At some moment in the past five years I realized that math, in addition to being the way I earn my living, is something even more important for my happiness: it’s a consolation for living in a world without magic.
By “magic” I don’t mean make-believe magic, the sort you’ll see in a stage-show; I mean real magic — the kind that you read about in books when you were a child, and hoped to encounter someday, but gradually stopped believing in as, year after year, the world you inhabit failed to deliver any evidence that magic existed.
And by “math” I don’t mean the inflexible cookbook math that too often is the only kind children learn; I mean the game that research mathematicians get to play, where they break those cookbook rules, adding new spices to old recipes and inventing entirely new ones.
Lots of people (most notably Martin Gardner and more recently Arthur Benjamin, Persi Diaconis, Ron Graham, and Colm Mulcahy) have written and talked about the links between math and magic tricks, but hardly anyone talks about the way that math, for many people who do research in it, satisfies a craving for the fantastic that most of us haven’t outgrown (even if we’ve persuaded ourselves that we have). Indeed, I think that most children get glimpses, all too easily forgotten, of math as a wondrous ticket to other worlds.
My goal in Mathematical Enchantments is to reawaken in my readers this childlike relationship to the subject, and to make this view of math enticing and even natural. And if you are an actual child, or an actual mathematician, and your sense of mathematical wonder is already awake and active, all the better! There’ll be lots of new games you can play. These things are fun, and fun is good.
I’ll do my best not to assume that you have much mastery of math beyond number-sense and basic algebra. But I’ll make heavy demands on your imagination, and your willingness to bend and stretch your mind into new shapes, or even to try to split your mind into two parts (more on this soon!). And I’ll assume that you’re curious about new ways of looking at familiar things, and bold enough to dance with new ideas that initially seem quite crazy, temporarily letting go of old ideas that get in the way of the dance (the letting go is usually harder than the dancing).
Do you want to see a geometry in which there’s a single point at infinity? Or another geometry that has infinitely many of them? Do you want to see a number system in which infinity is equal to its own negative? And another number system in which plus infinity and minus infinity are two different numbers? Are you intrigued by the prospect of a logic in which “It is false that P is false” is subtly different from the proposition “P is true”? Do you want to see a way of thinking about turn-taking games in which it makes sense to say you’re ahead by exactly half a move? Or a geometry in which you have to make two full turns to be back in the orientation you started in? Or a geometry in which you change size, Alice-like, as you move through space? Do you want to learn how to count the elements of infinitely large sets, and learn why some of them, in a certain sense, have a negative number of elements? Then you’ll probably enjoy this blog.
I may occasionally touch on some ways in which math is relevant to our day-to-day world, such as the shape of snowflakes (why do the symmetrical ones all have six arms, never eight?), or beer cans (why did Budweiser redesign their can?), or traffic jams (what simple step could all of us take to make traffic jams evaporate more quickly?), or converting between miles and kilometers (how can memorizing the first few Fibonacci numbers help you do these conversions in your head?) — or the way that some of the mind-bending possibilities I mentioned in the last paragraph are actually covertly at work in our world.
But really, my true passion is not the ways mathematics helps us in this world, but the ways mathematics takes us out of it, expanding our imaginations beyond the pull of gravity and even the shackles of space and time. If you want to know why math is useful, read the recent books of Jordan Ellenberg and Steven Strogatz, and the many fine books by authors who preceded them. But if you ever get tired of living in just one world, and our world in particular, and secretly pine for a passport to other, weirder realms, I hope to be your monthly tour-planner.
In this blog you’ll meet some kids’ games that are as fun (and challenging) for grownups as they are for children, a forgotten fractal from the nineteenth century, a simple model of sand piles that will probably perplex researchers for decades to come, high-dimensional spaces and the things that live there, and dozens of other fruits of the mathematical enterprise. And if you’ve ever wondered what the deal is with .999… (with infinitely many 9’s after the decimal point, whatever that means!), well, I’ve got a whole bunch of ways to look at that one to offer you.
I’ll also tackle some deeper issues, such as, what is the nature of this wondrous mathematical realm? If it’s “out there”, why can’t we see it? Or, if the facts of mathematics are all in our heads, why are they so obdurately unaffected by our wishes? And why do so many mathematical heads contain the same mathematical facts, if these facts are just figments? What do mathematicians mean by “proof”, and why do they care so much about it? How does the human mind engage in mathematics? How should mathematics be taught? How do researchers in pure mathematics decide what projects to spend their time on, and when they are engaged in those projects, what the heck are they actually doing?
One model for what I’m trying to accomplish is the writings of Martin Gardner. Some other models are … well, actually, I’m not going tell you; I’d much rather imitate these writers in hope that you’ll notice the resemblance and figure it out. That’s a game I’ll be playing with you over the next few years. In the meantime, I’ll mention that not all of my role-models wrote about mathematics or even about science.
I’ll close with a 1948 poem by Clarence Wylie whose final line compresses into ten syllables a surprisingly large chunk of what I’ll be trying to convey in these many dozens of blog posts:
PARADOX by Clarence Wylie:
Not truth, nor certainty. These I forswore
In my novitiate, as young men called
To holy orders must abjure the world.
“If …, then …,” this only I assert;
And my successes are but pretty chains
Linking twin doubts, for it is vain to ask
If what I postulate be justified
Or what I prove possess the stamp of fact.
Yet bridges stand, and men no longer crawl
In two dimensions. And such triumphs stem
In no small measure from the power this game,
Played with the thrice attenuated shades
Of things, has over their originals.
How frail the wand, but how profound the spell!
P.S. Go to http://mathenchant.org for audio versions of the blog, as well as a half-hour video of mine that briefly treats many of the themes I’ll be exploring here in the years ahead.