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

by Steven H. Strogatz

Without calculus, we wouldn’t have cell phones, computers, or microwave ovens. We wouldn’t have radio. Or television. Or ultrasound for expectant mothers, or GPS for lost travelers. We wouldn’t have split the atom, unraveled the human genome, or put astronauts on the moon. We might not even have the Declaration of Independence. It’s a curiosity of history that the world was changed forever by an arcane branch of mathematics. How could it be that a theory originally about shapes ultimately reshaped civilization?

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.” For reasons nobody understands, the universe is deeply mathematical. Maybe God made it that way. Or maybe it’s the only way a universe with us in it could be, because nonmathematical universes can’t harbor life intelligent enough to ask the question. In any case, 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. Isaac Newton was the first to glimpse this secret of the universe. He found that the orbits of the planets, the rhythm of the tides, and the trajectories of cannonballs could all be described, explained, and predicted by a small set of differential equations. Today we call them Newton’s laws of motion and gravity.

By inadvertently discovering this strange language, first in a corner of geometry and later in the code of the universe, then by learning to speak it fluently and decipher its idioms and nuances, and finally by harnessing its forecasting powers, humans have used calculus to remake the world. That’s the central argument of this book. If it’s right, it means the answer to the ultimate question of life, the universe, and everything is not 42, with apologies to fans of Douglas Adams and The Hitchhiker’s Guide to the Galaxy. But Deep Thought was on the right track: the secret of the universe is indeed mathematical.

Calculus is one of humankind’s most inspiring collective achievements. It isn’t necessary to learn how to do calculus to appreciate it, just as it isn’t necessary to learn how to prepare fine cuisine to enjoy eating it.

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.

Take the story of wireless communication. It began with the discovery of the laws of electricity and magnetism by scientists like Michael Faraday and André-Marie Ampère. Without their observations and tinkering, the crucial facts about magnets, electrical currents, and their invisible force fields would have remained unknown, and the possibility of wireless communication would never have been realized. So, obviously, experimental physics was indispensable here. But so was calculus. In the 1860s, a Scottish mathematical physicist named James Clerk Maxwell recast the experimental laws of electricity and magnetism into a symbolic form that could be fed into the maw of calculus. After some churning, the maw disgorged an equation that didn’t make sense. Apparently something was missing in the physics. Maxwell suspected that Ampère’s law was the culprit. He tried patching it up by including a new term in his equation — a hypothetical current that would resolve the contradiction — and then let calculus churn again. This time it spat out a sensible result, a simple, elegant wave equation much like the equation that describes the spread of ripples on a pond. Except Maxwell’s result was predicting a new kind of wave, with electric and magnetic fields dancing together in a pas de deux. A changing electric field would generate a changing magnetic field, which in turn would regenerate the electric field, and so on, each field bootstrapping the other forward, propagating together as a w...

Maxwell’s prediction of electromagnetic waves prompted an experiment by Heinrich Hertz in 1887 that proved their existence. A decade later, Nikola Tesla built the first radio communication system, and five years after that, Guglielmo Marconi transmitted the first wireless messages across the Atlantic. Soon came television, cell phones, and all the rest.

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.

If we’re lucky and skillful enough — if we transform the equations in just the right way — we can get them to reveal their hidden implications. To a mathematician, the process feels almost palpable. It’s as if we’re manipulating the equations, massaging them, trying to relax them enough so that they’ll spill their secrets. We want them to open up and talk to us.

In Maxwell’s case, there were countless ways to transform his equations, all of which would have been logically acceptable but only some of which would have been scientifically revealing.

At that point the linguistic function of calculus took over again. When Maxwell translated his abstract symbols back into reality, they predicted that electricity and magnetism could propagate together as a wave of invisible energy moving at the speed of light. In a matter of decades, this revelation would change the world.

It’s eerie that calculus can mimic nature so well, given how different the two domains are. Calculus is an imaginary realm of symbols and logic; nature is an actual realm of forces and phenomena. Yet somehow, if the translation from reality into symbols is done artfully enough, the logic of calculus can use one real-world truth to generate another.

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.”

Pythagoreans to the mystical belief that all is number. They are said to have believed that even the planets in their orbits made music, the music of the spheres.

Fortunately, a single big, beautiful idea runs through the subject from beginning to end. Once we become aware of this idea, the structure of calculus falls into place as variations on a unifying theme.

Infinity Principle.

In a nutshell, calculus wants to make hard problems simpler.

Calculus can’t help how it looks. Its bulkiness is unavoidable. It looks complicated because it’s trying to tackle complicated problems. In fact, it has tackled and solved some of the most difficult and important problems our species has ever faced. 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. Instead of cutting a big problem into a handful of bite-size pieces, it keeps cutting and cutting relentlessly until the problem has been chopped and pulverized into its tiniest conceivable parts, leaving infinitely many of them.

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.

Notice the act of creative fantasy here. Soup and steel are not really continuous. At the scale of everyday life, they appear to be, but at the scale of atoms or superstrings, they’re not. Calculus ignores the inconvenience posed by atoms and other uncuttable entities, not because they don’t exist but because it’s useful to pretend that they don’t. As we’ll see, calculus has a penchant for useful fictions.

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.

mathematics is so deeply woven into the fabric of the universe,

No one could figure out how much surface area a sphere had or how much volume it could hold. Even finding the circumference and area of a circle was an insurmountable problem in the old days. There was no way to get started. There were no straight pieces to latch onto. Anything that was curved was inscrutable. So this is how calculus began.

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 quest to solve the mystery of curves reached a fever pitch when it became clear that curves were much more than geometric diversions. They were a key to unlocking the secrets of nature. They arose naturally in the parabolic arc of a ball in flight, in the elliptical orbit of Mars as it moved around the sun, and in the convex shape of a lens that could bend and focus light where it was needed, as was required for the burgeoning development of microscopes and telescopes in late Renaissance Europe.

And so began the second great obsession: a fascination with the mysteries of motion on Earth and in the solar system.

From a more secular perspective, the patterns reinforced the claim that nature was deeply mathematical, just as the Pythagoreans had maintained. The only catch was that nobody could explain the marvelous new patterns, at least not with the existing forms of math. Arithmetic and geometry were not up to the task, even in the hands of the greatest mathematicians.

The trouble was that the motions weren’t steady. A ball rolling down a ramp kept changing its speed, and a planet revolving around the sun kept changing its direction of travel. Worse yet, the planets moved faster when they got close to the sun and slowed down as they receded from it.

The Infinity Principle came to the rescue, just as it had for curves. This time the act of wishful fantasy was to pretend that motion at a changing speed was made up of infinitely many, infinitesimally brief motions at a constant speed. To visualize what this would mean, imagine being in a car with a jerky driver at the wheel. As you anxiously watch the speedometer, it moves up and down with every jerk. But over a millisecond, even the jerkiest driver can’t make the speedometer needle move by much. And over an interval much shorter than that — an infinitesimal time interval — the needle won’t move at all. Nobody can tap the gas pedal that fast. These ideas coalesced in the younger half of calculus, differential calculus. It was precisely what was needed to work with the infinitesimally small changes of time and distance that arose in the study of ever-changing motion as well as with the infinitesimal straight pieces of curves that arose in analytic geometry, the newfangled study of curves defined by algebraic equations that was all the rage in the first half of the 1600s. Yes, at one time, algebra was a craze,

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.

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.

With the mysteries of curves and motion now settled, calculus moved on to its third lifelong obsession: the mystery of change. It’s a cliché, but it’s true all the same — nothing is constant but change. It’s rainy one day and sunny the next. The stock market rises and falls. Emboldened by the Newtonian paradigm, the later practitioners of calculus asked: Are there laws of change similar to Newton’s laws of motion? Are there laws for population growth, the spread of epidemics, and the flow of blood in an artery? Can calculus be used to describe how electrical signals propagate along nerves or to predict the flow of traffic on a highway?

By pursuing this ambitious agenda, always in cooperation with other parts of science and technology, calculus has helped make the world modern.

For example, in 1917 Albert Einstein applied calculus to a simple model of atomic transitions to predict a remarkable effect called stimulated emission (which is what the s and e stand for in laser, an acronym for light amplification by stimulated emission of radiation). He theorized that under certain circumstances, light passing through matter could stimulate the production of more light at the same wavelength and moving in the same direction, creating a cascade of light through a kind of chain reaction that would result in an intense, coherent beam. A few decades later, the prediction proved to be accurate. The first working lasers were built in the early 1960s. Since then, they have been used in everything from compact-disc players and laser-guided weaponry to supermarket bar-code scanners and medical lasers.

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. It governs the evolution of probability waves through something called the Schrödinger equation. It’s incredible but true: Even in the subatomic realm where Newtonian physics breaks down, Newtonian calculus still works.

Along with numbers, shapes mattered too. In ancient Egypt, the measurement of lines and angles was of paramount importance. Each year surveyors had to redraw the boundaries of farmers’ fields after the summer flooding of the Nile washed the borderlines away. That activity later gave its name to the study of shape in general: geometry, from the Greek gē, “earth,” and metrēs, “measurer.”

Circles, spheres, cylinders, and cones fascinated the early geometers, but they found them much harder to analyze than triangles, rectangles, squares, cubes, and other rectilinear shapes made of straight lines and flat planes. They wondered about the areas of curved surfaces and the volumes of curved solids but had no clue how to solve such problems. Roundness defeated them.

Calculus began as an outgrowth of geometry.

One of its first and finest was the solution of a long-standing enigma: how to find the area of a circle.

The strategy is to reimagine the circle as a pizza. Then we’ll slice that pizza into infinitely many pieces and magically rearrange them to make a rectangle. That will give us the answer we’re looking for, since moving slices around obviously doesn’t change their area from what they were originally, and we know how to find the area of a rectangle: we just multiply its width times its height. The result is a formula for the area of a circle.

Because the shapes keep getting closer and closer to that rectangle, we’ll call it the limiting rectangle.

This result for the area of a circle, A = rC/2, was first proved (using a similar but much more careful argument) by the ancient Greek mathematician Archimedes (287–212 BCE) in his essay “Measurement of a Circle.”

Everything becomes simpler at infinity.

Fortunately, in calculus, the unattainability of the limit usually doesn’t matter. We can often solve the problems we’re working on by fantasizing that we can actually reach the limit and then seeing what that fantasy implies.

Logical, no. Imaginative, yes. Successful, very.

A limit is a subtle concept but a central one in calculus. It’s elusive because it’s not a common idea in daily life. Perhaps the closest analogy is the Riddle of the Wall. If you walk halfway to the wall, and then you walk half the remaining distance, and then you walk half of that, and on and on, will there ever be a step when you finally get to the wall?

The Parable of the Infinite Polygon As a chastening example, suppose we put a certain number of dots on a circle, space them evenly, and connect them to one another with straight lines. With three dots, we get an equilateral triangle; with four, a square; with five, a pentagon; and so on, running through a sequence of rectilinear shapes called regular polygons.

In this way, infinity is bridging two worlds again. This time it’s taking us from the rectilinear to the round, from sharp-cornered polygons to silky-smooth circles, whereas in the pizza proof, infinity brought us from round to rectilinear as it transformed a circle into a rectangle.

Should we take the plunge and say that a circle truly is a polygon with infinitely many infinitesimal sides? No. We mustn’t do that, mustn’t yield to that temptation. Doing so would be to commit the sin of completed infinity. It would condemn us to logical hell.