Who Set the Dials on the Universe? For Four Laws, Nobody.

A chess knight moves in an L: two squares one way, then one square sideways.

Why?

Because somebody decided. The board doesn't care. Let the knight go three squares and then one sideways, and you'd still have a game. (Some chess variants have a piece that moves exactly like that. It's called a camel.) A different game, with different openings and different grandmasters, but a game.

Now take a stool with three legs. Put it on a tile floor, a gravel path, a lumpy lawn. It doesn't rock.

Why?

Nobody decided that, and nobody tuned it. Any three points that aren't in a line sit on one flat plane, so three feet always land together and there's nothing left to wobble. A four-legged stool is another story. Four points don't always sit on one plane, so one foot floats, the stool rocks, and somebody saws a leg until it stops.

Those are two kinds of rule. A knight's move is a rule somebody chose: change the choice and you change the game. A three-legged stool's steadiness is a rule nobody chose, because there's nothing to choose. Three points are on one plane whether you like it or not.

Now look at the laws of physics. Physics hands you a list of numbers: how heavy an electron is, how strong the electric force is, how many directions space has. Somebody measured or counted each one and wrote it down, and the list doesn't say why any of them isn't a different number. They look like dials. Sooner or later everyone asks the question in the title: who set them?

So which kind of rule is each law? A knight, or a stool?

The short answer

Some of them are stools. Four of them you can check by hand, and each is the only law that does its job.

The job is keeping a record of differences, and you're doing the first step of it right now: reading this tells one letter from another. Every measurement and every perception does the same. Telling two things apart is the smallest thing that can happen, and reality is made of differences that get compared and recorded. Four parts of that job each leave exactly one option:

1. One price for a difference. J(x) = (x + 1/x) / 2 - 1, where x is how many times bigger one thing is than the other.
2. One ratio between levels built by joining earlier levels. 1.618..., the golden ratio, written φ.
3. One number of dimensions in which a record holds. Three.
4. The shortest full cycle through three yes-or-no directions. 2 × 2 × 2 = 8 ticks, where a tick is one step in the writing of the record.

There's no dial behind any of these four. Each is what's left when every alternative that fails its job is crossed out, so there was nothing to set. (One convention sits inside the first: the size of its unit, like meters or feet. Step 2 shows where.)

What about the mass of the electron and the strength of the electric force? Those look like dials too, and they aren't on this list. Not yet derived: the step from this structure to the particular forces, their strengths and the masses of the particles. Derived, and shown below: the four laws.

Why this matters: a law usually arrives as a fact. Here's the rule, now see what follows. This post runs it backwards and asks why the rule is this one and not another. For these four, the answer is short enough to check on paper. It's also a method you can reuse: write down what the job requires, then cross out every law that fails. When one law is left, there was nothing to choose.

Here's the route. I'll stop after each step to zoom out, so you always know where we are, what we've got, and why the next step comes next.

Step 1: The job

Every law in this post answers the same question: what does it take to keep a record of differences?

The starting point is three plain facts, which What You Are develops:

1. Things can be told apart. Hot isn't cold. Here isn't there.
2. Comparing two things needs information about both. To know your coffee got colder, you need the earlier sip and the later one.
3. Checking a note is a new event. Remembering the earlier sip doesn't change it. It adds a new moment after it.

Telling two things apart takes information about both, and the least information there can be is one yes or no. So the smallest thing that can happen is one thing told apart from another: a single yes or no.

The record. One thing registering a difference in another is called recognition, and Recognition Science works out what it requires. Every recognition is written into one shared record of every event, and nothing in it is ever erased. Known physics already says this for the part it covers: its basic laws never destroy information. In quantum mechanics that is unitarity: the present fixes the past exactly, and running the laws backward recovers every earlier state. Our own work adds a second reason: the record only grows. It keeps every entry, which our work proves: writing a new entry never alters an earlier one.¹ What an entry is rests on a separate fact: each new entry is a comparison, and a comparison needs information about both sides, so every entry is written against what came before.

Counting comes out of the first fact too. Telling two things apart, again and again, is where the counting numbers and their rules come from, and Is Math Invented or Discovered? builds them.

Those three facts don't pick the four laws by themselves. Each step below states what a record of differences needs, in plain words, shows the one law that meets it, and then takes one requirement away to see what gets in.

Two words do the work in this post.

A dial is a number you can set to anything and still have a working world.

A requirement is a condition a law has to meet, written in words. You can't set a requirement to 1.7. Either a law meets it or it doesn't.

When many laws meet the requirement, somebody has to pick one, and that's a knight. When exactly one law meets it, there's nothing to set, and that's a stool.

Zoom out. Step 1 is done. The job is keeping a record of differences, and each step adds the requirements of one part of that job. A dial is a number you can set to anything. A requirement is a condition in words. Next, and why: the first part of the job is comparing. Before the record can write down how different two things are, it needs a price for the difference.

Step 2: One price for a difference

Comparing two things has a price, and exactly one formula meets the requirements.

Compare two things of the same kind, like two weights or two lengths. Let x be how many times bigger one is than the other. Equal things give x = 1. One twice the other gives x = 2, and read from the other end, x = 1/2.

The price of a comparison measures the imbalance between the two things: 0 when they match, growing as they differ. The price is

J(x) = (x + 1/x) / 2 - 1

Try it. At x = 1, the price is 0. At x = 2 it's (2 + 1/2) / 2 - 1 = 1/4. At x = 1/2 it's 1/4 again, so doubling and halving cost the same. At x = 4, which is two doublings, it's 9/8: 4½ times the price of one. Only equal things are free, and the price has no ceiling. Meet in the Middle: What Love Does works through the whole price list and uses it to price a lopsided friendship.

Where does the formula come from? Not from taste. Three requirements fix it.

1. Prices combine one way. Comparisons chain: A is twice B and B is three times C, so A is six times C. When two ratios x and y combine, by multiplying or by dividing, the prices of the two results are tied to the prices of x and y by one recipe, the same for every pair. The recipe is: price(x × y) + price(x ÷ y) = 2 × price(x) × price(y) + 2 × price(x) + 2 × price(y). Check it with x = 2 and y = 2. The product is 4 and the quotient is 1. The left side is 9/8 + 0 = 9/8. The right side is 2 × 1/4 × 1/4 + 2 × 1/4 + 2 × 1/4 = 1/8 + 1/2 + 1/2 = 9/8. They match.
2. No jumps. A slightly different ratio has a slightly different price.
3. One scale, and it's positive. A tiny mismatch is a small imbalance and costs a small positive amount, and we say exactly how small: a ratio of 1.01 costs about 0.00005, which is half of 0.01 squared. Part of this is a choice of unit, the way meters and feet are. Measure prices in units half the size and every price doubles: a ratio of 1.01 then costs about 0.0001, and the 2 in front of the product in the recipe becomes 1. The other three laws are a pure number and two counts, with no unit to choose.

Those three leave exactly one price. It's proved, and it's published in the journal Mathematics.² Two more things look like requirements, and they come free. Equal things cost nothing. And doubling costs the same as halving: put x = 1 into the recipe and it says price(y) + price(1/y) = 2 × price(y), so price(1/y) = price(y).

Take a requirement away and other prices get in.

No scale. The price J(x²) obeys the first two requirements. So does J(√x). They're the same curve, steeper or shallower. At x = 2, J(x) costs 1/4, J(x²) costs 9/8 and J(√x) costs about 0.061. These are different prices, not one price in other units: J charges 4½ times as much for two doublings as for one, and J(x²) charges 6¼ times as much. A change of unit can't do that. It multiplies every price by the same number, so it leaves those ratios alone, and it changes the 2 in the recipe, which J(x²) keeps. So the choice between J(x) and J(x²) isn't a choice of unit. With the recipe's 2 in place, the scale settles it: J(x²) charges about 0.0002 for a ratio of 1.01, four times the scale's 0.00005. The rule for combining prices, plus the scale, leaves exactly one price.

No positive scale. A wavy price that dips below zero obeys the first two requirements as well. (In symbols it's cos(ln x) - 1.) At x = 2 it comes to about -0.23: a negative price would pay you to be mismatched. The positive scale rules it out.

No combining rule. Here's a simpler price: take the gap between x and 1, square it, and halve it. That's (x - 1)² / 2. It never goes negative, it has no jumps, and it's right on the scale: a ratio of 1.01 costs exactly 0.00005. It fails the recipe at x = 2 and y = 2: the price of 4 plus the price of 1 is 9/2 + 0 = 9/2, but the recipe gives 2 × 1/2 × 1/2 + 2 × 1/2 + 2 × 1/2 = 5/2. And it charges 1/2 for doubling but only 1/8 for halving, so its price depends on which end you read from.

That's the price of comparing two things. Why Your Coffee Never Un-Stirs uses it to show that cost alone can't tell the past from the future.

Zoom out. Step 2 is done. Three requirements (prices combine one way, no jumps, one positive scale) leave one price, J(x) = (x + 1/x) / 2 - 1. Drop the scale, its sign or the combining rule and a different price gets in. Next, and why: the record builds bigger levels out of smaller ones, and building is comparing, so each build carries this price. That pins down how levels relate.

Step 3: One ratio between levels

Levels built by joining earlier levels climb by 1.618..., the golden ratio, and by no other number.

Reality has levels of structure, each built from earlier ones. Give each level a size, a measure of how much there is in it, and draw the levels as a ladder, one rung per level. Here are the requirements.

1. One rule. A size is how much there is, so sizes add: a stack of things of size 2 and size 3 has size 5. Going up a rung is the same step every time, and it respects that: stepping up a stack gives the same as stepping up each thing and then stacking them. It also keeps every comparison: if one thing was bigger than another before the step, it's bigger after. A step like that multiplies everything by the same number, r. (Squaring keeps every comparison but isn't such a step: a stack of 2 and 3 squares to 25, but 2 and 3 squared separately give 4 + 9 = 13.) Say rung 0 has size 1. Then rung 1 is r, and rung 2 is r × r. Any positive r makes a ladder, so this rule picks nothing by itself.
2. A join. Rungs 0 and 1 are the starting rungs. Reality builds each new rung out of rungs that already exist. The first new rung, rung 2, can only be built from rungs 0 and 1, since they're the only rungs there are, and joining means adding their sizes. A rung can be used twice, so there are three ways to pick the pair: rungs 0 and 0, rungs 1 and 1, or rungs 0 and 1.
3. The join is real. A join compares the two rungs it uses, and what the record writes is a difference: one thing told apart from another (Step 1). Comparing two different things has a price (Step 2), so joining two different rungs costs something. Joining a rung to a copy of itself has x = 1, so the price is exactly 0: nothing is told apart, and there is nothing to write. Two of the three picks join a rung to itself. Only rung 0 with rung 1 is real.

Now do the algebra. Rung 2 is r × r by the ladder's rule. Built from rungs 0 and 1, it's 1 + r. So

r × r = 1 + r

Only one positive number does that: r = (1 + √5) / 2 = 1.6180339... Check it: 1.618 × 1.618 = 2.618, and 1 + 1.618 = 2.618. The ladder is 1, 1.618, 2.618, 4.236, and from rung 2 up, each rung is the two below it added. Every join compares two rungs in the ratio 1.618, so each one costs J(1.618) = 0.118.

Nobody picked that number. We listed the ways to build rung 2, crossed out the free ones, and one way was left.

Take away "real" and two more ladders get in. Build rung 2 from rung 1 twice: r × r = r + r, so r = 2, and the ladder is 1, 2, 4, 8. Build it from rung 0 twice: r × r = 1 + 1, so r = 1.414..., and the ladder is 1, 1.414, 2, 2.83. Both joins are of equal parts. They cost 0, nothing is told apart, and the record writes nothing.

It isn't the cheapest ladder, and cheapness isn't the requirement. A cheaper ladder exists. It climbs by the plastic number, 1.3247..., and a join on it costs about 0.040. But its rung 2 isn't two earlier rungs added, so it breaks the join rule. The requirement is a real join, not a cheap one: a price doesn't pick what gets built, since writing nothing is cheaper than any entry and entries get written anyway.

Why Does the Golden Ratio Keep Showing Up? takes this step slowly, with every rule removed one at a time, and shows why any list of positive numbers that adds its last two lands on 1.618... from any start.³

Zoom out. Step 3 is done. A ladder with one rule, whose first new rung is a real join of two earlier rungs, climbs by exactly 1.618... Allow a join of equals, which costs nothing, and doubling gets in. Next, and why: a record has to hold, and whether a mark in the record can hold depends on how much room there is.

Step 4: One number of dimensions

Two loops can hook through each other in three dimensions and in no other number.

A record that holds needs a connection that no smooth motion can undo, and linking is that connection. Whether two loops can link depends on how much room there is, so the number of dimensions comes out of what the record needs. Here are the requirements.

1. A record holds. Nothing you do by sliding, bending or turning things, short of cutting one or passing it through another, wipes out what's written. That's Step 1's permanence, written as a condition on shape.
2. A mark and its reader are loops. A mark is something written in the record, and its reader is what registers it. Each one is a cycle that comes back to where it started, and in space a cycle is a loop. Lasting loops pair by linking, and that's the premise this step rests on.
3. Space is the plain kind. No edge and no handles, the way a balloon's skin has neither and a doughnut's has a handle.

Two links of a chain are the picture. Each is a loop, each passes through the other, and you can't part them without cutting one open. A hooked pair of loops is a mark that holds.

Now count the room.

Two dimensions. Picture the skin of a balloon. A loop drawn on it cuts it into two patches. A second loop can't cross the first, so it sits in one patch and shrinks to a dot. It was never hooked.

Four dimensions. Lift one loop of a hooked pair into the spare direction, slide it clear of the other, and set it back down. They're apart, and nothing was cut. (A dot inside a circle on a page can't get out without crossing the line. Lift it off the page and it can. That page is a drawing of the real mechanism, one dimension down.)

Three dimensions. There's no spare direction. Every move that frees one loop runs it through the other. Hooked stays hooked.

Here's the rule behind all three cases: two shapes hold each other when their dimensions add up to one less than the dimension of the space they sit in. A loop has dimension 1, so two loops hold when 1 + 1 = D - 1. That's D = 3: three dimensions, and no other number.

Take a requirement away and other numbers get in. Drop "a record holds" and four dimensions pass too, since nothing has to hook. Drop "a mark and its reader are loops" and let the reader be a bubble, the skin of a ball, with dimension 2. A loop and a bubble hold each other when 1 + 2 = 4 - 1, so it takes four dimensions. Every size of space has its own kind of hold. The loop requirement is what picks three.

Space has forward and back, left and right, up and down, and nothing else. The 3 isn't an input to the argument. It's what the requirements leave. Why Space Has Three Dimensions walks through every step.⁴

Zoom out. Step 4 is done. A record holds, its marks and readers are loops, and space is the plain kind. Then two loops can hook in exactly three dimensions. Let the reader be a bubble and four would do. Next, and why: three directions leave one more thing to count. Give each direction a yes-or-no and ask how long it takes to run through every case.

Step 5: One cycle

A full pass through three yes-or-no directions takes 8 ticks, and no fewer.

Here are the requirements. Each direction of space carries one yes-or-no, like left or right: the smallest event is one yes or no (Step 1), and space has three directions (Step 4). A tick is one step in the writing of the record, and on a pass each tick flips one yes-or-no. And a full pass is complete when it has touched every combination. That's what "full" means: no combination is left out.

Count the combinations. Three directions, three yes-or-no's: 2 × 2 × 2 = 8. Write yes as 1 and no as 0: 000, 001, 010, 011, 100, 101, 110, 111. Draw each as a corner of a cube and join two corners when they differ in one digit. A tick is one step along one edge.

A pass lands on one corner per tick. There are 8 corners and one landing per tick, so a pass takes at least 8 ticks. Is 8 enough? Yes. The cube has a route through every corner that comes back home: 000, 001, 011, 010, 110, 111, 101, 100, and back to 000. Eight ticks.

The 8 isn't a separate fact. It's Step 4's three directions with one doubling each: 2 × 2 × 2. With D directions the count is 2 × 2 × ... × 2, D twos in all: 4 ticks for two directions, 16 for four. It's 8 because space has three dimensions. The 8 counts ticks. It doesn't say how many seconds one tick lasts.

The same three sets how many electrons fill an outer shell. A filled outer shell is one s orbital plus a p orbital along each direction of space, and each orbital holds two electrons: 2 × (1 + 3) = 8. With D directions that count is 2 × (1 + D), while the cycle takes 2^D ticks. The two agree at D = 3 and at no other number: 2 × (1 + D) runs 4, 6, 8, 10 for D = 1 to 4, and 2^D runs 2, 4, 8, 16 and keeps doubling. Why Chemistry Runs on Eight works through the meeting.⁵

Take away "every combination" and a shorter cycle gets in. Flip one yes-or-no and flip it back, and that's a cycle of 2 ticks. It touches 2 corners out of 8.

Zoom out. Step 5 is done. One yes-or-no per direction, one flip per tick, and a pass that touches every combination: that's 2 × 2 × 2 = 8 ticks, the shortest full pass. Skip the "every" and 2 ticks would do. Next, and why: we have four laws and four sets of requirements. Now we can see what they have in common.

Step 6: Take one requirement away

Here are the four steps side by side.

Read the right-hand column. Every row is one requirement away from a different law. That's what "forced" means here. It doesn't mean nothing else can be written down. It means nothing else meets the requirement.

The world already shows two of these numbers. Space has three dimensions, and Step 4 gets three from the requirements. A filled outer shell holds eight electrons, and Step 5 shows that the shell's eight and the cycle's eight agree at three directions and at no other number.

Two of the steps also build on earlier ones. The real join in Step 3 is real because of the price in Step 2. The 8 in Step 5 is the 2 × 2 × 2 of the three in Step 4.

That's the difference between a dial and a requirement. A dial is a number you set to anything and still get a working world. Each number in the table is pinned by a sentence you can read and check. Change the sentence and you change the law, openly, and you know exactly which job you changed. Leave it alone and there's nothing to set. The size of the price's unit is a convention, and it's named as one.

The stool is a drawing of this. "Stand on any floor without rocking" is the requirement, and three legs are what it leaves, with nothing to tune. Allow rocking and four legs will do, or five, and now somebody has to choose. The knight is a rule with no requirement behind it.

Zoom out. Step 6 is done. Take any one requirement away and a different law gets in, so each of the four laws is what's left when its requirement is kept. A dial is a number set to anything. A requirement is a sentence that leaves one law. Next, and why: one thing a law can't do is say where the record starts.

One thing the laws leave alone: where the record starts

The laws don't say where the record starts. The rules for how the record grows work the same from any starting count. Think of an odometer: its gears run from any starting number, and the miles on it are a fact about the car, not about the gears. That's a drawing of the real split between laws and record. The laws are the gears. The record is the miles. Where the record starts is a separate fact, and Did Time Have a Beginning? takes it up.

The whole trip, in one breath

Reality is made of differences that get compared and recorded. Comparing needs a price, and the rule for combining prices, plus the scale, leaves exactly one price: J(x) = (x + 1/x) / 2 - 1. Levels built by real joins climb by one ratio, 1.618..., and a join of equals would have given doubling instead. A record that holds needs loops that hook, and space has exactly three dimensions where they do. A full pass through three yes-or-no directions takes 2 × 2 × 2 = 8 ticks. For each of the four, take the requirement away and a different law gets in, so there was nothing to set. A knight's L needs somebody to have decided it. A stool's three feet are on one plane whether anyone likes it or not. These four laws are the stool's kind.