A New Game with Infinity

Aleph-null bottles of beer on the wall,
Aleph-null bottles of beer!
Take one down, pass it around,
Aleph-null bottles of beer on the wall!

— Math nerd drinking song

You may already know the standard story about infinite sets like {1,2,3,…} and {2,3,4,…}. Even though the second set seems to be smaller (it’s missing one of the elements in the first set), Cantor taught us that the two sets are the same size (in the sense that there’s a one-to-one correspondence between them). The two sets have the same “number” of elements (namely aleph-null), and aleph-null minus one equals aleph-null. For many students, that anomaly takes some getting used to.

Cartoon by Ben Orlin, now the author of https://mathwithbaddrawings.com/2018/05/23/math-with-bad-drawings-the-book/. (I urge you to buy a copy rather than steal one, since there are only finitely many copies.)

But there’s a perfectly respectable mathematical sense in which the two sets do not have the same number of elements. With a suitable notion of what it means to “count” the elements of an infinite set of numbers, different from Cantor’s, the size of {2,3,4,…} is smaller than the size of {1,2,3,…}; in fact, it has one fewer element. Likewise, in this alternative way of measuring how big sets of numbers are, the set {1,3,5,…} is slightly bigger than the set {2,4,6,…}. How much bigger? Half an element! (Though see Endnote #2.)

Knots and Narnias

Say you’re walking north across a meadow surrounded by hills when you come across a solitary doorframe with no door inside it. Stranger still, through the doorway you see not the hills to the north of the field but a desert vista. Consumed by curiosity and heedless of danger, you cross the threshold into the desert. The sun beats down on your bare head; you see a vulture off in the distance. In sudden panic you spin around; fortunately the doorway is still there. You run through the doorway back into the field, grateful that the portal works both ways.

Now what?

You cross through the doorway into the desert again, and turn around, and once more you see the doorframe behind you, and through it, the southern hills surrounding the meadow. But now a question occurs to you: what would you see if, staying in the desert world, you went around the doorframe, and looked through the portal from the other side? What kind of new world would you see? Arctic tundra, maybe?

A pair of shorts

This month I wrote two short essays for The Aperiodical‘s Big Internet MathOff: “The Mystery of the Vanishing Rope Trick” and “Cantor’s Paradise Meets Skolem’s Paradox”. Whittling an essay down to a thousand words is hard but it’s good exercise!

THE MYSTERY OF THE VANISHING ROPE TRICK

Have you ever done something impossible?

About twenty-five years ago I invented an impossible rope trick by accident, and afterwards I could never figure out what I’d done or get the trick to work again. It wasn’t the rope that had vanished, but the trick itself. The incident took place at a math conference in Amherst, Massachusetts, and no, I hadn’t been drinking, though I admit that it was late at night and I was tired.

Why Does Exploding Dots Work?

A few months from now, if James Tanton and his Global Math Project co-conspirators have their way, ten million schoolchildren will take a huge mathematical step from the twenty-first century all the way back to the Bronze Age: instead of using a gadget with a state-of-the-art interface (say, a telepathic smartphone that tells you the answer to an arithmetic problem when you merely think the question), these kids will solve arithmetic problems by moving counters around on boards, the way people did thousands of years ago.¹

But if you think Tanton is a back-to-basics reactionary, you’ve got him all wrong: he’s a math-Ph.D.-turned-math-educator possessed by the conviction that math can be made understandable to, and exciting for, everyone. Tanton’s “Exploding Dots” approach to precollege math is designed to bring illumination and joy to a subject that students all too often associate with mystery and misery, and the Global Math Project’s aim is to carve out one week each year (“Global Math Week”) from the grade-K-through-12 academic calendar, in which every student gets a chance, if only for an hour, to experience that illumination and joy.

Time and Tesseracts

“The fourth dimension became a thing you talked about, without knowing what it meant.”

— Marcel Duchamp

“The fourth dimension!” The mere phrase makes some small part of my brain shiver. And it’s not just me that feels that there’s something awe-full, and maybe awful, about the fourth dimension. 16th-century scholar Gerolamo Cardano¹ said it was contrary to Nature. 17th-century thinker John Wallis² found the very idea of it monstrous. And now 21st century virtual-reality pioneers are about to bring it into your home, if you want it there.

One of the most famous monsters inhabiting the fourth dimension is the tesseract, which you might have heard mentioned ever-so-briefly in the recent film “A Wrinkle in Time”. The movie is based on the classic book by Madeleine L’Engle, which has introduced generations of children to the idea of dimensions beyond the three that we see around us.

Who Knows Two?

All the work they made them do was rigorous.

— Exodus 1:14

As I write this, it’s April 5, midway through the eight-day festival of Passover. During this holiday, we Jews air our grievances against the ancient Pharaoh who enslaved and oppressed us, and celebrate the feats of strength with which the Almighty delivered us from bondage — wait a minute, I think I’m mixing up Passover with Festivus. In any case, on the first two nights of Passover, Jews tell the story of the Exodus from Egypt before dinner, reading from individual copies of a book called the haggadah (“the telling”), and after dinner they sing a few extra songs, one of which is called “Ekhad Mi-Yodeah?”, or “Who Knows One?” It’s a cumulative song, with each verse containing one more line than the verse before, and in my family the game is to try to get through the increasingly long middle part of each verse without taking a breath. That’s tricky when you get to the last verse, whose English translation is

Who knows thirteen? I know thirteen!
Thirteen are the attributes of God;
Twelve are the tribes of Israel;
Eleven are the stars in Joseph’s dream;
Ten are the commandments;
Nine are the months to childbirth;
Eight are the days to circumcision;
Seven are the days of the week;
Six are the order of the Mishnah;
Five are the books of the Torah;
Four are the mothers of Israel;
Three are the fathers of Israel;
Two are the tablets of the covenant;
One is our God in heaven and earth.

But what does it mean to know a number?

The Genius Box

Mr. Jabez Wilson laughed heavily. “Well, I never!” said he. “I thought at first that you had done something clever, but I see that there was nothing in it after all.”

“I begin to think, Watson,” said Holmes, “that I make a mistake in explaining.”

— Arthur Conan Doyle, “The Red-Headed League”

When I was a kid living in the Long Island suburbs, I sometimes got called a math genius. I didn’t think the label was apt, but I didn’t mind it; being put in the genius box came with some pretty good perks. For one thing, the kids who thought I was a genius at math had lower expectations of me when it came to other things, so to the extent that I was endowed with a normal measure of certain traits (like a healthy sense of humor), people were impressed by how normal I was — and to the extent that I had deficits of athleticism or common sense, people tended to be forgiving: “What do you expect? He’s a genius!” (See Endnote #1, though.)

There were other kids I heard about, most of them living in New York City and attending Stuyvesant High School and the Bronx High School of Science, whose mathematical achievements were described in awed tones on the high school math grapevine, and I envied their endowments and their successes, but I didn’t think they were geniuses either. They knew lots of things that I didn’t know — things I wanted to know — but they didn’t know things in a different way than I did. The things those kids knew weren’t facts, but rather habits of thought: the kind of habits that won you top honors in math competitions and science fairs.

I got acquainted with some of those kids in my junior and senior years of high school, and they were awesomely good at solving math problems, but there was no uncanny aspect of their performance that seemed worth explaining away by appealing to some mystical inborn attribute like “genius”.