In today’s episode, I’d like to share some thoughts on a new book by the author and mathematician Steven Strogatz titled, Infinite Powers: How Calculus Reveals the Secrets of the Universe.
Sometimes it’s hard to remember how intelligent humanity can be. How great our ideas can be. How capable we are of transcending tribalism and other parochial nonsense. It’s hard to remember humanity’s greatness when we’re relentlessly bombarded by bad news. Relentlessly reminded of the evils of our history. Relentlessly exposed to the worst of ourselves. The quality of our national discourse today seems like high school gossip on a grown up budget. Mired in group think, morally panicked, fetishizing the distribution of power.
There is so much more to the human story than this, if only we chose to focus on it.
Considering how cynical our national discourse has become, I’ve been searching for more uplifting examples of human reason to fortify my perspective on our species’ potential. Mathematics came to mind. It just feels so clean in this moment. Mathematics is the most precise system of reasoning we’ve ever devised, with its symbolic operations directly expressing the rules of logical inference. It’s hard to find a discourse about how the world works more objective, more rigorous, and more apolitical than mathematics. At the same time, the language of mathematics turns out to be fantastically creative, elegant, and even mysterious. Math simultaneously displays both the order and the wonder of nature. So, it’s not only refreshingly objective, it’s deeply satisfying. Throughout history, people who have had the deepest understanding of mathematics consistently describe it as the language of God.
Calculus, in particular, is the mathematical language that’s best enabled humanity to discover the fundamental character of reality. Time and again, calculus has revealed new truths about the universe on paper that are verified out in the real world decades or even centuries later. No other intellectual discipline better confirms that truths about how the world works are discovered, not invented. Or, as Steven Strogratz put it in his book, “there seems to be something like a code to the universe, an operating system that animates everything… Calculus taps into this order and expresses it.”
Published in 2019, Infinite Powers is a compelling reminder that humanity is capable of discovering simple solutions to overwhelming complexity in ways that can dramatically improve human wellbeing. As Strogatz states in the opening paragraph, without calculus, countless critical technologies would be impossible, from cell phones and ultrasounds to GPS. But, beyond enabling the physical trappings of modern life, calculus offers a much more profound legacy. As Strogatz writes, “the logic of calculus can use one real-world truth to generate another. Truth in, truth out. Start with something that is [already known to be] empirically true… apply the right logical manipulations, and out comes another empirical truth, possibly a new one, a fact about the universe that nobody knew before. In this way, calculus lets us peer into the future and predict the unknown. That’s what makes it such a powerful tool for science and technology.”
The value of calculus derives from how it takes a typical problem solving strategy to an extreme. Most problem solving involves at least some aspect of breaking complex problems down into simpler parts that are easier to deal with. But calculus breaks complexity down completely, cutting big problems into an infinite number of infinitely small parts. Isolated, infinitesimal bits of anything tend to be easier to understand than the larger problem they comprise. As Strogatz explains, calculus simply reimagines continuous natural phenomena, such as the perimeter of a circle or a hot cup of coffee cooling off or the passage of time as an infinite series of discrete bits of information comprising the whole, which mathematicians can analyze, and then they can add the results back together to make sense of the whole. As Strogatz explains, “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 is the uber problem solving language; a tool that leverages the power of infinity to interpret the analog nature of reality digitally, which makes reality much easier for us to understand and work with.
In our time, the concept of infinity as a bridge between the analog nature of reality and our digital representations of it should be pretty familiar. It’s the basis of computer technology, after all. But for a simpler example, just consider movies. Since the days of celluloid, movies have displayed a series of discrete images in sequence to create the illusion of continuous action. Generally speaking, the more discrete images—or frames per second, the better the illusion. Once you get above around 20 frames per second, our brains can no longer distinguish the discrete images, and the illusion of seamless flow is achieved. Get your popcorn. In theory, the more frames per second, the better the illusion, HD, Ultra HD, going, in theory, all the way up to infinitely many frames per second, in which case you’d end up with something that wasn’t even really a movie anymore, but something that truly was analog; a true two dimensional replication of reality. Or something like that. The point is, as Strogatz writes, “for many practical purposes, the discrete can stand in for the continuous, as long as we slice things thinly enough. In the ideal world of calculus… Anything that’s continuous can be sliced into infinitely many infinitesimal pieces. That’s the Infinity Principle. With infinity, the discrete and the continuous [merge into] one.”
Calculus is an incredible success story. It’s taken thousands of years for humanity to learn how to leverage the mathematical concept of infinity to better understand how the world works. Conventional histories of mathematics suggest that calculus was invented by Gottfried Leibniz in Germany and Isaac Newton in England in the 1600s. Newton certainly made tremendous progress in modernizing calculus to describe the universal laws of nature, but mathematicians have been toying with infinity far longer to decipher the world. The dynamics of linear phenomena like squares and pyramids and constant rates of change have been well understood for millennia, but anything non-linear, like growth and decay, metabolism, infection, waves, the acceleration of a falling rock, or even something as simple as the circumference of a circle, perplexed history’s smartest humans for a long, long time. The first person to even come close to calculating the correct ratio between a circle’s diameter and its circumference, for example, was the Greek mathematician Archimedes, who used the infinity principle to discover that ratio—what we now call pi—barely 2300 years ago, thousands of years after mathematicians first began puzzling over the problem.
Every high schooler today knows that to find the circumference of a circle, you just multiply its diameter by pi. But in Archimedes’ day, no one knew what pi was. Determining the perimeter of straight-edged shapes like triangles and squares was easy—you just added up the lengths of each side. And determining those lengths was pretty straight forward. But circles were a lot harder, and the geometrical relationship between a circle’s diameter and circumference was mysterious. Clearly, that ratio wasn’t a whole number, it didn’t even appear to be rational because you couldn’t express it as a fraction. Mathematicians knew that it was something close to 3, but every time they attempted to pinpoint it, it eluded their grasp, it remained a mystery. For centuries.
Around the year 250 BC, Archimedes managed to solve the mystery by engaging one of our species’ greatest strengths: creativity. He reimagined a circle as a straight-edged approximation of a circle instead, starting with a six sided hexagon. A hexagon is sort of circular, more so than a triangle or a square or a pentagon. Archimedes imagined placing a hexagon flush inside of a circle with a known radius, so all six of the hexagon’s points touched the circle’s circumference. He started with a hexagon because it was comprised of six equilateral triangles whose edges were the same length as the circle’s radius. So, the hexagon’s outer edges were the same length as the circle’s radius too—equilateral—and adding all of those six edges together gave him the hexagon’s perimeter. Starting with a hexagon made things easy: the perimeter of Archimedes’ hexagon simply equaled six times the circle’s radius. The next steps took more imagination and attention to detail. Because Archimedes could clearly see that the perimeter of the actual circle was greater than the perimeter of his hexagon—since the circle’s edges were curved while the hexagon’s edges were straight-edged short cuts—he determined that the circumference of the circle had to be greater than 6 times the radius, which is to say greater than 3 times the diameter. This established the lower bound of the value of pi for Archimedes—it proved that whatever the ratio was between the diameter and circumference of a circle, it had to be greater than 3.
To hone in on precisely how much greater, all Archimedes had to do was increase the number of edges of his straight-edged circle approximation and repeat these steps, proceeding from a six sided hexagon, to a 12 sided dodecagon and so on. Each time he did this, his approximation came closer and closer to a true circle, and so his estimation of the value of pi came closer and closer to pi’s true value. Moving beyond the easy equilateral triangles of his original hexagon made the math a bit harder, but he could still calculate the length of each new polygon’s edges all the same. As Strogatz puts it, “a hexagon is obviously a very crude caricature of a circle, but Archimedes was just getting started. Once he figured out what the hexagon was telling him, he shorted the [edges]… Then he kept doing that, over and over again up to 96 edges, [which revealed] that pi is [barely] greater than 3.14.” This was an exceptional feat not only of creativity, but also of patience. What Archimedes demonstrated was a fundamental principle of calculus: you could keep using known artificially discrete values to approximate closer and closer to unknown, naturally continuous values, in this case approximating a circle using a straight-edged polygon with more and more sides, in theory up to an infinite number of sides to get infinitely close to a true circle, revealing an infinitely close approximation of the true ratio between a circle’s circumference and diameter, the true measure of pi—a simple but nonetheless stunning historical example of how humanity learned to use the infinity principle to reveal the fundamental structure of reality. The only thing that stopped Archimedes from carrying on past a 96-sided polygon in practice was the sheer difficulty of manual calculation. Incidentally, as of 2019, Google supercomputers had crunched the value of pi all the way out to over 30 trillion decimal places, an inconceivably precise approximation pi. The rest of us remain content to use Archimedes estimation of 3.14, but the infinite potential of pi’s precision is part of what makes calculus so awesome and so deeply mysterious.
Mathematicians went on to leverage the principle of infinity to discover countless other truths about the structure of the universe. The kind of truths we’re talking about in the case of calculus, of course, might seem esoteric. But they’re not—they support the infrastructure that supports practically everything else we care about. For example, when Galileo discovered the mathematical relationship between the length of a pendulum and the period of its swing, that simple insight opened the door to a whole new world of mathematical revelations based in calculus that would transform virtually every aspect of modern life. As Strogatz writes, “In mathematics, pendulums stimulated the [further] development of calculus…[as] physicists and engineers learned to see the [entire] world in a pendulum’s swing…[for] the same mathematics applied wherever oscillations occurred. The worrisome moments of a bridge, the bouncing of a car with bad shock absorbers, the thumping of a washing machine with an unbalanced load, the fluttering of Venetian blinds in a breeze, the rumbling of the earth in the aftershock of an earthquake, the sixty-cycle hum of fluorescent lights—every field of science and technology today has its own version of to-and-fro motion, of rhythmic return. The pendulum is the grandaddy of them all. Its patterns are universal.”
The same calculus that interprets oscillations occurring throughout nature have enabled humanity to develop accurate clocks, quantum mechanics, non-invasive epilepsy treatments, power grids, glasses and contact lenses, a map of the universe, supercomputers, and satellite communications, just to name a few. But again, technology, no matter how impressive, no matter how life saving, doesn’t actually represent calculus’ greatest legacy. The consequences of discovering truth go much further than gadgetry. Calculus empowered us to discover that we are not the center of the universe, and that nature operates according to universal laws rather than according to the will of a God who believes that some people are better than others. Calculus catalyzed our rational investigation of everything, including ourselves, which led us to discover the self-evident truth of Universal Human Rights to life, liberty, and the pursuit of happiness. This is not a stretch—like most Enlightenment thinkers, Thomas Jefferson was a huge fan of Isaac Newton, and he modeled his argument for self-evident human equality in The Declaration of Independence on Newton’s argument for the self-evident mathematical laws of nature. Calculus did not create the modern world, but we would never have been able to create the modern world without it.
When I read Strogatz’s book, I can’t help but think of calculus as a model for our pursuit of modern prosperity in general. Although Strogatz offers quite a technical history of mathematics, his engaging style nonetheless evokes larger philosophical themes that inspire appreciation for how modern society has achieved so much progress. We have progressed from our ancestors’ nasty, brutish, and short lives not by insisting on utopian terms of engagement, but by working with what we have, and this includes the brains we have evolved to help us survive in hunter-gatherer tribes of no more than a few dozen families. Given that our brains evolved in this context, our capacity to transcend tribal nonsense might seem miraculous, but it’s based on a simple principle. Although we may be capable of imagining perfection—in ourselves, in our institutions, in our knowledge of how the world works—we progress only when we accept that we can never attain perfection. The United State’s is humanity’s most explicit political effort to enact this principle—America’s mission statement is not to perfect a union, but to form an ever more perfect union. The power of calculus comes from leveraging the very same idea. As Strogatz emphasizes, throughout human history, “we could only approach the state of perfection. 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. In fact, many of the greatest pioneers of the subject did precisely that and made great discoveries by doing so. Logical, no. Imaginative, yes. Successful, very.”
The best leaders in history have intuitively grasped the infinity principle at the heart of calculus, ever striving toward their ideals while accepting the practical impossibility of ever reaching them. The worst leaders in history have been too ignorant, too stubborn, or too evil to grasp this principle, and have ensured destruction by insisting on perfection. The history of calculus may seem irrelevant to most of our going concerns, but as Steven Strogatz shows, the spirit of calculus expresses one of the best ideas humanity has ever had: greatness is not to be found in the end, but in the effort. I can’t think of a better antidote to nihilism, and I can’t think of a more refreshing message in this moment.
I’m Brad Harris. So long.