Ben Orlin: Change is the Only Constant: The Wisdom of Calculus in a Madcap World (Black Dog & Leventhal, 2019)

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:

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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 …
An integral.

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.

The Math Book even a Mathphobe Can Love: Ben Orlin’s Math with Bad Drawings

This is a book about math. That was the plan, anyway.
Somewhere, it took an unexpected left turn. Before long, I found myself without cell phone reception, navigating a series of underground tunnels. When I emerged into the light, the book was still about math, but it was about lots of other things, too: Why people buy lottery tickets. How a children’s book author swung a Swedish election. What defines a “Gothic” novel. Whether building a giant spherical space station was really the wisest move for Darth Vader and the Empire.
That’s math for you. it connects far-flung corners of life, like a secret system of Mario tubes.

Ben Orlin, Introduction, Math With Bad Drawings: Illuminating the Ideas that Shape our Reality

As a lifelong mathphobe, I rarely buy math books; no matter how highly recommended they are, I get stuck in the pages of formulas, equations, and sample problems that require translating, much as a text in an unknown language requires, resulting in a Google-translate version of the math. I neither learn nor enjoy it. But I’ve been following Ben’s blog for about five years now, so I knew I was going to buy this book, I knew I’d enjoy it – AND I knew I’d learn something.

To be sure, I had a couple of concerns. First, I thought it might be what the Tumblr-turned-book market cranks out, merely a “greatest hits” reprint of his blog. Nope; all the material in the book is brand-new, though he references his blog a few times. And by new, I mean new: I’ve seen many explanations of the triangle inequality theorem that turned into the Charlie Brown Teacher wah-wah but I will remember Ben’s triangle struggling to bring its arms together and falling… short.

My second concern was that, while starting out as a “fun math for everyone” book, it would soon turn into formulas and equations. Let’s face it, most math books (even “friendly” math books that start out with lots of reassurance that “anyone can learn math”) read like rule books: “The ball shall be a sphere formed by yarn wound around a small core of cork, rubber or similar material, covered with two strips of white horsehide or cowhide, tightly stitched together. It shall weigh not less than five nor more than 5¼ ounces avoirdupois and measure not less than nine nor more than 9¼ inches in circumference.” But again, my faith in Ben was rewarded: This book is more like the 7-year-old next door tossing a ball in the air and calling out, “Hey, wanna come play?” The only things that resemble math-book formulas are a couple of endnotes that are pre-declared to be esoteric. Oh, there is a bit about methods of calculating certain baseball and football stats, but even that is presented in a non-scary way (at least as far as the math is concerned; the sports stuff is still lingo, but I suspect most will consider that a plus rather than a minus).

The first thing that struck me about the book, before I’d read anything, was the high quality of the physical object. The dust jacket is appealing and representative, including some of Ben’s “bad drawings”, also reproduced on the endpapers and flyleaves. My practice is to remove and put aside dust jackets while I’m actively reading a book, lest it get torn or dirty (I’m super-destructive, I am I am), and recover the book when I’ve completed the first read and shelve it. I was surprised, and pleased, to see one reproduced drawing on the cover of the book itself, a nice detail. The book felt heavy, and I soon realized that’s not just because it’s a 400 page book, but because the pages are of unusually thick paper, presumably to prevent bleed-through of the many color drawings on nearly every page. In fact, I had to learn how to turn pages all over, the feel was so different.

And color! On every page, color! The Bad Drawings are all in color, sometimes monochrome but often a mixture. Even the running titles (vertically set!) and colored boxes enclosing page numbers are in colors that match the topic (red is probability, purple is statistics). I’ve become more appreciative of books as physical objects as I’ve encountered more truly well-produced books; this one keeps the bar high.

But what about content? No problem. You can find an excerpt at Popular Science, and another at Vox; I find these difficult to read online, and like the book layout far better, but then I’m an old fart. And by the way, you can find a bunch of reviews, interviews, and other goodies on Ben’s blog, if you want more.

The opening division – How to Think Like a Mathematician – presents the playful, investigative approach to math: “Creativity born from restraint.” That tickled me, because it’s in many respects the basis of poetry with its forms of meter and rhyme, not to mention Oulipo, who delight in such things as writing entire books without the letter “e”. And the Ultimate Tic Tac Toe is pretty cute.

I’d like you to meet this chapter’s star: the triangle.
It’s not your typical protagonist. Snooty literary types may dismiss it as two-dimensional. Yet this atypical hero will embark on a typical hero’s journey: rising from humble origins, learning to harness an inner strength, and ultimately serving the world in a time of crisis.

Chapter 6, We Built This City on Triangles

In the Geometry division, I was surprised by how captivating I found bridge trusses, and the reasons they are made with triangles instead of some other shape. The chapter on European paper sizes was maybe my favorite of the book; they make sense, like the metric system, unlike the American way of remembering how many ounces in a pound and feet in a mile and make it stop! The stories of brownies and the Colossus at Rhodes made scaling more understandable than memorizing formulae, and we even got into some biology with explanations of the differences between ant and elephant legs. I found the chapter on dice asked a question I’d never considered: why are dice typically shaped as cubes? I had encountered pig-shaped dice in a logic mooc, and I’ve seen a few non-cuboid novelty dice in my travels, but why are standard run-of-the-mill dice always cubes? Then there’s Chapter 10, An Oral History of the Death Star, told from the POV of various participants: the Imperial Geometer, Imperial Physicist, Imperial Engineer, and a few others in conversation with Grand Moff Tarkin and Darth Vader, trying to outline the difficulties involved in building the most advanced killing machine ever.

The Probability section included the hilarious chapter “10 People You Meet in Line for the Lottery”, as well as chapters on DNA (yes, more biology), and weird insurance and how companies determine what to charge for, say, multiple-birth insurance or Extraterrestrial Kidnapping Insurance. Some of this chapter gets into economics more than I’m comfortable with, since, despite how much I enjoyed the terrific Oxford mooc on economics, I still find anything to do with money to be boring as snot.

I admit that there is something reductive about the whole project of statistics, of taming the wild, unpredictable world into docile rows of numbers. That’s why it’s so important to approach all statistics with skepticism and caution. By nature, they compress reality. They amputate. They omit. They simplify.
And that, of course, is the precise source of their power.
….By condensing the world, statistics give us a chance to grasp it.
And they do more, too. Statistics classified, extrapolate, and predict, allowing us to build powerful models of reality. Yes, the whole process depends on simplification. And yes, simplification is lying by omission. But at its best, statistics are an honest kind of lying. The process calls upon all the virtues of human thought, from curiosity to compassion.
In that way, statistics are not so different from stick figures. Their bad drawings of reality, missing hands and noses, yet speaking a peculiar truth all their own.

IV: Statistics: The Find Art of Honest Lying

I had some troubles with the Statistics section, possibly because it’s the mathiest in the book, but that doesn’t mean there weren’t fun spots, such as the history of baseball statistics. Anyone who liked the movie Moneyball will find a similar storytelling approach that makes a niche subject interesting to the outsider. There’s also a chapter on corpus linguistics, the statistical analysis of language use and a particularly hot topic in literary circles these days. But I confess: I still don’t understand p-hacking beyond the most elementary level.

The final section – On the Cusp – is about the difference between what is continuous and what is discrete. The camel example reminds me of the sorites paradox, aka the Problem of the Heap (from another logic mooc), but I’m improvising wildly here; as a mathphobe, I don’t quite grasp the connection in this section as we veer from electoral math to measurement of coastlines (which is in itself another fascinating paradox from, you guessed it, another mooc). I’m sure those with greater in-depth understanding will see a discipline that’s lost on me. In any case, it’s all fun, and if there’s anything that can convince a mathphobe that math matters, it’s election math.

After I’d read the book – and I couldn’t believe I’d read a math book, cover to cover – I went back to connect the endnotes with the text to which they referred. I’m going to guess this was a “de-academicizing” decision, avoiding cluttering pages with footnotes, but I like to know when I read a text that hey, there’s more info about this in the back. So I spent a couple of hours flipping back and forth, making notes in the margins (I told you, I’m destructive). Sometimes it’s just a simple citation; sometimes it’s an expansion on the topic; and sometimes it’s a funny comment. Don’t skip the end notes. Ok, you can skip the citations (unless you want to look something up; I looked up Poe’s prose poem “Eureka” but TLDR’d it), but the rest are worth reading.

I’ve had a complicated relationship with math all my life. It wasn’t until about five years ago, thanks to a brilliant mooc on mathematical thinking, that I saw a different way to approach math, a more investigative, playful way that viewed mistakes as part of the game. It was in that course that I also was referred to Ben’s breakout essay, “What it Feels Like to be Bad at Math”, and started following his blog. He’s always been patient with my stupidest questions, encouraging with my painfully slow progress, and generous with his time and talents.

I hope this book introduces him to a wider circle of mathphobes (and those of us slowly recovering from same); there are lots of us out there, and we all could benefit from seeing the fun side of math.