bookstand

Change Is the Only Constant

The Wisdom of Calculus in a Madcap World

Ben Orlin

First annotation on . Last on .

35 quotes

mathematics

Chapter 1

  • Philosophers in ancient China played similar mind games. “The dimensionless cannot be accumulated,” one wrote. “Its size is a thousand miles.” In the mathematical sense, a moment is dimensionless: It possesses no length, no duration. It’s zero seconds long. But since two times zero is still zero, two moments will also amount to zero time. The same holds for moments, or a thousand, or a million. In fact, any countable number of moments will still total up to zero seconds.Dec 5 2022 2:07PM
  • This bread-and-butter mathematical concept is oddly like a poet’s fancy. The derivative is “instantaneous change”: movement captured in a moment, like lightning in a bottle. It’s the repudiation of Zeno, who said that nothing can happen in a single instant, and the vindication of Hladik, who believed that anything can.Jun 9 2023 12:35AM

Chapter 3

  • Psychologists call this process habituation. It means that I’ve got vision like a dinosaur’s: keenly trained on whatever moves, I overlook anything that stands still, even if it comes buttered. Perhaps evolutionary psychology can explain the phenomenon, or perhaps I’m an ungrateful louse, but either way, you can frame habituation mathematically. We grow accustomed to the function, whatever its height. Before long, it takes a derivative—a nonzero rate of change—to draw our attention. Only a newer newness can catch our eye.Jun 9 2023 1:35AM
  • “The derivative is your rate of improvement. The second derivative asks: Are you improving faster and faster? Or is the improvement slowing down?”Feb 13 2023 4:26AM
  • “Poetry begins in trivial metaphors,” Robert Frost once wrote, “pretty metaphors, ‘grace’ metaphors, and goes on to the profoundest thinking that we have.” I’m not sure Frost would have found much poetry in derivatives—they are hopelessly direct, saying one thing only, and with lamentable precision—but the soil here is fertile with metaphor.Jun 9 2023 1:37AM

Chapter 4

  • Alas, this is yet another way in which Gottfried Leibniz’s achievements exceed mine, because he gave to the mathematical lexicon words such as: • Constant, for a quantity that doesn’t change; • Variable, for a quantity that does; • Function, for a rule associating inputs to outputs; • Derivative, for an instantaneous rate of change; and • Calculus, for a system of calculation, like the one he developed.Jun 9 2023 1:38AM
  • The purpose of mathematical symbols is to let us project our thoughts onto paper. Well-chosen ones feel so natural that you forget how artificial the whole process is. Make no mistake: mathematical symbolism is a technological feat, the extension of the brain by other means, as eerie and profound as a robotic limb.Jun 9 2023 1:39AM

Chapter 7

  • Fads follow similar logic. The more people a fad recruits, the more people they recruit. A feedback loop leads to exponential growth—at least, for a little while. But sooner or later, the fad begins to exhaust its targets. Lots of recruiters, and no one left to recruit. A surplus of catalysts, and nothing left to catalyze. You might say, if you’re a fan of ostentatious chemistry jargon, that a viral fad is an autocatalytic reaction of humans.Jun 9 2023 1:48AM

Chapter 9

  • The jig of the particles—dubbed “Brownian motion”—exhibits puzzling features. It is indiscriminate, with particles favoring no direction over any other; independent, with each particle dancing alone, showing no correlation with its neighbors; and unpredictable, with past movements offering no hint of future ones. But perhaps strangest of all is the nature of those direction changes. They are, in our mathematical model, nondifferentiable.Jun 9 2023 1:49AM

Chapter 10

  • “The battle between geometry and algebra is like the battle between the sexes,” mathematician Sir Michael Atiyah once said. “It’s perpetual… The dichotomy between algebra, the way you do things with formal manipulations, and geometry, the way you think conceptually, are the two main strands in mathematics. The question is what is the right balance.”Feb 13 2023 4:29AM

Chapter 12

  • As in Aesop’s fable “The Tortoise and the Technological Singularity,” the moral is clear: don’t build an unstoppable agent indifferent to your survival. “The AI does not hate you,” says philosopher Eliezer Yudkowsky, “nor does it love you, but you are made out of atoms which it can use for something else.”Feb 13 2023 9:19AM

Chapter 13

  • Soon, with a bold and mysterious leap of imagination, he begins to interpret the Laffer curve as a metaphor for everything—say, a father disciplining a son. “Harsh penalties for violating both major and minor rules” is like a high tax rate, and “only invites sullen rebellion, stealth, and lying (tax evasion, on the national level).” The permissive father, meanwhile, is like a low-tax state, and “invites open, reckless rebellion”: his son’s “unfettered growth comes at the expense of the rest of the family.”Jun 9 2023 1:52AM

Chapter 14

  • “Elvis started off as a dog who was a really good friend,” Pennings told me. “By the time he died, he was a good friend who just happened to be a dog.”Jun 9 2023 1:54AM

Chapter 17

  • What’s true of cannonballs also holds for the folks firing them: strength is about more than just size. “In warfare,” says Tolstoy, “the force of armies is the product of the mass multiplied by something else, an unknown x.”Jun 7 2023 7:08AM
  • The dopey historian looks for one-dimensional explanations of infinite-dimensional effects. It’s a failure to understand the multiplicity, the thickness, of history, like plucking out a few grains of sand as the “cause” of the dune.Jun 7 2023 11:43AM
  • Well, Tolstoy knew where history must begin: with the tiny, fleeting data of human experience. A surge of courage, a flash of doubt, a sudden lust for nachos—that interior, spiritual stuff is the only kind of reality that matters. Furthermore, Tolstoy knew where history must end: with grand, all-encompassing laws, explanations as tremendous as what they seek to explain.Jun 7 2023 11:44AM
  • There’s a fissure in Tolstoy. On one side is his knack for detail, his gift for capturing the effervescent data of daily life. On the other side is his yearning for big, bold answers. What steers human events? Why war? Why peace? The integral is the bridge between Tolstoy’s gift and his dream. It’s supposed to reconcile the world he knows (a jumble of details) and the world he craves (a well-governed realm), to fuse infinite multiplicity into perfect oneness.Jun 7 2023 9:18PM

Chapter 18

  • “Squares after squares of flame, set and cut into the ether,” wrote poet Ezra Pound of a New York evening. “A city of geometric heights,” wrote essayist Roland Barthes, “a petrified desert of grids and lattices.” Just like a skyline, a Riemann sum is an aggregate, built of rectilinear units.Jun 7 2023 9:22PM

Chapter 19

  • For mathematicians, “inverse processes” are actions that undo each other, counteracting opposites.Jun 7 2023 9:25PM
  • But for Agnesi, mathematics wasn’t about practicality. It was a sacred project, a pathway to God. Pure logical thought gave humanity its closest experience of divine cognition, of eternal truth. For someone as devout as Agnesi, that meant everything. Why sully the holy with the earthly, the geometric with the physical?Jun 7 2023 9:28PM
  • Agnesi understood the unity of opposites better than anyone. Just look at the identities she embodied: mathematician and mystic, Catholic traditionalist and proto-feminist, disciple of science and of religion alike. She even bridged the starkest opposition of all, the bitter Newton/Leibniz feud, which was still radioactive when she set out to write. As no one else had, Agnesi managed to unify the Englishman’s “fluxions” with the German’s “differences,” achieving such a perfect fusion that one Cambridge mathematics professor learned Italian expressly so that he could translate her masterwork into English.Jun 7 2023 9:30PM

Chapter 20

  • Teaching himself integration in the back corner of his high school physics classroom, Feynman never learned some of the standard techniques. Instead, he gathered tools from off the beaten path: nifty yet little-taught maneuvers like “differentiating under the integral sign.”Feb 13 2023 9:27AM
  • Feynman would solve them, leaning often on that one powerful trick. “I got a great reputation for doing integrals,” he wrote, “only because my box of tools was different from everybody else’s.” With derivatives, everyone dances the same choreography, but integrals lend themselves to personal style.Jun 7 2023 9:32PM
  • I’m sure Feynman would approve. It’s the defeat of math class and the triumph of math club—of the trickster approach that, for him, encompassed everything in lifeJun 7 2023 9:34PM
  • To Feynman no educational philosophy could have been more wrong. The answer is all that does matter, he said… Better to have a jumbled bag of tricks than any one orthodox method.Jun 7 2023 9:34PM

Chapter 21

  • “Matter tells space-time how to curve,” said physicist John Wheeler, “and curved space tells matter how to move.”Jun 8 2023 8:36AM

Chapter 23

  • In Jevons’s model, a stellar two-minute back rub can somehow “equal” a pretty-good five-minute one. Two hours of kinda needing to pee might be “equal” in some sense to minutes of desperately needing to go. Robert Frost wrote, in one poem’s title, that “happiness makes up in height for what it lacks in length.” Jevons makes that relation explicit, mathematical.Jun 8 2023 11:27AM
  • A century after Jevons, a team of psychologists led by Daniel Kahneman set out to study a particular experience of pain: forcing people to hold their hands immersed in icy water. (Psychology: it’s sociology for sociopaths.) One hand was submerged in °F water for a minute. At another time, the other hand underwent the same experience, followed by an additional seconds in the water, during which the temperature gradually rose to °F. Later, subjects were asked: Which trial would you rather repeat?Jun 8 2023 11:30AM
  • Jevons’s theory tells us that nobody should choose the latter trial. It has all the chilly pain of the first, plus a little extra. Unless you’re an Arctic mammal, a masochist, or both, bonus icy hand time should not appeal to you. And yet that’s exactly what most subjects chose. Looking back on an experience, people tend to ignore how long it lasted. Instead, they focus on extremes and endings—the maximum of pain, and the final pain level. Because the second trial hits the same extreme and ends on a slightly less painful note, subjects recall it more fondly.Jun 8 2023 11:30AM
  • Emotion, as retained in human memory, is not a Jevons-like integral. It overweighs finales. I’m reminded of Ray Bradbury’s insight: “A bright film with a mediocre ending is a mediocre film. Conversely, a medium-good film with a terrific ending is a terrific film.” What makes a story happy or sad, cynical or hopeful, tragic or comic? It’s the ending, and nothing else. That’s why we rush to visit deathbeds, why we dwell on final words, why the last minutes of a lifetime can redefine the eight decades prior.Jun 8 2023 11:31AM
  • Even so, utilitarianism remains a booming and necessary voice in our moral sphere. Sure, we may debate what ranks as “the greatest good” (“better to be Socrates dissatisfied than a fool satisfied,” said th-century economist John Stuart Mill), or who counts among that “greatest number” (“most human beings are speciesists,” warns philosopher Peter Singer), or how to aggregate billions of subjective experiences into a single sum (maybe Tolstoy can help?). We probably reject the specifics of Jevons’s moral calculus. But every time we posit a moral calculus of our own—a new model better matched to the complexities of emotional reality—we tread in Jevons’s footsteps. Explicit or not, consistent or not, we live our lives by a kind of felicific calculus.Jun 8 2023 11:34AM

Chapter 26

  • The trouble with college math classes—which… consist almost entirely in the rhythmic ingestion and regurgitation of abstract information… is that their sheer surface-level difficulty can fool us into thinking we really know something when all we really “know” is abstract formulas and rules for their deployment. Rarely do math classes ever tell us whether a certain formula is truly significant, or why, or where it came from, or what was at stake.Jun 9 2023 12:26AM

Chapter 27

  • Douglas Hofstadter, author of Gödel, Escher, Bach, goes further. “The drive to eliminate paradoxes at any cost…” he writes, “puts too much stress on bland consistency, and too little on the quirky and bizarre.”Jun 9 2023 12:30AM

28. SCENES FROM AN IMPOSSIBILITY, in which calculus vexes and thrills

  • As Albert Einstein put it: “God does not care about our mathematical difficulties. He integrates empirically.”Feb 23 2023 11:29PM

Chapter 28

  • I love Euler’s identity the way I love the Beatles: with the sheepish knowledge that it probably gets too much attention. But the Gaussian integral! Look at this beauty, folks! It’s the Moody Blues of equations involving e and pi!Feb 13 2023 9:44AM

Other posts in mathematics