Infinite Powers: The Story of Calculus - The Language of the Universe

by Steven H. Strogatz

It tempts us with the prospect of forbidden knowledge, with insights unavailable by ordinary means.

For those who can’t resist and want to understand why infinity lurks in the shadows, imagine dividing 6 by a number that’s small and getting close to zero, but that isn’t quite zero, say something like 0.1. There’s nothing taboo about that. The answer to 6 divided by 0.1 is 60, a fairly sizable number. Divide 6 by an even smaller number, say 0.01, and the answer grows bigger; now it’s 600. If we dare to divide 6 by a number much closer to zero, say 0.0000001, the answer gets much bigger; instead of 60 or 600, now it’s 60,000,000. The trend is clear. The smaller the divisor, the bigger the answer. In the limit as the divisor approaches zero, the answer approaches infinity. That’s the real reason why we can’t divide by zero. The faint of heart say the answer is undefined, but the truth is it’s infinite. All of this can be visualized as follows. Imagine dividing a 6-centimeter line into pieces that are each 0.1 centimeter long. Those 60 pieces laid end to end make up the original.

Likewise (but I won’t attempt to sketch it), that same line can be chopped into 600 pieces that are each 0.01 centimeter or 60,000,000 pieces that are each 0.0000001 centimeter. If we keep going and take this chopping frenzy to the limit, we are led to the bizarre conclusion that a 6-centimeter line is made up of infinitely many pieces of length zero. Maybe that sounds plausible. After all, the line is made up of infinitely many points, and each point has zero length. But what’s so philosophically unnerving is that the same argument applies to a line of any length.

In the context of chopping a line into pieces, potential infinity would mean that the line could be cut into more and more pieces, as many as desired but still always a finite number and all of nonzero length. That’s perfectly permissible and leads to no logical difficulties. What’s verboten is to imagine going all the way to a completed infinity of pieces of zero length. That, Aristotle felt, would lead to nonsense — as it does here, in revealing that zero times infinity can give any answer. And so he forbade the use of completed infinity in mathematics and philosophy. His edict was upheld by mathematicians for the next twenty-two hundred years.

In their telling, Zeno was trying to prove that change is impossible. Our senses tell us otherwise, but our senses deceive us. Change, according to Zeno, is an illusion.

Three of Zeno’s paradoxes are particularly famous and strong. The first of them, the Paradox of the Dichotomy, is similar to the Riddle of the Wall but vastly more frustrating. It holds that you can’t ever move because before you can take a single step, you need to take a half a step. And before you can do that, you need to take a quarter of a step, and so on. So not only can’t you get to the wall — you can’t even start walking.

In these first two paradoxes, Zeno seemed to be arguing against space and time being fundamentally continuous, meaning that they can be divided endlessly. His clever rhetorical strategy (some say he invented it) was proof by contradiction, known to lawyers and logicians as reductio ad absurdum,

To solve this equation, subtract t from both sides. That gives 9t = 10. Then divide both sides by 9. The result, t = 10/9 seconds, is the same as we found with infinite decimals. So from the perspective of calculus, there really is no paradox about Achilles and the tortoise. If space and time are continuous, everything works out nicely.

In a third paradox, the Paradox of the Arrow, Zeno argued against an alternative possibility — that space and time are fundamentally discrete, meaning that they are composed of tiny indivisible units, something like pixels of space and time. The paradox goes like this. If space and time are discrete, an arrow in flight can never move, because at each instant (a pixel of time) the arrow is at some definite place (a specific set of pixels in space). Hence, at any given instant, the arrow is not moving. It is also not moving between instants because, by assumption, there is no time between instants. Therefore, at no time is the arrow ever moving. To my mind, this is the most subtle and interesting of the paradoxes. Philosophers are still debating its status, but it seems to me that Zeno got it two-thirds right.

Consider how music is recorded and played back. My younger daughter recently received an old-fashioned Victrola record player for her fifteenth birthday. She’s now able to listen to Ella Fitzgerald on vinyl. This is a quintessential analog experience. All of Ella’s notes and scats glide just as smoothly as they did when she sang them; her volume goes continuously from soft to loud and everywhere in between, and her pitch climbs just as gracefully from low to high. Whereas when you listen to her on digital, every aspect of her music is minced into tiny, discrete steps and converted into strings of 0s and 1s. Although conceptually the differences are gigantic, our ears can’t hear them.

For many practical purposes, the discrete can stand in for the continuous, as long as we slice things thinly enough. In the ideal world of calculus, we can go one better. Anything that’s continuous can be sliced exactly (not just approximately) into infinitely many infinitesimal pieces. That’s the Infinity Principle. With limits and infinity, the discrete and the continuous become one.

Tunneling is real. Alpha particles tunnel out of uranium nuclei at the predicted rate to produce the effect known as radioactivity. Tunneling also plays an important role in the nuclear-fusion processes that make the sun shine, so life on Earth depends partially on tunneling. And it has many technological uses; scanning tunneling microscopy, which allows scientists to image and manipulate individual atoms, is based on the concept.

By applying calculus and quantum mechanics, physicists have opened a theoretical window on the microworld. The fruits of their insights include lasers and transistors, the chips in our computers, and the LEDs in our flat-screen TVs.

Although quantum mechanics is conceptually radical in many respects, in Schrödinger’s formulation, it retains the traditional assumption that space and time are continuous. Maxwell made the same assumption in his theory of electricity and magnetism; so did Newton in his theory of gravity and Einstein in his theory of relativity. All of calculus, and hence all of theoretical physics, hinges on this assumption of continuous space and time. That assumption of continuity has been resoundingly successful so far. But there is reason to believe that at much, much smaller scales of the universe, far below the atomic scale, space and time may ultimately lose their continuous character. We don’t know for sure what it’s like down there, but we can guess.

In 1899, the father of quantum theory, a German physicist named Max Planck, realized that there was one and only one way to combine these fundamental constants to produce a scale of length. That unique length, he concluded, was a natural yardstick for the universe. In his honor, it is now called the Planck length. It is given by the algebraic combination

When we plug in the measured values of G, ħ, and c, the Planck length comes out to be about 10–35 meters, a stupendously small distance that’s about a hundred million trillion times smaller than the diameter of a proton. The corresponding Planck time is the time it would take light to traverse this distance, which is about 10–43 seconds. Space and time would no longer make sense below these scales. They’re the end of the line.

Take the largest possible distance, the estimated diameter of the known universe, and divide it by the smallest possible distance, the Planck length. That unfathomably extreme ratio of distances is a number with only sixty digits in it. I want to stress that — only sixty digits. That’s the most we would ever need to express one distance in terms of another. Using more digits than that — say a hundred digits, let alone infinitely many — would be colossal overkill, way more than we would ever need to describe any real distances out there in the material world.

And yet in calculus, we use infinitely many digits all the time. As early as middle school, students are asked to think about numbers like 0.333 . . . whose decimal expansion goes on forever. We call these real numbers, but there is nothing real about them. The requirement to specify a real number by an infinite number of digits after the decimal point is exactly what it means to be not real, at least as far as we understand reality through physics today.

From the beginning, calculus has stubbornly insisted that everything — space and time, matter and energy, all objects that ever have been or will be — should be regarded as continuous. Accordingly, everything can and should be quantified by real numbers. In this idealized, imaginary world, we pretend that everything can be split finer and finer without end. The whole theory of calculus is built on that assumption. Without it, we couldn’t compute limits, and without limits, calculus would come to a clanking halt.

Reality is too hard to model any other way. With infinite decimals, as with the rest of calculus, infinity makes everything simpler.

Archimedes for his principle of buoyancy (a body immersed in a fluid is buoyed up by a force equal to the weight of the fluid displaced) and his law of the lever (heavy objects placed on opposite sides of a lever will balance if and only if their weights are in inverse proportion to their distances from the fulcrum).

what really puts him in the pantheon is what he did for mathematics. He paved the way for integral calculus. Its deepest ideas are plainly visible in his work, but then they aren’t seen again for almost two millennia. To say he was ahead of his time would be putting it mildly. Has anyone ever been more ahead of his time?

He began with a hexagon because it was a convenient base camp from which to embark on the more arduous calculations ahead. The advantage of the hexagon was that he could easily calculate its perimeter, the total length around the hexagon. It’s six times the radius of the circle. Why six? Because the hexagon contains six equilateral triangles, each side of which equals the circle’s radius. Six of the triangle’s sides make up the perimeter of the hexagon. So the perimeter equals six times the radius; in symbols, p = 6r. Then, since the circle’s circumference C is longer than the hexagon’s perimeter p, we must have C > 6r. This argument gave Archimedes a lower bound on what we would call pi, written as the Greek letter π and defined as the ratio of the circumference to the diameter of the circle. Since the diameter d equals 2r, the inequality C > 6r implies Thus the hexagon argument demonstrates π > 3.

Once he figured out what the hexagon was telling him, he shortened the steps and took twice as many of them.

Then he kept doing that, over and over again. A man obsessed, he went from six steps to twelve, then twenty-four, forty-eight, and, ultimately, ninety-six steps, working out their ever-shrinking lengths to migraine-inducing precision.

Unfortunately, it got progressively harder to calculate the step lengths as they shrank, because he had to keep invoking the Pythagorean theorem to find them. That required him to calculate square roots, a nasty chore to do by hand.

By using a 96-gon inside the circle and a 96-gon outside the circle, he ultimately proved that π is greater than 3 + 10/71 and less than 3 + 10/70. Forget about math for a minute. Just savor this result at a visual level:

The squeeze technique that Archimedes used (building on earlier work by the Greek mathematician Eudoxus) is now known as the method of exhaustion because of the way it traps the unknown number pi between two known numbers. The bounds tighten with each doubling, thus exhausting the wiggle room for pi.

Circles are the simplest curves in geometry. Yet, surprisingly, measuring them — quantifying their properties with numbers — transcends geometry.

Euclid’s omission was a signal that something deeper was needed. To come to grips with π’s numerical value required a new kind of mathematics, one that could cope with curved shapes. How to measure the length of a curved line or the area of a curved surface or the volume of a curved solid — these were the cutting-edge questions that consumed Archimedes and led him to take the first steps toward what we now call integral calculus. Pi was its first triumph.

We no longer make this distinction between magnitude and number, but it was important in ancient Greek mathematics. It seems to have arisen from the tension between the discrete (as represented by whole numbers) and the continuous (as represented by shapes). The historical details are murky, but it appears that sometime between Pythagoras and Eudoxus, between the sixth and the fourth centuries BCE, somebody proved that the diagonal of a square was incommensurable with its side, meaning that the ratio of those two lengths could not be expressed as the ratio of two whole numbers. In modern language, someone discovered the existence of irrational numbers.

Numbers couldn’t be trusted anymore. They were inadequate as a foundation for mathematics. To describe continuous quantities and reason about them, the ancient Greek mathematicians realized they needed to invent something more powerful than whole numbers. So they developed a system based on shapes and their proportions. It relied on measures of geometrical objects: lengths of lines, areas of squares, volumes of cubes. All of these they called magnitudes. They thought of them as distinct from numbers and superior to them.

As of this writing, twenty-two trillion digits have been computed by the world’s fastest computers.

There’s something so paradoxical about pi. On the one hand, it represents order, as embodied by the shape of a circle, long held to be a symbol of perfection and eternity. On the other hand, pi is unruly, disheveled in appearance, its digits obeying no obvious rule, or at least none that we can perceive. Pi is elusive and mysterious, forever beyond reach. Its mix of order and disorder is what makes it so bewitching. Pi is fundamentally a child of calculus. It is defined as the unattainable limit of a never-ending process.

circle. A slightly steeper cut produces an ellipse. A cut that has the same slope as the cone itself produces a parabola.

It took a kaleidoscopic leap of artistic imagination to see a smooth, gently curving parabolic segment as a mosaic of jagged shapes. If he had been a painter, Archimedes would have been the first cubist.

He came up with a brilliant idea — brilliant because it established a rule, a consistent pattern that held from one level of the hierarchy to the next. He imagined sliding the oblique line at the base of the segment upward while keeping it parallel to itself until it just barely touched the parabola at a single point near the top.

He proved that each newly created triangle had one-eighth as much area as its parent triangle. Thus, if we say that the first, biggest triangle occupies 1 unit of area — that triangle will serve as our area standard — then its two daughter triangles together occupy ⅛ + ⅛ = ¼ as much area.

In other words, the parabolic segment has 4/3 the area of the big triangle.

“It is easier to supply the proof when we have previously acquired, by the method, some knowledge of the questions than it is to find it without any previous knowledge.” In other words, by noodling around, playing with the Method, he gets a feel for the territory. And that guides him to a watertight proof.

This is such an honest account of what it’s like to do creative mathematics. Mathematicians don’t come up with the proofs first. First comes intuition. Rigor comes later. This essential role of intuition and imagination is often left out of high-school geometry courses, but it is essential to all creative mathematics. Archimedes concludes with the hope that “there will be some among the present as well as future generations who by means of the method here explained will be enabled to find other theorems which have not yet fallen to our share.”

What is the Method, and what is so personal, brilliant, and transgressive about it? The Method is mechanical; Archimedes finds the area of the parabolic segment by weighing it in his mind. He thinks of the curved parabolic region as a material object — I’m picturing it as a thin sheet of metal carefully trimmed into the desired parabolic shape — and then he places it at one end of an imaginary balance scale. Or, if you prefer, think of it as being seated at one end of an imaginary seesaw. Next he figures out how to counterbalance it against a shape he already knows how to weigh: a triangle. From this he deduces the area of the original parabolic segment.