With what I’m writing today, I’m reminded of Demitri Martin’s comment on bumper stickers. To paraphrase (but still use quotation marks because relax, English teachers): “A lot of people hate bumper stickers, but I kind of like them because they simplify things for me with people. I see bumper stickers on a person’s car, and they all say this: ‘Hey, let’s never hang out.’” I feel like getting all psyched about all of this and nerding it up has been the two-part equivalent of slapping a bunch of activist bumper stickers all over myself. But what are you going to do? I thought it was cool.
I mentioned when I wrote about Everything and More, this book on infinity by David Foster Wallace, that it was challenging me in good ways and making me think. I thought I’d jump in this time with some specific things that I thought were sort of head-exploding and interesting and—dare I say it?—fun for me to read and think about. Feel free to pull a Lifetime1 and read something else if this sounds sort of blech to you. Don’t worry, we’ll still hang out as long as you’re willing to.
Now then. We start with Zeno. Wallace describes Zeno2 as “the most fiendishly clever and upsetting Greek philosopher ever,” and even though the number of personal relationships I have with classical Greek philosophers is pretty small (and shrinking what feels like daily), I get the sense that this might be true. Zeno devised a few of these great and very famous paradoxes, by all appearances for the sole reason of annoying everyone. One of them looked very much like this, only pre-modern, probably more verbose, and in Greek.
Suppose you are on the sidewalk somewhere—let’s say the huge intersection at Shibuya in Tokyo, just for kicks (but here’s the fun part: you can use your street if you want!)—and you’re waiting for the walk signal so you can cross the street. When the signal changes and all the cars stop, you have half a minute or so to make your way to the other side. If it takes much longer than that, you’re going to either annoy a bunch of people, or get smacked by (a) car(s), or both. Here’s the problem. Before you can get all the way across the street, you have to get to the halfway point, right? Now the trip can be thought of as two “subtrips,” if you will. One from Sidewalk1 to the median (assuming there is one) and one from the median to Sidewalk2. But the kicker is that the same problem arises when you try to make the first subtrip (the one to the median): you have to get halfway there first, and halfway to that point. And so on, into infinitely smaller and smaller subtrips until it’s basically impossible for you to cross the street at all.3 (The upshot, of course, being that it’s also impossible to get a fraction of the way there and get in the way of a speeding black German sedan, which situation would likely be fatal. The additional down side being that you also can’t go anywhere to eat or get out of the rain, which situation Maslow reminds us is also fatal.) Of course, we know from many, many experiences crossing streets that it’s not actually impossible. In fact, the synpases successfully firing in Zeno’s brain to come up with this in the first place all tell us that, as Wallace puts it, “there has got to be something fishy about Zeno’s argument.” However, as he (Wallace, I mean) goes on to point out, and then explain, “[f]inding and articulating that fishiness is a whole other matter.” It’s also a whole other level of nerd which I won’t go into.4 Suffice it to say that this is one of the parts of this book that put some new wrinkles on my brain in the best possible way.
Here’s another. Remember repeating decimals? Things like 1⁄3 being represented as 0.333 or 2⁄3 as 0.666? Well, remember those, and then remember also that any digit is ten times larger than the one to its right. For example, take the number 555.55. The third digit is five, and that is ten times larger than the fourth digit, which is one half. OK. Armed with these things, we we enter the next puzzle, in which we will prove that .999 is the same thing as 1. Which practically, yes, is basically true as far as any of us actually care. But mathematically, that’s unsettling, right? (I assume that if you’ve read this far, then you’re engaged enough to at least humor me here.) It goes like this. Assume that x = .999 and that, therefore, 10x = 9.999. Then subtract x from 10x. 9.999 – .999 = 9, which means 9x = 9.0 and, therefore x = 1.0. But we just said at the beginning of this part that x = .999. Math strikes again!
The problem in both of these cases, of course, is that language can very easily come close to describing mathematical truths but can only sometimes really tighten down all the bolts, as it were, and make an airtight formula without being just incredibly and off-puttingly verbose (or what mathematicians, both amateur and professional, call “inelegant,” which is about the worst insult there is for them. Not the most creative bunch in the world…)
One more. And this isn’t so much a contradiction or anything as it is a cool illustration of a property of lines, points, and infinite quantities. Remember from early math (or don’t; whatever you like) that a point is a specific location with no size whatsoever and that a line goes through infinite points along a single plane. “Infinite points” means that a line one inch “long” goes through, or contains, just as “many” points as one that is one light year “long.”5 And here’s a cool illustration of that. If you were to make a perpendicular line from the base of a triangle (QR), up through the hypotenuse PQ, like this…

…then that line would go through only one point on the base and only one point on the hypotenuse. No leftovers or shortcomings or anything. Just one and one. And then suppose you drew a bunch of these lines, like this:

You can see how the distance between points on the hypotenuse starts to look longer than the distance between points on the base, right? But the thing is, if there are infinite points to draw the line from on the base (and there are), and every one of those points corresponds to one and only one point on the hypotenuse (which it must, or all sorts of nerdy people will come and find you and beat you with pocket protectors, psychological afflictions, and all manner of other cliches), it must mean that the number of points on both of these lines is equal. And that number, in fact, is the enigmatic ∞ that we’ve been concerned with for these couple of posts. Hopefully this gets itself into your head just enough that I don’t feel like such a weirdo. What do you think? Let me know.
I’d like to tell you that I’m done throwing this stuff at you and that I’ll just eventually finish this and you’ll never hear another word about it. And that might be true. But if this book keeps dropping things like this on me and ruining my stuff… Well, I can’t make any promises, I guess. Chances are, though, you’re safe, so keep reading. See you next time.
1 This is what it’s called when you’re flipping through channels and stop on something and then realize after a few seconds that ohmygoodness! you’re watching a Lifetime movie and then panic and flip to a different channel—any other channel!— faster than you ever thought possible.
2 In this case, Zeno of Elea; I guess there were several significant Zenos in classical Greece. I was actually disappointed to look it up and find this out, because I really like some (not all) of what Zeno of Citium had to say re: worrying about stuff, and I thought it’d be cool if it was the same guy. The problem is that Zeno wasn’t too unusual of a name in Greece at the time, so it’s sort of like saying, “I really like William’s writing!” Sure, the most famous William is probably Shakespeare, but you could be—and really are about as likely to be—talking about, for example, Jennings Bryan, Wordsworth, or F. Buckley, or any of a gaggle of other Williams that weren’t among the first three to pop into my head just now.
3 This applies to any motion (touch your finger to your nose, or kick something, for example), and it brushes up against a series that you end up seeing a lot when you’re reading about infinity in general, which is [1⁄2, 1⁄4, 1⁄8, 1⁄16, 1⁄32...].
4 “More nerdy than this?” you say. Yes. Remember that we’re dealing with the infinite here, and just like arithmetic tells us that for every integer n there must always exist another, greater integer n+1 (and here’s another infinity-related thing to deal with), I would remind you that for every quantity of nerdy—I’ll use a capital N here—there will always exist another, greater quantity N+1. If you don’t believe me, come to Redmond some time and I’ll give you a tour of my workplace.
5 I’m using quotations marks here because there is a math-enabled person somewhere who will come after me for being inaccurate or “not telling the whole story” with the words I choose, and I’m making a command decision to initiate a pre-emptive strike, thus.