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

by Steven H. Strogatz

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.”

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.

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.

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

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.”

But it was only in the limit of infinitely many slices that it became truly rectangular. That’s the big idea behind calculus. Everything becomes simpler at infinity.

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.

What I’m trying to say is that his calculation of π was heroic, both logically and arithmetically. By using a 96-gon inside the circle and a 96-gon outside the circle, he ultimately proved that π is greater than 3 + 10/71 and less than 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. That latter result, 3 + 10/70, reduces to 22/7, the famous approximation to π that all students still learn today and that some unfortunately mistake for π itself.

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.

Descartes badmouthed Fermat mercilessly and to some extent succeeded in diminishing him. Fermat’s work was never properly published until 1679. His results trickled out through word of mouth or in copies of his letters, but he was not truly appreciated until long after his death. Descartes, however, hit it big. His Discourse became famous. The next generation learned analytic geometry from it. Even today, our students learn about Cartesian coordinates, even though Fermat came up with them first.

The pivotal moment in the story of calculus occurred in the middle of the seventeenth century when the mysteries of curves, motion, and change collided on a two-dimensional grid, the xy plane of Fermat and Descartes. Back then, Fermat and Descartes had no idea what a versatile tool they’d created. They intended the xy plane as a tool for pure mathematics. Yet from the start, it too was a crossroads of sorts, a place where equations met curves, algebra met geometry, and the mathematics of the East met that of the West. Then, in the next generation, Isaac Newton built on their work as well as on the work of Galileo and Kepler and brought geometry and physics together in a great synthesis. Newton’s spark set off the fire that lit the Enlightenment and caused a revolution in Western science and mathematics.

There’s no mathematical reason for preferring 10 over any other base. It’s a traditional favorite because of an accident of biological evolution: we happen to have ten fingers. Accordingly, we have based our system of arithmetic, the decimal system, on powers of ten.

exponential functions are for modeling growth that feeds on itself, and power functions are for modeling less violent forms of growth. Logarithms are useful for the same reason that staple removers are useful: they undo the action of another tool. Specifically, logarithms undo the actions of exponential functions, and vice versa.

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

Doctors use the same word for gallstones, kidney stones, and bladder stones. 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’s secret fountain was the fundamental theorem of calculus. Although he and Leibniz weren’t the first to notice this theorem, they get the credit for it because they were the first to prove it in general, recognize its overwhelming utility and importance, and build an algorithmic system around it. The methods they developed are now commonplace.

At barely twenty-two, Isaac Newton had found a path to the holy grail. By converting curves to power series, he could find their areas systematically.

In a way it makes sense that power series should have been anticipated in India. Decimals were also developed in India, and as we’ve seen, Newton regarded what he was doing for curves as an analog of what infinite decimals had done for arithmetic.

Actually, Newton’s series were infinitely more powerful than Ar-chimedes’s in that they enabled him to find the areas of not just one but a whole continuous infinity of circular and hyperbolic segments. That’s what his abstract symbol x did for him. It let him change his problems continuously and effortlessly. It enabled him to tune the shape of segments by sliding x to the left or right so that what appeared to be a single infinite series was in fact an infinite family of infinite series, one for each choice of x. That was the power of power series. They let Newton solve infinitely many problems in a single stroke.

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.

An infinitesimal is a hazy thing. It is supposed to be the tiniest number you can possibly imagine that isn’t actually zero. More succinctly, an infinitesimal is smaller than everything but greater than nothing.

So the recipe is this: to study infinitesimal change, keep terms that involve dx to the first power and ignore the rest.

The only thing wrong with infinitesimals is that they don’t exist, at least not within the system of real numbers. Oh, and one other thing — they are paradoxical. They wouldn’t make sense even if they did exist. One of Leibniz’s disciples, Johann Bernoulli, realized they’d have to satisfy nonsensical equations like x + dx = x, even though dx isn’t zero. Hmmm. Well, you can’t have everything. Infinitesimals do give the right answers once we learn how to work with them, and the benefits they provide more than make up for any psychic distress they may cause. They are like Picasso’s lie that helps us realize the truth.

In relativity, as in quantum mechanics, the laws of nature are still written in the language of calculus, with sentences in the form of differential equations. That, to me, is Newton’s greatest legacy. He showed that nature is logical. Cause and effect in the natural world behave much like a proof in geometry, with one truth following from another by logic, except that what is following is one event from another in the world, not one idea from another in our minds.

By assuming his laws of motion and gravity as axioms and using his calculus as a deductive instrument, Newton proved that all three of Kepler’s laws followed as logical necessities. The same was true for Galileo’s law of inertia, the isochronism of pendulums, the odd-number rule for balls rolling down ramps, and the parabolic arcs of projectiles. Each of them was a corollary of the inverse-square law and F = ma. This appeal to deductive reasoning shocked Newton’s colleagues and disturbed them on philosophical grounds. Many of them were empiricists. They thought that logic applied only within mathematics itself. Nature had to be studied by experiment and observation. They were dumbfounded by the thought that nature had an inner mathematical core and that phenomena in nature could be deduced by logic from empirical axioms like the laws of gravity and motion.

In the centuries after Newton, his system was refined by many other mathematicians, physicists, and astronomers. It was so trusted that when the motion of a planet disagreed with its predictions, astronomers assumed they were missing something important. This was how the planet Neptune was discovered in 1846. Irregularities in the orbit of Uranus suggested the presence of an unknown planet beyond it, an unseen neighbor that was perturbing Uranus gravitationally. Calculus predicted where the missing planet should be, and when astronomers looked, there it was.

Katherine Johnson was a computer for the National Aeronautics and Space Administration at a time when computers were women, not machines. She was there near the start, when she helped Alan Shepard become the first American in space, and she was there near the end, when she worked on the trajectory for the first moon landing. For decades, her work was unknown to the public. Thankfully, her pioneering contributions (and her inspiring life story) have now been recognized. In 2015, at age ninety-seven, she received the Presidential Medal of Freedom from President Barack Obama. A year later, NASA named a building after her. At the dedication ceremony, the NASA official reminded the audience that “millions of people around the world watched [Alan] Shepard’s flight, but what they didn’t know at the time was that the calculations that got him into space and safely home were done by today’s guest of honor, Katherine Johnson.”

The secrets of heat were unraveled by a man who often felt cold. Orphaned at the age of ten, Jean Baptiste Joseph Fourier was a sickly, dyspeptic asthmatic as a teenager. As an adult, he believed heat was essential to health. He kept his room overheated and swathed himself in a heavy overcoat, even in the summer. In all aspects of his scientific life, Fourier was obsessed with heat. He originated the concept of global warming and was the first to explain how the greenhouse effect regulates the Earth’s average temperature.

The unifying power of Fourier’s idea is that the sound of any musical instrument can be synthesized by an array of infinitely many tuning forks. All we need to do is strike the tuning forks with the right strengths and at the right times and, incredibly, out pops the sound of a violin or a piano or even a trumpet or an oboe, although we’re using nothing more than colorless sine waves. This is essentially how the first electronic synthesizers worked: they reproduced the sound of any instrument by combining a large number of sine waves.

The larger significance of Fourier’s work is that he took the first step toward using calculus as a soothsayer to predict how a continuum of particles could move and change. This was an enormous advance beyond Newton’s work on the motion of discrete sets of particles.

On her third try, Germain was awarded the prize, the first woman ever to be so honored by the Paris Academy of Sciences.

What is the use of these results? The answer is that I don’t know. They will almost certainly produce some theorems in the theory of partial differential equations, and some of them may find application in imaging with MRI or ultrasound, but that is by no means certain. It is also beside the point. Quinto and I are studying these topics because they are interesting in their own right as mathematical problems, and that is what science is all about.

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.

Kovalevskaya was born in 1850 and grew up in an aristocratic family in Moscow. When she was eleven she found herself surrounded by calculus, literally — one wall of her bedroom was papered with notes from a calculus course her father had attended in his youth. She later wrote that she “spent whole hours of my childhood in front of that mysterious wall, trying to make out even a single sentence and find the order in which the pages ought to have followed one another.” She went on to become the first woman in history to earn a PhD in mathematics.

Yet many things in nature are more complicated than this. Whenever parts of a system interfere or cooperate or compete with each other, there are nonlinear interactions taking place. Most of everyday life is spectacularly nonlinear; if you listen to your two favorite songs at the same time, you won’t get double the pleasure. The same goes for consuming alcohol and drugs, where the interaction effects can be deadly. By contrast, peanut butter and jelly are better together. They don’t just add up — they synergize.

Sofia Kovalevskaya helped us understand how different the world appears when we finally face up to nonlinearity. She realized that nonlinearity places limits on human hubris. When a system is nonlinear, its behavior can be impossible to forecast with formulas, even though that behavior is completely determined. In other words, determinism does not imply predictability. It took the motion of a top — a child’s plaything — to make us more humble about what we can ever hope to know.

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.

When Einstein needed four-dimensional geometry for curved space and time in general relativity, he was pleased to learn it already existed, thanks to Bernhard Riemann, who had created it decades earlier for the purest of mathematical reasons.

If we are to have any hope of making sense of cell biology using differential equations, we need to be able to solve those equations with formulas (which Sofia Kovalevskaya showed we cannot) or picture them (which our limited brains won’t allow).

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.

This, to me, is the greatest mystery of all: Why is the universe comprehensible, and why is calculus in sync with it? I have no answer, but I hope you’ll agree it’s worth contemplating.

QED is the quantum theory of how light and matter interact. It merges Maxwell’s theory of electricity and magnetism with Heisenberg’s and Schrödinger’s quantum theory and Einstein’s special theory of relativity. Feynman was one of the principal architects of QED, and after looking at the structure of his theory, I can see why he had such admiration for calculus. His theory is chock-full of it, both in tactics and in style. It’s teeming with power series, integrals, and differential equations and includes plenty of hijinks with infinity. More important, it’s the most accurate theory anyone has ever devised . . . about anything.

The theory also predicted that gravity could have a strange effect on time: The passage of time could speed up or slow down as an object moves through a gravitational field. Bizarre as this sounds, it really does occur. It needs to be taken into account in the satellites of the global positioning system as they move high above the Earth. The gravitational field is weaker up there, which reduces the curvature of space-time and causes clocks to run faster than they do on the ground. Without correcting for this effect, the clocks aboard the GPS satellites wouldn’t keep accurate time. They’d get ahead of ground-based clocks by about 45 microseconds per day. That may not sound like much, but keep in mind that the whole global positioning system requires nanosecond accuracy to work properly, and 45 microseconds is 45,000 nanoseconds. Without the correction for general relativity, errors in global positions would accumulate at about ten kilometers each day, and the whole system would become worthless for navigation in a matter of minutes.