The Calculus Diaries - Jennifer Ouellette [15]
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