Infinite Powers: How Calculus Reveals the Secrets of the Universe

by Steven H. Strogatz

For reasons nobody understands, the universe is deeply mathematical.

it’s a mysterious and marvelous fact that our universe obeys laws of nature that always turn out to be expressible in the language of calculus as sentences called differential equations. Such equations describe the difference between something right now and the same thing an instant later or between something right here and the same thing infinitesimally close by.

As should be obvious by now, I’ll be giving an applied mathematician’s take on the story and significance of calculus. A historian of mathematics would tell it differently. So would a pure mathematician. What fascinates me as an applied mathematician is the push and pull between the real world around us and the ideal world in our heads. Phenomena out there guide the mathematical questions we ask; conversely, the math we imagine sometimes foreshadows what actually happens out there in reality. When it does, the effect is uncanny.

Calculus, like other forms of mathematics, is much more than a language; it’s also an incredibly powerful system of reasoning. It lets us transform one equation into another by performing various symbolic operations on them, operations subject to certain rules. Those rules are deeply rooted in logic, so even though it might seem like we’re just shuffling symbols around, we’re actually constructing long chains of logical inference.

And it’s what Eugene Wigner meant in his essay “On the Unreasonable Effectiveness of Mathematics in the Natural Sciences” when he wrote, “The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.”

the music of the spheres.

Might be a good book title

Thus, calculus proceeds in two phases: cutting and rebuilding. In mathematical terms, the cutting process always involves infinitely fine subtraction, which is used to quantify the differences between the parts. Accordingly, this half of the subject is called differential calculus. The reassembly process always involves infinite addition, which integrates the parts back into the original whole. This half of the subject is called integral calculus.

The Infinity Principle To shed light on any continuous shape, object, motion, process, or phenomenon—no matter how wild and complicated it may appear—reimagine it as an infinite series of simpler parts, analyze those, and then add the results back together to make sense of the original whole.

The Infinity Principle organizes the story of calculus around a methodological theme. But calculus is as much about mysteries as it is about methodology. Three mysteries above all have spurred its development: the mystery of curves, the mystery of motion, and the mystery of change.

The breakthrough came from insisting that curves were actually made of straight pieces. It wasn’t true, but one could pretend that it was. The only hitch was that those pieces would then have to be infinitesimally small and infinitely numerous. Through this fantastic conception, integral calculus was born. This was the earliest use of the Infinity Principle. The story of how it developed will occupy us for several chapters, but its essence is already there, in embryonic form, in a simple, intuitive insight: If we zoom in closely enough on a circle (or anything else that is curved and smooth), the portion of it under the microscope begins to look straight and flat. So in principle, at least, it should be possible to calculate whatever we want about a curved shape by adding up all the straight little pieces. Figuring out exactly how to do this—no easy feat—took the efforts of the world’s greatest mathematicians over many centuries. Collectively, however, and sometimes through bitter rivalries, they eventually began to make headway on the riddle of curves. Spinoffs today, as we’ll see in chapter 2, include the math needed to draw realistic-looking hair, clothing, and faces of characters in computer-animated movies and the calculations required for doctors to perform facial surgery on a virtual patient before they operate on the real one.

And then a child was born on Christmas Day. This young messiah of calculus was an unlikely hero. Born premature and fatherless and abandoned by his mother at age three, he was a lonesome boy with dark thoughts who grew into a secretive, suspicious young man. Yet Isaac Newton would make a mark on the world like no one before or since. First, he solved the holy grail of calculus: he discovered how to put the pieces of a curve back together again—and how to do it easily, quickly, and systematically. By combining the symbols of algebra with the power of infinity, he found a way to represent any curve as a sum of infinitely many simpler curves described by powers of a variable x, like x2, x3, x4, and so on. With these ingredients alone, he could cook up any curve he wanted by putting in a pinch of x and a dash of x2 and a heaping tablespoon of x3. It was like a master recipe and a universal spice rack, butcher shop, and vegetable garden, all rolled into one. With it he could solve any problem about shapes or motions that had ever been considered.

Mathematically, circles embody change without change. A point moving around the circumference of a circle changes direction without ever changing its distance from a center. It’s a minimal form of change, a way to change and curve in the slightest way possible.

the idea of using infinity to solve difficult geometry problems has to rank as one of the best ideas anyone ever had.

That’s the big idea behind calculus. Everything becomes simpler at infinity.

To see why, suppose we entertain the thought, just for a moment, that a circle is indeed an infinite polygon with infinitesimal sides. How long, exactly, are those sides? Zero length? If so, then infinity times zero—the combined length of all those sides—must equal the circumference of the circle. But now imagine a circle of double the circumference. Infinity times zero would also have to equal that larger circumference as well. So infinity times zero would have to be both the circumference and double the circumference. What nonsense! There simply is no consistent way to define infinity times zero, and so there is no sensible way to regard a circle as an infinite polygon.

Somewhere in the dark recesses of prehistory, somebody realized that numbers never end. And with that thought, infinity was born. It’s the numerical counterpart of something deep in our psyches, in our nightmares of bottomless pits, and in our hopes for eternal life. Infinity lies at the heart of so many of our dreams and fears and unanswerable questions: How big is the universe? How long is forever? How powerful is God? In every branch of human thought, from religion and philosophy to science and mathematics, infinity has befuddled the world’s finest minds for thousands of years. It has been banished, outlawed, and shunned. It’s always been a dangerous idea. During the Inquisition, the renegade monk Giordano Bruno was burned alive at the stake for suggesting that God, in His infinite power, created innumerable worlds.

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.

Another paradox, called Achilles and the Tortoise, maintains that a swift runner (Achilles) can never catch up to a slow runner (a tortoise) if the slow runner has been given a head start in a race. For by the time Achilles reaches the spot where the tortoise started, the tortoise will have moved a little bit farther down the track. And by the time Achilles reaches that new location, the tortoise will have crept slightly farther ahead. Since we all believe that a fast runner can overtake a slow runner, either our senses are deceiving us or there is something wrong in the way that we reason about motion, space, and time.

His clever rhetorical strategy (some say he invented it) was proof by contradiction, known to lawyers and logicians as reductio ad absurdum, reduction to an absurdity.

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.

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.

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.

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 Planck length = √ ħG / c3 .

In a more serious vein, all students of science and engineering remember 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).

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 π = C / d = C / 2r > 6r / 2r = 3. Thus the hexagon argument demonstrates π > 3.

3 + 10 / 71 < π < 3 + 10 / 70. The unknown, and forever unknowable, value of π is trapped in a numerical vise, squeezed between two numbers that look almost identical except that the former has a denominator of 71 and the latter of 70.

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.

Archimedes would not have approved of the legerdemain above. He arrived at the same result by a different route. He resorted to a subtle style of argumentation often described as double reductio ad absurdum, a double proof by contradiction. He proved that the area of the parabolic segment could not be less than 4/3 or greater than 4/3, so it must equal 4/3. As Sherlock Holmes later put it, “When you have eliminated the impossible, whatever remains, however improbable, must be the truth.”

Any smooth surface can be convincingly approximated by triangles.

So Galileo’s odd-number rule seems to be implying that the total distance fallen is proportional to the square of the time elapsed. This charming relationship between odd numbers and squares can be proved visually. Think of the odd numbers as L-shaped arrays of dots:

As he put it, the speed increases in proportion to the time of falling.

Kepler’s First Law: Elliptical Orbits

the ancient Greeks had defined ellipses as the oval-shaped curves formed by cutting through a cone with a plane at a shallow angle, less steep than the slope of the conical surface itself.

Kepler’s Second Law: Equal Areas in Equal Times

In a nutshell, the second law says that the planets do not move at a constant speed. Instead, the closer they get to the sun, the faster they move. The statement about equal areas in equal times is a way of making this precise.

Kepler’s Third Law and the Sacred Frenzy

As the writer Arthur Koestler astutely observed, “Johannes Kepler became enamored with the Pythagorean dream, and on this foundation of fantasy, by methods of reasoning equally unsound, built the solid edifice of modern astronomy. It is one of the most astonishing episodes in the history of thought, and an antidote to the pious belief that the Progress of Science is governed by logic.”

From a modern perspective, there are two sides to calculus. Differential calculus cuts complicated problems into infinitely many simpler pieces. Integral calculus puts the pieces back together again to solve the original problem.

Remarkably, all other equations involving quadratic terms in x and y but no higher powers give curves of just four possible types: parabolas, ellipses, hyperbolas, or circles. And that’s it. (Except for some degenerate cases that yield lines, points, or no graph at all, but these are rare oddities that we can safely ignore.)

For example, the quadratic equation xy = 1 gives a hyperbola, while x2 + y2 = 4 is a circle and x2 + 2y2 = 4 is an ellipse. Even a quadratic as beastly as x2 + 2xy + y2 + x + 3y = 2 has to be one of the four possibilities above. It turns out to be a parabola.

So the basic idea is to choose the four parameters in the sine wave in such a way that they minimize the total squared error of the fit to the data. Accordingly, this approach is called the method of least squares. It works best when the data follow a pattern, as they do here.

Generally speaking, things can change in one of three ways: they can go up, they can go down, or they can go up and down. In other words, they can grow, decay, or fluctuate. Different functions are suitable for different occasions. Since we’re going to be meeting various functions in the pages ahead, it’s helpful to recall some of the most useful ones.

Power function, exponential function, multiples of 10

This reasoning explains why 100 is defined as 1 (and has to be defined that way), a definition that many people find puzzling. Any other choice would break the pattern. It’s the only definition that continues the trends established farther down in the two columns.

The log of a product is the sum of the logs.

The important point about e is that an exponential function with this base grows at a rate precisely equal to the function itself.

In the jargon of calculus, ex is its own derivative. No other function can say that. It’s the fairest of them all—at least as far as calculus is concerned.

Although base e is uniquely distinguished, other exponential functions obey a similar principle of growth. The only difference is that the rate of exponential growth is proportional to the function’s current level, not strictly equal to it.

From a twenty-first-century vantage point, calculus is often seen as the mathematics of change. It quantifies change using two big concepts: derivatives and integrals. Derivatives model rates of change and are the main topic of this chapter. Integrals model the accumulation of change and will be discussed in chapters 7 and 8.

Derivatives answer questions like “How fast?” “How steep?” and “How sensitive?” These are all questions about rates of change in one form or another. A rate of change means a change in a dependent variable divided by a change in an independent variable.

us. It defines a rate of change as a function. It specifies a rate at a given point or at a given time, even if that rate is variable.