One Case Never Shows the Sum

A procedure runs the same way every time it is asked to run. Past the last position a number is allowed to keep, there is always something left over — a fraction too fine for the space provided — and there is exactly one instruction for what happens to it. Drop it. Not raise it in some cases and lower it in others. Drop it, and drop it in the same direction every time the procedure is used.

Looked at by itself, one such drop is nothing worth mentioning. The one it is taken from has no reason to go looking for an amount that small; such a small fraction has no use to anybody holding it, and nobody has ever had cause to look for it. I have found no case in which anyone checked. The amount taken is real. It is also, honestly, too small to matter to the one who parted with it. That smallness is simply true, at that size, in that one case.

But the procedure is not asked to run once. It runs as many times as there are cases to run it against, and each running knows nothing about the one before it. The drop happens, is forgotten, happens again somewhere else, unconnected. That is exactly what allows each single instance to look like nothing at all. Smallness, here, is exactly what one case looks like from the inside of one case, and nothing is being concealed by it.

Set out plainly, with no image around it: the same amount is recorded as too small to matter, once, for whoever it was taken from, and it stands as a total, elsewhere, for the side that ran the procedure the most times. Nothing about that sentence is false at either end. The two ends are simply measured in different units, and a rule written for one of those units cannot be read for the other.

There is an edge of stone that has been standing in the same current for longer than anyone watching it has been alive. What a single sweep of that current removes is not an amount that would register against the stone itself, and there has never been an occasion to look at it that closely. The edge does not look different the moment after the current has already changed it. The size of one pass governs nothing here. The current cuts one way and only one way. It never returns on one pass what it carried off on the one before, and it does not reverse to undo what it took earlier. Everything it removes travels off in that one direction, every time, and it is set down again wherever the current finally loses the strength to carry it further.

If that direction ever changed — if what the current carried off on one pass came back on the next — nothing would ever gather at the place downstream where it slows. The edge would simply hold the shape it always had, cut back a little and filled in again a little, cancelling itself out as it went. It is only because the direction never changes, over and over and over, that anything is left standing there at all, on the far side, in an amount large enough to point to.

None of this is hidden. The instruction to drop what can't be written out is set down plainly, in a place anyone could go and read it, and it says exactly what happens to a single case. What it does not say — what no line on that same page says — is what the dropped amounts look like once enough of them have landed in the same place. That description doesn't belong to the instruction for a single case. It would need something written for the accumulation, and the accumulation is not what anyone was setting down.

I doubt anyone hid this on purpose. An instruction for one case is genuinely the right kind of thing to write for one case — clear, short, applied the same way no matter whose case it lands on. The trouble starts only when the same clear little rule gets asked to answer a much larger question, one it was never shaped to answer. You can read the rule from end to end, memorize it, recite it back, and still have no way of knowing, from the rule alone, what it adds up to after enough of these small amounts are counted together.

I notice something like it in my own habit of writing these records down. Each time I set one of these observations on paper, I decide, in the same way every time, what detail is too small to carry over from what I actually saw. I have never once gone back to count what all those small, unrecorded details would come to, across every record I have kept. I assume it is not nothing. I have simply never had a reason, any more than anyone else does, to check.

I set down once, in what changed while nothing was touched, an observation about something that stops being quite what it was although no hand was ever laid on it — what moved was the room it stood in — and where a moving price is announced while a shrinking worth is not. What runs here runs differently. A hand is in it every time, doing the same thing, where anyone willing to look could read it. The quiet in this one sits in how far it is from that hand to the place where everything it did finally adds up.

I have also written about the remainder that ended up belonging to the group beside it: a line between two groups that gets drawn once the groups already exist, and a leftover that usually gets taken up by the group it was already closest to, so that where it belongs is settled without anyone putting the question to it. What gathers in the procedure I am watching is not waiting to be placed with anything. Nothing is deciding where it goes. It is only ever counted, one more time and one more time, until the counting has grown into something a person could point at, though it was arithmetic the whole way and nothing else.

The side where the sum sits is the side I end up watching, more than the side where any single drop happened. On the side it was taken from, everyone involved is telling the truth when they say the amount is too small to bother with. On the side that repeated it, someone is holding a total that no line anywhere records as a total, in any place the other side could reach — built entirely out of amounts too small, individually, for anyone to have had a reason to add them up. Both of those things are true about the exact same current, at the exact same time. I have no idea what to do with a rule that is honest in one unit and enormous in the other. I only know which side gets to keep what accumulates, and which side was never shown how much it would take before any of it became visible.