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.1Continue reading
For Ted Propp, 1923 −
When I think about dividing by zero, I usually think about my father and his New York City law firm back when I was young.
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
dedicated to the memory of Elwyn Berlekamp
The mistaken formula (x+y)2 = x2 + y2 is sometimes called the First Year Student’s Dream, but I think that’s a bad name for three reasons. First, (x+y)2 = x2 + y2 is not exactly a rookie error; it’s more of a sophomoric mistake based on overgeneralizing the valid formula 2(x+y) = 2x + 2y. (See Endnote #1.) Second, most high-school and college first-year students’ nocturnal imaginings aren’t about equations. Third, the Dream is not a mere dream — it’s a visitor from a branch of mathematics that more people should know about. The First Year Student’s Dream is a formula that’s valid and useful in the study of fields of characteristic two.
Let’s start with something uncontroversial: a valid mathematical assertion like 3+3+3=9 can be “wrong” if it’s been dragged into a situation in which it just doesn’t belong. Consider the Sufi tale1 of Mullah Nasrudin and his wife.
Three months after Nasrudin married his new wife, she gave birth to a baby girl.
“Now, I’m no expert or anything,” said Nasrudin, “and please don’t take this the wrong way-but tell me this: Doesn’t it take nine months for a woman to go from child conception to childbirth?”
“You men are all alike,” she replied, “so ignorant of womanly matters. Tell me something: how long have I been married to you?”
“Three months,” replied Nasrudin.
“And how long have you been married to me?” she asked.
“Three months,” replied Nasrudin.
“And how long have I been pregnant?” she inquired.
“Three months,” replied Nasrudin.
“So,” she explained, “three plus three plus three equals nine. Are you satisfied now?”
“Yes,” replied Nasrudin, “please forgive me for bringing up the matter.”
Here’s a small puzzle that opens the door to a surprisingly tricky general problem: How can a teacher divide 24 muffins among 25 students so that everyone gets the same amount to eat but nobody gets stuck with any tiny pieces?
To get a clearer sense of what counts as a good answer, let’s consider a bad answer. You could remove 1/25th of each muffin, give an almost-complete muffin to each of the first 24 students, and give the 24 slivers to the last student. Then everyone gets 96% of a muffin, but it‘s a pretty crumby scheme for the student who gets nothing but slivers. We’d like to do better. Can you find a scheme in which the smallest piece anyone gets stuck with is bigger than 1/25 of a muffin? Can you find a solution in which the smallest piece is a lot bigger? After you’ve found the best solution you can and you can’t improve it, how might you try to prove that it’s the best solution anyone could ever find? And how would you solve the problem if there were a different number of muffins and/or a different number of students trying to share them? Puzzles of this kind can be challenging and addictive, and the general solution wasn’t found until last year.
In memory of Ron Graham, 1935-2020
Bear with me if I seem to be veering out of my lane (as they say nowadays), but let me ask: What is chess? If you play with a chess set in which a lost pawn has been replaced with a button, you’re violating tournament regulations but most people would say you’re still playing chess; the button, viewed from “inside” the game, is a pawn. Likewise, if you’re playing against your computer, the picture of a chessboard that you see on your screen is fake but the game itself is real. That’s because chess isn’t about what the pieces are made of, it’s about the rules that we follow while moving those pieces. Asking “Do pawns exist?”, meaning “Are there real-world objects that behave in accordance with the rules of chess?”, misses the point. If one of your pieces has been shoddily manufactured and spontaneously fractures, that doesn’t mean that your mental model of how chess pieces behave is flawed; it’s reality’s fault for failing to conform to your mental model.
You’ve probably already guessed the agenda behind my rambling about chess, but here it is explicitly: I claim that math (pure math, anyway) is as much a game as a science. The objects of mathematical thought, like the pieces in chess, are defined not by what they “are” but by the rules of play that govern them. The fact that in math the pieces exist only in our imaginations and the moves are mental events doesn’t make the rules any less binding. And even though the rules are human creations, once we’ve agreed to them, the answer to a question like “Is chess a win for the first player?” or “Is the Riemann Hypothesis true?” aren’t matters of individual opinion or group consensus; the answers to our questions are out of our hands, irrespective of whether we like those answers or even know what they are.1