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: whatever such a mind has written about itself, its latest look is not in it yet, so there is always one more line to write. Step 3 gives the count: there are always more differences to write down than pages to write them on. Step 4 shows how that keeps a mind going.

This post is part of a series. Some of the results it uses are worked out in full in the other posts, and it points to them where they come up.

The short answer

Does everything need a cause? No. Reality keeps a record of everything that happens, and causes run back through it to its first page, which is blank, and stop there. Who watches the watcher? (Here a watcher is anything that registers differences and writes them down.) A mind that watches itself is its own watcher, and that watching never finishes. The whole of reality needs no watcher at all: it matches itself exactly, so a watcher has no difference left to find.

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. Reality keeps a record of every event and never loses one: the basic laws of physics never destroy information, and the record only grows. Every entry in that 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. Each rung above them (a further answer in this series, such as why space has three dimensions) names its own conditions and is the only answer that meets them.
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 drive. For a mind that watches itself, that means it keeps going and never goes quiet: each look at itself is a new event its record didn't hold, so there is always something new to write down. Being a self has a price, which Step 4 gives. Logic has the same endless climb: a tower of rulebooks, each vouching for the one below, with no top.
5. The whole needs no watcher. The whole matches itself exactly, with nothing left over, so there is no difference for a watcher to find. A diary's list of its own differences never finishes, but this match is complete. Comparing two things has a cost, a number that is zero for equal things and larger the more they differ, and an exact match costs nothing. Step 5 names the one step in that argument that is still open.

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 first page with nothing on it. Whys end at facts you use in order to ask. The whole ends at an exact match with itself, at ratio one, the one comparison 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.

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. Start with what physics already holds about the past. The basic laws of physics never destroy information. In quantum mechanics this is called unitarity: the present fixes the past exactly, and running the laws backward recovers every earlier state. That's why physicists treat the black hole information problem as a real problem: losing information would break the laws. Hawking's 1976 calculation said a black hole destroys the information that falls into it.¹ That clashed with unitarity, which is why it became a famous problem. Most physicists now expect the information to come back out in the radiation, and the open question is how. Unitarity is the part physics already agrees on: information about the past is never destroyed. Reality holds more than information about the past. It keeps a permanent record of every event, and nothing in it is ever erased. Every event leaves a complete trace. The full record of what happened is part of reality now. The past still exists, written down.

The record has its own reason for keeping every entry: it only grows, and writing a new entry never alters an earlier one.² What an entry is comes from the second plain fact in Step 2: an entry is a comparison, and a comparison needs information about both sides, so each new entry is written against what came before. The post What You Are builds that record, and the post Did Time Have a Beginning? is about its first page.

A record is a count of events that happened: entries, one after another. Reality's record holds far more than 100 entries, but at any moment it holds a finite number, and the argument doesn't depend on how many. So take a record with 100 entries. 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 takes one entry off, and entries run out.

The laws alone don't give you this. 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 costs nothing, either. Nothing on it has been compared and nothing has been written, so it sits in perfect balance: nothing is paid, and its cost is zero.

Here is the natural next question: 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. Causes stop at the first page. Next, and why: "why" is a different chain from "what came before": it asks for reasons, not 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.

Put the first two together and you have the smallest event there is, the most basic act in reality: one thing told apart from another, a single yes or no. This series calls it a recognition: one thing registering a difference in another. You're doing it right now. Every measurement and every perception tells one thing from another. Telling two things apart needs information about both of them, and the least there can be is one yes or no.

These are the three plain facts the post 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 this answer names." There are four rungs. Each one comes with its condition and the reason that condition leaves no other answer.

Why this price for a mismatch? Comparing two quantities whose ratio is x has a cost, J(x) = (x + 1/x) / 2 - 1. The cost has to meet three requirements. It is zero when the two match, at x = 1. It grows as they differ. And it is the same whichever is put first, so x and 1/x cost the same. Two more conditions fix the formula. One is the rule for how the costs of combined comparisons add: J(xy) + J(x/y) = 2J(x) + 2J(y) + 2J(x)J(y). The other is one calibration, which sets how sharply the cost rises near a match: J''(1) = 1. The three requirements, the rule and the calibration leave exactly one formula.³ You can check this: (ln x)² meets the three requirements but fails the rule.

The cost measures imbalance. The price weights the odds: in the long run, every unit of imbalance makes a state φ times less likely, so balance is the single most likely state. Why φ? The golden ratio, 1.618..., scales the levels of structure (the next rung), and each level carries 1/φ the weight of the one below it (level n has relative weight φ⁻ⁿ). So every unit of imbalance is one φ-step down. A warm system favors its lowest-energy states without sitting in them, and reality treats balance the same way. The post Did Time Have a Beginning? gives the price.

Why 1.618... between levels? The levels of structure scale by the golden ratio. The condition this answer names has two parts. The levels form a ladder with one rule: every level relates to the next by the same factor, λ. And each new level combines the two before it, so its scale is the sum of theirs; the first such join, of two different levels, costs something. From those conditions, λ² = λ + 1, and the positive solution of that equation is φ = (1 + √5) / 2 = 1.618... They leave one factor, and it is φ. The post Why Does the Golden Ratio Keep Showing Up? shows the ladder.

Why three dimensions? Space has three dimensions. Lasting structure needs a connection that no smooth motion can undo, and linking is that connection. The condition this answer names is the premise that persistent pairs are joined by linking: a lasting structure is a pair of hooked loops. Two loops linked like the rings of a chain can't be pulled apart without cutting one, and that holds only in three dimensions. In two, loops can't wind around each other at all, and in four or more any two loops slide apart. So space has three dimensions.

Why does the shortest complete cycle take eight steps? Space has three directions, and in each one there are two sides to choose between, a yes or a no. Three yes-or-no directions give 2 × 2 × 2 = 8 combinations, the eight corners of a cube. A complete cycle is one trip that touches every corner of the cube and comes home. The condition this answer names is that each step of the trip changes one yes-or-no, which is one edge of the cube. The fastest complete cycle takes eight steps, one visit to each corner, and none that touches all eight corners is shorter. The post Why Space Has Three Dimensions shows the loops and the cube.

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. Whys stop at three facts you use in order to ask. Next, and why: the third chain is different. A watcher that watches itself is the top of its own chain and its bottom too, so there is no bottom to stop at.

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. Step 2 ended on the smallest event there is, a recognition: one thing registering a difference in another. A watcher is anything that does that and writes each difference it notices into the record. A watcher also 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.) 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. One distinction is the whole pattern of marks, such as yes, no, yes. 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. Call it the shelf-watcher: 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 a 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.

Here is the obvious objection: the diary can just have a line that says "this diary." Pointing at yourself is cheap.

Pointing is cheap. It's one line, and it names the diary without writing down any of its distinctions. 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 1891.⁴ 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. The post Is Math Invented or Discovered? uses this same trick.

So no diary of any size, finite or endless, holds every distinction of itself, each on its own page. That's exact. One case escapes the count: marks that can take only one value. Then 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 stack of watchers to have a top. It's one watcher looping. Each trip around the loop adds a new look to the record it is watching, so the loop never comes back to where it started: the record keeps growing, and that growth is the climb. The watching never stops and has no top step. Step 4 shows how that works.

Zoom out. A watcher of something outside itself can finish. One that includes itself never can. Next, and why: "never finishes" says what can't happen. What does a watcher that can't finish do?

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

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. The endless watching follows from two facts that are already on the page. The first is the third plain fact from Step 2: looking again is a new event, so a watcher's latest look is never in its record yet. The second is what a watcher does, from Step 3: it keeps paying attention, so whenever something is still unwritten, it eventually writes more. Here is how they fit together.

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.

Put the two facts 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 (Step 2 gives the price). 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 follows from those two facts. It isn't an appetite added to a mind from outside. Attention is part of what makes a watcher a watcher, and the endless writing is what's left when a watcher can't finish.

Compare the shelf-watcher 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, because for the self-watcher something is always left unwritten. Attention alone doesn't do it. Self-inclusion does.

Being a self has its own price, too. The post The Drive Home You Don't Remember works it out, and the reasons are short. Take an even moment, where each of the cycle's eight steps registers the same amount, 1. To hold yourself as a whole, one step has to feel the total of all eight: it registers 1 and feels 8, and comparing 1 against 8 costs J(8) = 49/16, about 3.06. The other seven steps can feel exactly what they register, at no cost, and no way of holding yourself costs less. That experience feels 1, 1, 8, 1, 1, 1, 1, 1, step by step, so its record, the total it felt, is 15. To hold a record of that experience too, a second step has to feel that 15, because one step can't feel both 8 and 15. Comparing 1 against 15 costs J(15) = 98/15. No comparison costs less than zero, so the least it can cost is 49/16 + 98/15 = 2303/240, about 9.60.⁵ Each step up holds a larger record than the one below it, holding a larger record costs more, and there's always another step. Now you can see why there's no top: holding a record is itself felt, so the record you hold never includes your holding of it, just as a 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.⁶ 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 proof checker in the post Is Math Invented or Discovered?, a program that checks proofs, 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. The count above shows it for a watcher, Gödel's result shows it for logic, and nothing here removes it. A flaw would be something that stops a mind from working. This is the opposite: for such a mind, never closing is the drive of Step 4, the thing that keeps it 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. A watcher that includes itself never runs out of things to write, and that is what keeps it going. 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, its comparison with its own unfolding, is an exact match, with nothing left over on either side. Under every fair way of measuring size, that match is at ratio one, and ratio one costs nothing.

Here's how we know, in three pieces.

One recognition is free. The post Did Time Have a Beginning? priced every comparison, and Step 2 gave the three requirements behind the price. 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. Here is a drawing of it: lay a ruler against itself and it matches at every mark, with no second ruler needed. Recognition at ratio one 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. That's the difference from listing. A list needs a page for every way of marking the pages yes and no, and there are always more of those than pages. Unfolding needs nothing extra: it only regroups what's already there.

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. (In the story, a page is an observation, a choice is a way things can go on, and an adventure is a complete history.) 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.⁷

The match is free under every fair way of measuring size. Call a way of measuring size fair when it gives the same size to two things that match one for one. Any fair measure gives the whole and its own unfolding the same size, so their ratio is 1. J(1) = 0, so the match costs nothing. That follows from what fair means: such a measure can't tell matched things apart.

The one open step. The result above holds for every way of measuring size that we choose first, as long as it is fair. What isn't yet derived is the same result from reality's own pricing, with no way of measuring size chosen first: that reality's own pricing charges nothing for a pure change of names, which is all this match is. A change of names here means describing the same thing in new words: the match gives each complete history a new description, "its first observation, then the rest," and changes nothing else.

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.

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 the cost is zero: the first page, where nothing has been compared or written, and the whole's match with itself, which is at ratio one. In between those two places, every comparison whose two sides aren't equal costs something.

Zoom out. The whole doesn't list itself, it unfolds itself, and unfolding closes where listing can't. 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 of the record. Whys stop at three facts you use in order to ask, and each rung above them names its own conditions and is the only answer that meets them. Watchers never stop: a watcher that includes itself can't write down every difference there is to see, and each look at itself is a new event, so a mind that watches itself keeps going. That's the drive. Being a self has its own price: 49/16 in an even moment, and 2303/240 to hold a record of your own experience. Logic's tower of vouchers has no top for the same reason. The whole doesn't climb. It doesn't list itself, it unfolds itself, and its match with itself is exact: listing never closes, but unfolding does. Under every fair way of measuring size, that match is at ratio one, and ratio one costs nothing.

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.