Making Sense of Things
Morning. The trees peering sleepily into their mirrors to see how they look in their new leaves.
There are six ways to order three different objects; twenty-four ways to order four. I had forgotten how to figure that out – I came late to that branch of mathematics, and it doesn't stick very well – but I was able to think my way to the algorithm, using my old programmer habits (start with the simple cases, and look for the patterns) and visualizing the objects as a row of buttons. I still can't remember the names for these things. But the formula is enchantingly simple. The number of ways to order 3 things is 3 x 2; the number of ways to order four objects is 4 x 3x 2, five is 5 x 4 x 3 x 2, etc. The solution came to me so easily that I must have remembered something about it. I wonder what. Those ghostly memories fascinate me. Did I remember only that the solution was simple and easy to remember? Did I remember an image of a row of buttons? Or did I only remember that the problem had a solution, that I once knew? That would be enough to keep me confidently gnawing at it till I got it.
This is how I thought about it. Two buttons – lets call them Fred and George – can only be ordered two ways: either Fred comes first or George comes first. So now we're going to introduce a third button. Call her Amy. Say Fred and George are standing like this:
Button Amy has three different places she can stand: before Fred, in between Fred and George, or after George.
(Amy?) (Fred) (Amy?) (George) (Amy?)
So there are three possible orders with Fred and George standing this way. But Fred and George could also stand the other way around, with George first. In that case Amy would, again, have three different choices about where to stand, making six possible orders (3 x 2) in all. And whenever you're going to add a button, it will be just the same. If you have a 20 buttons, there will be 21 slots the new button can take, so the formula for the number of orders of 21 buttons can take will be 21 x (however many orders the other 20 can take.)
Easy to fill in the parenthesis. You'd figure out 20 buttons the same way: 20 x (however many orders the other 19 can take): so it's 21 x 20 x (however many orders the other 19 can take.)
And you figure out 19 the same way, of course. 19 x (however many the other 18 can take), which gives us 21 x 20 x 19 x (however many the other 18 can take). At this point of course the pattern is obvious. And now you've solved the problem for any number of buttons, be it billions, or trillions, or even a large number, such as the number of pennies in the national debt. Lovely!
I didn't learn this stuff until I was in my thirties – I have a rather odd educational history – so I remember clearly my delight in it when I first came across it. And the counter-intuitive knack, which you use all the time in calculus, of deferring the working out of the actual solution until after you've understood the pattern. The pattern is (the number of slots) times (the number of orders of the rest). Once you understand that clearly, the only problem you actually have to solve, at the end of the series, is: how many orders can two objects take? And even my soggy old over-fifty brain can do that by dead reckoning. It's Fred first and then George, or George first and then Fred. No other option. Two different orders.
This all started because I was walking down the street and saw a business logo – four squares arranged in a big square, with one letter in each square – and I wondered which way you were supposed to read them: right to left and then top to bottom? Or top to bottom and then right to left? And then my eye started playing with different paths, and that brought me to wondering: how many different paths through these letters are possible? How many orders are there, anyway?
I frowned, thinking, I know I knew this once. So I set to work to recover the knowledge, and after walking a couple blocks I had it back.
My mind does this sort of thing all the time. It's always ferreting about for intelligibility, playing with patterns. Letters and numbers magnetize me, fascinate me. There are few pleasures more intense to me than that of figuring out a simple solution to something apparently confusing and complex. Knowing enough of Old English, Old Norman French, and the laws of phonetic change that English spelling actually makes sense is a daily gratification. I find the sort of mind that can rest easy with a spelling such as “knight” baffling. How can you accept such a thing? How can your mind not ache? How can it not ask: what 'k'? why is the 'i' long? what are that 'g' and 'h' doing there? How can this have seemed like the right way to spell this word to somebody?
Even more baffling to me: how can you be proud of knowing how to spell the damn word, without understanding why it's spelled that way? How can you lord it over someone who tries to spell it in a way that actually makes sense? Someone who writes “nite” is at least using systematic rules, is bringing his or her intelligence to bear on the problem. Every day I read people sneering at examples “illiteracy” which seem to me far more admirable than the obedient memorization of a set of opaque facts.
I don't want to just know the right answers. I want to understand the answers.
It all comes down a respect and tenderness for other human beings. An Anglo-Norman monk sat in his English scriptorium, on a day when the new leaves were just coming out, faced with an impossible task. He was keeping legal records, depositions: he was supposed to write down, for the sake of our Lord Christ, the noises the Saxons made in saying their word for the seigneur -- that click at the beginning of the word, the cat-like hiss in the middle. What to do? Well, 'k' would do for the click, but what about that horrible throat-clearing sound? 'g' at least put it at the back of the throat. Throw in an 'h' to suggest the hissing. What else can you do? He dipped his pen in the ink and carefully began to write.
The man did his best with what he had. A few centuries later the click and the hiss were gone, worn away with the rub of the years: the only sign of them left was the lengthening of the 'i' into a diphthong – what we call “long 'i'” – to compensate for the lost hiss. But the nonsensical spelling was enshrined in writing now. It was “correct,” and everybody had to use it, other people just as hapless, children sitting in schoolrooms on days when the new leaves were just coming out, and they ached to be outside playing.
It's not the rules, not the rules that count. It's not the correctness. It's trying to make sense of things. And the new leaves. Those are what count.