Infinite Powers: How Calculus Reveals the Secrets of the Universe

by Steven H. Strogatz

There are three central problems in calculus. They are shown schematically on the diagram below. The forward problem: Given a curve, find its slope everywhere. The backward problem: Given a curve’s slope everywhere, find the curve. The area problem: Given a curve, find the area under it.

the area beneath the curve, and so on. In the twenty-first century, we are more interested in the function that produced the curve, which models some natural phenomenon or technological process that manifested itself in the curve.

Can protein structures be modeled as curves? Are they already?

we zoom in closer and closer, that piece of the parabola should begin to look straighter and straighter. In the limit of infinite magnification (which amounts to zooming in on an infinitesimal piece of the curve around the point of interest), that magnified piece should approach a straight line. If it does, that limiting straight line is defined as the tangent line at that point on the curve, and its slope is defined as the derivative there. Notice that we are using the Infinity Principle here—we are trying to make a complicated curve simpler by chopping it into infinitesimal straight pieces. This is what we always do in calculus. Curved shapes are hard; straight shapes are easy, even if there are infinitely many of them and even if they are infinitesimally small. Calculating a derivative in this way is a quintessential calculus move and one of the most fundamental applications of the Infinity Principle.

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. Integrals have been defanged and turned into homework exercises for teenagers.

No matter how erratically something moves, the area accumulated under its speed curve up to time t always equals the total distance it has traveled up to that time. That’s one version of the fundamental theorem. It seems too easy to be true, but it is true.

The diagram below summarizes what we’ve just learned. A(x) derivative→ y(x) derivative→ dy/dx area under curve curve slope of curve It shows the three functions we’re interested in and the relationships between them. The given curve is in the middle, its unknown slope is on the right, and its unknown area is on the left. As we saw in chapter 6, these are the functions that occur in the three central problems of calculus. Given the curve y, we are trying to figure out its slope and its area.

A solution to the backward problem would also have much larger implications, for the following reason: An area is, from an Archimedean standpoint, an infinite sum of infinitesimal rectangular strips. As such, an area is an integral. It’s the integrated collection of all the pieces put back together, an accumulation of infinitesimal change. And just as derivatives are more important than slopes, integrals are more important than areas. Areas are crucial to geometry; integrals are crucial to everything, as we’ll see in the chapters ahead.

This is why it was so important to be able to find the area under an arbitrary curve. Because of its intimate connection to the backward problem, the area problem is not just about area. It’s not just about shape or the relationship between distance and speed or anything that narrow. It’s completely general. From a modern perspective, the area problem is about predicting the relationship between anything that changes at a changing rate and how much that thing builds up over time. It’s about the fluctuating inflow to a bank account and the accumulated balance of money in it. It’s about the growth rate of the world’s population and the net number of people on Earth. It’s about the changing concentration of a chemotherapy drug in a patient’s blood and the accumulated exposure to that drug over time. That total exposure affects how potent the chemo will be, as well as how toxic. Area matters because the future matters.

The reason why integration is so much harder than differentiation has to do with the distinction between local and global. Local problems are easy. Global problems are hard. Differentiation is a local operation. As we’ve seen, when we are calculating a derivative, it’s like we’re looking under a microscope.

Logarithms excited Newton for two reasons. First, they could be used to speed up calculations enormously, and second, they were relevant to a controversial problem in music theory he was working on: how to divide an octave into perfectly equal musical steps without sacrificing the most pleasing harmonies of the traditional scale. (In the jargon of music theory, Newton was using logarithms to assess how faithfully an equal-tempered division of the octave could approximate the traditional tuning of just intonation.)

Thanks to the marvels of the internet and the historians at the Newton Project, you can travel back to 1665 right now and watch young Newton at play. (His handwritten college notebook is freely available at http://cudl.lib.cam.ac.uk/view/MS-ADD-04000/.)

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.

Incidentally, similar reasoning shows that for any positive integer n, the derivative of y = xn is = dy/dx=nxn-1, a result we’ve mentioned earlier. With a little more work, we could extend this result to negative, fractional, and irrational n.

Thus the answer to Huygens’s original question about the infinite sum becomes clear: As N approaches infinity, the term 1/(N+1) approaches zero, and so S approaches 1. That limiting value of 1 is the answer to Huygens’s puzzle. The key that allowed Leibniz to find the sum was that it had a very particular structure: it could be rewritten as a sum of consecutive differences (in this case, differences of consecutive unit fractions). That difference structure caused the massive cancellations we saw above. Sums with this property are now termed telescoping sums because they call to mind one of those old collapsible telescopes you see in pirate movies, the kind of spyglass that can be stretched out or contracted at will. The analogy is that the original sum appears in its stretched-out form, but, because of its difference structure, it can be telescoped down to a much more compact result. The only terms that survive the collapse are the terms without partners to cancel them, the ones at the very ends of the telescope.

Within pure mathematics, the evolution of calculus has been a story of crossbreeding and its benefits. Older parts of math were invigorated after they were crossed with calculus. For example, the ancient study of numbers and their patterns was revitalized by an infusion of calculus-based tools like integrals, infinite sums, and power series. The resulting hybrid field is called analytic number theory. Likewise, differential geometry used calculus to shed light on the structure of smooth surfaces and revealed cousins they never knew they had, unimaginable curved shapes in four dimensions and beyond. In this way, the Cambrian explosion of calculus made mathematics more abstract and more powerful. It also made it more like a family. Calculus exposed a web of hidden relationships tying all parts of mathematics together.

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: a = dv⁄dt. Here dv is the infinitesimal change in the body’s velocity v during an infinitesimal time interval dt. So if we know the force F on the body, and if we know its mass m, we can use F = ma to find its acceleration via a = F/m. That acceleration in turn determines how the body will move. It tells us how the body’s velocity will change in that next instant, and its velocity tells us how its position will change. In this way, F = ma is an oracle. It predicts the body’s future behavior, one tiny step at a time.

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.

As we will see shortly, there are differential equations for describing how this function changes in space and time. Such a differential equation is not an ordinary differential equation. It can’t be, because it doesn’t depend on just one independent variable. In fact, it depends on four of them: x, y, z, and t. It’s a new kind of beast—a partial differential equation, so called because each of its independent variables plays its own “part” in causing change to occur. Partial differential equations are much richer than ordinary differential equations. They describe continuous systems moving and changing in space and time simultaneously or in two or more dimensions of space.

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.

To a physicist, what’s remarkable about sine waves (in the context of the vibration and heat flow problems) is that they form standing waves. They don’t travel along the string or the rod. They remain in place. They oscillate up and down but never propagate. Even more remarkably, standing waves vibrate at a unique frequency. That’s a rarity in the world of waves. Most waves are a combination of many frequencies,

Chladni patterns allow us to visualize standing waves in two dimensions.

To my mind, that vision of calculus is far too blinkered. Calculus is not just the work of Newton and Leibniz and their successors. It started much earlier than that and it’s still going strong today. 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.

New applications of calculus to the social sciences, music, the arts, and the humanities Ongoing applications of calculus to medicine and biology Coping with the randomness inherent in finance, economics, and the weather Calculus in the service of big data, and vice versa The continuing challenge of nonlinearity, chaos, and complex systems The evolving partnership between calculus and computers, including artificial intelligence Pushing the boundaries of calculus in the quantum realm.

Laplace’s naive conflation of determinism with predictability.

You may have heard that time is the fourth dimension. Indeed, in Einstein’s special and general theories of relativity, space and time are fused into a single entity, space-time, and represented in a four-dimensional mathematical arena. Roughly speaking, ordinary space gets plotted on the first three axes and time gets plotted on the fourth. This construction can be viewed as a generalization of the two-dimensional xy plane of Fermat and Descartes.

Today, mathematicians regularly think about abstract spaces having arbitrary numbers of dimensions. We speak about n-dimensional space, and we have developed geometry and calculus in any number of dimensions. As we saw in chapter 10, Allan Cormack, the inventor of the theory behind CT scanning, wondered how CT would work in four dimensions, purely out of intellectual curiosity. Great things have come from this spirit of pure adventure. 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.

The scariest point is that AlphaZero showed insight. It played like no computer ever has, intuitively and beautifully, with a romantic, attacking style. It played gambits and took risks. In some games it paralyzed Stockfish and toyed with it. It seemed malevolent and sadistic. And it was creative beyond words, playing moves no grandmaster or computer would ever dream of making. It had the spirit of a human and the power of a machine. It was humankind’s first glimpse of a terrifying new kind of intelligence.

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.

To Einstein, gravity was no longer a force acting instantaneously at a distance. Instead, it was an almost palpable thing, a warp in the fabric of the universe, a manifestation of the curvature of space and time. Curvature—an idea that goes back to the birth of calculus, to the ancient fascination with curved lines and curved surfaces—in