Infinite Powers: How Calculus Reveals the Secrets of the Universe

When I was in college I majored in mathematics, and calculus wasn’t as big a challenge to me as it is for others (statistics, however, is another matter entirely). Calculus was the backbone of my major and it was foundational when it came to most of the other courses I took, and this included advanced algebra and more abstract mathematics. Now, that said, in many cases I understood the “how” but not the “why”. In a few cases, I saw some parallels with algebra and how calculus could short-cut certain processes (sort of) to arrive at the same result, but even so, the subject remained a black art even though it was clear [to me] what to do.

What this book does is explain some of the why. It’s not a dry, dull read that throws theorems and formulas at you with brief explanations. Instead, there are real-world examples that show why calculus is the means to the end. Instead of gagging you with straight theory and practice like a class textbook, it brings calculus down to a level that makes it much more interesting and, in a tangible way, fun. Now, it does not offer comprehensive coverage, of course; my calculus textbook from college is a full 1,000 pages and covers a lot of more obscure stuff that this book doesn’t mention, but that’s not this book’s intention. The idea here is to give a more high-level coverage with application.

I sure wish I had this book when I was struggling with the subject; knowing the “why” makes the “how” a lot easier to work through.

I’m still working my way through this book, but so far it has been very enjoyable and thought provoking. What I have read so far has made the subject of calculus a lot more interesting, even though I already have a handle on the mechanics of it. If you have struggled with calculus, this book is a way to build more understanding and appreciation. If you’re more curious and just want to know what it’s about, it’s a good starting point.

I need to psyche myself up to do some math for work. And I have a math sherpa and I arranged to meet him so he can take me through the paper I must tackle. But I’m old and only really remember my high school math well, so there is a genuine task at hand here.

So I duck and dive between the paper and my notes from my MSc thesis from at least fifteen years ago and I work out the answer to lesser problems and I write out my questions for my sherpa and I also need to be thinking math the whole time; I need to be in a mood, basically.

That’s the task.

So I did the sensible thing and went on a bit of a binge and bought a whole bunch of popular math books in one go to read in the tube. “Infinite Powers” I read first, because it looked like it would not challenge me at all and it gets good writeups.

It’s bloody awesome!

It’s more than an anthology of results and it’s more than a series of mini-portraits of mathematicians, it’s almost got a plot. Surprisingly often, even the obligatory corny applications of the math are (somewhat) related to what the author’s talking about.

Huge caveat: I knew both the math and even many of the stories upfront, so perhaps it’s not very well explained. I have no way of knowing. But I bet you it is. Perhaps not well enough that you could hope to learn calculus from here, Jordan Ellenberg’s praise on the back cover notwithstanding. (For that I can refer you to “Quick Calculus” by twin gods Kleppner and Ramsey.) But probably well enough to be a companion to anybody taking calculus for the first time.

Steven Strogatz had me from “hello,” of course, because he starts with the Greeks, on whom he lavishes immense praise. He could have left it there and I’d still be basking in the warm glow of my ancestors’ work. Needless to say, it does not stop there, he takes you from them to Fermat and Descartes, before introducing you to Newton and Leibniz, a couple words on Fourier and from him straight to Einstein, taking special care to erase all traces of evil men like the unspeakable inventor of delta-epsilon proofs. You won’t find the C-word here.

So there’s a massive hole in the nineteenth century, somewhere, but I’m sure you can buy another book to find out about that. Here you’ll discover a decent definition of e, an intuitive explanation of general relativity, the common cause of death of Leibniz and Newton, a fun game to play with your microwave oven, the first and second derivative of the sine wave, the dimension of the three-body problem, a strong defense of infinitesimals, WHAT’S NOT TO LIKE?

Enough from me, I’ll now go buy some extra copies for a few boys and girls I know. If one of them likes it, my job is done.

Oh, sorry, one more thing. About the plot: it’s a history of how mathematicians throughout time have sliced hard problems into infinite infinitely-thin slices where the problem has a clear answer and then dealt with infinity to sum up the solutions to the easy problem in order to come up with an answer to the hard problem.

Whenever you do that, you’re doing calculus, you’re putting together the answer granule by granule.

Steven Strogatz has written an outstanding book that presents an overview of Calculus that students, and even professors, often fail to learn. His writing style is both informal and yet precise; the book is very reader friendly. The content covers 2200 plus years of the history of calculus from Archimedes to the present, and beyond - speculating about the future. But this is not a dry history. Strogatz, I believe uniquely connects stories from the past to today's applications. An example would be a section entitled "Archimedes Today: From Computer Animation to Facial Surgery." Here he connects Archimedes polygonal approximation to pi to the construction of the character of Shrek.
I teach Honors Calculus and Differential Equations at Pasadena City College. Most of my students are not math majors; they are typically engineering majors. They tend to be practical people. Learning calculus for them tends to be learning the "nuts and bolts" of a 1000 pages of textbook. Mathematics is at best a tool, and at worst a "speed bump" on their way to an engineering degree. They do not see the large view of calculus or how is applies to so much of our world. "Infinite Powers" remedies this situation beautifully. I will try to get this book into the hands of as many of my students as possible. But it's not only for students; professors would greatly benefit from this amazing book. It enriches our teaching.
There are many more examples of the wealth of material in the book. For example, Strogatz explains the distinction between "local" and "global" operations with great clarity. We see why integration is hard. He also gives the best explanation of the "birth" of partial differential equations I've ever read. The riches in this volume fill every page. In addition the combination of Endnotes and Bibliography constitute a treasure of wisely chosen pointers to further study.
Finally , Strogatz considers the vibrant future of Calculus: applications to biology, increasing work in nonlinear dynamics, and the ramifications of mathematical chaos. The last several pages of the book are written in an eloquent and almost lyrical tone - his passion for his subject and explaining it are evident.

A special challenge for popular mathematics writing, or for that matter any popular science writing is smoothing out the complexity of the subject without removing all meaning and content. A friend with a Ph.D./M.D. and a graduate from MIT once said that you didn't really understand any subject if you couldn't state a clear explanation understandable to a somewhat intelligent 12 year old, and Steven Strogatz has done that for the calculus.

He boils this complex area of study down to a single principle (page xvi in the "advance readers copy"); The Infinity Principle, which states: "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 add the results back together to make sense of the original whole."

He also provides examples from recent history and the most current topics in science, as well as those of Leibniz and Newton in the 17th century. These include applications in Radar, Lasers, String Theory, DNA and Artificial Intelligence.

This is a thoroughly enjoyable read where you will actually learn some real content.

Cornell applied math professor Steven Strogatz is awed by the role of calculus in the world. As he says in his introduction “there seems to be something like a code to the universe, an operating system that animates everything from moment to moment and place to place. Calculus taps into this order and expresses it.” In Infinite Powers Strogatz goes back in history all the way to 250 BCE and explores what calculus is, the importance of the concept of infinity, and the role of differential equations in the world in a way he hopes will be enjoyable and intelligible to “a very thoughtful, curious, knowledgeable person with little background in advanced math.”
Does he succeed? Well, yes, but that person needs to be willing to work at it, and I think he often assumes the reader will immediately grasp some assertion that probably needs more explanation. For example, Strogatz tells the fascinating tale of how Archimedes developed an estimate for pi by taking smaller and smaller steps around the circumference of a circle. He informs the reader that it got harder and harder to do the calculations as the steps got smaller because Archimedes had to keep invoking the Pythagorean theorem and calculate square roots by hand. I really think it would help many readers to spell out the relevance of the Pythagorean theorem here.
Once the reader figures out the math reasoning or gives up and takes it on faith, though, for each topic Strogatz brings us to the modern day and math in the world. In the case of Archimedes’ work above, for example, he tells us how Archimedes’ work is used by surgeons doing reconstructive facial surgery. Taking a historical perspective, he follows the various milestones towards calculus and for each relates it to the real world, sometimes in surprising and fun ways, such as when he analyzes Usain Bolt running a 100-meter dash in the chapter on derivatives.
Strogatz makes a convincing demonstration of how calculus helps us understand and enjoy our universe, from animated films to DNA and gravity waves. Math geeks will love this book. That “very thoughtful curious, knowledgeable person” may find it challenging but rewarding.

I suppose a book on calculus was a bit ambitious for an English major (at least for this one!), but I was lured in by the encomia suggesting its readability. The anecdotes about historical figures were, indeed, engrossing, but somewhere around page 100, my ancient exposure to mathematics (I took Trigonometry from Archimedes.) let me down. My math knowledge, apparently, was stored in the brain cells I have already buried in the last decade or two (or 6!). I do not want to take away from any estimate of the quality of the book, but not just anyone will be able to enjoy it. However, I persevered until the end.

Calculus has an image problem, now more than ever. It was never a crowd-pleaser to begin with, but the old student refrain “when will I ever use this?” has lately been gaining considerable traction among senior academics as well. An October 2019 Freakonomics episode joined a growing chorus that would be happy to see dusty old calculus yield space in the curriculum for more “21st-century skills” such as “data fluency.” Calculus is supposedly antiquated and irrelevant: when Freakonomics polled their listeners, they found that only 2% used calculus in their daily lives.

Infinite Powers can be read as a rebuttal of this wave of ungratefulness. It comes out swinging in its very first sentence: “Without calculus, we wouldn’t have cell phones, … ultrasound for expectant mothers, GPS for lost travelers,” and a host of other scientific breakthroughs and ubiquitous technological conveniences. The book is bursting with such triumphs throughout, as it weaves history and present together into a unified story of the calculus that is both timeless and topical. Let us hope that history will not recall this book as a eulogy for a doomed art, but as a tribute to its ongoing vitality.

It is a story told for general readers with no mathematical background, who prefer analogies to equations. But the book aims higher than palatable edutainment and vivid anecdotes: it aspires to convey the essence of “the world according to calculus.” Calculus is a language, a way of thinking — a state of mind, one might even be tempted to say. Or at least mathematicians view it that way. Readers who want an accessible path to this almost quasi-spiritual vision of calculus with a minimum of technicalities will find this book their best bet.

The episode that makes the most compelling utilitarian case for the value of a calculus education is perhaps that describing how David Ho — “a former physics major at Caltech and so, presumably, someone comfortable with calculus” — revolutionised HIV treatment with such success that he was named Time magazine’s Man of the Year in 1996. This treatment went diametrically against physicians’ intuitions but drastically improved life expectancy. Anyone with hazy memories of studying exponential functions in school will be amazed at how much its discovery reads like a word problem from a calculus textbook. Many a student has been asked to calculate coefficients, constants, and intercepts of population formulas, but few have come away with a sense that this could be a tool for discovery and exploring the unknown. Yet David Ho used much the same techniques to understand the virus population in a person infected with HIV. The numbers were a surprise: the calculus formulas suggested that even in patients in early stages of the disease, outwardly stability of symptoms belied a tenuous balance of the virus reproducing in huge numbers, while also being defeated by the body in equally large quantities. Understanding this life cycle of the virus made it possible to coordinate drugs that would attack the virus without giving it a window to mutate and develop drug resistance. It is hard to imagine what more calculus can do to convince slash-happy curriculum reformers of its value than help the terminally ill.

But Strogatz is not an agitator making a curricular argument. The range of modern examples he draws upon is more aimed at showing the unity of calculus than its utility. One section goes “from [Galileo’s] swinging chandelier to the Global Positioning System” in a matter of pages, since they are linked by the same mathematical idea. Such organic connections between past and present are a recurrent theme in the book. One of Strogatz’ most evocative metaphors is that “calculus was the Cambrian explosion for mathematics”: just as biodiversity skyrocketed quickly when multicellular life became complex enough to enable specialised speciation, so the calculus was a seminal core idea that soon proved itself remarkably adaptable to a wide range of circumstances. Yet despite the multitude of its niche successes, all the evolutionary offshoot branches of the calculus still have a prominent and unmistakable “calculusness” at their heart.

As a historian of mathematics I have a few quibbles. Strikingly, I found that some historical inaccuracies would only make Strogatz’ overall story all the more compelling if corrected. In my opinion, Strogatz overestimates the work of Galileo. It is not true that Newton “knew [the law of inertia] from studying Galileo” (231), because Galileo did not have the law of inertia. Galileo’s reasoning is very confused on this point, arguably precisely because he lacked the calculus point of view. Likewise, Newton could not get a usable value of the gravitational acceleration g from Galileo and “his inclined-plane experiments” (233). Galileo’s measurement of g is off by a factor of 2 and is quite useless for serious physics. Newton instead drew upon Huygens’ work on pendulums for an excellent value of g. A calculus treatment of pendulum motion readily shows that the period of a pendulum making small oscillations is related to g in a very simple way that enables g to be measured with much greater precision than the crude direct approach of trying to time falling or rolling objects. This equation for pendulum motion is still a staple of calculus courses today. It was unknown to Galileo though, precisely because he did not know calculus, just as Strogatz correctly points out elsewhere (72). So these historical inaccuracies are if anything a missed opportunity to highlight precisely the “infinite powers” of the calculus that the book is about.

Similarly, Strogatz undermines one of his core messages in a discussion of Descartes. It is true that Descartes “had an ego as big as his genius,” but as evidence for this Strogatz adds: “of the Greek approach to geometry, which all other mathematicians had revered for two thousand years, [Descartes] wrote dismissively: ‘What the ancients have taught us is so scanty and for the most part so lacking in credibility that I may not hope for any kind of approach toward truth except by rejecting all the paths which they have followed.’” (99) In fact, this quotation has nothing to do with geometry. It is from Descartes’ treatise on the passions and refers only to Greek philosophical writing on emotions. Descartes would not say such a thing about Greek geometers. For all his arrogance, even Descartes revered Archimedes, just as every other mathematician has before or since. Strogatz himself writes with great empathy and warmth of the kinship he feels with Archimedes. This astonishing power of mathematics to make brethren of us all and unite minds across centuries and oceans is a thread beautifully highlighted throughout the book. It is a real missed opportunity, therefore, that Strogatz erroneously characterises Descartes as an exception to the rule, when the rather more heartwarming truth is that the force that unites all mathematicians in their love of Archimedes is so strong that even Descartes could not bring himself to be a contrarian on this point.

Infinite Powers is popularisation, but with a message. It is rich in enjoyable set pieces, such as measuring the speed of light by melting cheese in your microwave oven, as well as historical and biographical stories highlighting the humanity and struggle of mathematics. But it is its expansive vision of calculus that is Infinite Powers’s real selling point. Written primarily to convey this big picture to humanists, it will also serve as an emphatic counterpoint to the bleak view of the subject painted by the technocrats who have turned on calculus in recent years.

First of all, let me say that I am not a deep student of math; getting out of trig with a B was one of my great accomplishments, and I was glad to stop there. Then I ended up with two daughters who kept telling me that if I didn't understand calculus, I didn't understand anything; I made a couple of mild attempts to learn it and decided that I was OK with not understanding anything.

This book is a wonderful eye-opener as well as a smooth and interesting read. It is a book _about_ calculus and not a calculus textbook. It helps understanding of the principles with little pain and a lot of enlightenment without a stern requirement to "do these ten problems and hand them in tomorrow." I understood it! At its biggest, calculus is about turning possible infinity into a usable format .

But--once a humanities person, always a humanities person; the most fascinating thing about the book was the history of the steps that led to this point in mathematics. As someone who was not totally interested in math, I had been left with the general impression that the calculus sprang full-formed from the brain of Newton (or Leibnitz, who cared?) somewhat like Athena from the mind of Zeus. But this friendly book takes us far back to the basics. Early on, problems were recognized that could be solved only by approximation. And the urge to be able to achieve greater and greater precision led to more and more new methods and attempts over time. Calculus was a great breakthrough, but it was also the result of a centuries-long evolutionary process fueled by a desire for mathematics as a more nearly perfect instrument.

To me, this all sounds excessively obsessive; on the other hand, back then you had nothing to do with your spare time like watch TV or stay in touch with the world through your mobile phone, or go to the moon, etc. , so you might as well work out new mathematical concepts. But--if it were not for calculus, we wouldn't have any of those fascinating things to occupy our minds.

I liked this book because it opened up a new area for me. From my daughters' point of view I still don't understand anything, but I don't understand anything _better_ than I did before.

His global, historical treatment of the uses of infinite processes is interesting and helpful, but in the crucial section of his book, chock-a-block with exasperatingly prolix obfuscation, Strogatz tries to substitute enthusiasm for clarity. He wants to make the Fundamental Theorem of Calculus available to the layman, apparently. He puffs up his narrative with examples from the sports pages, sometimes gratuitously and inappropriately—anything but—God forbid!—present a simple algebraic formula. It's condescension disguised as good will. (Also--he casually and without comment dismisses any notion of infinity as a completed quantity, as if Georg Cantor and transfinite math did not exist!)

I ordered the book from a local book store before I saw it on Amazon. I bought the book based on the reputation of the author even though I have no books written by Steven Strogatz. Although I was looking forward to learning more about the history of calculus and its applications, the historical examples in the book were quite limited with little new information presented. In addition, I was also quite disappointed in the book’s technical content which was minimal and more fitting for someone in high school.