I don’t like arithmetic, maybe in part because I’m not especially good at it. But what I dislike more than the feeling of doing arithmetic is the fact that so many people think math is nothing but arithmetic.
So let’s start with arithmetic — because if I’m trying to undermine the view of math as a mule train, there’s no better place to start than the place where the tethers feel tightest.
Suppose someone asked you to compute 997 + 998 + 3 + 2. How might you do it? You could use the standard one-size-fits-all procedure for adding up a list of nonnegative integers illustrated below (I’m omitting the cross-outs and carries).
But if the niceness of the final answer leads you to suspect that there’s a slicker way, you’re right: 997 + 998 + 3 + 2 equals (997 + 3) + (998 + 2), which equals 1000 + 1000, which equals 2000. This alternative path will lead you to the right answer because addition satisfies laws: the commutative law, which guarantees that changing the order of the terms doesn’t change the sum (so that 997 + 998 + 3 + 2 equals 997 + 3 + 998 + 2) and the associative law1, which guarantees that how you group the terms doesn’t change the sum (so that 997 + 3 + 998 + 2 equals (997 + 3) + (998 + 2)).
Unlike human laws, which constrain the behavior of people, number laws constrain the behavior of numbers and thereby free people to solve problems in flexible ways.
Abbott: Funny thing, Lou: there’s an infinite number of numbers.
Costello: How many?
Abbott: An infinite number.
Costello: What infinite number?
Abbott: Oh no, there is no infinite number.
Costello: But you just said there was!
Abbott: No I didn’t. I said there’s an infinite number of —
Costello: There! You said it again!
— Abbott and Costello, in a number of their movies (specifically, the number zero)
Earlier this year I dug a mathematico-linguistic rabbit hole on Twitter when I wrote:
Dozens of people posted on that thread with very different takes on my question. One of them, Fred Klingener, reported to me the following actual example of infinity in real life:
Another respondent, Akiva Weinberger, dug a secondary rabbit-hole of his own for his linguistics pals on Facebook, who have their own brand of nerdiness distinct from, but parallel to, the nerdiness of math folks.
After jumping down both holes and crawling through all the tunnels, I emerged blinking into the light of day with renewed appreciation of the way different people can use different words to talk about the same thing.
One reason negative integers can be confusing is that their resemblance to counting numbers makes us think we should understand them through counting. And you can’t use negative numbers to count things – or can you?
Here’s a setup that gives negative integers the opportunity to count things. It bears some resemblance to dangerous experiments you could (in principle) perform with particles and antiparticles, but it’s a lot safer because it doesn’t involve all those annoyingly lethal gamma rays that result from actual annihilation of matter and antimatter. It’s a pastime you can play with (real or imagined) bags and small objects of two easily distinguished colors, which I’ll call dots and antidots.1
A physicist, a biologist and a mathematician are sitting in a street café one morning watching an empty store on the other side of the street. They see someone unlock the store and go in. Time passes. Someone else goes in. More time passes. Then three people come out.
The physicist says, “Our measurements weren’t accurate.” The biologist says, “The two people who went in must have reproduced.” The mathematician says, “If one more person enters the store, it will be empty.”
Of course the joke here is that the mathematician is holding the absurd belief that there are a negative number of people in the store.1 The joke is built on our shared knowledge that although negative numbers make sense in some contexts, they aren’t sensible in the present context. But the knowledge that negative numbers do make sense in some contexts shouldn’t be taken for granted. Centuries after Chinese and Indian mathematicians figured out how to use negative numbers with comfort and ease, Europeans were still struggling to wrap their minds around the concept. It’s a shame that this story isn’t taught more broadly in the West, and that the real number system we teach to students ends up being viewed by many as a European invention. The truth is more interesting.
It’s significant when an old problem gets solved, but it’s even more significant when the intellectual landscape shifts so thoroughly (albeit slowly) that an old problem ceases to seem like a problem at all. A good example of this phenomenon is what happened to the problem of zero. And if you’re thinking “What problem?”, that just shows how thoroughly the winning side of the zero war carried the day.1
So there I was in outer center field, trying to figure out what twenty-six times twenty-six times twenty-six is. (As one does.)
You may not have heard of “outer center field”; come to think of it, I haven’t either. It just seems like the right name to give to the position I always played in kickball (called “soccer baseball” in my town) back in elementary school. The captain wanted me far from the action where I couldn’t do any harm, and I was happy with the arrangement; it left me free to daydream about whatever I wanted — such as the product 26 × 26 × 26.
My interest stemmed from initials. I was JGP, my best friend was SBC, my siblings were WHP, STP, and DJP, and everyone I knew had three-letter initials. Even my dad, who had no middle name, had been assigned a middle initial on his birth certificate, perhaps because the blank field on the form had cowed his immigrant parents into improvising one. I wondered how many different possibilities there were. Of course some were unfortunate and a few were even unprintable, but leaving that issue aside, how many first initial / middle initial / last initial combinations could be constructed?
It’s gratifying that a few thousand of you read this blog each month (hi, whoever you are!), but the way to really have an impact in this century is to have a YouTube channel. One mathematician who’s figured this out is Burkard Polster, whose Mathologer channel reaches more than a hundred times as many math-interested folks as my blog. Last month he made a video that was viewed by over 100,000 people just in its first week. I was glad to see it get so many views, both because it’s about a subject close to my heart and because it mentions my name and discusses work I did back in the 1990s. So this month I invite you to watch the video (maybe not all at once though; it’s almost an hour long!) and find out some of the backstory of Aztec diamonds and the arctic circle theorem.
Traveling between worlds isn’t as simple, or as dramatic, as falling down rabbit holes, walking into wardrobes, or getting snatched up by cyclones. I should know: I frequently nip off to a parallel world, often without anyone realizing I’ve left this one.
Before you can visit that other world — my second home — you have to imagine it, which is trickier than it sounds: sometimes you think you’re imagining it but in fact you aren’t. It helps to describe what you see in your mind’s eye to someone who’s been there, who can help you determine whether you’re genuinely imagining the other world or deluding yourself. (For instance, it may look like you’ve squared the circle, but someone with a good grasp of geometry can help you see that you haven’t.) Ultimately, when your imaginings of the other world are properly calibrated, you can go there — though “go there” is a misleading phrase, since all of us who visit the other world are engaged in nothing more than calibrated imaginings. But surely the place is real, for how else can you explain why two visitors, independently exploring precincts of that world that no one has ever visited, will see the same things?
Of course, I am talking about the world of pure mathematical form. One recent visit I took to that world prompted me to write the following in a succession of tweets:
“I wish you hadn’t just told me not to touch it, because I don’t want to get into trouble and I didn’t even want to touch it, but your telling me not to makes me want to touch it!” my five-year-old exclaimed in frustration, apropos of something or other I’d asked him not to touch. Children are like that. Or, as the song “Never Say No” puts it: “Children, I guess, must get their own way the minute that you say no.”1
Adults are like that too. Being told what we can’t do takes us back to the time when we were powerless children, and sometimes we grownups respond to prohibitions in childish ways. Consider how many supposedly grown-up people have tantrums when they’re told they can’t enter a certain establishment unless they’re wearing a face mask! I sometimes wonder whether I’ve really matured as much as my change in station over the past half-century (from snotty pre-teen to tenured professor) would indicate; maybe I only seem more mature because, in my present life circumstances, fewer people tell me what I can’t do.
SQUARING THE CIRCLE
Among the adults who don’t like being told “You can’t do that” are many adults who enjoy math as a hobby, and the most common thing they’re told they can’t do is square the circle. Continue reading →