Calculus takes the most vexing and mysterious things imaginable — motion, change, the flow of time — and boils them down to ironclad rules of computation….It inspired Tolstoy, Borges, and David Foster Wallace. It shaped visions of history, ethics, and the powers of the human mind. Calculus is the canonical example of turning the impossible into the routine, and its ideas have nourished not only science, but economics, philosophy, and even literature, too.
That’s the case I wanted to make in this book…. an exploration of the human side of calculus, what it has meant over the years to everyone from scientists to poets to philosophers to dogs. If calculus is going to remain a fixture of math education—even for those not pursuing STEM careers—then we need to bring out its humanity, to find a version of calculus that speaks to everyone.
Ben Orlin, Ars Technica interview with Jennifer Ouellette
First, the important stuff: I’M IN A MATH BOOK! And a calculus book no less. Ok, it isn’t a calculus textbook – it’s a history/philosophy/literature/science/mythology/puzzle book that shows how concepts of calculus exist in all those disciplines – and it’s just my name, but still, if you flip back to page 319, the last page, I’m listed as one of the people who “gave excellent feedback at various stages”. I considered myself honored to receive an early draft of some of the chapters, and while I’m not so sure my feedback was excellent, I’m thrilled to be right there in print.
And now that It’s All About Me time is over, what about the book?
Last year, Ben Orlin’s first book, Math With Bad Drawings (also the title of his ongoing math-humor blog, completely charmed me despite the persistent mathphobia I periodically try to overcome. And now, a year later, his second book takes on the same challenge but focuses on calculus. After three years and five moocs (two of which I actually passed) trying to learn calculus, I’ve felt pretty traumatized by derivatives and, especially, integrals. Could Change is the Only Constant: The Wisdom of Calculus in a Madcap World charm even me?
Spoiler alert: Yes!
I want to be clear: this object in your hands won’t “teach you calculus .” It’s not an orderly textbook, but an eclectic and humbly illustrated volume of folklore, written in non technical language for a casual reader. That reader may be a total stranger to calculus, or an intimate friend; I’m hopeful that the stories will bring a little mirth and insight either way.
While this book won’t teach you calculus, it will teach you all sorts of other interesting things about interesting people, events, and ideas from literature, history, and, yes, math. Because the chapters are short, self-contained and cover individual topics, it’s possible to skip over something that seems confusing and move on to something completely different a few pages later. I’ll be honest: I’m not sure how this book would strike someone with no experience whatsoever in calculus. I’d love to find out; any volunteers?
Writers know that all writing is rewriting, and this book underwent extensive editing. Ben helpfully wrote about the process, from his recognition that “my book was not working” to his use of a mathematical model to fix it. I read a pre-revision draft, so I saw the murdered darlings. I am quite sad that a section on Adrienne Rich ended up minimized to a single epigraph (“The moment of change is the only poem”) but I have to admit, the rewrite was an improvement, and far closer in style to his first book.
Also similar to his first book is the physical object: clever dust jacket and thematic echo on the hardcover and endpapers, great page design allowing lots of room for notes and doodles, heavy paper preventing bleed-through of colors (though, unlike the first book, the only color used throughout is red). And just so you don’t think I’m some groupie who’d applaud anything Ben did, another reader, book artist Paula Beardell Krieg, also gave it high praise.
Some of my favorite chapters:
Chapter 3: The Fleeting Joys Of Buttered Toast
One day, cradling a fresh mug of tea and munching a piece of wheat toast (ugh – I thought I grabbed white), I plopped onto a sofa next to my friend James, an English teacher. “How’s it going?“ I greeted him.
James took this placeholder question like he takes everything: in complete and utter earnest.
“I’m happy this week,“ he reflected. “Some things are still hard, but they’ve been getting better.“
Evidently, I’m a math teacher first and a human being second, because this is how I responded to my friend’s moment of openness: “So your happiness function is at a middle sort of value, but the first derivative is positive.“
James could have slapped the toast from my hand, dumped his tea over my head, and screamed, Friendship annulled! Instead, he smiled, leaned in, and said – I swear this is a true story – “That’s fascinating. Explain to me what it means. “
And he does. Don’t be scared, there aren’t really any nasty equations, just a lot of graphs, and if you can tell up from down, slash from backslash, you’ll be all set. My takeaway: if you’re talking about a good thing (like being happy), a positive first derivative is what you want. And, for that matter, a positive second + derivative, though at some point we get into the philosophy of too-much-of-a-good-thing. And if you’re talking about a bad thing, you definitely want the first derivative to be negative. But there are lots of combinations, and Ben explains which ones are preferable. Assuming you want to be happy (hey, I just read a short story about a masochistic robot, I take nothing for granted).
Chapter 6: Sherlock Holmes and the Bicycle of Misdirection
You know how Holmes always had a brilliant way, unknown to anyone else, to figure out his mysteries?
Turns out he didn’t always get it right. Don’t get me (or any of the logic professors I’ve taken moocs from) started about deduction vs induction, but here we’re talking about a specific story, “The Adventure of the Priory School”, in which the tracks of a bicycle are analyzed to figure out which direction the bike is moving. This is one of those cases where I’m not completely sure I fully understand the analysis, but it’s so much fun to read, I don’t mind.
Chapter 11: Princess On The Edge Of Town
This is a wonderful chapter for those of us who would rather read about Phoenician legends than math equations. It features Pygmalion and his sister Elissa (aka Dido when Virgil got around to writing the Aeneid), and has absolutely nothing to do with My Fair Lady (different Pygmalion myth) and everything to do with getting the most out of an oxhide. Or, in calculus terms, maximization. In calculus class, this often gets turned into the sheep-pen problem; this is way more fun.
Chapter 15: Calculemus
This might be my favorite chapter. It’s a debate about making math easier for people to use, versus keeping math in the realm of specialty knowledge only a few can access.
As 20th-century mathematician Vladimir Arnol’d explains, Gottfried Leibniz made sure to develop calculus “in a form specially suitable to teach …by people who do not understand it to people who will never understand it.”
….The point of “calculus” – a word Leibniz coined – was to create a unified framework for calculation. Centuries later, mathematician Carl Gauss would write of such methods: “One cannot accomplish by them anything that could not be accomplished without them.“ In my darker moments, I have said the same of forks. But just as I continue to dine with times, Gauss saw the profound value of calculus: “anyone who masters it thoroughly is able – without the unconscious inspiration of genius which no one can command – to solve the respective problems, yea to solve them mechanically …”
This surprised me. Every math course I’ve taken now in my adulthood (which means moocs) has stressed the importance of understanding what the notation means and has gone through extensive proofs to show that, yes, the sum of the derivatives is the derivative of the sum and how the power rule works instead of just moving, multiplying, and subtracting the exponent. I would have been happy to take it for granted, but noooooo. And here’s Leibniz, saying the point of his system is to take the understanding out of it:
For all inquiries that depend on reasoning would be performed by the transposition of characters and by a kind of calculus…. And if someone would doubt my results, I should say to him: `let us calculate [Calculemus], Sir,’ and thus by taking to pen and ink, we should soon settle the question.
I asked Ben, via email (one of the many things I appreciate about Ben is that he’s so patient with fools like me), to clarify for my own edification: Have math teachers been overcomplicating things for us poor students? No, not really.
It’s important to understand mathematics deeply, but it’s a pain if you constantly have to draw on your deep understanding.
Take arithmetic. It’s important to know how our numeral system works (i.e., the meaning of place value), and why the standard algorithms (e.g., “carrying” and “borrowing”) do what they purport to. You don’t want arithmetic to be a collection of black-box procedures beyond the reach of your understanding.
But also, once you know the procedures, it’s okay to execute them a bit mindlessly. In fact, it’s preferable!
The chapter goes on to explain that Leibniz was imagining calculus as part of a greater system, where all reasoning, particularly mathematical, could be reduced to symbol manipulation, making it more accessible so that more problems could be solved without constantly reinventing the wheel to figure out a derivative.
The first Calculus mooc I took (one I actually passed, and that made me so happy I took it again) this kind of accessibility was described as democratization:
This is an example of the way in which mathematics is a democratizing force: problems that at one time would have only been accessible to the geniuses on earth are now accessible to everyone. At one time in history, you would have had to have been the smartest person on earth to have calculated the area of some curved object. But now, armed with the Fundamental Theorem of Calculus, we can all take part in these area calculations.
—Jim Fowler, Calculus 1 (Coursera/OSU), Winter/Spring 2013
I have a feeling a lot of calculus students would settle for a little tyranny of genius, particularly around the time the AP Calc tests get started.
Chapter 17: War And Peace And Integrals
Back in 2014, Ben wrote on his Math With Bad Drawings blog: “Forget the history of calculus. Write me a paper on the calculus of history.” He suggested seeing history as an integral, as Tolstoy did; or as an infinite series (converging or diverging?); or as a set of partial differential equations (this is where I flunked out of most calculus classes, so don’t ask me) or as various other mathematical structures. In this chapter, he expands on Tolstoy’s vision of history as a giant Riemann sum (don’t worry, he explains those in terms we can all understand).
Tolstoy knew where history must begin: with the tiny, fleeting data of human experience. A surge of courage, a flash of doubt, a sudden lust for nachos – that interior, spiritual stuff is the only kind of reality that matters. Furthermore, Tolstoy knew where history must end: with grand, all encompassing laws, explanations as tremendous as what they seek to explain.
The only question is what comes between. How do you get from the infinitely small to the unimaginably large? From tiny acts of free will to the unstoppable motions of history ?
Though he couldn’t fill the gap himself, Tolstoy sensed what kind of thing should go there. Something scientific and predictive; something definite and indisputable; something that aggregates, that unifies, that binds tiny pieces into a singular whole; something akin to Newton’s law of gravitation; something modern and quantitative … something like … oh, I don’t know …
This doesn’t quite work out, since systems made up of many very. small pieces can become unpredictable pretty quickly. But it’s a wonderful journey, and, as Ben says, “Tolstoy’s integral fails as science but succeeds as metaphor…. History is the sum of the people living it.” To us today – and I mean today, this very day, these days when the story of the decade happens every couple of hours – it may not seem like we contribute much, given the way power has been working lately. But we are still affecting history. At least, let’s hope we are.
Chapter 19: A Great Work Of Synthesis
One way I know I don’t really understand calculus is that to me, the Fundamental Theorem of Calculus is just another ho-hum thing to remember, a not-very-exotic thing at that. In every course or video, its introduction is heralded with pomp and circumstance. It seems pretty straightforward to me: the derivative and the integral are inverse functions: what you do with one, you can undo with the other (there’s a lot of ‘sort of’ in there). The chapter explains how this works in simple terms, because it’s fairly simple. Why it’s such a big deal, I still don’t know. Some things, even Ben can’t explain to me.
While this chapter is indeed about the FT of C, I include it in my favorite chapters because it stars an unlikely player: an 18th century woman, Maria Gaetana Agnesi. Wikipedia describes her as a philosopher-mathematician-theologian-humanitarian. Her mathematical achievement was something like what Euclid did for geometry or Fibonacci for algebra: at the age of 30, she wrote the first comprehensive calculus text for students. And she positioned the FT of C prominently.
She received an appointment to the University of Bologna, only the second woman so honored, but changed course and spent the rest of her life serving the poor and running various charities and institutions. And there’s also a fun story about the mistranslation of her book that generated a curve still called “the Witch of Agnesi”. I’m always up for fun stories.
Chapter 26: A Towering Baklava Of Abstractions
This is a chapter about a two-page endnote published in 1996. Perhaps that sounds arcane, so let me dispel any doubt: it is arcane. Fantastically so. The endnote in question imports a prickly, cactus-like topic from one arid setting to another – from the desert of introductory calculus to the bizarre greenhouse of experimental fiction. The book in which the endnote appears – Infinite Jest by David Foster Wallace – has been dubbed “a masterpiece,” “forbidding and esoteric,” “the central American novel of the past thirty years,” and “a vast, encyclopaedic compendium of whatever seems to have crossed Wallace’s mind.“
My question is this: why, in a work of fiction, a dream of passion, would Wallace force his soul to this odd conceit? Why devote two breathless pages to – of all things – the mean value theorem for integrals?
What’s the MVT to him, or he to the MVT ?
In this chapter, Ben juggles the MVT, its “elder cousin” the IVT, Infinite Jest, DFW’s view of math, a quick view of 18th century mathematics history, and Sierpinski triangles. It might be the most fascination-dense nine pages in the book. And it all hangs together, because parts of it aren’t supposed to make sense.
So first he lays out the MVT, which is really pretty simple and intuitive when it’s just explained with a real-life example, like taking a car trip and figuring out your average speed. No problem.
Then we go to Infinite Jest. No, I haven’t read it. I did try: I was on page 6 when news of DFW’s suicide broke, and that ended the book for me. But apparently there’s bit on page 322 about Eschaton that has something to do with tennis balls “each representing a thermonuclear warhead,” and that points to an endnote about nuclear weaponry that requires the MVT. And just when I’m ready to throw the book in a corner, Ben tells me: “Now, if none of this is making sense to you, fear not. The fact is that none of this makes any sense to anyone.” Couldn’t you have told me that before I started crying over how bad I suck at calculus?
Much of the rest of the chapter discusses why DFW would have done this, and involves his fascination with a certain type of math, his degree in analytical philosophy, and the shift in the 18th century from intuitive descriptions to symbolic notation of concepts like the MVT. Buried in there are two gems.
The first relates directly to the influence of mathematics on Infinite Jest: “In one interview, [Wallace] explained that Infinite Jest borrows its structure from a notorious fractal called the Sierpinski gasket.” By the way, that 1996 interview with Michael Silverblatt of NPR’s Bookworm – who recognized the fractal structure and asked specifically about it – is available online. This almost makes me ready to pick up the book again. But… no, not yet.
The second is a math book DFW wrote, Everything and More: A Compact Hisory of Infinity. I have always wanted to read more of his nonfiction, and at first I thought this would be a great place to start, but Ben describes it with phrases like “a dense, technical treatise” and “a thornbush of forbidding notation” so I think I’ll stick with A Supposedly Fun Thing I’ll Never Do Again. But I’m happy to know about it.
These are only a few of the twenty-eight chapters; maybe the one that most grabs your fancy lies in one of the others, like the time the Church tried to ban paradoxes, or the medical researcher who ran afoul of Math (it’s a good thing this happened in 1994, before Twitter), or the chapter that borrows from Flatland (another wonderful book; I have an annotated edition that’s historically, sociologically, and mathematically enlightening), or how Ben finally finds a real purpose for Clippy, that annoying MS-Word helpbot from years past.
An end section titled “Classroom Notes” lists chapters according to topics as they would be covered in a calculus class. Since, as Ben made clear, this is not a textbook, this makes it easier for students who are using a textbook and/or class to find the material pertaining to, say, limits or optimization. As such, it’s far more useful than an index. A thorough bibliography for each chapter is also helpful.
This storybook is by no means complete – missing are the tales of Fermat’s bending light, Newton’s secret anagram, Dirac’s impossible function, and so many others. But in an ever-changing world, no volume is ever exhaustive, no mythology ever finished. The river runs on.
Could this mean there’s a Volume 2 in the future? I have no idea, but I’m betting there’s something lurking in Ben’s idea kit that will someday result in another book on my shelf. For now, I can say that this one helped to ease some of my lingering anxiety and shame about calculus, and generated just a little more motivation to try again than I had before. Not now, not yet; but maybe someday, and that makes it a valuable door re-opener for me.