Infinite Powers: How Calculus Reveals the Secrets 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.

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.

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.

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.

every inanimate thing in the universe bends to the rule of differential equations.

If anything deserves to be called the secret of the universe, calculus is it.

I’ve written Infinite Powers in an attempt to make the greatest ideas and stories of calculus accessible to everyone. It shouldn’t be necessary to endure what Herman Wouk did to learn about this landmark in human history.

I’ve written Infinite Powers in an attempt to make the greatest ideas and stories of calculus accessible to everyone. It shouldn’t be necessary to endure what Herman Wouk did to learn about this landmark in human history.

Calculus, like other forms of mathematics, is much more than a language; it’s also an incredibly powerful system of reasoning.

we’re actually constructing long chains of logical inference.

It’s eerie that calculus can mimic nature so well,

“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.”

a single big, beautiful idea runs through the subject from beginning to end.

Alas, most calculus courses bury the theme under an avalanche of formulas, procedures, and computational tricks.

the Infinity Principle.

what calculus wants . . . and how it gets what it wants. In a nutshell, calculus wants to make hard problems simpler. It is utterly obsessed with simplicity. That might come as a surprise to you, given that calculus has a reputation for being complicated. And there’s no denying that some of its leading textbooks exceed a thousand pages and weigh as much as bricks. But let’s not be judgmental. Calculus can’t help how it looks. Its bulkiness is unavoidable.

Calculus succeeds by breaking complicated problems down into simpler parts.

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.

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.

In every case the strategy remains the same: split a complicated but continuous problem into infinitely many simpler pieces, then solve them separately and put them back together.

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.

In the story of calculus, more than in other parts of mathematics, logic has always lagged behind intuition. This makes the subject feel especially human and approachable, and its geniuses more like the rest of us.

Three mysteries above all have spurred its development: the mystery of curves, the mystery of motion, and the mystery of change.

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. It grew out of geometers’ curiosity and frustration with roundness.

Like explorers climbing Mount Everest, geometers wanted to solve curves because they were there.

The breakthrough came from insisting that curves were actually made of straight pieces. It wasn’t true, but one could pretend that it was.

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.

Motion was proving to be as much of a conceptual Mount Everest as curves were.

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.

the younger half of calculus, differential calculus.

Which we actually teach first. Differential calculus, the younger half.

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.

Hence the Declaration of Indepemdence.

He even influenced Thomas Jefferson and the writing of the Declaration of Independence, as we’ll see.

calculus moved on to its third lifelong obsession: the mystery of change.

Yet even when geometry was fixated on straightness, one curve always stood out, the most perfect of all: the circle.

Calculus began as an outgrowth of geometry. Back around 250 BCE in ancient Greece, it was a hot little mathematical startup devoted to the mystery of curves.

The ambitious plan of its devotees was to use infinity to build a bridge between the curved and the straight.

Given all the discoveries and technologies that ultimately flowed from calculus, the idea of using infinity to solve difficult geometry problems has to rank as one of the best ideas anyone ever had.

The new shape looks bulbous and strange with its scalloped top and bottom. It’s certainly not a rectangle, so its area is not easy to guess. We seem to be going backward. But as in any drama, the hero needs to get into trouble before triumphing. The dramatic tension is building.

This is the allure of infinity. Everything becomes better there.

Dividing by zero summons infinity in much the same way that a Ouija board supposedly summons spirits from another realm. It’s risky. Don’t go there.

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.” That almost brings a tear to my eye. This unsurpassed genius, feeling the finiteness of his life against the infinitude of mathematics, recognizes that there is so much left to be done, that there are “other theorems which have not yet fallen to our share.” We all feel that, all of us mathematicians. Our subject is endless. It humbled even Archimedes himself.

I hope I’ve managed to convey what an acid trip of an argument this is. Instead of a potter reassembling shards, here Archimedes is more like a butcher. He takes the tissue of the parabolic region apart, one vertical strip at a time, and hangs all these infinitesimally thin strips of flesh from a hook at S.

Although Archimedes seems embarrassed by his dalliance with infinity, he is brave enough to own up to it.

Coping with curved shapes means coping with infinity, one way or another.

His favorite result, which he loved so much that he asked that it be carved on his tombstone, concerned the surface area and volume of a sphere. Picture a sphere sitting snugly in a cylindrical hatbox. Using the Method, Archimedes discovered that the sphere has ⅔ the volume of the enclosing hatbox, as well as ⅔ of its surface area (assuming the top and bottom lids are also counted in the hatbox’s surface area). Notice that he didn’t give formulas for the volume or the surface area of the sphere, as we would today. Rather, he phrased his results as proportions. That’s classic Greek style.

As the Infinity Principle teaches us, the straight and the jagged can impersonate the curved and the smooth.

When Avatar was released nearly a decade later, in 2009, the level of polygonal detail became more extravagant. At director James Cameron’s insistence, animators used about a million polygons to render each plant on the imaginary world of Pandora.