Something very much like nothing anyone had ever seen before came trotting down the stairs and crossed the room.
“What is that?” the Duke asked, palely.
“I don’t know what it is,” said Hark, “but it’s the only one there ever was.”
— James Thurber, “The 13 Clocks”
Why was a Cambridge University Fellow and Lecturer named John Conway, on an unremarkable day in the late 1960s, lying on his back, waving his feet in the air, and giggling?
To be fair, it was a decade in which many people did crazy things. In the U.S., Conway’s fellow-academic Timothy Leary was giving LSD to Harvard undergradutes, while some of Conway’s fellow-Liverpudlians, talented lads who called themselves The Beatles, were causing musical mania on both sides of the Atlantic. But Conway’s performance was for an audience of none (not counting himself), and the thing that had caused him such hilarity was neither a drug nor a catchy melodic hook but a psychedelic mathematical insight — specifically, the realization that, in an arcane but rigorous sense, four times four equals not sixteen but six.
I just went through my lesson plan for an upcoming lecture on number-sequences and replaced the name “Fibonacci” by the name “Hemachandra”. By the time you finish reading this essay, you’ll know why I did it, and if you’re a teacher, I hope you’ll do it too. [Note added on November 19: I might now go back again and change “Hemachandra” to “Virahanka”; see the Endnotes.]
from historical sources, the tale starts with the ancient Indian poet and mathematician Pingala (a contemporary of Euclid’s, give or take a century). For Pingala, these numbers arose from exhaustive consideration of the rhythmic possibilities of Sanskrit poetry. If you want a six-beat poetic phrase built out of short (1-beat) syllables and long (2-beat) syllables, how many possibilities are there? The answer turns out to be 13, so that’s the sixth term of Pingala’s sequence. Likewise, if one is playing the tabla, there are 13 different six-beat drumming patterns one can build from 1-beat and 2-beat components. (The 1-beat and 2-beat components are often rendered vocally as “dhin” and “dha” respectively, so that the two most dissimilar six-beat patterns would be the leisurely “dha, dha, dha” and the rapid-fire “dhin-dhin-dhin-dhin-dhin-dhin”.)
Last week my daughter asked me about weird bases. “Do bases have to be integers?” “Do they have to be real?” (She’s heard about complex numbers such as i, the infamous square root of minus one.) Then she asked “How would you write 256 in base i+1?”
She swears that she made up the question on the spot, but the answer is suspiciously nice, as we can see by starting with i+1 and repeatedly squaring. If we multiply i+1 by itself we get (i+1)·(i+1) = i·i + i·1 + 1·i + 1·1, or –1 + i + i + 1; the –1 and the 1 cancel, so we get (i+1)2 = 2i. Square both sides: (i+1)4 = (2i)(2i) = (2)(2)(i)(i) = (4)(–1) = –4. Square again: (i+1)8 = (–4)(–4) = 16. Square one last time: (i+1)16 = (16)(16) = 256. So 256 equals 1 times (i+1)16, plus 0 times (i+1)15, plus 0 times (i+1)14, plus …, plus 0 times (i+1)1, plus 0 times (i+1)0, and we conclude that the base i+1 representation of 256 is 1 followed by sixteen 0’s: 10000000000000000. (What are the chances?)
I posted this on Twitter, and someone wrote “Impressive; how old is she?”. My daughter just turned 13, and 13 is (16)+(–4)+(1) which equals (i+1)8 + (i+1)4 + (i+1)0, so I wrote back “She just turned 100010001.”
But then I got to thinking: do I really know how to write every positive integer in base i+1? Or did I just get lucky?
This past summer the Journal of Humanistic Mathematics published a revised version of my essay “The Genius Box“. In the original 2018 version I had asked “What are we doing when we call someone a genius?” and I had tried to show the ways in which having a special category of people called geniuses is harmful. At the journal’s request I added some new material to the published version, and put in a short new section called “Myth and Countermyth” that showed how one version of The Genius (the lightning-fast thinker) can give way to an antithetical version (the slow, deep thinker) without really fixing the problem with certain people being called geniuses in the first place.
While putting the finishing touches on the published version, I came across a relevant quote from a major twentieth-century physicist who was on a first-name basis with most of the people hailed as geniuses in the twentieth-century physics community. Here’s what he said about the pioneers of quantum physics and about himself:
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