*Infinite Powers* by Steven Strogatz

WAIT! COME BACK! The word “calculus” is enough to send many people running, and for good reason. Calculus is based on a paradoxical way of thinking. Its intimidating alchemy of symbols underwrites dauntingly complex concoctions of science and engineering. Because of its difficulty, calculus is often a gateway to elite colleges and professions, and so a form of cultural capital even for those who never use it directly.

If calculus has turned you off, left you behind, shoed you away, or beaten you down, Steven Strogatz — a professor of applied mathematics at Cornell University — wants to help. (If you and calculus are on friendly terms, you will find much to learn and enjoy here, but you are not the target audience.) Strogatz is a specialist in the mathematics of chaotic interacting systems, the topic of his first crossover book,

When modern calculus debuted in the late 17th century, only a handful of people could claim to understand it well. In the 18th century, scientists and philosophers came to see it as both a powerful tool for reasoning and a way of marking the bounds of rational inquiry: a rational scientific question was, by definition, one that could be addressed with calculus. In the 19th century, it became the foundation of science and engineering education. We now live in a scientific and technological age made by those who learned, with great effort, to see the world through calculus-tinted glasses.

So it is no wonder that people like Strogatz, highly trained to see calculus all around them, can conjure wonder and excitement at the wages of calculus in the world. With this book, he joins a long succession of scientific eminences aiming to carve a royal road to calculus appreciation that bypasses the rigors of calculus training, stretching from David Acheson (

Strogatz’s prose frequently verges on the mystical, abounding with “secrets” and “mysteries” that calculus “unlocks” or “reveals” and waxing philosophically about human nature and “the language of the universe.” Such passages tell us a lot about his enthusiasm for calculus and very little about calculus itself. A better guide to calculus — one that distinguishes Strogatz’s applied take on the subject from other contributions to this genre — comes when he sets aside the panegyrics and starts explaining with analogies and illustrations and thought experiments and examples how to adopt a calculus point of view.

Most often, taking such a perspective is less a matter of recognizing universal principles of nature and more of finding appropriate simplifications, approximations, and accommodations to the situation at hand. Instead of following inevitably from some cosmic connection between calculus and nature, the power of calculus in these situations comes from its users’ ability to ply it flexibly. Its methods work when skilled practitioners interpret them in context and adjust their view of the world to match their methodological needs.

Strogatz explains the concept of instantaneous speed, for instance, by examining Usain Bolt’s record-setting 100-meter sprint at the Beijing Olympics in 2008. He starts with a coarse measure of Bolt’s average speed, then proceeds through split times, smooth approximations, and precise radar measurements. A standard telling would show how increasingly refined subdivisions of time converge to approximate the sprinter’s true instantaneous speed, but Strogatz takes a different tack. The “meaningful trend” appears only when the applied mathematician steps back from the fine-grained irregularities of the sprinter’s stride and looks instead at a smoothed approximation that doesn’t “try to push the resolution of our measurements too far.”

The key to understanding calculus is what Strogatz calls the Infinity Principle. This is the paradoxical core of the method that vexed and thrilled millennia of thinkers. To analyze something complicated, the Infinity Principle says you should first break it down into an infinity of simpler parts and analyze those. Putting those infinitely many analyzed parts back together into an analyzed whole can be difficult, but it can be easier than analyzing the complicated whole directly.

As a figure of speech, squaring the circle means reconciling the irreconcilable. As a geometric problem, it was among the first challenges to fall to the Infinity Principle. (Much later, mathematicians proved that it could never be solved by finite means.) A circle may not seem like a complicated whole, but its very roundness confounds the rectilinear principles of measurement needed to find its (square) area. Bit by bit and with plenty of pictures, Strogatz shows how to find the area of a circle by cutting it into an infinity of wedges and putting them back together as a rectangle.

Even this seemingly straightforward ancient method, associated with Archimedes, harbors paradoxes as old as the Infinity Principle itself. The circle’s component wedges (rendered here as decidedly non-ancient slices of pizza) are all rounded at their exterior edge, and those curves do not go away when the wedges are rearranged. But take an

Strogatz then skips to the Scientific Revolution in early modern Europe, and to the period’s new mathematical descriptions of mechanical phenomena like pendular motion. Isaac Newton in England and Gottfried Wilhelm Leibniz on the Continent each invented a version of calculus before locking horns in a rancorous priority dispute. While both built on long mathematical traditions, the two are credited with inventing calculus because each, in his own way, demonstrated a crucial relationship between the two core parts of the Infinity Principle: differentiation (breaking something into its infinitesimal parts) and integration (putting them back together). Strogatz uses the dispute about who invented calculus to explain these concepts, as well as the fundamental theorem of calculus that establishes mathematically that these are not just a complementary tandem but really two sides of the same coin.

The story then continues toward the present (and some speculative futures involving AI, data, chaos, and other motifs) with a selection of topics, drawn most notably from the calculus of waves and of dynamical systems. Ideas about regular oscillation, for example, link theories of heat and vibration to medical imaging and the changing seasons. Strogatz lards his chronological narrative with vignettes illustrating key ideas through recent applications. If some seem contrived (such as counting calories from slices of bread or optimizing the dimensions of a carry-on bag), others are genuinely eye-opening.

Doctors treating HIV infections were initially reluctant to use harsh antiretroviral drugs during times when patients’ viral loads appeared stable, saving them for when the virus multiplied out of control. Strogatz explains how immunologists used calculus to model and test how the virus behaved during these stable periods, leading to a vital new understanding of the give-and-take between the virus and patients’ immune systems. The virus was not dormant, but instead, “like what happens if you turn on the faucet and open the drain at the same time,” constantly multiplying and being cleared. The response was the multi-drug cocktail therapy that has turned HIV from a seeming death sentence into a manageable chronic condition.

While

Like the 18th-century math historians who wrote monumental narratives to make sense of the mathematics of their own time, culminating in calculus’s new efflorescence, Strogatz’s aim is to use historical vignettes to show his readers how to see calculus in the present. For instance, his imaginative speculations about the thought processes and opinions of Archimedes invite readers to develop concepts and perspectives that arguably have more to do with computer modeling than with ancient Greek mathematical philosophy. The early-17th-century astronomer Johannes Kepler’s approach to data analysis only superficially resembles the tidy story Strogatz tells of it; but Strogatz’s story serves effectively to communicate key ideas about using models, approximations, and calculations to make sense of observations.

In other words, his stories teach us to see calculus in recognizable places — the stars, clocks, or microwave ovens — and challenge us to extend that vision to new domains — computer programs, DNA, MRI. This is one version of the power of calculus. It belongs to those like Strogatz who use calculus with confidence. If you are one of Strogatz’s imagined readers, then you are not one of those confident users. This book can show you how scientists and engineers experience the power of calculus, but it has less to say about

A broader account of the power of calculus must contend with other kinds of power in history: the power of educational systems to shape citizens and nations, of states and markets to allocate authority and resources, of imagination and conviction to challenge or define what is obvious, right, or natural. These kinds of power can explain how calculus became a gateway to other kinds of knowledge and action. They explain how calculus has sorted people as much as it has sorted the natural world, shaping opportunities for those who use it regularly or not at all. They help us grapple with the fact that calculus remains as distant and mystifying and elitist as ever to large segments of society, who by virtue of gender, class, race, location, luck, and circumstance are not enabled or expected to pursue the hard road to calculus mastery.

Yet, in a roundabout way,

If calculus has turned you off, left you behind, shoed you away, or beaten you down, Steven Strogatz — a professor of applied mathematics at Cornell University — wants to help. (If you and calculus are on friendly terms, you will find much to learn and enjoy here, but you are not the target audience.) Strogatz is a specialist in the mathematics of chaotic interacting systems, the topic of his first crossover book,

*Sync*(Hyperion, 2003). He followed with a mix of memoir and differential equations in*The Calculus of Friendship*(Princeton, 2009) and a*New York Times*series that grew into an exuberant mathematical primer,*The Joy of X*(Houghton Mifflin Harcourt, 2012). His latest,*Infinite Powers*, aims to explain the fundamental ideas of calculus, some of its history, and a few of its applications. Even those with no need or desire to*do*calculus, Strogatz contends, should be able to appreciate it.When modern calculus debuted in the late 17th century, only a handful of people could claim to understand it well. In the 18th century, scientists and philosophers came to see it as both a powerful tool for reasoning and a way of marking the bounds of rational inquiry: a rational scientific question was, by definition, one that could be addressed with calculus. In the 19th century, it became the foundation of science and engineering education. We now live in a scientific and technological age made by those who learned, with great effort, to see the world through calculus-tinted glasses.

So it is no wonder that people like Strogatz, highly trained to see calculus all around them, can conjure wonder and excitement at the wages of calculus in the world. With this book, he joins a long succession of scientific eminences aiming to carve a royal road to calculus appreciation that bypasses the rigors of calculus training, stretching from David Acheson (

*The Calculus Story*, Oxford, 2017) to Lancelot Hogben (*Mathematics for the Million*, Allen & Unwin, 1936) to Augustus De Morgan (*Elementary Illustrations of the Differential and Integral Calculus*, Baldwin and Cradock, 1842) and beyond. Many of Strogatz’s historical and conceptual set-pieces, pinned to famous names from Archimedes and Zeno to Galileo, Leibniz, and Newton, are mainstays of the genre. Like those before him, Strogatz cannot resist a substantial dose of numbers, symbols, and graphs, but uses them with a light touch and makes it easy for readers to reenter the narrative if they are derailed by an irruption of calculation.Strogatz’s prose frequently verges on the mystical, abounding with “secrets” and “mysteries” that calculus “unlocks” or “reveals” and waxing philosophically about human nature and “the language of the universe.” Such passages tell us a lot about his enthusiasm for calculus and very little about calculus itself. A better guide to calculus — one that distinguishes Strogatz’s applied take on the subject from other contributions to this genre — comes when he sets aside the panegyrics and starts explaining with analogies and illustrations and thought experiments and examples how to adopt a calculus point of view.

Most often, taking such a perspective is less a matter of recognizing universal principles of nature and more of finding appropriate simplifications, approximations, and accommodations to the situation at hand. Instead of following inevitably from some cosmic connection between calculus and nature, the power of calculus in these situations comes from its users’ ability to ply it flexibly. Its methods work when skilled practitioners interpret them in context and adjust their view of the world to match their methodological needs.

Strogatz explains the concept of instantaneous speed, for instance, by examining Usain Bolt’s record-setting 100-meter sprint at the Beijing Olympics in 2008. He starts with a coarse measure of Bolt’s average speed, then proceeds through split times, smooth approximations, and precise radar measurements. A standard telling would show how increasingly refined subdivisions of time converge to approximate the sprinter’s true instantaneous speed, but Strogatz takes a different tack. The “meaningful trend” appears only when the applied mathematician steps back from the fine-grained irregularities of the sprinter’s stride and looks instead at a smoothed approximation that doesn’t “try to push the resolution of our measurements too far.”

The key to understanding calculus is what Strogatz calls the Infinity Principle. This is the paradoxical core of the method that vexed and thrilled millennia of thinkers. To analyze something complicated, the Infinity Principle says you should first break it down into an infinity of simpler parts and analyze those. Putting those infinitely many analyzed parts back together into an analyzed whole can be difficult, but it can be easier than analyzing the complicated whole directly.

As a figure of speech, squaring the circle means reconciling the irreconcilable. As a geometric problem, it was among the first challenges to fall to the Infinity Principle. (Much later, mathematicians proved that it could never be solved by finite means.) A circle may not seem like a complicated whole, but its very roundness confounds the rectilinear principles of measurement needed to find its (square) area. Bit by bit and with plenty of pictures, Strogatz shows how to find the area of a circle by cutting it into an infinity of wedges and putting them back together as a rectangle.

Even this seemingly straightforward ancient method, associated with Archimedes, harbors paradoxes as old as the Infinity Principle itself. The circle’s component wedges (rendered here as decidedly non-ancient slices of pizza) are all rounded at their exterior edge, and those curves do not go away when the wedges are rearranged. But take an

*infinity*of curved wedge edges and you get the straight edge of a rectangle. Each of those infinitesimally small wedges, in turn, has no area, but adding up that infinity of zero areas gives a definite area for the circle. Zeno’s paradoxes showed the absurdity of dividing space infinitely, which implied the impossibility of motion because there would always be infinitely many points between you and where you want to go. And yet, we move, and the Infinity Principle worked, paradoxes notwithstanding.Strogatz then skips to the Scientific Revolution in early modern Europe, and to the period’s new mathematical descriptions of mechanical phenomena like pendular motion. Isaac Newton in England and Gottfried Wilhelm Leibniz on the Continent each invented a version of calculus before locking horns in a rancorous priority dispute. While both built on long mathematical traditions, the two are credited with inventing calculus because each, in his own way, demonstrated a crucial relationship between the two core parts of the Infinity Principle: differentiation (breaking something into its infinitesimal parts) and integration (putting them back together). Strogatz uses the dispute about who invented calculus to explain these concepts, as well as the fundamental theorem of calculus that establishes mathematically that these are not just a complementary tandem but really two sides of the same coin.

The story then continues toward the present (and some speculative futures involving AI, data, chaos, and other motifs) with a selection of topics, drawn most notably from the calculus of waves and of dynamical systems. Ideas about regular oscillation, for example, link theories of heat and vibration to medical imaging and the changing seasons. Strogatz lards his chronological narrative with vignettes illustrating key ideas through recent applications. If some seem contrived (such as counting calories from slices of bread or optimizing the dimensions of a carry-on bag), others are genuinely eye-opening.

Doctors treating HIV infections were initially reluctant to use harsh antiretroviral drugs during times when patients’ viral loads appeared stable, saving them for when the virus multiplied out of control. Strogatz explains how immunologists used calculus to model and test how the virus behaved during these stable periods, leading to a vital new understanding of the give-and-take between the virus and patients’ immune systems. The virus was not dormant, but instead, “like what happens if you turn on the faucet and open the drain at the same time,” constantly multiplying and being cleared. The response was the multi-drug cocktail therapy that has turned HIV from a seeming death sentence into a manageable chronic condition.

While

*Infinite Powers*may be read as a guide to the history of calculus, it ought not to be. Strogatz is an expert mathematician but an amateur historian, and this combination shows in his patchy choice of subjects and occasionally dubious attributions. Some towering figures in calculus history are missing, like Leonhard Euler in the 18th century and Augustin-Louis Cauchy in the 19th. In other cases, he speculates ahistorically about his characters’ views while omitting the messy connections and contingencies that preoccupy historians.Like the 18th-century math historians who wrote monumental narratives to make sense of the mathematics of their own time, culminating in calculus’s new efflorescence, Strogatz’s aim is to use historical vignettes to show his readers how to see calculus in the present. For instance, his imaginative speculations about the thought processes and opinions of Archimedes invite readers to develop concepts and perspectives that arguably have more to do with computer modeling than with ancient Greek mathematical philosophy. The early-17th-century astronomer Johannes Kepler’s approach to data analysis only superficially resembles the tidy story Strogatz tells of it; but Strogatz’s story serves effectively to communicate key ideas about using models, approximations, and calculations to make sense of observations.

In other words, his stories teach us to see calculus in recognizable places — the stars, clocks, or microwave ovens — and challenge us to extend that vision to new domains — computer programs, DNA, MRI. This is one version of the power of calculus. It belongs to those like Strogatz who use calculus with confidence. If you are one of Strogatz’s imagined readers, then you are not one of those confident users. This book can show you how scientists and engineers experience the power of calculus, but it has less to say about

*your*experience of that power.A broader account of the power of calculus must contend with other kinds of power in history: the power of educational systems to shape citizens and nations, of states and markets to allocate authority and resources, of imagination and conviction to challenge or define what is obvious, right, or natural. These kinds of power can explain how calculus became a gateway to other kinds of knowledge and action. They explain how calculus has sorted people as much as it has sorted the natural world, shaping opportunities for those who use it regularly or not at all. They help us grapple with the fact that calculus remains as distant and mystifying and elitist as ever to large segments of society, who by virtue of gender, class, race, location, luck, and circumstance are not enabled or expected to pursue the hard road to calculus mastery.

Yet, in a roundabout way,

*Infinite Powers*helps us understand this side of the power of calculus, too. Set aside the philosophical bluster and the march of insights and there remains in Strogatz’s book a telling trail of confusion, contradiction, and struggle — in the competing formulations of Descartes and Fermat, for example, or the skepticism that met proposals for CT scanning. We can see the power of calculus in the panoply of domains where its principles have displaced, supplemented, or enabled theories and applications, and we can understand that this power has been hard-won. Part of its success has been how its skilled users have learned to regard it as ordinary, natural, and fundamental. Overlooking the systemic social roots and consequences of the power of calculus is part and parcel of its success, and*Infinite Powers*is a masterful lesson in how that is done.¤

*Michael J. Barany teaches history of science at the University of Edinburgh and writes on the history and culture of modern mathematics.*LARB Contributor

**Michael J. Barany teaches history of science at the University of Edinburgh. His essays on the history and culture of modern mathematics can be found at http://mbarany.com.**

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