Present Value: Steven Strogatz on the importance of calculus and the power of mathematical lies
Contributing Authors: Michael Brady and Jack Moriarty
Present Value, an independent editorial project produced and hosted by Johnson students, had the pleasure of interviewing Steven Strogatz, Jacob Gould Schurman Professor of Applied Mathematics and Stephen H. Weiss Presidential Fellow at Cornell University, about his forthcoming book, Infinite Powers: How Calculus Reveals the Secrets of the Universe, scheduled to be released in April 2019.
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The ancient origins of calculus
In his upcoming book, Infinite Powers: How Calculus Reveals the Secrets of the Universe, Professor Steven Strogatz takes readers through the multi-millennial history of calculus while instilling a reverence for the crucial role calculus played in shaping our modern world. The goal of the book is to help people, whether they have taken calculus or not, “understand what [calculus’s] big ideas are, where it came from, and how it changed the world.” In the podcast, Strogatz points out that although about a million students take a calculus course every year, courses are typically overloaded with so many techniques and little theorems that the human side of the story is often left out. His new book tells the human side of story: who invented calculus, why did they invent it, what were their struggles, and why does it matter?
Chapter two of Infinite Powers, “The Man Who Harnessed Infinity,” is about Archimedes, the ancient Greek polymath. With his nearly limitless (calculus joke!) brilliance, Archimedes harnessed infinity to solve a problem that had eluded great minds for centuries: how to find the area of a circle. He rigorously approximated the area of a circle, and more importantly approximated π, by using an infinite approximation method similar to methods used today. Strogatz explains in the podcast that, amazingly, “Archimedes has got all the ideas of the half of calculus that we call integral calculus,” even though he was severely limited by Greek numerals (similar to the familiar Roman numerals) and didn’t have algebra! Archimedes “paved the way for integral calculus,” which would be developed nearly 2,000 years later by Newton and Leibniz in the 17th century. Of course, Infinite Powers does include the stories of Newton and Leibniz, two “fine gentleman” as Strogatz jokes, but the book also includes a wealth of modern stories about landmark developments in which calculus played a significant role such as CT scans, gravitational waves, and the revolution in HIV treatment.
Breakthrough discoveries in HIV treatment
The role calculus played in revolutionizing HIV treatment is “one of the most compelling examples of where calculus has made a difference in the world … it really did save countless lives.” Strogatz takes listeners and readers back to the early 1990’s when the HIV / AIDS epidemic was ravaging the U.S. and the world; hundreds of thousands of people were dying from HIV / AIDS every year globally. Before patients developed full-blown AIDS, they typically experienced an asymptomatic period which could last upwards of 10 years. Patients in this asymptomatic stage showed a fairly stable amount of HIV in the blood with no apparent HIV-related symptoms other than a low T-cell count. The lack of symptoms and stable viral load led researchers to think that HIV was dormant during this multiyear period. That changed when HIV / AIDS researcher David Ho and mathematical biologist Alan Perelson teamed up and figured out what was really happening.
Strogatz explains the story in fascinating detail in his book and in the podcast. But in short, by using calculus (differential equations to be specific) to model the effects of a new HIV treatment called a protease inhibitor, Ho and Perelson discovered that HIV was absolutely not lying dormant during the asymptomatic stage. Instead, HIV-infected cells released about a billion new virus particles every day, and concurrently, the immune system cleared about a billion particles every day. They discovered that HIV was in an “all out raging war with the immune system,” Strogatz explains. It was a stalemate which the immune system would eventually lose.
These breakthrough findings soon led Ho to champion triple-combination therapy, or “three-drug cocktail” therapy, which became the standard of care and has enabled HIV-positive patients with access to treatment to live long lives and avoid full-blown AIDS. Perelson’s math showed that triple-combination therapy is effective at staving off HIV because, although the virus mutates incredibly quickly, the probability of it mutating to become immune to all three drugs at the same time is extremely low. Strogatz closes this section of the book by giving readers a deep appreciation for the profound impact that Ho and Perelson’s work had on the world. “It gave hope where almost none had existed before.”
Pablo Picasso and the power of mathematical lies
Midway through Infinite Powers, Strogatz quotes Pablo Picasso: “Art is a lie that makes us realize truth.” In the Present Value episode, Strogatz connects Picasso’s quote to mathematics by arguing that a large part of the spirit of math is that “you can arrive at beautiful truths by making approximations that amount to lies.” A straightforward example of a lie in art is a pointillist painting; it may appear to be a beautiful scene of a park from afar, but close examination reveals it to be merely an array of dots.
Strogatz illustrates in the podcast that, in math, there are lies everywhere. One of math’s most fundamental concepts, a straight line with no width, is indeed a lie. Though straight lines are easily imagined in our heads, “There are no straight lines in nature.” Another lie, critical for real-world physical applications of calculus, is that you can divide time into infinitely small units as if there is no smallest unit of time; that, too, is a lie (google “Planck time” or “Planck units” for more on this).
“Math is full of stuff like this, where we imagine a world that is more perfect than our world, where circles are perfectly round and lines are perfectly straight,” Strogatz explains in the podcast. “These are lies that help reveal the truth.”
Six degrees of Kevin Bacon: it’s a small world
Strogatz is perhaps most renowned for co-authoring a pioneering paper with Duncan Watts, “Collective dynamics of ‘small-world’ networks,” published in Nature in 1998. They were inspired by the common “it’s a small world” phenomenon strangers experience because they often have a friend, or a friend of a friend, in common. In their paper, they model and characterize the types of networks in which people are only a few handshakes away from one another. These ideas were in the zeitgeist at the time: The movie Six Degrees of Separation had come out in 1993, and many people were playing the popular parlor game “Six Degrees of Kevin Bacon.” Their model, now called the Watts-Strogatz model, had implications that went far beyond social networks and Footloose.
In the Present Value episode, Strogatz explains that Watts’s vision was that “this seemingly ridiculous question about actors in the Kevin Bacon game is the question at the heart of all of modern science, whether it’s economics, or finance, or neuroscience, or epidemiology. It’s all about big networks and the things that propagate on them.” Their model of the “small-world” phenomenon, therefore, has direct applications to how neurons are connected, cascading failures and contagion in banking systems, and the spread of epidemics like HIV.
Strogatz is keen to point out that their model is a lie. “We made a model of what was going on in ‘small-worlds’ that was definitely a lie. It wasn’t a realistic model of social networks, or of epidemics spreading, or of banking failures.” Twenty years later, Watts’s original vision has been validated. Their lie and their model proved to have broad implications and, with over 38,000 citations, is one of the most cited papers of all time.
To Infinite Powers, and beyond
Infinite Powers is a captivating tour through the broad contributions calculus has made to humanity. What will be perhaps most striking to readers and to listeners (it certainly was to our team at Present Value), is how Strogatz makes complex mathematics easily accessible to nonexperts. By integrating together a broad range of stories—Archimedes, Usain Bolt, HIV treatment, gravitational waves, and much more—Strogatz empowers readers to understand why calculus is “one of the greatest ideas of all time” and how it has changed the world.
For more, check out the full-length Present Value podcast, Steven Strogatz — Calculus and the Power of Mathematical Lies. Listen, subscribe, and share!
About Steven Strogatz
Steven Strogatz earned a bachelor’s degree in mathematics from Princeton, went on to study as a Marshall Scholar at Trinity College, Cambridge, and earned a PhD in applied mathematics from Harvard University, where he published work on the dynamics of the human sleep-wake cycle. His research focuses on dynamical systems applied to physics, biology, and social science and he has worked on a variety of problems, including the geometry of supercoiled DNA and the collective behavior of biological oscillators, such as swarms of synchronously flashing fireflies. He is well known for his 1998 paper with Duncan Watts, “Collective Dynamics of Small-World Networks,” and for his popular books, New York Times columns, and frequent appearances on “Science Friday” and “Radiolab.”