Ben Orlin’s charming new book Math for English Majors: A Human Take on the Universal Language is a welcome addition to the growing fold of books about math for non-mathematicians – though I have to say, speaking here as an ally of English majors everywhere, that I vehemently protest the cover art’s implication that majoring in English is solely about finding accidentally omitted words. It’s about more than that. I mean, it’s also about spelling the words correctly! And putting the right punctuation marks in between them!
Of course I’m kidding. The study of English isn’t about fastidiously following the rules of English grammar, spelling, and punctuation; it’s about using written language as a vehicle for conveying understanding and imparting beautiful ideas. The same is true of math, but the news usually doesn’t reach people whose math journeys stop too soon, which is almost everybody.
Your typical mathematician has stories about encounters with people whose experiences of math were very different from their own. Orlin’s editor told him “I just feel like school math was full of formulas. Formulas I had to memorize.” Orlin’s uncle expressed surprise at the idea – commonplace among mathematicians – that at bottom, math is about structure. “I don’t get that,” said the uncle. “To me, math is instructions.”
Math is instructions. Also formulas. But it’s lots of other things too, including – and this is Orlin’s unique contribution to the field of mathematical expositionology – bad drawings. Just as instructions and formulas can be exit ramps from the enjoyment of mathematics for those who don’t like them, bad drawings like Orlin’s can be an entrance ramp, demonstrating a spirit of whimsy with the capacity to animate anatomically inaccurate (yea, biologically impossible) collections of lines, circles, and other simple shapes – a spirit that can make math come alive too.
MATH AS LANGUAGE
Instructions – “Don’t play in the street!” – are made of language. Formulas – “The speed limit on this street equals 25 miles per hour” – are also made of language. But the language of mathematics is used for different purposes than language-on-the-street.
One way this difference shows up is in the way extreme cases are treated. If you called a chessboard a rectangle, people would think you confused if not a bit strange. (Similarly, while it’s apt and even sweet to call your spouse “Lover” in private, try referring to your spouse as your “lover” at a party you’re attending without your spouse and see what kinds of looks you get.) When a shape is a square, why not call it a square? Carrying this logic a step further, you might want to define the word “rectangle” specifically so as to exclude squares, and if your purpose is purely descriptive, this is a sensible thing to do. But in math we want to know All The Truths, and we want to understand them in the most economical way possible. For instance, in every square the two diagonals bisect each other (cut each other perfectly in half), and the same is true in every rectangle. Do we really want to have two separate theorems telling us this? Ditto for all the other square-facts that correspond to rectangle-facts. So if our game is not categorizing individual shapes in as specific a way as possible but deducing the properties of all shapes of a given kind, it’s sensible to subsume squares within the category of rectangles.
But this choice, like the thousands of others like it that mathematicians have made over the millennia, exacts a cost. When I ask you to picture a rectangle, you must picture neither a square nor a rectangle that isn’t a square. You have to hold in your mind a shimmery conception that somehow contains both possibilities.
A similar shimmer surrounds the concept of number. Consider the proposition “If p is a prime number, p + 7 cannot be prime.” The claim isn’t about any particular prime; it’s about all of them. So when you picture p, you had better not imagine any one prime in particular, except by way of example.
To compensate for this inherent shimmer in the way we picture the objects of mathematical thought, mathematical language has had to become super-precise. We have to be rigid in how we define flexible terms like rectangle and prime because no single exemplar can stand in for the infinite class of mathematical objects (rectangles, primes) that our words signify, and because it’s just as important to make it clear which objects aren’t included under our definition. (If you think picturing a rectangle that’s neither a square nor not-a-square is hard, try picturing a typical shape that isn’t a rectangle.)
Sometimes the habit of careful use of mathematical language leads to over-literal understanding of non-mathematical language. Facetiously exaggerated literalism becomes a form of humor among the mathematical. “I thought you said you weren’t in bed!” “I’m not in bed; I’m on bed.”
The final chapter of Math for English Majors is a phrase book subtitled “A Local’s Guide to Mathematical Vocabulary”. Orlin catalogues several dozen words and phrases that, for many of us who do math, have shaped our ways of thinking so deeply that those bits of language leak out into our non-mathematical lives. One good example is the word “derivative” (noun). This is a way mathematicians measure the rate of change, and it’s a word I use every month or so with my non-mathematician wife, though she took her own path to the place where my math background led me.
My wife is a psychologist and sometime dog-trainer, and she found a marvelous book that we read together called What Shamu Taught Me About Life, Love, and Marriage: Lessons for People from Animals and Their Trainers by Amy Sutherland. In it, Sutherland talks about the similarities between training an orca to do a back flip and training a husband to deposit his dirty underwear in and not merely near the hamper. In both cases, the trick is to provide positive reinforcement for improvements even if the behavior being offered still falls short of the desired goal. My wife and I have applied this lesson during the often vexing process of raising two children, though I phrase the lesson of Shamu differently: “Reward the derivative.”
But “derivative” can mean different things in different contexts. If an English professor calls a novel derivative, that doesn’t mean that the novel represents a change from what’s already out there – quite the opposite! What’s most interesting to me are the ways in which mathematics uses some familiar words for unfamiliar purposes and the confusion that can result. That’s not Orlin’s main point but I want to explore it a bit.
FOUR FALSE FRIENDS
The term “false friends” (faux amis if you’re a French major) refers to terms can mislead us because they sound like one thing but mean something else. What are the false friends of math-speak? I’ll single out four of them.
Italics and quotation marks and the phrase “so-called”: Sometimes we mathematicians need a way to signal that we’re about to use a term whose meaning the reader doesn’t know yet. If we’re writing for each other, we may use italics to signify that a new word or phrase is being used for the first time, with a meaning that we’re about to assign to it. We’re not shouting; we’re just saying “You’ll probably want to remember this.” If we’re writing for students, we’re more likely to use boldface to make it easier for a student-in-a-hurry – a student cramming for an exam, say – to find the place in the text where the term was defined. In popular writing, mathematicians sometimes use “quotation marks”, but this can cause confusion, because that kind of punctuation often conveys a whiff of authorial sass that isn’t intended. This is even more true with the phrase “so-called”. If someone uses the phrase “my so-called friend” they’re insinuating that the person in question is not a true friend. But if a mathematician writing for the public refers to “the so-called Taniyama-Shimura conjecture”, they’re not throwing shade at Taniyama or Shimura for stealing unjustified credit – they’re trying to reassure you that it’s okay if you have no idea yet what they’re talking about.
The phrase “in general”: In general, mathematicians use the phrase “in general” to mean “always”. But not always; you have to attend to context. If a mathematician writes “In general, primes are odd”, they’re saying that the exceptions are few. (In this case, the sole exception is 2, which some mathematicians cheekily call “the oddest prime”.) But if a mathematician writes “5 can be written as 12 + 22, 13 can be written as 22 + 32, and in general, every prime that’s one more than a multiple of four can be written as a sum of two squares,” they’re saying that there are no exceptions.
The phrase “the following”: We use this a lot in math to cope with the fact that some mathematical thoughts don’t fit comfortably within the confines of a single sentence. This is especially the case when we’re asserting that three or more conditions all stand or fall together (that is, in any given situation either all hold true or none hold true); the phrase “The following are equivalent” is so common in the rhetoric of mathematics that it had its own abbreviation (“TFAE”) even before text messaging made abbreviations cool. I once had a friend1 who thought me pompous for using the phrase “the following” in ordinary speech. To me, saying “Well, consider the following situation: …” was just a way of opening a somewhat complicated thought and signaling in advance that the thought might require more than a single sentence. Then again, I have to admit that warding off interruptions with this kind of pre-disclaimer is itself pompous. And instructing one’s interlocutor to “consider” something is a bit pompous too.
But the falsest friend of all is the utterance “”, which is to say, saying nothing at all. If you pass a non-mathematician friend and say “Hi!” and they remain blank-faced and say “”, it could mean some flavor of “I’m mad at you,” but if your friend is a mathematician – me, say – it’s likelier that their silence means something closer to “Jim’s not here right now.” I already mentioned that the objects of mathematical thought, unlike the objects of our world, have a shimmer of universality to them, and the sensory world can only get us so close to those objects. French mathematician Henri Poincaré nodded at the disconnect between the sensory world and the realm of pure mathematical form by calling geometry “the art of correct reasoning from incorrectly drawn pictures,” and that’s not just applicable to geometry; it’s true of all pure mathematics. When it comes to depicting the things that live in the place pure math describes, all drawings are bad, though some are useful.2 Some people call this place “the Platonic realm”; we might call it “The There” for short, or just “There” for shorter, as in the following exchange: “There exists a polyhedron with twelve regular pentagonal faces.” “Where does it exist?” “I just told you: There.”
Beyond a certain point, if we want to work in The There, the physical world gets in our way, and we have to ignore it if we’re going to make progress. I try not to do this around others, but when I’m in hot pursuit of some result I care about, it can be hard to stay fully in The Here. Or as my wife put it to me recently when I made the mistake of gently teasing her about her habit of going around wearing headphones in our house: “You go around wearing headphones too; it’s just that yours are inside your head.”
HIGH AND LOW MATH CULTURE
Although I’ve described math culture as if it were a monolith, it isn’t. For one thing, what I’ve written so far is more relevant to pure mathematics. There are plenty of mathematicians who bring things between Here and There; they’re called applied mathematicians.
A less familiar distinction than pure versus applied is the gap between high math culture and low math culture, analogous to the divide between Great Books and beach books. Mathematics has its own Great Books – the Elements of Euclid and the Principia Mathematica of Newton (though nowadays we mostly read adaptations of them). Our version of low culture consists mostly of math puns and Pi Day. Actually, Pi Day usually brings math puns along for the ride, inasmuch as eating pie usually plays a role in the festivities.
Imagine a bunch of lit majors telling their department chair “We’d like you to sponsor a spelling bee to raise student morale about literature. And we’d like to serve everyone honey, because bees. What do you think?” The chair would reply “We are here to study lit-er-a-ture, not to compete at orthography or consume the products of apiculture.”
Part of me has similar feelings about Pi Day. The number pi runs like a bright crimson thread through the fabric of mathematics, and if you do any kind of math, even one far removed from geometry, you’re bound to run into pi there sooner or later. There are many orderly formulas involving pi, such as this:
There are also many disorderly formulas for pi, such as this:
Here the boldface numbers are just the decimal digits of pi, and they follow no known pattern. Pi Day is the day when students honor pi by rattling off those unruly boldface numbers instead of celebrating the many orderly formulas for pi and the many corners of the mathematical tapestry that the crimson thread adorns. So math-snobs can get a bit snooty about Pi Day, and even those of us who aren’t math-snobs (or who maybe are but don’t like to admit it) may privately sympathize just a bit with Orlin’s Pi Day Grinch.
“But . . . isn’t getting people in the department door a good thing, whether the bait is honey or pie or something else entirely?” I’ll come back to that question at the end of the essay.
ORLIN’S CHALLENGE FOR THE INTREPID AND FOOLHARDY
A different divide is the distinction between conceptual math and procedural math. Here I want to repurpose a story that Orlin tells, to illustrate a point of my own: that, properly employed, the two kinds of math support one another.
Orlin writes about how hard it is for even seasoned math teachers to correctly multiply 2573 by 389 on paper or on the board, because there are too many places where a person could slip up. In trying to meet the “Orlin Challenge” myself, I decided to exploit the fact that Orlin doesn’t specify a time limit. This freed me to take things slow to reduce the chance of my making a mistake, and even more importantly, it gave me the freedom to use strategies that would help me catch mistakes if I made them – strategies that someone who lacks a conceptual understanding of math might never think of.
I started out by performing the familiar tabular procedure I’ve known for fifty-plus years, a recipe that calls for multiplying 2573 by the single-digit multipliers 3, 8, and 9 respectively, shifting those products appropriately, and adding them up:
Here the exclamation points form a place-holder for the final answer; we’ll get that answer by adding the three numbers indicated by question marks, which are the values of the products 2573 × 9, 2573 × 8, and 2573 × 3, respectively.
According to my calculations3, those values are 23157, 20584, and 7719. But could I have goofed somewhere? I could re-do the calculations to increase my confidence in those answers, but brains tend to fall into ruts; if I goofed the first time I computed 2573 × 9, I’m likely to commit the same goof the second time through.
A better approach is to be a number-detective.
Just as real-life detectives try to sift the guilty from the innocent by seeing whose stories hang together and whose stories don’t, number-detectives can detect mistakes when “something just doesn’t add up”, in this case in a quite literal sense. We may not know in advance what 2573 × 8 and 2573 × 9 are equal to, but we can be certain beforehand that the correct value of 2573 × 9 must equal exactly 2573 more than the correct value of 2573 × 8. Why? Some would say “it’s just common sense”, but others would say “it’s just the distributive law”:
2573 × 9 = 2573 × (8 + 1) = 2573 × 8 + 2573 × 1
Is it common sense or the distributive law? Orlin would say it’s both; mathematical laws are just a formalization of common sense, though sometimes the laws will help us in contexts that are too complicated for common sense. Or, as mathematician and math popularizer Jordan Ellenberg puts it, “Mathematics is the extension of common sense by other means.” In any case, we can put the compound claim “2573 × 9 = 23157 and 2573 × 8 = 20584” to the test by adding 2573 to 20584 and seeing if we get 23157. Sure enough, we do!
What about the third product I computed, namely 2573 × 3? I got 7719, but how can I be sure, or at least surer? We haven’t interrogated Suspect 7719 yet, but here’s one way to trip it up if it’s lying: multiply it by 3. We may not know in advance what 2573 × 3 and 2573 × 9 are equal to, but we can certain beforehand that if you triple the former, you get the latter. Why? You could say “it’s just common sense”, or you could say “it’s just the associative law for multiplication”:
2573 × 9 = 2573 × (3 × 3) = (2573 × 3) × 3
So we can put the compound claim “2573 × 9 = 23157 and 2573 × 3 = 7719” to the test by multiplying 7719 by 3 and seeing if we get 23157. Sure enough, we do!
This doesn’t mean that the stories are true (that is, that the computed values of the products are correct), but the way they corroborate each other increases our confidence that they are.
Now we can fill in those question marks with greater confidence than before we played number-detective, with the numbers 23157, 20584, and 7719. Then it’s time to add those three products. I got 1000897, but is there a way to be surer? Yes, and it was developed by number-detectives in India over a thousand years ago. It’s often called casting out nines, and I wrote about it in my essay The Magic of Nine. See Endnotes #4 for details of how to apply that method here.4
WHY BOTHER?
At this point you’re probably thinking, That’s an awful lot of work for a simple arithmetic problem that a calculator can solve with a punch of nine buttons. Why work so hard to rule out potential mistakes? I grant that most of the time it isn’t worth it. But if you’re taking an exam that counts for 25 percent of your final grade, and each question is worth 10 points out of 100, and you still have some time to spare after you’ve worked out your answers, sure, you could leave the exam early and grab some extra sunshine, or you might want to spend a few minutes corroborating your results.
In my case, a bit of ego was on the line when I took up the Orlin Challenge. I’m not especially good at calculating with pencil and paper, but I like to think that when I approach a calculation the way I would approach a research problem, using intelligent methods of checking my work along the way for signs of a mistake, I can compensate for my calculational deficits with conceptual understanding, perseverance, and occasional craftiness, much as Orlin compensates for his lack of drafting skills by bringing other attributes to the drawing table.
In this instance, the relevant conceptual understanding is just the basket of basic laws satisfied by addition and multiplication: the commutative, associative, and distributive properties. Orlin says a bit about them, using pictures to show how these laws are just a codified form of common sense. For instance, the preceding picture gives pictorial meaning to the double distributive law
(a+b) × (c+d) = a×c + a×d + b×c + b×d
through the example (20+7) × (30+8), shedding light on why the tabular multiplication method gives the right value of 27 × 38.
In my own classroom teaching, I stress the role of checking one’s own work in creative ways, but students seem resistant. Too many years of pre-college math have given them the latent conviction that math is about getting the right answer (or at least some answer) as quickly as possible and moving on. I beg my students to develop the habit of solving problems in two different ways and comparing the results, offering an incentive: “If you derive two different answers to an exam problem and one of them is right, I’ll give you full credit.” But hardly anyone takes me up on it. It’s too different from what they think math is. My students have failed to develop a “debugging mindset” in which a key skill is finding your own mistakes – despite the fact that the course I’m teaching was designed for computer science majors!
This makes me sad, because one of the gifts math can offer the soul is a powerful elixir that combines a humbling awareness of our own individual fallibility and our collective ability, as a species, to stumble towards the truth over the course of centuries. Making mistakes and catching them, then making more mistakes and catching them, as individuals, gives us a microcosmic model of humanity’s collective journey into the light.
And, in a less lofty but much more urgent vein, our civic discourse would be a lot more bearable if more people made it a regular practice to ask themselves “How do I know?” and “How can I be sure, or surer?”
THE NEED FOR DIFFERENT KINDS OF BOOKS
It’s not surprising that people come to the classroom with different backgrounds and different expectations for what math is and can be. What surprises me more, even though it shouldn’t, is that the same applies to the editors who publish reviews of books about math. A case in point is the New York Times review of the new book The Secret Lives of Numbers by Kate Kitagawa and Timothy Revell; the review appeared under the blurb “The Fascinating Story of Math in a Book You Can Actually Understand”. (Although the review was written by Alec Wilkinson, the blurb was probably written by some uncredited editor.)
My first reaction was, Don’t we have enough messages out there in the culture already, telling people that math is incomprehensible if not downright scary? Does the New York Times really need to reinforce this all-too-common conception of what math is?
But thinking it over further, I had to concede that the blurb could lead some I’m-just-not-a-math-person person to read the book-review and maybe even read the book itself. The strategy of baiting readers by saying “Look, I know you’re confused but I can help” goes back at least as far as Maimonides and his Guide for the Perplexed.5 So if there’s a demographic of people who’ve tried to read popular books about math and found them confusing, a blurb that says “Look, I know you’ve tried other books about math and been burned, but this one really delivers the goods” (if it’s true!) seems a valid tactic for reaching this audience, and it’s probably the only one that stands a chance of working. The pool of people whose lives could be enriched by a deeper appreciation of what math is about is a diverse one, so the math popularizer community needs to adopt what game theorists would call a “mixed strategy” for getting out the word that math is about more than formulas and instructions.
I think a big selling point of Orlin’s books is that they’re #NotLikeOtherMathBooks, in large part because of the prominently displayed silly drawings. The very crudeness of Orlin’s drawing skills is part of what draws a crowd. Here’s a bad-at-art guy who does it anyway because he loves doing it. This invites readers who identify as being bad-at-math to stop worrying about being bad at it and have some fun under the guidance of an author who knows how to give readers a good time, through both playful pictures and deftly comical prose. His latest book is no exception.
So, if you read only n books about mathematics this year, let Math for English Majors be one of them.
ENDNOTES
#1. Well, ex-girlfriend actually. After our breakup I demoted her to “friend” when referring to her in conversation. One of my friend-friends teased me about this for years.
#2. Here I’m riffing on the saying “All models are wrong but some are useful.” The quote is often attributed to statistician George Box, though the sentiment is surely older.
#3. I’d be curious to know when the prefatory phrase “According to my calculations” and its twin “If my calculations are correct”, now staples in pop-culture representations of geekiness, first got pushed into the meme-pool (presumably bleating “Wait, I don’t have my inhaler!”).
#4. The casting-out-nines rule is, whenever you see a number with more than one digit, replace it by the sum of the digits. In this way, 23157 becomes 18 which becomes 9, 20584 becomes 19 which becomes 10 which becomes 1, and 7719 becomes 24 which becomes 6. If you apply this same procedure to the sum of the boldface numbers (9, 1, and 6) you get 16 which becomes 7. Meanwhile, the putative final answer to Orlin’s problem, 1000897, becomes 25 which becomes 7. The fact that those underlined numbers (the 7’s) were equal is no accident, and it provides further corroboration that we did our arithmetic correctly. Taking our fastidiousness a step further, we can apply the method to the two numbers we were originally tasked with multiplying, 2573 and 389: 2573 becomes 17 which becomes 8, while 389 becomes 20 which becomes 2. 8 times 2 is 16, which becomes 7 again – further corroboration! If our suspects are lying, they’re super lucky to be getting away with it.
#5. I wonder if God (widely known to be a mathematician) found the title A Guide for the Perplexed offensive. “Look, folks, my laws are spelled out very clearly in the five Books of Moses and the thirty-seven tractates of the Talmud; what could anyone possibly find confusing?”