Summer
1.
When I finally reached calculus, I was somewhere I had never been. I could have said that algebra introduces abstractions into arithmetic, and geometry studies forms, but I couldn’t have said what calculus does. I expected it to be beyond my capabilities and was gratified when only some of it was.
Calculus departs almost completely from arithmetic, by which I mean that numbers do not really explain it. Nor does the logic of geometry. The definition of calculus that I find most often is that calculus is a means of describing instantaneous change. In A Tour of the Calculus, David Berlinski writes that calculus measures “how far an object has been going fast,” and “how fast an object has gone far.”
Another description I have heard is that calculus uses linear forms to bring precision to nonlinear forms—straight lines to measure curved lines such as circles, ellipses, and parabolas, which depicted on a graph might represent, among other things, movement or distance or the rate at which something is increasing or decreasing or time, all of these being cases of change. A turning line on a graph can’t be measured precisely, for example, but a straight line, technically a line segment, can. Likewise, a turning path can be approximated by dividing it into segments of straight lines. Adding the lengths of the segments describes the path. The smaller the segments, the more precisely they portray the curving line. Calculus then is a series of approximations that become more and more accurate as the measures are refined. Anything that changes demonstrably, both actual and imaginary things, including markets and machines and designs for potential machines, is within the range of calculus.
The measurements can never be exact, because of a concept called a limit. A limit is an endpoint in an investigation, a destination that you can get arbitrarily close to but usually never reach. In the case of a turning path, the limit is where you would arrive if you could take an infinite number of steps, which you can’t. A limit is exemplified in Zeno’s paradoxes, the most famous of which probably is the race between Achilles and the tortoise, in which the tortoise gets a head start. Zeno’s argument is that Achilles can never pass the tortoise, because every place he reaches is somewhere that the tortoise has already left. He can get half as close, a quarter as close, and so on.
Approaching a limit, measurements cluster as finer and finer reductions are made. In measuring distance, consider the diminishing sequence 1, 1/2, 1/4, 1/8, 1/16, 1/32,… The figures do not reach their limit, which is zero distance, because eventually they would approach 1/∞, which is undefined, meaning that it does not exist, division by infinity being impossible. Even so, the sequence approaching zero has an infinite number of members.
If speed is being measured, the time considered cannot be reduced to zero, because at zero there is no movement. An object moves only between two moments.
A limit allows precise rather than average descriptions of change. Average speed is simply distance divided by time. Driving 60 miles over 2 hours means an average speed of 30 miles per hour. Calculus determines the speed over an interval—how fast the car was going between 48 and 49 miles, say. Between 48 miles and 48 miles and 5 feet. Between 48 miles and 48 miles and 6 inches and so on.
There are two kinds of calculus, differential and integral. Differential calculus divides up time—how fast an object has gone far (or how fast a process is accumulating). Integral divides up distance—how far an object has gone fast (or how much is accumulating). To assess the progress of a car trip, say, you could plot the trip on a graph on which the x axis is time, and the y axis is distance. The trip would begin where the axes intersect, at the point 0,0, and proceed as a line on the graph. On the x axis the trip’s endpoint would represent how long the trip took, and on the y axis the endpoint would represent how far you had gone. (If you had ended where you started, left home and returned, perhaps, the endpoint on the x axis would be 0.)
The line would curve, because sometimes you sped up and sometimes you slowed down, and if you stopped it would be flat until you started again. To reveal speed, you find the slope of the line between two points—the difference, that is, between the heights at each point—and divide it by the time the line encloses; this is called rise over run, a phrase with its own kind of poetry and one of the few things in mathematics that I remembered easily. As the distance between the points gets closer to zero, the slopes approach the derivative. A version of the derivative is what a speedometer provides.
Knowing how fast you’d been going, you can make a second graph where the x axis also represents time but the y axis represents speed instead of distance. The measurement of the area between two points on the line and their coordinates on the x axis tells how far you had traveled between any two moments. This is the integral. Finding the derivative uses mainly algebra, in the form of difference in ratios, and finding the integral uses geometry in the form of computing area. One operation reverses the other. The first case used distance as a function of time, and by means of differentiating—finding the derivative, that is—figured out how fast you were going. Measuring the area beneath the curve on the graph, by means of integration, revealed how far you had gone. By differentiating and then integrating, one returns to where one began. This is elegant and surprising and exemplifies the fundamental theorem of calculus, which says that differential and integral calculus are inversions of each other. This much about calculus I was able to absorb after much effort.
2.
I don’t want to say that Amie was mistaken, but word problems return in student calculus. A car stops, a man or a woman gets out. The car drives away. How fast is it moving when it is twenty feet from him or her? A quarter mile? Ten miles and thirty-four inches? And how long has it taken to get there? I have also encountered calculus as the means for solving problems such as, What is the price at which an item will draw the largest number of buyers and make the most money? How long should you own a car before the depreciation and the repairs make it no longer sensible?
When I came to these, I did something I didn’t do in algebra: I turned the page. I thought, Life is too short.
3.
Algebra and geometry are earthbound and practical, and there doesn’t seem to be anything mysterious about their simpler versions. They have their origins in the immediately visible world. How to determine the boundaries of a piece of land or establish the characteristics of forms. Geometry is a bridge to abstract thinking in being bound up with suppositions about the world’s design, but calculus is about being made intimate with unseen things, about making the difficult to observe comprehensible. Algebra and geometry imply time, in the unfolding of their procedures; calculus embodies time.
I found starter calculus formidable in a way that starter algebra and geometry aren’t. Calculus contains powerful tools that are not so easy to understand, although (some of them) manageable by means of persistence. The pleasure of using calculus, though, compared with algebra and geometry, is greater even at the simple level I am acquainted with. One feels engaged with larger powers and occasionally capable of impressive gestures; at least they impressed me. I had as much difficulty with calculus as I had with algebra and geometry, maybe more, but I thought I was cool while I was doing it.
CALCULUS WAS CREATED to provide answers for problems in physics that beforehand could be done only painstakingly or by improvised methods. Isaac Newton is regarded as having discovered calculus, but Gottfried Leibniz worked on similar material and published before Newton, who for roughly ten years had kept his work to himself. Newton was in England, and Leibniz was in Germany, and while Newton is given more credit than Leibniz is, Leibniz’s notation is the one that is used.
Newton organized disparate procedures into a formal order. He was intending to provide a working method, rather than prove the theoretical basis for one. In this sense, Deane says, it was as if he had begun in midair rather than on the ground the way someone such as Euclid did. Unlike Euclid, also, Newton wasn’t specifically engaged in extending the range of mathematics. When his suppositions were correct, he pressed forward instead of working backward to establish a foundation, as another mathematician might have. Newton’s influence, partly, was to favor intuition as a source for mathematical ideas, for finding problems in the physical world and inventing mathematics for solving them. Before him, most mathematics involved solving problems presented by earlier work. An attitude about pure mathematics begins with Newton.
Physicists tend to take on a concept that perhaps has an intuitive meaning and use it to develop a tool or a theory. When they see it describe something in nature, they figure it works. Mathematicians examine the concept and demonstrate that it actually has a rigorous basis in mathematical axioms. This is called axiomatizing physics.
Physics is descriptive and not generally concerned with logic. A physicist cares about calculations that describe and predict observed phenomena. There are no external phenomena in mathematics. There are only assertions and axioms and their consequences. Things are pursued, but they are abstract things, and one can never be completely sure that the sought-after thing will appear. There is also the possibility that the mathematics will refute it.
THE ELEGANCE, THE ambition, the sweep of calculus signify a moment when the world stilled and a spectacular truth came into being, although only for one person, or maybe two, to give Leibniz his due. Einstein said of calculus that it was “the greatest advance in thought that a single individual was ever privileged to make.”
Since change is pervasive and everlasting, calculus applies widely; in mathematics it is a small part in a lot of fields, and in the real world it applies in engineering, commerce, markets, stock trading, statistics, and medicine, among many other fields. The things that calculus describes—the gradients of change in a world that fluctuates constantly and permanently at all degrees of its scale—are so intrinsic and essential to understanding movement and time that it seems like it would have been found sooner. The Greeks had possession of some of the means necessary. Archimedes’s measuring the area of a circle by drawing straight lines is conceptually connected, but the Greeks conducted mathematics in the context of proofs, to which calculus didn’t initially lend itself, and furthermore, while the Greeks were aware of irrational numbers, they didn’t commonly use them or for a while even accept them, and irrational numbers are required in order to express fine degrees of measurement. Maybe, also, the world changed less dramatically in front of their eyes. The progress of a ship toward the horizon or from the horizon to the harbor or the transits of the sky and the heavens happened so gradually that perhaps they didn’t suggest a subject for investigation.
4.
Newton’s law of universal gravitation describes a mathematical pattern that is real but invisible. The circumstances resemble the abstract quality of numbers and mathematical objects in that you can see them only when they are represented by something else, when they are practical metaphors (AAAA). Pure mathematics must be invoked to be seen, but Newton described something ubiquitous that is both concrete and hidden.
In A Tour of the Calculus Berlinski writes that with Newton one has the impression of math’s being used to explore parts of the world that are beyond what the senses explain. With The Principia, the proper title of which is Philosophiæ Naturalis Principia Mathematica, Newton ends the version of the universe as being without governing principles. Before Newton, the planets and the stars had been assumed to behave according to laws known by the spirits or gods or God, depending on the period. There were celestial laws for the heavens and terrestrial laws for earth, and they were assumed to be different from each other. The Pythagorean and Euclidean notions of the world’s being built from mathematical forms were speculative and mystical, whereas Newton established that the portrayal of the physical world by mathematics is actual. Berlinski describes the universe after Newton as being “coordinated by a Great Plan, an elaborate and densely reticulated set of mathematical laws.”
For Christmas Day 1942, the tercentenary of Newton’s birth, the Royal Society of London planned a celebration, but the war prevented it. The economist John Maynard Keynes was to have addressed the ceremony, but he died a few months before it was finally held, in 1946, so his brother read the remarks he had written. Newton was widely regarded as a modern figure, the first figure of the Enlightenment, but he was also a transitional one, being, Keynes said, “the last of the magicians, the last of the Babylonians and Sumerians, the last great mind which looked out on the visible and intellectual world with the same eyes as those who began to build our intellectual inheritance rather less than ten thousand years ago.”
Newton thought of the universe and its contents as “a secret which could be read by applying pure thought to certain evidence, certain mystic clues which God had laid about the world to allow a sort of philosopher’s treasure hunt,” Keynes said. (These are remarks that might also be applied to Kepler.)
5.
I thought that if I could learn calculus I would begin to apprehend that behind ordinary appearances were patterns that were stable but also constantly enacting themselves. Or perhaps not behind but more closely at hand, so as to see relations that unfolded over discrete intervals, something at times like the rolling patterns of waves in deep water.
I am right up against melodrama here, but I needed romantic assumptions to sustain me sometimes, especially as the work got harder.
6.
Nearly a year in, after it was too late to do much about it, I realized that I had become separated from ordinary life. My wife would leave for work, dressed differently each day, and I wore the same clothes all week. There was something solemn and monk-like about my confinement. For hours I turned the pages of books, and everything took place in my head. I might as well have been working by candlelight. It wore on my spirits to be removed from the rest of the world. I had immersed myself in math, and I hadn’t expected to. I had thought it would be easier to learn. I thought it would be a laudable and high-minded pastime, but math became all that I did and thought about. I had an abstruse hobby, not a hobby, an obsession. Everyone was interested when it came up, but no one wanted to hear about it once they saw that I’d gone a little crackers with it. They were mostly math averse, and I struck them as zealous. I could listen when people spoke to me, I could nod and appear to be paying attention, but I wasn’t really. I had no idea how long some of my silences were.
7.
The attractive equation ∫ba f´(x)dx = f(b) − f(a) says that the integral from a to b of f´ is equal to f(b) minus f(a)—f of b, that is, minus f of a. It took me much more time than it should have to understand that the notation dx stands for change in x and that d is not a number to be multiplied by x, as it would be in algebra. It is a fundamental distinction, and trying to work the problem without understanding it is hopeless, but no one had explained it to me, and I had failed to grasp its meaning in whatever texts I’d encountered it in, which was unfortunate. I lost a painful amount of time trying to figure out why my answers were not only wrong but wildly wrong. Exotically wrong, comically wrong, I might have thought, if I had any tolerance left for being wrong.
dx is not an instruction to perform a task. It’s a phrase, or maybe an abbreviation, for change with respect to x. Also, x times dx is not dx2, but xdx. dx in algebra treats d as a number; xdx in calculus is a concept, not a procedure. To try to work out why none of this was clear to me, I turned to Calculus Made Simple, by H. Mulholland, and only became more deeply lost and also indignant at the title. Simple for who, I’d like to know. In Mathematics for the Nonmathematician, a non-mathematician appears to be someone with at least a college-level familiarity with, if not a higher degree in, mathematics. I tried Calculus Made Easy, by Silvanus P. Thompson, which was published in 1910, partly on the assumption that I ought to be at least as smart as people were a hundred years ago. On page 20, Thompson says that he is employing the binomial theorem, which, if I need to be reminded of it, he says is on page 116. I do need to be reminded of it. Actually, I need to be introduced to it.
Putting aside that I haven’t ever read a book where the writer tells me that the information I need appears much later in the book, I make the trip to page 116, where I find:
Accordingly, we will avail ourselves of the binomial theorem, and expand the expression 
in that well-known way.
The binomial theorem gives the rule that
Since Thompson died in 1916, I can’t ask him what he meant by “well-known way.” I’d like to rebuke him retroactively, though. Something valuable shouldn’t by ineptness be made more difficult to learn. That should be unacceptable. It awoke my resentment of haughty explainers who can’t explain things clearly.
At the worst moments with calculus, I try to write out what I am attempting to solve, as I had been able to do sometimes with algebra, and I find that often I can’t even frame the question. There are also periods when I spend my time unwisely. Accidentally dropping elements of equations, trying to get perfect scores, but making arithmetical mistakes, or losing track of positive and negative signs, which is my specialty and easier than you might think. You start at the beginning with algebra and geometry, but calculus is a compendium. Each time you nod your head in calculus, you are affirming three or four principles. It is a dense exchange.
Mathematics has a quality like a spiritual practice in that I can’t pretend to know what I don’t know. I can guess at an answer, as I had in high school, and of course sometimes I will be right but mostly I won’t. Math has taught me in its own severe way that I can’t trust the passing view, the glance. The world being as perilous as mathematics is, it’s helpful to be reminded that I need to attend the present, while also considering the past—that I need to pay attention. One consequential judgment made or action taken without the support of thought and reason is hazardous in life, as in math. The most reliable way for me to leave the trail of the correct answer is to take the figuring too fast or cavalierly. Even in an empty room, at the edge of old age, I feel the pressure of the classroom, and the requirement from my childhood to be the bright boy.
MATH AND LANGUAGE may be connected in the minds of some linguists, but learning math does not seem to me to be like learning another language. It is like learning a language when you don’t have one yet. There are rules and circumstances to grasp before you even guess that you can use this tool to express thoughts and feelings. I may be inclined this way, though, since I am doing this late and my learning is compressed. When we are children, the span of time between learning to count and learning algebra is long, and there are years to achieve some sort of facility with numbers before we are expected to use them in complicated ways.
ON A ROUGH day with calculus, opening a textbook, I see a blizzard of numbers and symbols, and think of other things that I might rather be doing. Somewhere late in algebra, or maybe early in geometry, is when I realized that new knowledge lingers fitfully in the mind, and that it isn’t possible to remember everything I am learning. An acquaintance is what I am managing. What I wish is that in high school, while studying geometry, say, I had also been taught the ways in which it figures in painting and perspective, in architecture and in the natural world and so on, so that I could see it perhaps the way the Greeks had seen it. If I had felt that the world was connected in its parts, I might have been provoked to a kind of wonder and enthusiasm. I might have wanted to learn. In approaching mathematics now through its separate disciplines, I wonder if I have misaligned my purposes. Perhaps I should have done what Deane suggested and looked more for mathematics in the world and less on the page.
What I might have seen as a child if I had paid closer attention or perhaps been more receptive is that behind the lessons I was learning was a form of instruction that would equip me to think for myself, a necessary condition of life. An advantage of this endeavor has been learning to think differently. It has been partly a project of renovation.
8.
To return to my parallel theme: believing in Platonism, at least in some degree, means accepting a companion reality that can’t be seen or located or have its dimensions made plain, and, except to say that it is entirely other, can’t be described beyond generalizations. Like numbers, human experience has abstract and practical sides, feeling and reason, and in each of us one or the other tends to dominate. Belief does not persuade a scientist, and science does not persuade a believer. Too ardent an embrace of reason leads to irrational thinking, and too ardent an embrace of feeling leads to madness. William James says that religious mysticism is only half of the possible mysticisms, the others are forms of insanity. These are the states in which mystical convictions circle back on a person and pessimistically invert notions of divinity into notions of evil.
Feeling responds to intuition, which arrives unbidden and suddenly. The suddenness by itself often persuades, Bertrand Russell says, but reason decides whether an intuition is reliable. In mathematics, as in art, intuition produces thoughts that reason enacts.
The boundary between these states is fluid and encloses the territory occupied by mysticism, of which there are two types. One type involves believing that there is more to life than we apprehend. The other includes people who have had a mystical experience. For as long as I can remember, I have been among the first type. If a mystical experience is one that makes a person aware of being a smaller part of a large design, then I guess I belong to the second type, too, if maybe weakly.
I’m not proud of this, especially. Having mystical inclinations seems a little simple. A means of thinking about the world that isn’t too taxing or penetrating, but is satisfying and makes one feel a connection to deeper forces and broad in one’s outlook, and, in a vain way, select, with finer capacities than many people have. It isn’t rigorous, though, and maybe it is even shallow. Mystics sometimes appear to be insufficiently tough-minded and indifferent to reason, to be credulous and unsophisticated versions of theists. Mysticism has the appeal of the express arrival at an end, rather than working one’s way patiently toward a position. We say, I know this is true, but what we mean is, I feel it to be true, or I believe it to be true, or, secretly, I wish it were true.
On the other hand, perhaps mysticism is an attempt to find a larger arrangement, a place where myth and science intersect, the boundary between the inner and outer lives. “The most beautiful thing we can experience is the mysterious,” Einstein said. “It is the source of all true art and science. He to whom the emotion is a stranger, who can no longer pause to wonder and stand wrapped in awe, is as good as dead.”
9.
The companion region in which Platonism resides has no significant divisions. According to Russell, all things being one, there is no past or future. The world, as Leonard Meyer describes it in Music, the Arts, and Ideas, is “a complex, continuous, single event.” This is perhaps an Existential mysticism as much as a Platonic one.1
In Naming Infinity, Loren Graham and Jean-Michel Kantor write that mathematics has been the means of approaching the Absolute in all ages and cultures from “the classical Greek period, to the pre-Socratics, the Egyptians and the Babylonians,” and in “China, India, and in Muslim, Jewish, Christian, and Buddhist tradition.” Another means of considering mystical inclinations is to regard what we see as the surface of something else, as the outskirts of a further reality, areas of which mathematics describes. This can suggest the inner life and the psyche or the larger one represented by the cosmos. Animism, astrology, spirit thinking, the Greek and Roman gods, gods of all kinds and races are descriptions of an engagement with the ineffable. “The insight into the mystery of life, coupled though it be with fear, has also given rise to religion,” Einstein says. “To know what is impenetrable to us really exists, manifesting itself as the highest wisdom and the most radiant beauty, which our dull faculties can comprehend only in their most primitive forms—this knowledge, this feeling is at the center of true religiousness.”
I WOULD PREFER not to regard my receptivity to mysticism as a flaw. I could try to rid myself of it, to grow up, as it were, but this seems like trying to learn to throw with one’s opposite hand. And what would be the point. Trying not to think about something leads to thinking about it more, in my experience. I would rather believe that it reminds me that human life contains layers of consequence and meaning, and can be contemplated endlessly without necessarily arriving at an end, a point of view that is itself mystical; apparently this type of thinking is circular. Perhaps, so far as thinking is concerned, there is no end to arrive at, or we are not yet anyway equipped to arrive at one.
Russell says that mysticism is better embraced as an attitude than a creed. “By sufficient restraint, there is an element of wisdom to be learned from the mystical way of feeling, which does not seem to be attainable in any other manner,” he writes. Overblown mysticism is a misled product of human feeling. As an element among thought and feeling, though, mysticism “is the inspirer of whatever is best in Man,” Russell says. “Even the cautious and patient investigation of truth by science, which seems the very antithesis of the mystic’s swift certainty, may be fostered and nourished by that very spirit of reverence in which mysticism lives and moves.” The Russian mathematician Nikolai Luzin wrote that while there are disciplines such as physics and biology that rely on the senses for insight, and logic and mathematics, which rely on reasoning, there was a third means of understanding, generally overlooked by scholars, which was “intuitive-mystical understanding.”
By temperament I am more inclined to observe than to appraise. I haven’t really got any critical faculties. I’m an enthusiast, a species of surveyor, the sort of person who cannot pass a hole in the ground without stopping to look into it.
10.
A religious mystic believes that God is understood better through illumination than through proposition and appraisal. An example of a non-mystic religious thinker would be Descartes, who thought that he could prove God’s existence by logical means.
Some people who use mathematics to try to find God are irrationally excited, but there are also great mathematicians who feel that mathematics exemplifies something elementary and elusive about existence. Pythagoras, Blaise Pascal, Georg Cantor, and Hermann Weyl all had mystical inclinations. In God and the Universe, Weyl writes that mathematics “lifts the human mind into closer proximity with the divine than is attainable through any other medium.” Pascal believed that the two infinities, small and large, were enigmas offered by nature that, as Alain Connes observes, could be admired but not understood. We are poised between the two infinities surrounding either end of our lives the way mysticism is poised between reason and feeling.
I believe the mystical realm exists, but this is a declaration of faith. Julian Jaynes proposes in The Origin of Consciousness in the Breakdown of the Bicameral Mind that human consciousness as we know it develops in the period between The Iliad and The Odyssey. In The Iliad the instructing voices come from elsewhere, from fogs and hallucinated figures, he says, whereas in The Odyssey, they reside at least partly in the minds of the characters. The impulses behind creative thought have never been easy to locate. An artist doesn’t always know the sources of his or her work. Perhaps the images were created in their minds, but perhaps artists have a capacity or a receptivity that is greater than in more conventional sensibilities or perhaps they resist it less than others do. I cannot say that this view is not sentimental. I prefer to think that there is more to life than we apprehend, but clearly it isn’t something I can prove. Arguing the existence of something that is immaterial is difficult, even though people try to all the time.
IN The Varieties of Religious Experience, William James writes that someone who has had a mystical experience can convey its quality only vaguely and indirectly. Moreover, someone not receptive to obscure states of being cannot appreciate what one actually is and is likely to think that the person describing one is being ridiculous.
Mystic states “are states of insight into depths of truth unplumbed by the discursive intellect,” James writes. They appear to be transitory and transcendent positions between thought and feeling. Moreover, they have a successive effect in that ones that return seem to deepen the meanings of earlier ones. Finally, they are independent of conscious control. They might be facilitated by preparations “as by fixing the attention, or going through certain bodily performances, or in other ways which manuals of mysticism prescribe,” but basically they arrive when they arrive.
The capacity to entertain mystical feelings perhaps exists in all of us but is elevated only in some of us. Whatever it is, “We are alive or dead to the eternal inner message of the arts according as we have kept or lost this mystical susceptibility,” James writes. We are separated from mystical experiences “by the filmiest of screens,” he continues. “We may go through life without suspecting their existence; but apply the requisite stimulus, and at a touch they are there in all their completeness, definite types of mentality which probably somewhere have their field of application and adaptation. No account of the universe in its totality can be final which leaves these other forms of consciousness quite disregarded. How to regard them is the question.” James quotes a man receptive to such experiences who wondered if they were the same as what the saints describe and are not “the undemonstrable but irrefragable certainty of God?”
Mystical states, like occasions of faith, are “absolutely authoritative over the individuals to whom they come,” James writes, but they don’t often persuade anyone else. They demonstrate, though, that there is more to consciousness than the form that we accept as dominant. Kantor and Graham believe that the ineffable states that James describes arise during mathematical work, “moments when mathematicians refer to ‘marvelous intuitions.’” In mathematics, according to Plato, “one seems to dream of essence.”
I wonder how in our times a God-evoking experience might be obtained, but I think it would also be a terrifying one. It would be difficult not to think that one was going crazy. The description of God that I find most persuasive is the one by the southern evangelist who told the writer Philip Hamburger that God was a “luminous, oblong blur.”
11.
With calculus I learned that I had to do even more than I had become accustomed to doing. I had to teach myself to follow an explanation to its end instead of seeking flaws as it was unfolding or looking for exits. I had to suppress my anxieties.
I haven’t encountered a border with mathematics that is precisely defined, but if I go farther, and maybe not all that much farther, I will find material I simply can’t comprehend even if it’s explained to me patiently. That, I suppose, is where I will have to stop.
12.
Artists can quote other artists, but only for effect. They can quote straightforwardly or stealthily or ironically. Mathematicians quote each other constantly, by applying theorems and methods, but always in full and without irony.
Controversy exists in mathematics, but it is not general or common. When it arises it tends to be specific to a case—is this or that piece of work new, or was it borrowed from someone else? Even when something is discovered to be borrowed, it is not overthrown, the way, appropriation art aside, a piece of plagiarism in the other arts, even if accidental, might be; it is merely given its proper attribution. In The Weil Conjectures, Karen Olsson describes how André Weil published a result that was called the Weil conjecture, although not the ones he published in 1949, which are still attached to his name. When it became known that Weil’s work had drawn heavily on conversations with Yutaka Taniyama and Goro Shimura, the name was changed to the Taniyama-Weil conjecture, then the Taniyama-Shimura-Weil conjecture, and now commonly, Olsson writes, to “the modularity conjecture for elliptic curves.”
Reading and rereading formulas and equations and theorems in the attempt to understand them is different from looking for meaning in difficult prose or poetry. A literary reader is seeking a writer’s intentions, which may be willfully concealed, or possibly not even clear to the writer. Mathematics prohibits obscurity, at least of a willful kind. In mathematics, the clearer something is, the more powerful it is, and the more useful. To obfuscate in math restricts the reach of the idea involved. In writing, in speaking, obfuscation has its place as a means of refinement, concealment, ornamentation, and even prestige. To write prose or poetry that is difficult to understand, to restrict intentionally one’s audience, is to elevate the importance of one’s work in the minds of some people. Sometimes, of course, it is merely pretension, a form of incompetence.
I can read a sentence over and over in prose or poetry and not be sure I understand what the writer intends, but I can continue reading, knowing usually that my uncertainty will not mean that I can’t later find something larger and deeper in the design. So far as I can tell, you can’t advance in math without understanding the design and its references. You don’t always have to have the end in sight, but you have to see a way to it or part of a way. You have to know where to begin. I don’t know if my mind doesn’t find this form of thinking sympathetic, or if it is simply a matter of the material’s being difficult. Still, it is a strange thing to confront something that I can almost but not quite understand, something that I know is not ambiguous, that has a single, universally understood meaning for those who grasp it, a remark that is a statement of fact, a logical extension of something I have been understanding, following, even if imperfectly, but am now lost with. An increase of thinking is required, and I don’t know why sometimes I can’t make it. Is there such a thing as thinking everything that one can, terminal velocity thinking? Or a state where no more thoughts are possible until some thoughts are shed? Once it became clear that Amie wouldn’t be able to steer me through all my difficulties, perhaps I should have engaged a tutor, but that’s against the rules that I had set for myself. It would be as if I had determined to build a house and was calling in a carpenter for the parts that were hard or seemed to be beyond my capacities. If I did that, I wouldn’t be able to regard my house as my own work. This is pedantic, but those are the terms I adopted.
13.
In shedding “Math is inconsistent,” for the awareness that the fault is usually mine, I exchange a confidence in my own judgment for a more vulnerable position. On the other hand, I am not unhappy to be reminded that it is better to ask a question than to assume an answer. Humility requires a denial of self, of a portion of self anyway, the flamboyant, the insecure, the part that fears being overlooked at the party, a phrase I am borrowing from William Maxwell. To allow another person, to allow everyone, the same regard, the same justification, is an achievement of psychic life, and a lot works within us to prevent it. And anyway, just coming up with a good question represents an advance. Cantor said that to ask the right question is sometimes more difficult than to answer it.
Humility has been forced on me by my engaging in a pursuit that I appear to be unfitted for. I had expected as I neared the end to feel pleased at succeeding where I’d once failed. To have enjoyed a small triumph in setting a private record straight, and I have, a little, but not to the degree I’d hoped for. If I had turned out to be capable at math, I suppose I could have blamed other people for my failing, but blaming others for one’s poor performance is never attractive.
14.
The days of difficulty accrue. As much as Amie tries to help, the problem of someone’s explaining something to you, especially something complex, is that they don’t know what you don’t know, or why it is hard for you to know it. To be taught a difficult subject intimately is sometimes to wonder how human beings manage to talk to each other at all.
Calculus occasionally became so challenging that I would think, Why do I need to know this? Why am I persisting in trying to learn a discipline that not a single ordinary civilian I know has possession of and that is of no use to me anyway? The questions were beside the point. I don’t need to know calculus. I need to make the as-if-pilgrimage of learning it as well as I can manage, because it asks that I practice patience and discipline and think clearly and try hard. I need to attempt a task that appears to be beyond my abilities and that insists that I think in ways I am not accustomed to thinking or feel comfortable attempting. Learning calculus also embodies my intuition, or perhaps only my desire, that dormant within me are capacities that might be enlivened and make me more than I am. I am hoping to meet a better version of myself that I haven’t yet become.
It seems important to continue enlarging ourselves for as long as we can and to claim new territory for our thinking. We are all given talents, and I don’t want to arrive at the end feeling that I have not made what I might have of my own, however modest they might be. From the lives that I have witnessed the ends of, I know that feeling disappointed by what one has made of oneself is no way to prepare for a decent finish.
15.
Rules in calculus about how to handle formulas and functions, such as one called the chain rule, fall in a territory between numbers and words, making them difficult for me. While being aware that math has an organic quality, that the materials I am studying have developed from one another, in no place more apparent, at least in these early stages, than in Euclid and in calculus, I nevertheless have difficulty discerning the designs. When I was lost among textbooks, I would sometimes watch calculus videos on the Khan Academy website, and not infrequently I would feel like I was observing a magic trick and thinking, How did that happen?
Calculus exemplifies patterns but not always, or at least that is how it seems to me. A function, for example, is a sort of machine into which you enter values, and it delivers a single response. f(x) = x2 is a simple function that delivers squares. An x you enter on the left-hand side of the equal sign becomes its square on the right-hand side. The variations in functions are endless. You can have a simple function, f(x) = x2 + 1. You can have a function f(x) = x2 + 1 if x is even, and x2 + 2 if x is odd or the other way around. Equations that have more than one solution cannot serve as functions: x2 + y2 = 8, for example, cannot be a function, because if you solve for y by entering a number for x, say, 2, you get 4 + y2 = 8, which becomes y2 = 4, which becomes y = 2, –2, since, while the square root of any positive integer is positive, it also has a negative duplicate, a companion on the other side of the looking glass, as it were.
Certain functions are named for their characteristics. f(x) = x2 is a square function. f(x) = x3 is a cube function. f(x) = √x is a square root function. f(x) = 1/x is a reciprocal function. Using f to denote a function is common but not a rule. Functions can also be combined. A combined function is called a composite function, which in beginner calculus is often written as f(g(x)). This means perform on x whatever operation g involves and enter that result into the function that f represents. The chain rule finds the derivatives of composite functions; that is, it finds the rates at which they are changing, it differentiates them. It says that the derivative of f(g(x)) is f′(g(x)).g′(x). The designation f′, which is read as f prime, signifies the derivative of f, and g′ is the result of differentiating g. (The function sin(x²) is the composition of f(x) = sin(x) and g(x) = x².) f′(g(x)) means take the derivative of f and evaluate it on a graph at the point g(x). The derivative measures the slope of the tangent line on a graph at whatever point is chosen, and represents the instantaneous rate of change, meaning the limit of lines that are nearly tangent to the point.
This is not especially difficult, but it is not easy, either. The simplest way I can think of to put it is to say that to differentiate a composite function, you identify the functions that compose it and apply the chain rule. I am aware that the transaction appears to involve invoking a clear-cut procedure, but it was not clear-cut for me. The only reason I raise it is to discuss my next difficulty, which was the case of integration.
In identifying the integral, that is, the accumulation, from the derivative using the fundamental theorem of calculus, you have to figure out what the integral is derived from—that is, you have your answer, but you have to determine what provided it. Amie says differentiating is like following a recipe, say, for pancakes, whereas integrating is having the pancakes and trying to determine what they are made from. A process called u substitution undoes the chain rule, by working backward, and I could follow it, sort of, but only in the simplest cases. U substitution is where I hit the rocks hard.
U substitution involves the notation du/dx, which looks like a fraction but isn’t. It means the derivative of u with respect to x, but there are times when you can pretend it’s a fraction, if that happens to make your problem easier to solve. Exactly when you can do that is something I don’t even now understand.
“We don’t think backwards naturally,” Amie said, trying to talk me down from the ledge. “We do think backwards naturally with murder mysteries, for example, we work backwards to the conclusion, but we don’t do that in math until we get to calculus. Math up to that point is applying a set of rules, and you hit integration, and you have to apply guesses. You have to deduce things indirectly and understand objects by the way they behave, not by what they are.” When she intuited how lost I was, she added, sympathetically, “It’s not well defined.”
Deane said, “The problem with integration is that there is no systematic way to do it, like there is with differentiation. Integration involves trying to guess which techniques might work, then doing the calculations and seeing if they provide a path to a solution. The first step, guessing what might work, can be challenging.”
I about quit when he said that.
ALL I COULD do was go over it again and again, in the hope that familiarity, even if merely in the form of a blunt determination, would cause it to reveal itself. I was a little consoled to find that my circumstances were not singular. In A Tour of the Calculus, David Berlinski writes, “The eye slows; a feeling of helplessness steals over the soul. At first, it seems as if the confident language of mathematical assertion constitutes a subtle form of mockery. There is no help for any of this save the ancient remedies of practice and a willingness to put pencil to paper.”
WHILE BEING SENTENCES, equations carry emphasis and meaning differently from the way that words do. Prose and poetry stimulate the memory, provoke layers of associations or form them, whereas an equation is freighted with rules that one must have command of as a means of translation. Formulas, equations, and functions are commands, not suggestions. In prose and poetry the literal meaning is sometimes a placeholder for other meanings. An equation, while also a placeholder, is typically a vessel for a single thought. It might include many thoughts in terms of its composition and reach, but it delivers a single meaning. The references, the associations, are, by requirement, particular and without ambiguity. It is, in that sense, an ideal form. The rules allow your memories to be attached only to these references and in circumscribed ways. Its instructions are explicit and unbending.
If your mathematical vocabulary is narrow and shallow, as mine is, you might struggle to manage the information a formula or an equation contains or to understand how to use it. Sometimes one has to perform a type of translation. For example, seeing a statement such as (ln(sinx))3, I had to remind myself that ln stands for the natural logarithm, something that, like u substitution, undoes a procedure, the raising of a number, called the base, to an exponent. 103 = 1000. The logarithm of 1000 with regard to a base of 10 is 3. The natural logarithm is something different. The natural logarithm involves a base with a number called e, which is also called Euler’s number and is approximately 2.718. ln(e3) = 3.
Then I had also to know that there are two procedures within what appears to be one: ln(sinx)3, which is to say, the natural log taken of the sin of x, and the natural log of sin(x) then taken to the third power. To differentiate this expression I perform the chain rule on it, which finds the elements that compose the expression and multiplies their derivatives by each other. I have to understand what sinx means, what a logarithm is and what a natural log is, what its derivative is, and what it means to find the derivative of the sin of the term represented by x and then to multiply it by the other term, not to mention using the power rule to find the derivative and also to know what the exponent means. Years of learning are involved, and it’s a simple destination, encountered during the first year of calculus, but a pilgrim has to have been on the road, undergoing trials, for a while to get there.
16.
I felt occasionally like a figure in a myth or a folk tale, attempting a scary or impossible task and having to compose myself in the face of doubts. More than in algebra and geometry, in calculus it seems as if a number of answers might apply, there appear to be a sea of answers, and I am being asked to locate one among them, while sifting water in my hands. Surely I am looking in the wrong part of the ocean, too, as how can I not be. In addition, I feel underequipped, in a knife-to-a-gunfight sense. How deeply I am persuaded of my inadequacies has to do with how many internal taunts I am willing to listen to and how long I will listen to them. Eventually I have to collect myself and reply, “You probably are going to get it wrong, but perhaps you will learn something.” This doesn’t quiet the objections, but it gives me somewhere to stand that is separate from them. I sometimes imagine myself as two different people. One entertains every anxiety provoked by the task, and the other, having handed off his distractions, tries to solve the problem with a less troubled mind. It gives me a little breathing room.
Occasionally at night there are formulas and equations in my dreams. These versions are likely nonsense, the thoughts of a man in solitary writing on a wall, at least I don’t seem to have advanced any when I wake. It is simply as if I couldn’t shake the concerns of the day. My mind appeared determined to keep after them, even if pointlessly.
Grasping something after long labor is sometimes a pleasure. Other times I feel indignant at having to spend so much time at a pursuit that might not be difficult for someone else and maybe even shouldn’t have been so difficult for me. I feel now and then as if I am living in a different world from the people around me. Mine is rich but also fierce and exclusive and remote. They seem to have the run of creation.
SO FAR AS I can tell, the point of an education is to be introduced to books and art and matters of science and history and thought sufficient to engage one for the rest of one’s life or to suggest a means of doing so. An education is a template for knowing where to look for the means to continue one’s learning. Among the men and women whose classrooms I sat in were some inspiring figures and also some duds. Nearly all of them, however, concealed the notion, if they knew it, that I was engaged in a starter course of cultural appreciation as a means of preparing for a life of thought and response, and by thought I mean the capacity to reason one’s way out of difficulties. One needs to know where to turn, to have direction. Children are weeded out from bright futures as often as creatures in the wild come to unhappy ends. No agency protects the innocent, Maxwell used to say. I understood late when I was young that within the approaching rush of the world, there was a path. I wish that I had understood it sooner. I might have learned more and felt confused less often. I might have seen opportunity instead of felt overwhelmed. This concern feeds an anxiety I have that I inhibited my interest, self-protectively, and that if I had challenged myself, instead of giving up, I might have more to show for it. There is only so much, though, that one can reasonably expect of one’s childhood self.
SOMETHING ABOUT THE orderliness of calculus is what I imagine might have appealed to Maxwell, its compactness, its concision, its descriptive powers, its sweep, and its elegance. Amie thinks that he might have said that he loved its harmony. “I also think he found the rigorous aspect of it pleasing. How it presented a complete story that fit together so neatly,” she said. Calculus resides near a border with a practical incoherence, the way language resides near a border with one’s thoughts before they are entirely formed. Calculus is a means of personifying an abstraction. The Happiness of Getting It Down Right is the title of Maxwell’s correspondence with the writer Frank O’Connor. It comes from a phrase Maxwell used to describe the pleasure of finding the right words to impose on a line of one’s thoughts or feelings. Calculus allows a mathematician to clarify a movement or a quantity with an ideal specificity.
Algebra and geometry felt to me like attempts to characterize the world we can see and put our hands on. The prosaic observations of algebra leading to a refined view of circumstances you could almost but not quite figure out on your own. The simple, hard-looked-at, and by now scuffed-up shapes of Euclidean geometry, triangles and circles and squares, placed on pedestals and closely examined, the primary sculptural forms embedded in ordinary structures.
Calculus seems to enclose the world in its logic. It makes, as Berlinski writes, a circle. Algebra and geometry describe things that you see plainly or at one remove, so to speak. How far you are from something, how tall something else is, the flight of a bird making a pattern like a line on a graph. Calculus, Berlinski says, describes “the world’s network of mathematical nerves.”