Sorry Doesn't Rewind the Tape

You promised to feed your neighbour's cat on Monday. You forgot.

On Tuesday you turn up with double food. By Friday the cat is glossy, enormous, and behaving as if Monday never happened, which is what cats do best. Your neighbour comes home to a perfect cat.

So: is Monday fixed? Is it, in the end, as if it never happened?

Most of us feel that it is, or nearly. "No harm done" is what people say after a repair, and the more thorough the repair, the more the whole thing feels like it never happened. A perfect repair feels like rewinding a tape and recording over the bad bit.

It isn't that. Reality isn't a tape. It's a ledger, and a ledger only takes new lines. The reason is exact enough to put numbers on.

The short answer

No. Monday is not fixed, and nothing you do on Tuesday can fix it. The past still exists. Monday's empty bowl is written down, and nothing that happens afterwards reaches back to change it.

What Tuesday does change is what happens next: whether the cat gets through the week. One more thing can change after the fact, and that's what you owe. If your neighbour says "forget it," the debt is gone. The miss is still there.

Here's the whole answer in five steps. The rest of this post takes them one at a time and shows how we know each one.

1. The ending forgets. Two histories can finish in exactly the same state, with the same payments made, and still differ in what happened on the way. Only the history has the difference.
2. Zero is not safe. Paying what you owe and finishing with nothing left is not the same as being fine. Going on takes a margin above zero.
3. Repair the future, not the past. After a miss, no way of finishing the story makes the whole story right. A repair that puts everyone back on their feet does exist, and the miss stays written beside it.
4. Forgiveness is an event. Two exchanges can have identical accounts and different debts: 1 owed in one, nothing owed in the other. Forgiveness is a new entry that changes what's owed, and in this exchange only the person who is owed can write it.
5. Making it right has one shape. When a correction is owed between two people, there is exactly one way to carry out just that correction, and nobody else is touched.

Why this matters: everyone has broken something and tried to put it right. "Making it right" sounds like a feeling with no edges. It has edges, and you can count them. It can change what each person has, what is owed, and how lopsided the relationship between two people is. It never changes what happened.

It also shows what forgiveness is. Forgiving has something to work on only because the past is still there. If sorry were a rewind, there would be nothing left to forgive.

If you haven't read Part 1 of The Universe Writes in Pen, here's the one idea you need. Reality keeps a permanent record of every event. Nothing in it is ever erased, and no new event reaches back and edits an old one. We call the record the ledger, and in the drawings it's a character called the Ledger: unflappable, extremely thorough, with one response to everything that has ever happened, Noted. This post puts that to work on the most ordinary thing people do, which is mess up and try to make up for it.

Here's the route. I'll stop after each step to zoom out, so you always know where we are and why the next step comes next. The Ledger keeps the map.

Step 1: The ending forgets

Two histories can end in exactly the same place and still be different histories. A rule that looks only at the ending can't tell them apart.

The cat story is the everyday version. The exact version needs a world small enough to check with a pencil, so: two people, Alice and Bob.

Alice has promised to feed Bob, so that he keeps going. Bob counted on her, so he's running low: he spent his last interval working instead of feeding himself. Here are the rules of their world.

• Each of them has a reserve: what they have left to keep going on. (The drawings show it as a fuel gauge.) Alice starts with 2. Bob starts with 1/4.
• Every interval costs each of them a quarter unit of upkeep, just to stay in working order.
• A meal costs Alice a quarter unit of work and gives Bob half a unit.
• At the end of every interval comes a checkpoint, where the promise is checked. To pass, both of them need more than zero left.
• The example runs for four intervals, and Alice can cook at most two meals in them.

Picture an interval as a day. In each one, Alice either feeds Bob or waits, which means she doesn't cook.

The numbers are this example's own, chosen so the arithmetic fits on a napkin. The results below are exact for this world. Real life has more people, more promises and numbers that come out of real bodies, and carrying the exact results over to all of that is work still ahead.¹

Run two histories. In the first, Alice feeds in interval 1 and waits in interval 2. In the second, she waits and then feeds.

Interval 1, Alice feeds. She pays a quarter for upkeep and a quarter for the cooking, so her 2 becomes 3/2. Bob gets half a unit from the meal and pays his quarter in upkeep, so his 1/4 becomes 1/2.

Interval 1, Alice waits. She pays only the upkeep, so her 2 becomes 7/4. Bob gets no meal and pays his quarter, so his 1/4 becomes 0.

Interval 2 works the same way. Here are both histories side by side:

Look at the last column. After interval 2 the two histories are identical: Alice has 5/4, Bob has 1/4, one meal has been eaten, and each of them has paid a quarter in upkeep twice.

Now look at the middle column. After interval 1, Bob has 1/2 in the first history and nothing in the second. Zero doesn't pass the checkpoint. The first history meets checkpoint 1 and the second misses it.

How we know. Check every number with a pencil. The rest is one plain fact about rules: a rule that reads only the ending gives the same answer to the same ending. These two endings are identical and the two verdicts are different, so no rule that reads only the ending can get both right. The same goes for a rule that reads only the list of upkeep payments, because that list is identical too. What separates the two histories is when Bob had what he needed.

The Ledger doesn't have this problem. It keeps every event in order, so it keeps the part the ending forgets.

Zoom out. Step 1 is done. A history holds facts its ending loses, and the Ledger keeps the history. Next, and why: the number that decided checkpoint 1 was Bob's reserve, and it read zero. Zero looks like a number you can shrug at. It isn't.

Step 2: Zero is not safe

Going on takes more than zero left. A paid bill isn't enough.

Look at Bob's first interval in the second history. He starts with 1/4. His upkeep is 1/4. He pays it, and he's back to even: nothing is out of balance, so there's no strain left anywhere. A meter that reads only strain says everything is fine. And he has exactly nothing left. (In this world, zero means nothing left for the next interval. The world says no more about it than that.)

Technically paid. Now look how thin that is. Run one more interval with no meal. The Bob who was fed in interval 1 has 1/2, pays his quarter and still has 1/4. The Bob who wasn't has nothing to pay with, so his upkeep goes unpaid and the strain stays on him: a quarter.

It doesn't even take a missing meal. Go back to Bob's first interval: he had exactly the 1/4 he owed. Say a hundredth of a unit had gone missing before the upkeep was due. He'd have been a crumb short, and the bill would have bounced. Give Bob a margin, some amount left over after the bill, and he can absorb a crumb up to the size of the margin. With a margin of exactly zero, no crumb is small enough.

How we know. It isn't a quirk of these numbers. In this world, going on through every checkpoint is exactly the same thing as having a buffer at every checkpoint: some positive amount you can lose, at every interval, and still pay. A reserve of exactly zero has no buffer, so any loss, however small, breaks it. That's why the checkpoint asks for more than zero and not just enough.

The everyday version: anyone who has ended a month at exactly zero knows this. Every bill paid, and one surprise fee away from a bounce.

Zoom out. Step 2 is done. Being fine isn't "the bill is paid." It's "something is left." So Bob at zero is a miss, even though every payment went through. Next, and why: a miss is written down. Can anything Alice does next fix it? That's the question in the title.

Step 3: Repair the future, not the past

After a miss, no way of finishing the story makes the whole story right. A repair that puts everyone back on their feet exists anyway.

Back to the second history. Alice waited in interval 1. Nothing in physics stopped her: waiting was possible. It was also a miss: the promise failed its first check. Bob reached checkpoint 1 with zero, and checkpoint 1 is written down.

Yeah-But, the small voice in the back of your head, has been clearing his throat. "But Alice fixed it in interval 2!" Look at what's actually on the table. There are two questions.

Can Alice finish the story so that all four checkpoints come out met? No. Checkpoint 1 is already written, and it reads zero. Whatever Alice does from here gets added after it, and nothing added after a line changes that line. So the list of ways to finish the story with all four checkpoints met is empty. Not short. Empty.

Can Alice finish the story so that Bob is back on his feet and every checkpoint still ahead is met? Yes: feed, feed, wait. Alice feeds in interval 2 and again in interval 3, then waits in interval 4. That's the exact version of "double food on Tuesday": two meals in a row. Run it:

Checkpoints 2, 3 and 4 are all met. And look at the ending: Alice 1/2, Bob 1/4, two meals eaten. That is exactly the ending of the history where nobody slipped.

So both of these are true at once, in one history. The repair worked, and the miss happened. Neither cancels the other. They're two lines in the Ledger: Noted. Also noted. If Alice says sorry, that's a third line, added after both. Noted.

Here's the everyday version. Gary the drummer has a good night. He misses beat one of a four-beat bar and nails beats two, three and four. Everyone in the room still heard the gap where beat one should have been.

How we know. The arithmetic is in the table. Here's the rule behind it. Counting the meals Bob has had so far, he needs at least 1 by checkpoint 1, at least 1 by checkpoint 2, and at least 2 by checkpoint 3 and again by checkpoint 4. Waiting first gives him 0 by checkpoint 1, and that's the miss. The repair gives him 1, 2 and 2 by checkpoints 2, 3 and 4, which is enough.

The empty list is plain logic: a later move can't change the reading at an earlier checkpoint. And the repair is a real, possible way to finish from where Alice and Bob stand. Repair looks forward from the state things are in. It never had a way to look back.

Zoom out. Step 3 is done. A miss is permanent and repair is still possible, and both sit in the same history. Repair changes what each person has from here on. Next, and why: what is owed is a different fact from what each person has, and it has its own rules. That's where forgiveness lives.

Step 4: Forgiveness is an event

Forgiveness is a new event that changes what is owed. It changes nothing about what happened.

Yeah-But again: "So forgiving is just forgetting, politely."

No. Forgetting would be an edit. Forgiving is an entry. Here's the exact version, in the same world, on a different afternoon.

Alice delivers a meal to Bob on credit. She offered it, Bob accepted, and she delivered it. The price is 1, to be paid later, so Bob now owes 1.

Each of them keeps their own account of it, which I'll draw as a notebook. Alice wrote "Bob: 1" in hers. Bob accepted the deal, but he never wrote anything in his.

Now two worlds. In World A nothing else happens. In World B Alice says "forget it" and releases the whole 1.²

Open the notebooks. They're identical in the two worlds, page for page: Alice's says "Bob: 1" and Bob's is blank. But what's owed is 1 in World A and 0 in World B.

How we know. A rule that reads only the notebooks has to give the same answer for two identical sets of notebooks, and the right answers here are different, so no such rule can know what's owed. That's exact: no rule, however clever, that reads only the notebooks gets the amount right in both worlds.

What does tell the worlds apart is what happened, written down event by event. In this exchange, five kinds of event decide what's owed: an offer, an acceptance, a delivery, a payment and a release. Once the meal is delivered, what's owed is the price, minus what's been paid, minus what's been released. In World A that's 1. In World B it's 1 minus the 1 that was released, which is 0. A release can be partial: if Alice releases half, what's owed is 1/2.

The release is a new line at the bottom of the Ledger. It doesn't edit the offer, the acceptance or the delivery. Alice still offered, Bob still accepted, and the meal was still delivered. Forgiveness changed the total at the bottom and left every line above it alone. The Ledger writes it in the same ink as everything else.

What it does to the bond. The ledger writes every relationship as a bond: a ratio that reads 1 when the two sides are even. Strain costs. A bond at 1 costs nothing, and a bond at 2 costs 1/4. A bond at 1/2 costs the same 1/4, because the cost doesn't care which direction you're lopsided.

Each notebook starts every account at 1, a baseline this example chooses. Alice's "Bob: 1" sits on top of that, so her side reads 1 plus 1, which is 2. Bob acknowledges nothing, so his side still reads 1. The bond is the ratio of the two sides, so it reads 2, and it costs 1/4. That's the same quarter as the upkeep: the ledger has one cost curve. The release itself leaves the notebooks alone, so Alice's still says "Bob: 1." Once she corrects it to match what's owed, the bond reads 2 minus the amount released: 3/2 after half, and 1 after the whole thing. After a full release the strain on it falls from 1/4 to nothing.

Who can write the release. In this exchange, only Alice can, because she's the one who is owed. If Bob writes "I release myself," the Ledger notes it and what's owed stays at 1. If Bob writes "I owe nothing" in his own notebook, his notebook changes and what's owed doesn't. That's a rule this exchange is built with. Deriving it from deeper physics is still being worked out.

"Release" here means one thing: the person owed wrote it. Whether they wrote it freely and knowing everything is a different question, and this result doesn't settle it.

The same shape holds for a promise. After Alice skips the first meal, Bob can release her from the rest of the promise. That clears what's required of Alice from then on. It doesn't cook the missed meal, and it doesn't erase the missed checkpoint: Bob's reserve stays where the omission left it.

Zoom out. Step 4 is done. Forgiveness is a new event that changes what is owed, even though the notebooks don't change when it's written, and in this exchange only the person owed can write it. It leaves what happened exactly as it was. Next, and why: the last piece. When a correction is owed between two people, what does carrying it out look like, and who else has to change?

Step 5: Making it right has one shape

When a correction is owed between two people, there is exactly one way to carry out just that correction, and nobody else's bonds change.

Take two people and the two bonds between them, one in each direction.³ Say Alice's bond to Bob reads 1/2 and Bob's bond to Alice reads 2. They're lopsided in opposite directions, and the gap between them is a factor of 4, because 2 divided by 1/2 is 4. Say what happened shows that this gap is owed a correction: it has to close.

What does closing it take? Multiply Alice's bond by some number a and Bob's by some number b. Two requirements decide a and b.

• Keep the pair's total. A correction moves things from one side to the other. If it also added to the pair or took something away from it, that would be a second act, with its own questions about who owes it. Bonds are ratios, so the pair's total is its two bonds multiplied together, not added: here 1/2 times 2, which is 1. Keeping that total means a times b is 1.
• Close the gap. The two bonds have to end up reading the same. Alice's starts at 1/2 and Bob's at 2, so Alice's has to grow by a factor of 4 more than Bob's. That means a divided by b is 4.

Only one pair of numbers does both. Multiply the two requirements together and you get a times a equals 4, so a is 2 and b is 1/2. Double Alice's bond, halve Bob's, and both read 1.

Meet Carol. She has a bond of her own, reading 2, that has nothing to do with this pair. Before the correction it reads 2. After it, it still reads 2. Carol notices nothing, and that's the point: nobody outside the pair is touched.

How we know. The two requirements leave exactly one answer, and the arithmetic above is the whole argument. It works the same way for a correction of any size: one bond takes half of it and the other takes the opposite half. (Halving a multiplication means taking a square root, so half of a factor of 4 is a factor of 2, because 2 times 2 is 4.)

Corrections behave like entries. Two corrections on the same pair add up. An opposite correction undoes one, as a new line after it, and both lines stay written. A correction of nothing changes nothing.

This is justice, in the exact sense: the correction that is owed, carried out exactly, and borne by no one else.

How much is owed comes from what happened, which is one more reason the past has to stay. How that amount turns into this correction is a link still being worked out. The shape itself is exact.

Carrying out just the correction that is owed has exactly one shape, and nobody else's bonds change.

The whole trip, in one breath

The past still exists, so nothing you do afterwards rewinds it. Not sorry, not a perfect repair, not forgiveness: each one is a new line at the end of the Ledger. A history holds facts its ending forgets, like whether Bob had enough at checkpoint 1. Zero is not safe: going on takes a margin above zero. After a miss, no way of finishing the story makes the whole story right, yet a repair that puts everyone back on their feet exists, and the miss stays written beside it. Forgiveness is an event: it changes what is owed while the notebooks stay the same, and in this exchange only the person owed can write it. And when a correction is owed, carrying out just that correction has exactly one shape, and no one else's bonds change.

So, is Monday fixed? No. Monday is exactly where you left it, in pen, and so is Tuesday. That's the bad news and the good news in one sentence: the miss is permanent, and so is the repair. Nothing you do right is erased either.

That's how reality is built. The Ledger has no rewind button. The past can't be edited, and the only thing the Ledger ever does is take a next line. Every apology, every repair and every "forget it" ever said is one of those lines, and every one of them is still there.

Next time: what does love do to a lopsided bond? The cheapest way to even two people out, while keeping what they share, has exactly one answer, and it changes no one outside the pair.

(The Ledger has noted that you read to the end. You can't un-read this post now. Sorry. That doesn't rewind it either.)