A Divine Language Read online

  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.

  4.

  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.

  5.

  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.

  6.

  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.

  7.

  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 figu
res 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.