Change Is the Only Constant

Book Preface

“What is,” said the philosopher Parmenides, not quite a million days ago, “is uncreated and indestructible, alone, complete, immovable and without end.” It’s a bold philosophy. Parmenides permitted no divisions, no distinctions, no future, no past. “Nor was it ever, nor will it be,” he explained; “for now it is, all at once, a continuous one.” To Parmenides, the universe was like Los Angeles traffic: eternal, singular, and unchanging

A million days later, it remains a very stupid idea.

C’mon, Parmenides. You can lull us with poetry and ply us with adjectives, but we’re not dupes. A million days ago, there were no Buddhists, Christians, or Muslims, because Buddha, Jesus, and Muhammad had yet to be born. A million days ago, Italians did not eat tomato sauce, because “Italy” wasn’t a concept and the closest tomatoes grew 6000 miles away. A million days ago, 50 or 100 million humans walked the Earth; now, that many people visit Disney-branded theme parks each year.
In fact, Parmenides, only two things were the same a million days ago as today: (1) the ubiquity of change, and (2) your philosophy’s profound and irredeemable wrongness.
That’s the last we’ll hear of Parmenides in this book (although his savvier disciple Zeno will pop up later). Good riddance to toga-clad stoners, I say. For now, we jump ahead, past his wiser contemporary Heraclitus (“you can’t step in the same river twice”), to arrive in the late 17th century, a mere 120,000 or 130,000 days ago. That’s when a scientist named Isaac Newton and a polymath named Gottfried Leibniz birthed this book’s protagonist. It was a fresh form of mathematics, a language of change, a stab at quantifying the flux and flow of this wobbling top called Earth.
Today, we call that math “calculus.”
The first tool of calculus is the derivative. It’s an instantaneous rate of change, telling us how something is evolving at a specific moment in time. Take, for example, the apple’s velocity precisely as it strikes Newton’s noggin. A second earlier, the fruit was moving a smidge slower; a second later, it will be moving in a different direction entirely, as will the history of physical science. But the derivative does not care about the second before, or the second after. It speaks only to this moment, to an infinitesimal sliver of time.

Calculus’s second tool is the integral. It is the sum of infinite pieces, each infinitesimally small. Picture how a series of circles, each shadow-thin, can unify to create a solid object: a sphere. Or how a group of humans, each as tiny and negligible as an atom, can together constitute a whole civilization. Or how a series of moments, each of them zero seconds in itself, can amount to an hour, an eon, an eternity.
Each integral speaks to a totality, to something galactic, which the panoramic lens of our mathematics can somehow capture.

The derivative and integral have earned a lofty reputation as specialized technical tools. But I believe they can offer more. You and I are like little boats, knocked by waves, spun by whirlpools, thrown by rapids. The derivative and the integral, I hold, are pocket-sized philosophies: extendable oars for navigating this flood-swollen river of a world.

Hence, this book, and its attempts to distill wisdom from mathematics.

In the first half, Moments, we’ll explore tales of the derivative. Each extracts an instant from the babbling stream of time. We’ll consider a millimeter of the moon’s orbit, a nibble of buttered toast, a dust particle’s erratic leap, and a dog’s split-second decision. If the derivative is a microscope, then each of these stories is a carefully chosen slide, a scene in miniature.
In the second half, Eternities, we’ll call upon the integral and its power to unify infinite droplets into a single stream. We’ll encounter a circle fashioned from tiny slivers, an army raised from myriad soldiers, a skyline built of anonymous buildings, and a cosmos heavy with a billion trillion stars. If the integral is a widescreen cinema, then each of these stories is a sweeping epic that you’ve got to see in theaters. The TV at home won’t do it justice.
I want to be clear: the object in your hands won’t “teach you calculus.” It’s not an orderly textbook, but an eclectic and humbly illustrated volume of folklore, written in nontechnical language for a casual reader. That reader may be a total stranger to calculus, or an intimate friend; I’m hopeful that the stories will bring a little mirth and insight either way.
This storybook is by no means complete—missing are the tales of Fermat’s bending light, Newton’s secret anagram, Dirac’s impossible function, and so many others. But in an ever-changing world, no volume is ever exhaustive, no mythology ever finished. The river runs on.

BEN ORLIN DECEMBER 2018