At Huygens’s urging, Leibniz read Blaise Pascal’s work on infinitesimals, as well as the work of René François de Sluse, who had made a rule for constructing tangents to a point on a curve. Leibniz realized that Pascal’s approach to infinitesimals could be combined with Sluse’s tangent rule and applied to any geometric curve. That same critical insight—the universality of the method—led him to create his own version of calculus independently of Newton.
Leibniz published his first account of differential calculus in 1684, followed by a discussion of integral calculus two years later. It caused a sensation, which rankled Newton’s pride; he became convinced that Leibniz had stolen his ideas from his earlier unpublished papers that had been circulating privately in academic circles over the years. (He used his new techniques in his scientific work long before the publication of Opticks.) There were rumblings of impending conflict in the ensuing years, as tensions brewed between those in Camp Newton and Camp Leibniz, but things didn’t erupt into outright war until Newton published his essay in Opticks.
The opening volley in the calculus wars was an anonymous review of “On the Quadrature of Curves” that appeared in a European journal early in 1704, implying that Newton had “borrowed” his ideas from Leibniz. While Leibniz denied it for the rest of his life, historians generally accept that he was the author. He also engaged in a form of “sock puppetry”: He penned numerous anonymous attacks on his archrival’s work and then reviewed those attacks (one assumes favorably) in his own signed papers. At the time, Newton was by far the more famous scientist, and a prominent member of the Royal Society of England. While he didn’t engage in sock puppetry, he wasn’t above using his considerable influence to crush the scientific competition. In addition to Leibniz, during his long scientific career he fought with John Flamsteed, with Huygens, and with Robert Hooke, and each proved to be an acrimonious battle. Newton was not a people person; no wonder he purportedly died a virgin.
In one letter to Leibniz, Newton offered his “proof ” that he had invented calculus—but he couched it in a sort of anagram of a Latin sentence. He took all the individual letters and put them in alphabetical order: six a’s, two c’s, one d, thirteen e’s, two f ’s, and so forth. To Newton, it was perfectly obvious: Anyone could simply rearrange all of the letters and find the proof they sought that he, Isaac Newton, had prior knowledge of the key concepts. Very few people felt inclined to go to all that trouble, and frankly, even decoded, the “proof ” wasn’t especially clear. Roughly translated, the sentence read, “Having any given equation involving never so many flowing quantities, to find the fluxions, and vice versa.” That was his stab at summarizing derivatives; Newton would have been a lousy math teacher.
The Royal Society of England sided with Newton on the controversy, crediting him in 1715 with the discovery of calculus. Leibniz wasn’t given shared credit until after his death a year later. Today, the consensus seems to be that the two men represent two complementary approaches to the discipline they co-invented. Leibniz was the more abstract of the two, and it’s his system of notation that modern scientists still use today, while Newton focused on the more practical applications of calculus. Leibniz can also claim credit for coining the word calculus, named for a type of stone once used for counting purposes by the Romans.
Calculus did not find immediate acceptance within the scientific community; there was one final missing piece. The method worked, in that it gave the right answer, but mathematicians found the notion of the infinitesimal deeply troubling. Once again, the problem of infinity raised its ugly head. For instance, Newton relied on a bit of magical hand-waving to make his method work: He argued that since his fluxion units were so small—infinitely close to zero but not exactly equal to zero—they could be ignored for all practical purposes. In his equations, they effectively vanish for no reason. A rigorous explanation for what happens to those fluxion units when an equation is solved would not be found for another hundred years.
Leibniz adopted a symbolic notation—Δx, which stands for a tiny increment—that preserved the infinitesimals yet still enabled mathematicians to manipulate them as if they were actual numbers. (In modern notation, scientists often use dx to represent an infinitesimal.) Yet this approach seemed to many mathematicians to be a bit of a cheat. Chief among the naysayers was an Irish bishop named George Berkeley, who in 1734 (seven years after Newton’s death) criticized Newton and Leibniz for their fudging of the method, calling infinitesimals “ghosts of departed quantities” and observing that if they were comfortable with that sort of thing, they “need not, methinks, be squeamish about any point in divinity.”
TAKE IT TO THE LIMIT
A fictionalized Albert Einstein (portrayed by the late Walter Matthau) plays mischievous matchmaker between his egg-head niece, Catherine Boyd, and a good-hearted auto mechanic named Ed Walters in the charming 1994 romantic comedy I.Q. Some might object to the considerable liberties taken with historical fact and illustrious personages, but there’s a lot to admire in the film, if for no other reason than its inclusion of Einstein’s real-life cronies, Kurt Gödel, Boris Podolsky, and Nathan Liebknecht, as supporting characters. As Ed introduces Einstein to Frank, one of his co-workers at the garage, he declares, “This is Albert Einstein, the smartest man in the world!” Intones Frank in his best Joisey accent, “Hey, how they hangin’?”
There is a lovely scene in a diner, where Catherine tries to explain to Ed the gist of one of Zeno’s paradoxes. Zeno was a Greek philosopher living in the fifth century B.C. who thought a great deal about motion. Specifically, he speculated that all motion is illusory, and came up with a famous set of arguments to “prove” it. Catherine explains it thus: If she takes one step forward, and then halves the distance traveled with her next step, then halves it again, and so forth, such that the progression goes on for infinity, she will never be able to reach Ed. The distance between them will get smaller and smaller but will never reach zero. The subtext here is Catherine’s belief that there is no way to bridge the gap between the couple’s intellectual differences and social status. But the practical-minded Ed simply steps over the imaginary line to close the gap: “So how did I do that?” A confused Catherine stammers, “I . . . I don’t know.” But if she knows her calculus (and she should), the “mystery” should be easy to solve.
Perhaps you’ve encountered some variation on Zeno’s paradoxes before; I certainly had. It pains me to admit this publicly, but I did not realize it was tied to the essence of calculus. In one paradox, Zeno used an arrow flying through the air toward a target9—say, your high school calculus teacher—to illustrate his points, rather than a young couple in a diner, but the basic idea is the same: To reach the target, the arrow must first cover half the distance, then half the remaining distance, and so on, moving an infinite number of times. By that logic, the distance between the arrow and the target would keep getting smaller and smaller, and yet the arrow could never close the gap completely in order to actually reach the target. Your calculus teacher lives to torment you another day.
There’s an equally paradoxical corollary: At any given moment in time, the arrow has a specific fixed position—it can only be in just one place at any given time—which means it is technically at rest (not moving) at that particular instant, even though, taken all together, those individual points add up to an arrow in motion. Motion, after all, is basically the measure of how an object’s position has changed over time. But break down motion into infinitely small increments—similar to the individual frames in a film reel—and you find yourself trying to determine how far it traveled in zero amount of time: instantaneous motion. Ergo, the paradox.
In the real world, this doesn’t happen, Eventually, the arrow will find its mark, and the calculus teacher will curse the limit with his or her last breath. Ed will close the distance with Catherine, and the two will live happily ever after. This makes the argument a little flimsy by the standards of common sense. But Zeno never intended his paradoxes to be
taken literally. The Greeks may have lacked strictly mathematical solutions to the problems, but they certainly recognized the need to reconcile the paradoxes.
Mathematically speaking, the problem is this: Zeno’s paradoxes rest on the assumption that the progression will go on for infinity and has no ultimate goal, or limit. But in physical reality, there can be some kind of limit even to an infinite series. That endless series can have a finite sum. In the case of the arrow’s tip and its target, as the distances between points become smaller, so does the elapsed time, even if speed remains constant.
The problem of infinity stumped the greatest mathematical minds for two millennia. The Greeks lacked the concept of zero and failed to grasp the idea that a finite distance between two points can be divided into an infinite number of pieces in between. For them, the continuous motion of an arrow in flight is divided into an infinite number of discrete steps, and because there must be an infinite number, the Greeks presumed the arrow would continue flying toward its target forever.
Aristotle tried to get around the difficulty by drawing a distinction between what he called the potential infinite and the actual infinite, arguing that the latter didn’t exist. It was fine if a line could always be extended—that would be potentially infinite. But an actual infinitely long line? That would be impossible. Archimedes followed Aristotle’s lead: He never claimed that the method of exhaustion would result in the exact value for the area of a curved object; this would require an actual infinite number of triangles or rectangles. He simply said one could refine the approximation as much as one liked—a concept he likewise called potential infinity.
Or so historians and mathematicians believed. That is why the rediscovery of the Archimedes palimpsest in the 1990s is so significant. Heiberg had been unable to transcribe the relevant pages in 1908 because they were so badly damaged. Modern analytic methods uncovered that long-hidden text. This time around, the task of transcribing the fully restored text fell to Reviel Netz, a professor of mathematical history at Stanford University. Netz’s transcription hints that Archimedes had a far more sophisticated understanding of the infinite than historians have generally credited him with. In particular, the Greek mathematician flirted with the notion of actual infinity while calculating the volume of a fingernail-shaped figure.
This is not the same as fashioning a rigorous mathematical proof to deal with infinity, however. The man who gets the credit for resolving the problem of infinitesimals is an eighteenth-century mathematician named Jean le Rond d’Alembert. D’Alembert’s life story is fodder for an Oscar-worthy biopic. He was a foundling who took the name of the church in Paris on whose steps he was found: Saint Jean Baptiste le Rond. He was raised by a glazier but later discovered his birth parents were a general and a noble-woman.
D’Alembert’s insight into Zeno’s problem of motion seems obvious in retrospect: that arrow shot from a bow is on a journey, and it makes that journey in an infinite number of smaller steps, but its travel does not continue indefinitely—it has a destination, namely, your calculus teacher’s heart. That ultimate destination is the limit, even though the arrow makes an infinite number of subjourneys before arriving. All those subjourneys, added together, mean that the arrow hits its mark—and that summing process is integral calculus.
Let’s go back to Catherine and Ed to illustrate. Ed moves closer and closer to Catherine, halving the distance between them with each step. We can add those numbers together: 1 + ½ + ¼ + ⅛ + , and so on, and notice that the sum of this “infinite series” gets closer and closer to exactly 2. We can check this by “taking the limit”: if 2 is the limit, then 2-1 = 1, 1-½ = ½, ½-¼ = ¼, and so on into infinity. With each iteration, the result gets closer and closer to 0. Ed still crosses a distance of 2 feet, but he does it in an infinite number of steps.
This gives rise to one of the most common stumbling blocks for the beginning calculus student: the notion that 0.9999 . . . is actually equivalent to 1. My mind, too, balked at accepting this mathematical fact when my spouse, Sean, patiently explained it to me late one night as I was struggling to understand the limit. Intuitively, we think of a fraction as being a finite sum. We have all eaten one-ninth of a pie, after all. But we also learn about irrational numbers—like π, or the so-called golden ratio, ø—where the string of numbers in the decimal expansion really does go on forever.
That was my mistake: assuming that 0.999 . . . is like an irrational number, and therefore represents a sequence of numbers that get closer and closer to 1 but never reaches it exactly. Such is not the case; it is a rational number, in which the same decimal number endlessly repeats, and thus can have a finite sum. Eventually the limit of the sequence equals 1, and that means 0.999 . . . has a fixed value, rather than being an infinite progression. It might seem paradoxical, but two very different mathematical expressions can nonetheless represent the same number. Ergo, to a calculus teacher, 0.999999 . . . is just another way of writing 1.10
One could argue that calculus itself was invented via tiny infinitesimal bits of accrued knowledge that, taken together, added up to a revolutionary new whole. But like the function, calculus is far more than the sum of its parts, making it possible to understand the world around us in dynamic, rather than static terms. “We live in a world of ceaseless growth and decay, with things in fretful motion on the surface of earth, planets wheeling in the sky,” mathematician David Berlinski writes in A Tour of the Calculus. “Geometry may well describe the skeleton, but the calculus is a living theory and so requires flesh and blood and a dense network of nerves.” Life is constantly moving and changing. Life, in short, is curvy.
2
Drive Me Crazy
We apprehend time only when we have marked motion . . . not only do we measure movement by time, but also time by movement because they define each other.
—ARISTOTLE
Abrooding shot of a long, straight desert highway, shimmering slightly in the heat and stretching far into the horizon, opens Ridley Scott’s classic 1991 film Thelma and Louise. It’s become an iconic image, foreshadowing the women’s glorious demise as they drive off a cliff in the Grand Canyon in their 1966 Thunderbird convertible, immortalized forever in celluloid history.
The portion of I-15 that runs between Los Angeles and Las Vegas doesn’t stretch quite so dramatically into infinity, but after three long hours of driving under a relentless midsummer sun, it’s starting to feel like it could go on forever, particularly since traffic has slowed to a crawl. We have road construction to thank for the delay: The state of California is adding a southbound lane just for trucks, and for some reason, this is also slowing down traffic on the northbound side. Thelma and Louise would have just floored it and blasted their way out, but we are wimpy, law-abiding citizens, and meekly accept our fate.
We’re on I-15 in our shiny red Prius because we’re hard-core Vegas fans: Taking a weekend jaunt now and then to play some poker, do some shopping, and perhaps indulge in a spot of fine dining or a spa treatment proves quite refreshing. But the road trip also provides an excellent example of calculus in action. Calculus deals with rates of change. Motion is, in essence, change in position with respect to time—however slowly that position is currently changing thanks to the impeded traffic. In fact, at this point, we’re barely moving at all, inching along at a scant 10 mph while our stomachs rumble in anticipation of savoring the world’s best gyros11 and falafel at the Mad Greek Cafe in the tiny town of Baker, California (population 600).
There’s precious little to do on a road trip, creeping along a desert highway while breathing in exhaust fumes, with nothing but dusty hills, tumbleweeds, and a long line of rear bumpers as scenery. So I figure it’s as good a time as any to muddle through the basics of derivatives and integrals; I’m already bored, hungry, and cranky. Also, it occurs to me that our predicament is reminiscent of Zeno’s paradox, outfitted in contemporary garb—exchanging Zeno’s trademark toga and sandals for acid-washed Levi’s and snazzy ostrich-skin boots
, if you will.
Think about it: If our motion is divided into infinitely smaller increments of time and distance—as it would be in a calculus class—in what sense can I claim we are “moving” at all? I can solve this modern paradox by using the tools of calculus to determine our instantaneous speed—how fast we are going at any brief, fixed moment in time—even though our position in time and space is constantly changing. Assuming I know our instantaneous speed (velocity) at every possible moment, can I then use that information to determine how far we’ve traveled—our position—without cheating and looking at our trusty odometer? Calculus says I can.
ROAD TO NOWHERE
Let’s start with a bit of precalculus to demonstrate the concept of instantaneous speed, using the simplest possible example with highly idealized conditions. In my mind’s eye, I-15 magically morphs into that endless, perfectly straight road in Thelma and Louise, except rather than stretching into eternity, it runs between our home in Los Angeles and the Luxor Hotel in Las Vegas, with an infinite number of points in between.
Imagine that a squad car pulls up as we drive into the Luxor entrance. The officers claim Sean ran a red light a few miles away. Sean denies it. As proof, they show us a time-stamped photograph taken by a traffic camera, showing the Prius just before its nose passes through the intersection. Fair enough, says Sean, but all that proves is that the Prius was at that particular point at that particular time. It merely shows our position, not our velocity. How can they prove that the car was actually moving at that point, rather than stopped at the light? He is a scientist. He demands evidence. He also doesn’t want to pay the imaginary fine. To prove he ran the red light, he insists, the officers need to offer compelling proof of the car’s instantaneous speed at the moment that photograph was taken.
The Calculus Diaries: How Math Can Help You Lose Weight, Win in Vegas, and Survive a Zombie Apocalypse Page 4
