Infinite Powers: How Calculus Reveals the Secrets of the Universe

by Steven H. Strogatz

Consider Toy Story, released in 1995. Back then, it took a single animator a week to sync an eight-second shot. The whole film took four years and eight hundred thousand hours of computer time to complete. As Pixar co-founder Steve Jobs told Wired, “There are more PhDs working on this film than any other in movie history.”

Before the work of Galileo and Kepler, natural phenomena had rarely been understood in mathematical terms. Archimedes had revealed the mathematical principles of balance and buoyancy in his laws of the lever and hydrostatic equilibrium, but those laws were limited to static, motionless situations.

The challenge for Galileo, Kepler, and other like-minded mathematicians of the early seventeenth century was to take their beloved geometry, so well suited to a world at rest, and extend it to a world in flux.

In mathematics, pendulums stimulated the development of calculus through the riddles they posed.

Like the line in William Blake’s poem about seeing the world in a grain of sand, physicists and engineers learned to see the world in a pendulum’s swing.

In some cases, the connections between pendulums and other phenomena are so exact that the same equations can be recycled without change. Only the symbols need to be reinterpreted; the syntax stays the same. It’s as if nature keeps returning to the same motif again and again, a pendular repetition of a pendular theme.

expect). That’s because period squared is proportional to distance cubed (not squared),

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 that cutting comes naturally before rebuilding, it seems sensible for a novice to learn differential calculus first. And indeed, that’s how all calculus courses begin today. They start with derivatives—the relatively easy techniques for slicing and dicing—and then work their way up to integrals, the much harder techniques for reassembling the pieces into an integrated whole. Students find it more comfortable to learn calculus in this order because the easier material comes first. Their teachers like it because the subject seems more logical this way. Yet, strangely enough, history unfolded in the opposite order. Integrals were already in full swing in ancient Greece in Archimedes’s work around 250 BCE, whereas derivatives weren’t even a gleam in anybody’s eye until the 1600s. Why

Although calculus culminated in Europe, its roots lie elsewhere. In particular, algebra came from Asia and the Middle East.

This is what changed after Newton and Leibniz. They independently discovered and proved a fundamental theorem that made such problems routine. The theorem connected areas to slopes and thereby linked integrals to derivatives. It was astonishing. Like a twist out of a Dickens novel, two seemingly distant characters were the closest of kin. Integrals and derivatives were related by blood.

I love this! Hahaha. Twist in a dickens novel = FTC

Right now, millions of students in high school and college all around the world are grinding away on their calculus problem sets, solving integral after integral with the help of the fundamental theorem. Yet many of them are oblivious to the gift they’ve been given.

Perhaps understandably so—it’s like the old joke about the fish who asks his friend, “Aren’t you grateful for water?” to which the other fish says, “What’s water?” Students in calculus are swimming in the fundamental theorem all the time, so naturally they take it for granted.

The simplest illustration of his idea takes us back to the familiar problem of a car moving at a constant speed for which distance equals rate times time. As elementary as this example may be, it still captures the essence of the fundamental theorem, so it’s a good place to start.

Imagine a car cruising down the highway at 60 miles per hour. If we plot its distance versus time and, beneath that, its speed versus time, the resulting distance and speed graphs look like this:

Having seen how speed manifests itself on the distance graph (as the slope of the line), we now turn the question around and ask: How does distance reveal itself on the speed graph? In other words, is there some visual or geometric feature of the speed graph that would allow us to infer how far the car has traveled up to any given time t?

That’s one version of the fundamental theorem. It seems too easy to be true, but it is true.

My calculus teacher in high school, Mr. Joffray, had a memorable metaphor for this fluid scenario, with its sliding x and its changing area. He asked us to imagine a magical paint roller moving sideways. As it rolls steadily to the right, it paints the region under the curve gray. The dotted line at x marks the current position of this imaginary roller as it rolls to the right. Meanwhile, to ensure that the region is painted neatly, the roller instantly and magically shrinks or stretches in the vertical direction,

Skinny, stooped, and pale, Leibniz might not have been much to look at, but his mind was beautiful. He was the most versatile genius in a century of geniuses that included Descartes, Galileo, Newton, and Bach. Although Leibniz found his calculus a decade after Newton did, he is generally considered its co-inventor for several reasons. He published it first, in a graceful and digestible form, and he couched it in a carefully designed, elegant notation that’s still used today. Moreover, he attracted disciples who spread the word with evangelical zeal. They wrote influential textbooks and developed the subject in luxuriant detail. Much later, when Leibniz was accused of stealing calculus from Newton, his disciples defended him vigorously and counterattacked Newton with equal fervor.

Being differentiated is a traumatic experience for most curves. But not for a sine wave.

Ever since Newton, mathematicians have avoided nonlinear differential equations wherever possible. They’re seen as nasty and recalcitrant. In contrast, linear differential equations are sweet and docile.

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.

I can imagine a day, not too far in the future, when artificial intelligence will give us proofs from the Book. What will calculus be like then, and what will medicine be like, and sociology, and politics?