stood for opposites and was ignorance, because it was separated, but it also implied wisdom, since ignorance leads to wisdom. According to Nichomacus, a Greek mathematician, in The Theology of Arithmetic, a compendium drawing on various sources—in this case, Anatolius, a philosopher and theologian—“The triad, the first odd number, is called perfect by some, because it is the first number to signify the totality—beginning, middle and end.” Four, the tetrad, was “the greatest miracle, a God after another manner (than the triad), a manifold, or rather, every divinity.” Five, the pentad, was “the privation of strife, and the unconquered, alteration or change of quality, light and justice, and the smallest extremity of vitality.” It was also immortal. The Pythagoreans believed that in addition to the nine planets, there was a tenth, the counter-earth, because ten, the decad, was a perfect number—being the sum of 1, 2, 3, and 4, the first numbers, and it was sensible that the heavens would have a perfect design.
After Pythagoras, mathematics is no longer regarded as merely practical, and it is never again separate from philosophical and spiritual speculations or the presumption of abstract qualities. When I read Sextus Empiricus I sense the frustration that ancient thinkers felt when they tried to get numbers to behave according to rules that humans imposed on them. Empiricus lived in the second or third century CE, exactly when isn’t known. He was a Pyrrhonian Skeptic, meaning that he believed that every argument had an opposite and equally persuasive argument. Someone who could hold both arguments in mind attained an objectivity that delivered him or her to a state of tranquility. Empiricus writes in Against the Mathematicians that numbers don’t exist. All numbers are multiples of 1, the monad, he says. Putting two monads together allows the possibilities of subtraction or addition. If one monad is subtracted from the other, a monad no longer exists. Adding one monad to another gets four monads, not two, because the dyad produced by the first operation consists of two single monads, meaning there are four monads altogether. This makes addition impossible, since “the same difficulty will exist in the case of every number, so that owing to this number is nothing.”
One morning in the Algebra Hotel, I wrote “Number is nothing” on a piece of paper and taped it to the wall above my desk, a small defiant act.
By week six, I was losing courage, a little. I had gone far enough, and I had foolishly told enough people what I was doing, that there was nothing to do but keep going. One foot in front of the other toward the hills in the distance until I saw the paths and patterns that everyone else saw.
My specialty of overthinking makes things harder. Also, my resistance to learning new ways of thinking. Also, my suspicion, especially with word problems, that I am being deceived. Every now and then, of course, this is true. When I was a boy, math teachers and writers of math textbooks loved trick questions, and to my astonishment they still do. It seems a shabby device, a wan practical joke. The rhythm of their appearing is more or less the same as the rhythm governing my resolutions to stop looking for trick questions and just answer what’s apparently being asked, so I keep falling for them.
Sometimes I feel that these problems are designed to exploit a weakness in my thinking, which is that I search too hard for a solution, which means that they have exploited a weakness in my thinking, the tendency to overthink.
It helps to remember the math professor Morris Kline’s observation that especially with arithmetic and algebra simple ideas often took thousands of years to arrive at. My slow, even laborious, progress seems less chastening then.
ANSWERING A WORD problem correctly sometimes feels like making a trick shot in pool. Certain mechanics apply generally, but each shot has specificities, and being good at one shot doesn’t mean being good at a different one, even if the difference between them is slight.
I wonder if the talents needed to solve word problems are not congenial to someone whose habits of mind are those of a daydreamer, except that I have been given the example of pure mathematicians who seem to live all but entirely in a dream world. Of course, that doesn’t mean that they are good at word problems.
I noticed that musicians seem to include mathematics, even if obscurely, in the way that they regard the world, in terms of melody, rhythm, and chord structure. Often they were comfortable as adolescents with math. I thought this was an observation of my own and was disappointed to learn that others had arrived at it first. Then I was further chagrined to learn that there was even a reaction against it, that musicians don’t care to be described as being good at math, they think it is reductive. Still, I think it applies. Harmony, the circle of fifths, the modeling of scales and modes, rhythms and odd time signatures all demand an ability to conceive of the world, or one’s part of it, as signified by numbers and mathematical intervals and to count. The harmonic structure of a piece of music is a progression of mathematical forms. The scales and modes and chords built from them are a series of mathematical relationships. In the key of C, which has no sharps or flats, the major scale is C, D, E, F, G, A, B, C. The initial triad is C major, built from the first, third, and fifth tones: C-E-G. C-E-G-B is C major seventh. C-E-G-B-D is C major ninth. C-E-G-B-D-F is C major eleventh, and C-E-G-B-D-F is C major thirteenth. Chords can be inverted: C-E-G-A, which is C major sixth, is also A-C-E-G, A minor seventh (A-C#-E-G# is A major seventh). Chords can also be substituted for one another: C major seventh, C-E-G-B, includes E minor, which is E-G-B. In both classical and improvisatory music, they can be made to resolve in a number of ways that involve the relations among their intervals. What we hear are mathematical arrangements in the form of tone and rhythm. Leibniz somewhere remarks that music is the pleasure the human mind receives from counting without knowing it is counting.
WORKING AS METHODICALLY as an accountant, I was eventually able to solve word problems more often than not. I didn’t have a breakthrough, so much as I just sort of wore them down. I managed more than thought my way through them. Still, I was pleased. I felt I had partly erased a flaw in my past.