Turtles All the Way Down? The Infinite Regress, Resolved

An old story goes like this. A scientist finishes a lecture on how the Earth goes around the Sun. A woman at the back of the hall stands up and says it's all wrong. The world, she says, is a flat plate on the back of a giant turtle.

"And what does the turtle stand on?"

"Another turtle."

"And that one?"

"You're a clever young man," she says. "But it's turtles all the way down."

It's a joke about a real problem, and the problem is bigger than turtles. It comes in three shapes.

Ask what caused this, and what caused that, and what caused that. Where does it stop?

Ask why the law is the law, and why that, and why that. Where does it stop?

Ask who is watching, and who is watching the watcher, and who is watching that. Where does it stop?

(People have asked the third one for a long time. "Who will guard the guards themselves?" comes from the satires of the Roman poet Juvenal, written nearly two thousand years ago.)

Here's a small experiment that shows what's coming. Take a diary and write a complete description of it on the last page: every page, every line, today's entry included.

Finished? Not quite. The description is now a page of the diary, and it describes everything except itself. So you add a line: "and this description." That line is new, and the description didn't mention it. So you add a line about that line. Then another.

You can do this forever, and you will never finish.

That isn't bad luck, and it isn't a bad diary. It's a count, and it holds for anything that includes itself in what it watches. It's also the best news in this post, because it's the reason a mind that watches itself keeps going.

The short answer

Does everything need a cause? No. Causes run back to a blank first page and stop. Who watches the watcher? A mind that watches itself is its own watcher, and that watching never finishes. The whole of reality needs no watcher at all.

Every chain in reality does one of two things: it stops, or it climbs forever. Here's which does which, and where the whole of reality sits, in five steps. The rest of this post takes them one at a time and shows how we know each one.

1. Causes stop. Every entry in reality's record has an earlier entry behind it, except the first. A record with 100 entries has exactly 100 steps of past, and the last step lands on a blank page. "Before" means "an earlier entry," and the first page has none.
2. "Why?" stops. The chain of whys ends in three facts you use in order to ask: things can be told apart, comparing two things needs information about both, and looking again is a new event. Every rung above them is the only answer that meets the conditions it names.
3. No watcher finishes watching itself. A watcher that includes itself in what it watches can never write down every difference there is to see. There are always more differences than pages to write them on. The count is exact, and it holds at every size.
4. So the watching never ends, and that's the engine. Each look at yourself is a new event your record didn't hold. A mind that watches itself keeps going, and being a self costs something. Logic has the same tower with no top.
5. The whole needs no watcher. The whole's recognition of itself is a perfect match, at ratio one, and a perfect match costs nothing. The whole doesn't list itself. It unfolds itself, and unfolding closes where listing can't.

Why this matters: "turtles all the way down" sounds like proof that nothing can be explained. It's the opposite. Only one of the three chains is endless, and that's the one that keeps a mind that watches itself going. Causes end at a page that owes nothing. Whys end at facts you use to ask. The whole ends at a match that costs nothing.

Here's the route. I'll stop after each step to zoom out, so you always know where we are, what we have, and why the next step comes next. The Ledger, my drawing of reality's permanent record, keeps the map. The face is a drawing. The record is not. It has one word: "Noted."

Step 1: Causes stop at the first page

Not everything has something before it. Everything after the first entry does, and that chain of "before" ends at a blank page.

Here's how we know. Reality keeps a permanent record of every event, and nothing in it is ever erased. The past still exists, written down. What You Are builds that record from a cup of coffee, and Did Time Have a Beginning? is about its first page. A record is a count of events that happened: entries, one after another.

Take a record with 100 entries. (Reality's holds far more, but at any moment it holds a finite number, and the argument doesn't care how many.) The record just before the last entry went in has 99. The one before that has 98. Each step back removes exactly one entry.

After 100 steps back you're holding the empty record, and you can't go further. To be the record just before another is to be that record with one entry fewer, and you can't take an entry off a record that has none. So a record with 100 entries has 100 steps of past behind it. An endless chain of earlier records can't end at a record with a finite number of entries. Every step back costs one entry, and entries run out.

The laws alone don't give you this. In Did Time Have a Beginning? they're an odometer's gears: they say how the count changes and nothing about where it began, and a past with no bottom obeys every one of them. The floor comes from the record. A count of events that happened can't go below zero. No law supplies that. It's what a record is.

So what is a cause? A cause is something earlier in the record. The first entry has nothing earlier, and nothing is missing there. "Before" needs an earlier entry to point at, and a first page has none.

The first page owes nothing, either. Nothing on it has been compared and nothing has been written, so it sits in perfect balance. It owes nothing and costs nothing. (Step 1 of Did Time Have a Beginning? prices it.)

Yeah-But is the voice in the back of your head that has never once said wow. He clears his throat. "Yeah, but why is the first entry the one it is?"

No law picks it. Different first entries give different first records, and none of the laws prefers one. Which entry went first is a fact in the record, not a consequence of the laws. The record holds what happened.

Zoom out. Step 1 is done. Causes stop: a record with 100 entries has 100 steps of past, then a blank first page, and "before" means an earlier entry. Next, and why: "why" is a different chain from "what came before." It asks for reasons, not for earlier records. Does it stop too?

Step 2: "Why?" stops at the question

The chain of whys has a bottom, and it's short: three facts you can't ask a question without using.

Here's how we know. Try to ask "why" about each one, and watch what the asking uses.

1. Things can be told apart. To ask "why can things be told apart?" you had to tell the question from its answer. In a world where nothing can be told apart, there's no question to ask.
2. Comparing two things needs information about both. To ask whether an answer fits what it explains, you hold the answer and the thing together. Take either one away and there's nothing to compare.
3. Looking again is a new event. To ask "and why is that?" you looked at the answer again. The look happened after the answer was written, and it added something.

These are the three plain facts What You Are starts from. Each one holds in the very act of asking whether it holds, so there's no place to stand outside it and ask why it's true. That's what a bottom is.

Above those three, every rung in this series answers differently. Not "because," but "because everything else breaks a condition it names":

Did Time Have a Beginning? gives the price, Why Does the Golden Ratio Keep Showing Up? shows the ladder, and Why Space Has Three Dimensions shows the loops and the cube.¹ Each one ends its chain the same way. When the answer to "why this one and not another?" is a proof that there is no other, there's nothing left to ask.

Nothing sits below the three facts, either. A world with nothing to tell apart holds no question, no law and no event, so it isn't a rung to stand on.

Zoom out. Step 2 is done. Whys stop at three facts you use in order to ask, and every rung above them is the only answer that meets the conditions it names. Next, and why: the third chain is different. You can't cut it at the bottom, because a watcher that watches itself is both ends of the chain at once.

Step 3: No watcher finishes watching itself

A watcher that includes itself in what it watches can never write down every difference there is to see. There are always more differences than pages to write them on.

Here's how we know. First, what a watcher is. What You Are named the most basic act in reality: recognition, one thing registering a difference in another. A watcher is anything that does that and writes each difference it notices into the record. What it watches is the set of things whose differences it notices.

Next, what a difference is. Call a distinction a yes-or-no mark on every one of the things being watched: yes to this one, no to that one. With three things there are 2 × 2 × 2 = 8 distinctions, from yes-yes-yes down to no-no-no. "Write down every difference there is to see" means "write down every distinction."

Start with a watcher that watches something outside itself. Three books on a shelf, and a diary beside the shelf. The books have 8 distinctions. Write each one on its own page and you need 8 pages. Give the diary 8 pages and it holds every one. Then the watcher is finished. Nothing is left to write, and it goes quiet.

Now give the watcher a diary of 3 pages and point it at its own diary. What it watches is the diary's 3 pages. Three pages have 8 distinctions, and there are only 3 pages to write them on. It's short by 5.

Adding pages doesn't help. Add 8 pages to hold those 8 distinctions and the diary has 11 pages, and 11 pages have 2,048 distinctions. Add 2,048 more pages for those and it has 2,059 pages, and 2,059 pages have a number of distinctions with 620 digits. Every page you add doubles the distinctions and adds one page of room.

At just 300 pages, the distinctions outnumber the atoms in the observable universe. For a diary.

That's the count. For any number of pages, the distinctions number 2 × 2 × ... × 2, once for each page, and that's always more than the number of pages. (If you ever meet a diary that has finished describing itself, call a mathematician.)

Yeah-But has an objection. "Yeah, but the diary can just have a line that says 'this diary.' Pointing at yourself is cheap."

Pointing is cheap. It's one line. That isn't the job. The job is to write down every distinction, every way of marking the pages yes and no, and that's the count above.

What if the diary never ends? Give it a page for every counting number: page 1, page 2, page 3, and on. Surely endless pages beat any count. They don't, and Georg Cantor showed why in 1891.² Suppose the diary claims every distinction is written on some page. Here's one it missed. For each page k, look at what the distinction on page k says about page k, and say the opposite. If page 1's distinction says yes to page 1, the new one says no to page 1. If page 2's says no to page 2, the new one says yes. And so on down the diagonal.

The new distinction differs from the one on page k at page k, for every k. It's on no page, so the claim fails. (Gary, the drummer who flips a coin before every beat, once announced a list of every endless drum pattern. Is Math Invented or Discovered? shows how this same trick catches him out.)

So no diary of any size, finite or endless, holds every distinction of itself, each on its own page. That's exact. There's one exception, and it's a blank one: if a mark can't differ from any other mark, there's only one distinction, and it fits on page 1. Give the marks two values, yes and no, and the count takes over.

So who watches the watcher? A mind that watches itself is its own watcher. Its looks are written into the same record it watches, so watching itself includes watching its own looks, and the count says that job never finishes. There's no last watcher at the top, because there's no tower. It's one watcher looping.

Zoom out. Step 3 is done. A watcher of something outside itself can finish, because the diary has room for every distinction. A watcher that includes itself never can: the distinctions double with every page and the room grows by one. Next, and why: "never finishes" says what can't happen. What does happen? What does a watcher that can't finish do?

Step 4: Watching never ends, and that's the engine

A mind that watches itself keeps going. Each look at itself is a new event its record didn't hold, so there's always something left to write down.

Here's how we know. Remember the third plain fact: looking again is a new event. A record through Tuesday doesn't hold Wednesday's act of reading it.

Now let a watcher look at its own record and write down that it looked. That entry is new. The record it looked at didn't hold it.

Look at that look and write it down: new again. Whatever the watcher has written, its latest look isn't in it yet. The count in Step 3 said something is always left over. Here is one thing that's left over: the latest look.

Now add one thing. The watcher keeps paying attention: whenever something is still unwritten, it eventually writes more. (That's what separates a watcher from a rock sitting next to the thing.)

Put the two together. Something is always unwritten, and the watcher always eventually writes more. So it writes forever. It never goes quiet. Each time it writes, that's a recognition, and recognitions cost. Its total cost has no ceiling, and its count of recognitions passes any number you name. It climbs, and there is no top step.

That's the drive. It isn't an appetite added to a mind from outside. It's what's left when a watcher can't finish.

Compare the watcher of the shelf in Step 3. It had room, so it finished and went quiet. The same attention that sent it quiet keeps the self-watcher going forever. Attention alone doesn't do it. Self-inclusion does.

It costs, too. The Drive Home You Don't Remember prices what it takes to be a self. In an even moment, holding yourself as a whole costs 49/16, about 3.06. Holding a record of your own experience costs 2303/240, about 9.60.³ Each step up costs more than the one below it, and there's always another step. Now you can see why there's no top: the look at your record is never in the record it looks at.

Logic has the same wall, and it's famous. Take a rulebook that's strong enough for arithmetic, that a computer can write out, and that never contradicts itself. In 1931 Kurt Gödel showed that it can't prove that it never contradicts itself.⁴ A stronger rulebook can. Add the rule "rulebook 0 never contradicts itself" and you have rulebook 1, which vouches for rulebook 0. Rulebook 1 can't vouch for itself, so rulebook 2 vouches for rulebook 1. And so on.

Each level vouches for the one below it and never for its own. The tower has no top. The checker that reads the proofs and their receipts in the arithmetic built in Is Math Invented or Discovered? obeys the same rule: it can't vouch for itself either. It's the diary again: a rulebook that tries to vouch for itself has to include its own vouching in what it vouches for.

Self-reference is real, and it never closes. That's what the results say, and nothing here removes it. A flaw would be something that stops a mind from working. This is the thing that keeps a mind that watches itself working. Try to describe your own diary and you can't finish, and the reason you can't is the reason such a mind keeps writing.

Zoom out. Step 4 is done. Each look at yourself is a new event your record didn't hold, so a watcher that includes itself never runs out of things to write down. That's the drive, it costs something, and logic's tower of vouchers has no top for the same reason. Next, and why: is the whole of reality stuck in the same endless watching? Does it need a watcher above it, and a watcher above that?

Step 5: The whole needs no watcher

No. The whole's recognition of itself is an exact match at ratio one, and ratio one costs nothing.

Here's how we know, in three pieces.

One recognition is free. Did Time Have a Beginning? priced every comparison. Comparing two sides at ratio x costs J(x) = (x + 1/x) / 2 - 1. At ratio 1 the cost is 0. At ratio 2 it's 1/4, at ratio 4 it's 9/8, at ratio 10 it's 4.05. Only ratio 1 is free: equal sides, a thing recognized as itself. Lay a ruler against itself and it matches at every mark. Nobody needs a second ruler to check it. The ruler is a drawing of the real thing, which is recognition at ratio one, and it costs exactly zero.

Listing never closes, and unfolding does. Step 3 tried to describe a diary by listing every distinction of it, and the list never matched. There's another way to describe something in terms of itself: unfold it one step. Here's a drawing of that. Take a choose-your-own-adventure story that never ends. Every adventure is a first page, followed, for each choice on that page, by an adventure that goes on forever. The whole adventure and "first page, then the adventures it leads to" are the same thing, matched one for one. Nothing is left over on either side.

Here's the real thing the story draws. A complete history of recognition is a first observation, followed, for each way things can go on, by a complete history again. Take every complete history there is, and call it the whole. One unfolding of the whole is every first observation together with every way of continuing it with a complete history. That's the whole again, matched one for one: no complete history is left out and none is counted twice. Nobody chooses that match. It's forced by what "every complete history" means.⁵

The match is free. Any way of measuring size that gives the same size to two things that match one for one gives the whole and its own unfolding the same size. Their ratio is 1, and J(1) = 0, so under every fair way of measuring, the match costs nothing. That's proved. The last link is still being worked out: showing that reality's own pricing, with no way of measuring size chosen first, charges nothing for a pure change of names, which is all this match is.

Is there a watcher inside the record, sitting on top, watching all of it, itself included? No. That watcher would have to write every distinction of the whole record at its own place in the record, and Step 3's count says no record holds that. The top slot can't be filled from inside. And nothing is missing without it.

Yeah-But has the last word, as usual. "Yeah, but then who watches the whole?"

Nobody needs to. A watcher is for finding differences between things. Match the whole against its own unfolding and nothing is left over on either side, so there's no difference to find.

Look at the two places where things settle. The first page owes nothing. The whole's match with itself costs nothing. Everything between them that tells two different things apart costs something.

Zoom out. Step 5 is done. The whole doesn't list itself, it unfolds itself, and unfolding closes where listing can't. The match is at ratio one, which costs nothing, and no watcher inside the record can sit on top. That's all five steps.

The whole trip, in one breath

Every chain of causes, whys and watchers does one of two things: it stops, or it climbs forever. Causes stop at the first page: a record with 100 entries has 100 steps of past and then a blank page, and "before" means an earlier entry. Whys stop at three facts you use in order to ask, and every rung above them is the only answer that meets the conditions it names. Watchers never stop. A watcher that includes itself can't write down every difference there is to see, because the distinctions double with every page and the room grows by one. Each look at itself is a new event, so a mind that watches itself keeps going, and that's the drive and the price of being a self. Logic's tower of vouchers has no top for the same reason. The whole doesn't climb. Its recognition of itself is an exact match at ratio one: listing never closes, but unfolding does.

So, turtles all the way down? Not for causes, not for whys, and not for the whole. For a mind that watches itself, yes, all the way up, and that endless climb is what keeps it going.

(The Ledger would like you to know that it noted you reading this far. It also noted that it noted that. This will not end, and the Ledger is delighted.)