Infinite Powers: How Calculus Reveals the Secrets of the Universe

by Steven H. Strogatz

The essence of the answer lies in a quip that the physicist Richard Feynman made to the novelist Herman Wouk when they were discussing the Manhattan Project. Wouk was doing research for a big novel he hoped to write about World War II, and he went to Caltech to interview physicists who had worked on the bomb, one of whom was Feynman. After the interview, as they were parting, Feynman asked Wouk if he knew calculus. No, Wouk admitted, he didn’t. “You had better learn it,” said Feynman. “It’s the language God talks.”

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. The details differ depending on what part of nature we’re talking about, but the structure of the laws is always the same. To put this awesome assertion another way, there seems to be something like a code to the universe, an operating system that animates everything from moment to moment and place to place. Calculus taps into this order and expresses it.

But why should the universe respect the workings of any kind of logic, let alone the kind of logic that we puny humans can muster? This is what Einstein marveled at when he wrote, “The eternal mystery of the world is its comprehensibility.” 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.”

In a nutshell, calculus wants to make hard problems simpler. It is utterly obsessed with simplicity.

Calculus succeeds by breaking complicated problems down into simpler parts. That strategy, of course, is not unique to calculus. All good problem-solvers know that hard problems become easier when they’re split into chunks. The truly radical and distinctive move of calculus is that it takes this divide-and-conquer strategy to its utmost extreme—all the way out to infinity

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.

So this is how calculus began. It grew out of geometers’ curiosity and frustration with roundness.

And so began the second great obsession: a fascination with the mysteries of motion on Earth and in the solar system. Through observation and ingenious experiments, scientists discovered tantalizing numerical patterns in the simplest moving things.

Then he cracked the code of the universe. Newton discovered that motion of any kind always unfolds one infinitesimal step at a time, steered from moment to moment by mathematical laws written in the language of calculus. With just a handful of differential equations (his laws of motion and gravity), he could explain everything from the arc of a cannonball to the orbits of the planets. His astonishing “system of the world” unified heaven and earth, launched the Enlightenment, and changed Western culture. Its impact on the philosophers and poets of Europe was immense. He even influenced Thomas Jefferson and the writing of the Declaration of Independence, as we’ll see.

According to quantum mechanics, trajectories become jittery, blurry, and poorly defined at the microscopic scale, so we need to describe the behavior of electrons as probability waves instead of Newtonian trajectories. As soon as we do that, however, calculus returns triumphantly.

To get a feel for the level of precision we’re talking about here, consider how many digits we would need to make one of the most extreme comparisons imaginable. 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.

It’s at this point that I begin to feel real affection for Archimedes, because he does something in one of his essays that few geniuses ever do: He invites us in and reveals how he thinks. (I’m using the present tense here because the essay is so intimate, it feels like he’s speaking to us today.) He shares his private intuition, a vulnerable, soft-bellied thing, and says he hopes that future mathematicians will use it to solve problems that eluded him. Today this secret is known as the Method. I never heard of it in calculus class. We don’t teach it anymore. But I found the story of it and the idea behind it enthralling and astounding.

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

I hope I’ve managed to convey what an acid trip of an argument this is.

As always, the ravages of time were cruel. Kingdoms fell and libraries were burned. Manuscripts decayed. Not a single copy of the Method was known to have survived the Middle Ages. Although Leonardo da Vinci, Galileo, Newton, and other geniuses of the Renaissance and the scientific revolution pored over what was left of Archimedes’s treatises, they never had a chance to read the Method. It was thought to be irretrievably lost. And then, miraculously, it was found. In October 1998 a battered medieval prayer book came up for auction at Christie’s and sold to an anonymous private collector for $2.2 million. Barely visible under its Latin prayers lay faint geometrical diagrams and mathematical text written in tenth-century Greek. The book is a palimpsest; in the thirteenth century, its parchment folios had been washed and scraped clean of the original Greek and overwritten with Latin liturgical text. Fortunately, the Greek was not completely obliterated. It contains the only surviving copy of Archimedes’s Method. The Archimedes Palimpsest, as it is now known, first came to light in 1899 in a Greek Orthodox library in Constantinople. It spent the Renaissance and the scientific revolution undetected in a prayer book in the monastery of St. Sabas near Bethlehem. It now lives in the Walters Art Museum in Baltimore, where it has been lovingly restored and examined using the latest imaging technology.

Amid this swirl of questions, an influx of ideas from Islamic and Indian mathematics offered European mathematicians a new way forward, a chance to go beyond Archimedes and break new ground. The ideas from the East would lead to fresh ways of thinking about motion and curves and then, with a thunderclap, to differential calculus.

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. Given

Why did differential calculus—the easier side of the subject—develop so much later than integral calculus? It’s because differential calculus grew out of algebra, and algebra took centuries to mature, migrate, and mutate. In its original form in China, India, and the Islamic world, algebra was entirely verbal.

For now, the main thing we need to know is that sine waves are connected to circular motion. To see the connection, imagine a point moving around a circle at a constant speed. If we track its up-and-down position as a function of time, the point traces out a sine wave. And because circles are intimately connected to cycles, sine waves come up wherever cyclic phenomena occur, from the cycle of the seasons to the vibrations of a tuning fork to the sixty-cycle hum of fluorescent lights and power lines.

In a little more detail, here’s how it works. For any given choice of the four parameters, we calculate the discrepancy (in other words, the error) between the sine-wave fit and the actual data at every one of the twenty-seven points recorded throughout the year.

makes the square of a negative error count as a positive discrepancy, as it should.) 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. All of which raises an extremely

But he did make one contribution to applied mathematics of lasting importance: he was the first person to deduce a law of nature from a deeper law by using calculus as a logical engine. Just as Maxwell would do with electricity and magnetism two centuries later, Fermat translated a hypothetical law of nature into the language of calculus, started the engine, and fed the law in, and out popped another law, a consequence of the first one. In so doing, Fermat, the accidental scientist,

But for people in technical professions, numbers are just the beginning. Scientists, engineers, financial quants, and medical researchers need to work with relationships between numbers, which show how one thing affects another. To describe relationships like that, functions are indispensable. They provide the tools needed to model motion and change.

In this way, logarithms enable us to write any positive number as a power of ten. Doing that makes many calculations easier and also reveals surprising connections between numbers. Look what happens if we multiply 90 by a factor of 10 or 100 and then take its log again: log 900 ≈ 2.9542 . . . and log 9000 = 3.9542 . . . . Observe two striking things here: All the logs here have the same decimal part, .9542 . . . . Multiplying the original number, 90, by 10 increased its log by 1. Multiplying it by 100 increased its log by 2, etc. We can explain both of these facts by appealing to a rule of logs: The log of a product is the sum of the logs. Here, log 90 = log (9×10) = log9+log10 = .9542 . . . +1 and log 900 = log (9×100) = log9+log100 = .9542 . . . +2 and so on. This

In other words, when we multiply two numbers together and then take their log, the result is the sum (not the product!) of their individual logs. In that sense, logarithms replace multiplication problems with addition problems, which are much easier. This is why logarithms were invented. They sped up calculations tremendously. Instead of having to deal with Herculean multiplication problems, square roots, cube roots, and the like, such calculations could be turned into addition problems and then solved with the help of a lookup table known as a table of logarithms. The idea of logarithms was in the air in the early seventeenth century, but much of the credit for popularizing them goes to the Scottish mathematician John Napier, who published his Description of the Wonderful Rule of Logarithms in 1614. A decade later, Johannes Kepler enthusiastically used the new calculational tool when he was compiling astronomical tables about the positions of the planets and other heavenly bodies. Logarithms were the supercomputers of their era.

Recondite as all this may sound, the natural logarithm is extremely practical, though often inconspicuously. For one thing, it underlies a rule of thumb known to investors and bankers as the rule of 72. To estimate how long it will take to double your money at a given annual rate of return, divide 72 by the rate of return. Thus, money growing at a 6 percent annual rate doubles after about 72/6 = 12 years. This rule of thumb follows from the properties of the natural logarithm and exponential growth and works well if the interest rate is low enough.

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

We know from the Infinity Principle that the way to make progress on complicated problems is to chop them into infinitesimal bits, analyze the bits, and then put the bits back together to find the answer. The little changes dx and dy are those infinitesimal bits in the context of differential calculus. Putting them back together is the job of integral calculus.

Two thousand years after Zeno, the founders of differential calculus solved the riddle of instantaneous speed. Their intuitive solution was to define instantaneous speed as a limit—specifically, the limit of average speeds taken over shorter and shorter time intervals.

In a cruel irony, both Newton and Leibniz, the pioneers of calculus, died in excruciating pain while suffering from calculi—a bladder stone for Newton, a kidney stone for Leibniz.

Newton was led to it by thinking of area as a flowing, moving quantity, not as a frozen measure of a shape, as was then customary in geometry. He brought time into geometry and viewed it like physics.

The upshot is that dA = y dx = f(x) dx. Boom—that’s the fundamental theorem of calculus. Or, as it is more politely phrased nowadays (in our misguided era when differentials have been forsaken for derivatives), dA⁄dx = y = f(x). This is exactly what we found in chapter 7 with the paint-roller argument. One last thing: When we regard the area

A(x) =∫x₀ f(x) dx. That long-necked, swan-like symbol is actually a stretched-out S. The S reminds us that a summation is taking place. It’s a summation of a peculiar kind, distinctive to integral calculus, involving a sum of infinitely many infinitesimal strips, all being integrated into a single, coherent area. As a symbol of integration, it’s called an integral sign. Leibniz introduced it in a 1677 manuscript and published it in 1686. It’s calculus’s most recognizable icon. The zero at the bottom of it and the x at the top of it indicate the endpoints of the interval of the x-axis over which the rectangles stand. Those endpoints are called the limits of integration.

The centerpiece of his theory is his differential equation of motion: F = ma. It ranks as one of the most consequential equations in history. It says that the force, F, on a moving body is equal to the body’s mass, m, times its acceleration, a. It’s a differential equation because acceleration is a derivative (the rate of change of the body’s velocity) or, in Leibnizian terms, the ratio of two differentials:

This is Galileo’s principle of inertia: In the absence of an outside force, a body at rest stays at rest, and a body in motion stays in motion and moves at a constant velocity. Its speed and direction never change. We have just deduced the law of inertia as a logical consequence of Newton’s deeper law of motion, F = ma.

Calculus and the Enlightenment Newton’s picture of a world ruled by mathematics reverberated far beyond science. In the humanities, it served as a foil for Romantic poets like William Blake, John Keats, and William Wordsworth. At a raucous dinner party in 1817, Wordsworth and Keats, among others, agreed that Newton had destroyed the poetry of the rainbow by reducing it to its prismatic colors. They raised their glasses in a boisterous toast: “Newton’s health, and confusion to mathematics.” Newton got a warmer

But why, you may be asking yourself, would anyone ever want to take two derivatives of something? Because nature does—and it does it all the time. Or, rather, our models of nature do it all the time. For example, in Newton’s law of motion, F = ma, the acceleration a involves two derivatives. To see why, remember that the acceleration is the derivative of speed and speed is the derivative of distance. That makes acceleration the derivative of the derivative of distance, or to put it more concisely, the second derivative of distance. Second derivatives come up everywhere in physics and engineering. Along with Newton’s equation, they also star in the heat equation and the wave equation. So

The story of radar and microwave ovens is a testament to the interconnectedness of science. Think of what went into them: physics, electrical engineering, materials science, chemistry, and good old serendipitous invention. Calculus played an important part too. It provided the language for describing waves and the tools for analyzing them. The discovery of the wave equation, which started as an outgrowth of music in connection with vibrating strings, was ultimately used by Maxwell to predict the existence of electromagnetic waves. From there it was a short hop to vacuum tubes, transistors, computers, radar, and microwave ovens.

Remember that circles are always associated with sine waves, and sine waves are the building blocks of Fourier series.

Ironically, given that wave theory and Fourier analysis began with the study of music, at a key moment in the development of computerized tomography, music proved indispensable again. Hounsfield had his breakthrough ideas in the mid-1960s when he was working for a company called Electric and Musical Industries. He had first worked on EMI’s radar and guided weaponry, and then he turned his attention to developing Britain’s first all-transistor computer. After that smashing success, EMI decided to support Hounsfield and let him do whatever he wanted for his next project. At that time, EMI was flush with money and could afford to take risks. Their profits had doubled after they’d signed a band from Liverpool called the Beatles. Hounsfield approached management with his idea of imaging organs with x-rays, and EMI’s deep pockets helped him take the first step.

Calculus, to me, is defined by its credo: to solve a hard problem about anything continuous, slice it into infinitely many parts and solve them. By putting the answers back together, you can make sense of the original whole. I’ve called this credo the Infinity Principle.

Chaotic systems are finicky. A little change in how they’re started can make a big difference in where they end up. That’s because small changes in their initial conditions get magnified exponentially fast. Any tiny error or disturbance snowballs so rapidly that in the long term, the system becomes unpredictable. Chaotic systems are not random—they’re deterministic and hence predictable in the short run—but in the long run, they’re so sensitive to tiny disturbances that they look effectively random in many respects.

Chaotic systems can be predicted perfectly well up to a time known as the predictability horizon. Before that, the determinism of the system makes it predictable.

This is sure to be a growth area for the applications of calculus. Mathematical biology is a no-holds-barred exercise in nonlinear differential equations. With the help of Newton-style analytical methods, Poincaré-style geometric methods, and an unabashed reliance on computers, mathematical biologists are looking for and starting to make headway on the differential equations that govern heart rhythms, the spread of epidemics, the functioning of the immune system, the orchestration of genes, the development of cancer, and many other mysteries of life. We couldn’t do any of it without calculus. Complex

But what about when we unleash AlphaInfinity on such problems? A machine like that would come up with beautiful proofs, as beautiful as the chess games that AlphaZero played against Stockfish. Its proofs would be intuitive and elegant. They would be, in the words of the Hungarian mathematician Paul Erdős, proofs straight from the Book. Erdős imagined that God kept a book with all the best proofs in it. Saying that a proof was straight from the Book was the highest possible praise. It meant that the proof revealed why a theorem was true and didn’t merely bludgeon the reader into accepting it with some ugly, difficult argument.

It’s a clear winter night as I write these last words. I’ve stepped out to look at the sky. With the stars up above and the blackness of space, I can’t avoid feeling awe. How could we, Homo sapiens, an insignificant species on an insignificant planet adrift in a middleweight galaxy, have managed to predict how space and time would tremble after two black holes collided in the vastness of the universe a billion light-years away? We knew what that wave should sound like before it got here. And, courtesy of calculus, computers, and Einstein, we were right. That gravitational wave was the faintest whisper ever heard. That soft little wave had been headed our way from before we were primates, before we were mammals, from a time in our microbial past. When it arrived that day in 2015, because we were listening—and because we knew calculus—we understood what the soft whisper meant.