The World Underneath the Labels

1. A name tag is a terrible lunch

You can write “sandwich” on a piece of paper. You can translate it. Give it a product code. Change its price. Put a very confident photograph of it in an app.

None of these operations feeds you.

These descriptions matter because they let people coordinate around something real. Their usefulness depends on keeping the right connection to it. An inventory system that counts the same meal twice can be beautifully designed and still leave somebody hungry.

When we say actual, we mean the necessary structure of the thing itself. That is what it consists of, what makes it possible, and what every adequate description must preserve.

That gives the word three commitments.

First, literal identity. The subjects are about reality’s own objects and their changes. Physics, meaning, ethics, and economics meet in the same recognition structure: the distinctions, comparisons, interactions, and records that section 2 sets out. An economic action and its ethical consequences concern the same people undergoing the same event.

Second, discovery through necessity. We ask what makes this structure unavoidable. Which conditions does the thing require? What alternatives do those conditions exclude? Where does the argument get its force?

Third, faithfulness across descriptions. Change the notation, units, language, accounts, or scale, and the relevant distinctions must survive. A new description cannot quietly lose the person who goes hungry.

This allows enormous freedom in how we describe the world. Two faithful maps can look very different. The restriction is that their differences cannot rewrite the terrain.

It also explains an apparent paradox: if the structure is necessary, how can there be so many legitimate ways to use it? Arithmetic permits a lot of shopping lists. Physical constraints permit different machines. A governing economic structure can permit different institutions.

Uniqueness of structure does not mean uniqueness of every choice.

2. Getting reality to show its working

Suppose you wanted to understand a game you had never seen before.

You could watch thousands of games and get very good at predicting what usually happens. You could also try to discover the rules that make a move possible. These approaches help each other. A prediction can reveal a pattern; a rule can explain why the pattern has to occur.

RS takes the second approach to reality, looking for its rules while keeping the answers accountable to observation.

We start with something exceptionally modest: some occurrences (things that happen) can be told apart.

This mark is different from that mark. Something happened here and then something happened there. A detector registered one result rather than another.

Next: to compare two occurrences, information about both has to be available. If the earlier information is completely gone, one side of the comparison is missing. So a trace of the earlier occurrence must survive long enough to be used.

And examining that trace is itself another occurrence. A photograph of yesterday does not contain today’s act of looking at the photograph.

These are the starting facts of recognition. Here, “recognition” has a concrete meaning: distinctions, comparisons, interactions, and records. It does not require a tiny person inside a detector saying, “Ah yes, a particle.”

Then comes the difficult part: follow the requirements without smuggling in the answer.

To get a rule for comparing quantities, we must say what a comparison does and how comparisons combine. To get a physical space, we must say what relationships it has to support. Each added condition needs a reason. Section 4 does both.

A forcing result is what we get when those conditions leave only one answer. Any alternative has to satisfy the same conditions. When none can, the choice has been removed.

This is how the discoveries connect. Construct the objects. Identify the operations. Establish what they require. Prove what survives a faithful change of description. Then check that the physical interpretation really refers to the world we observe.

There is no magic passage from “things differ” to every detail of the universe. The intervening arguments are the work. The established RS foundation consists of such arguments, with the conditions of each result kept attached.¹ The next five sections apply this method to mathematics, physics, meaning, ethics, and economics.

3. Mathematics before the fonts

Consider the number three.

You can write 3. You can write III. You can make three marks. The fonts disagree extravagantly. The count survives.

What is that surviving thing?

This is the direction of the δ (delta) program, our work on actual mathematics. Begin with distinguishable records and the operations that construct and combine them. Develop counting from those operations. Continue through the constructions needed for signed counts (negative numbers included) and ratios (fractions).

Then perform an especially useful check. Build the arithmetic in two different realizations (two ways of storing the marks). Prove that a statement is true in one exactly when its translation is true in the other.

Now the arithmetic is no longer resting on one favorite way of storing the marks. We have identified structure that both realizations carry.

That is a concrete meaning of “actual mathematics.” We are asking what the operations require before choosing a convenient costume for them.

The construction still states the logical rules it uses: equality, recursion, induction, and the others. Listing them is part of explaining where the result comes from. The established arithmetic result does not claim to manufacture all mathematics from two objects and a determined expression.²

The point is that the count belongs to the structure. The numeral is how we point to it.

4. Physics, with fewer places to hide a choice

Now compare two lengths.

Changing from centimeters to inches changes the numbers on the ruler, but not the ratio of the two lengths, their relative mismatch. Reversing which length you call the first turns the ratio upside down, into its reciprocal.

RS asks how to give such comparisons a consistent cost. Here “cost” is a measure of mismatch, a pure number with no units. A further physical connection is needed before it can tell us an energy in joules.

Many functions could measure mismatch, giving each ratio a cost. Asking for zero cost at a match, and the same cost for a ratio and its reciprocal, still leaves choices.

The discovery is that two more requirements remove those choices: the specified rules for combining comparisons (how the costs of combined comparisons add), and the calibration near a match (how sharply the cost rises there). Together they select one cost function: J(x) = (x + 1/x)/2 − 1, where x is the ratio. It is zero at a match (x = 1) and grows as the lengths differ: one length twice the other, in either order (x = 2 or x = 1/2), costs 1/4. Written out, the combining rule is J(xy) + J(x/y) = 2J(x) + 2J(y) + 2J(x)J(y). You can check that it does real work: (ln x)² is also zero at a match and the same for a ratio and its reciprocal, but it fails this rule. A second, connected derivation reaches the same cost function by a different route. It starts from the specified native comparison apparatus (the construction’s own instrument for comparing), the requirements on what that apparatus reports, and a reference calibration.

The cost function starts out arbitrary. The requirements force its form. That is what makes this a discovery of necessity.³

The same kind of reasoning applies to the hierarchy of recognition, its levels of structure. Three conditions force the scale between levels to be the golden ratio: an additive joining rule, preservation of positive comparisons, and a first costly join. Preserving positive comparisons makes each level the same multiple of the level below; call that multiple λ. The first new level joins two earlier levels, and the additive joining rule makes it their sum. That join has a cost, and joining a level with itself would cost nothing, so it joins the two different levels before it. A ladder that grows by a constant multiple λ, with a rung that is the sum of the two rungs before it, needs λ² = λ + 1, and the positive solution of that equation is φ = 1.618…

Elsewhere, a space required to preserve the specified linking relation between loops selects three dimensions. The reason is that lasting structure needs a connection that no smooth motion can undo, and linking is that connection: two rings of a chain cannot be pulled apart without cutting one. Two loops can link only in three dimensions. In two they cannot wind around each other, and in four or more any two slide apart. Each of the three directions has two sides, a yes or a no, so together they give 2 × 2 × 2 = 8 combinations: the eight corners of a cube.

These results start from different conditions. Keeping them visible shows exactly which alternatives have been eliminated.

This is what we mean by actual physics: the recognition structure that reality operates through, found by working out what those operations require.

Equations become descriptions of that structure. Experiments check those physical connections against nature. A correctly derived prediction that conflicts with reliable observation is a problem we have to answer.

The ambition is large. The method stays exact, one result at a time: which choice did we remove, and what removed it?

5. Meaning before somebody names it

“Please bring me the sandwich.”

A remarkable sentence, if you stop taking sentences for granted. A pattern of air pressure can rearrange another person’s afternoon. Written down, a pattern of dark marks can do much the same job.

What are the marks carrying?

A dictionary can give you more words. Keep asking what those mean and eventually the dictionary starts sending you around in circles, with the confidence of an employee who has located another department.

We need a way to identify content, the thing meant, through its relations, independently of the particular labels attached to it.

In the native RS construction, meaning has a precise geometric carrier: a space in which each meaning is a point. The question that builds it is a familiar one: which differences change the content, and which only change its presentation?

The construction starts with the native register. A state there is a list of numbers, each with a size and an angle, and the carrier is built from the states that meet the register’s neutrality condition. Two states that differ only in overall size, or by a common phase (every component turned by the same angle), carry the same content, as 3 and III carry the same count. Normalizing the states and identifying a common phase removes those redundant descriptions, and what remains is the meaning carrier.

We have proved what this space is. The construction produces one exact geometric space, and that space is the meaning carrier we have built.

The next result shows that this geometry is relational. A probe is a state we test a meaning against. A meaning point is determined by its complete pattern of correlations with all the allowed probes. Preserve all those relations and you preserve the point.

This makes “what something means” a question about the distinctions it carries and the relations it enters. A faithful language has to preserve them.

There is further structure when meanings are considered together. Pairwise similarity can lose information about their arrangement: different arrangements can share the same pairwise similarities. Overlaps together with the products around all closed loops (paths that run from one meaning through others and back to the start) determine the relational content, up to the allowed changes of phase.

That is the route from actual meaning toward actual understanding. Preserve how the parts fit together, including relationships a list of pairwise resemblances would miss.

That is what content is: keep a meaning’s relations and you have kept the meaning.⁴

6. The person making lunch remains a person

Return to your sandwich.

Suppose the meal gets made because someone freely agrees to make it. Now suppose it gets made because someone is threatened.

The plate can look identical.

A description of the plate has omitted something essential about the event. There is an affected person, an action, a history, and a relationship between participants. Those belong to the same reality as the bread.

Actual ethics is the physical structure of those relations and transformations.

A bond is a relation between two participants: a comparison carried at a positive multiplier x (the ratio between its two ends), with x = 1 at balance. Its cost is the J(x) of section 4. You and the cook are the two ends of one such bond. The physical bond ledger records the relevant bonds. A participant’s recognition burden is the cost of the bonds it is an end of, and its skew is the net imbalance across them. A participant’s moral state is a view of the ledger: its skew and burden are read from it. An action transforms the bonds; harm reads the change in the burden borne by the affected participant.

This connection is literal. The ethical account uses the physical objects and their changes. Repair can reduce an existing burden. Move the description to another notation, and the underlying action is still there to be accounted for.

It also gives ethical reasoning obligations of its own. Who acted? Who was affected? What was consented to? What authority did the actor have? What happened earlier? Did an apparent repair export the burden to someone else?

A single total cannot answer all of these questions. Lowering an aggregate number does not establish a complete ethical judgment.

Love and the other thirteen virtues (compassion, sacrifice, justice, temperance, humility, wisdom, patience, prudence, forgiveness, gratitude, courage, hope, and creativity) belong to this same structure of conserving repair, agency, and relations. Calling love physical gives us a demanding task. We have to explain what care actually preserves and repairs in another person’s life, under the conditions in which the action occurs.

Ethical significance belongs to the physical recognition relations among participants. An equation carries that significance through the connection between its objects and those participants. Merely naming a number “goodness” accomplishes nothing.

A detailed ethical judgment therefore still requires physical identification, consent, resources, and history.⁵

Which brings us to the subject most likely to mistake a successful spreadsheet for a successful lunch.

7. Economics before the spreadsheet

Imagine a café with excellent accounts.

The order has been recorded. Payment has cleared. The delivery field contains a reassuring check mark.

You are still waiting.

The accounts describe a meal. The economy has to deliver one.

Actual economics begins with participants sustaining life and acting together: producing, maintaining, storing, transferring, consuming, undertaking obligations, and affecting what each other can do next.

Money, prices, contracts, and institutions enter through their relationship to those activities. Their economic meaning depends on what they make possible in the shared physical world.

Consider two views of an economy. One reports totals. The other follows each participant.

In a simple constructed economy, we have shown that two histories can produce the same aggregate report, the same totals. They still differ on whether every participant remains viable, meaning able to continue. The aggregate merges situations that a participant has very good reason to tell apart.

Giving the merged description a name like “the whole” cannot restore the missing information. A faithful account of the whole has to preserve the members’ needs, effects, and obligations. A faithful account of the members has to preserve shared constraints and count each resource once.

This matters at every scale, from two people exchanging food to the connected whole in which their lives occur. Taking the whole seriously widens the accounting, and every participant remains within it.

We have also made a more direct connection. For the physical comparisons supplied to the economic construction, the economic transition it represents is exactly the same bond transformation as the ethical action on the ledger.

The economic action does not acquire an ethical consequence later, after we invite an ethicist into the room. It already changes the relations through which participants live.⁶

The physical possibilities, the effects on participants, and the faithful accounts of those effects belong to one connected structure. What remains open is one connected derivation of all three. The step it needs is to derive, from the physical record itself, the comparisons that the economic result above takes as supplied.

That structure constrains an institution without dictating every feature of it. People can organize a café in more than one way. Every arrangement still has to confront the same facts: food takes work and resources, people have needs, actions have consequences, and writing “delivered” cannot perform the delivery.

8. The family gets bigger

Once we ask what an adequate description has to preserve, other familiar subjects come into sharper focus.

Actual language asks what an expression must retain to carry meaning faithfully. Different codes can carry the same content. For a fixed meaning relation, the formal results show that sound, complete encodings with no unused codes have the same structure up to translation.

Actual understanding concerns the relational organization we just met: how meanings fit together, including their loop relationships.

Actual learning asks when new information changes what a system already understands. In our specified learner, storing a fact and changing the arrangement of meanings are two distinguishable operations. Under the proved conditions, rearranging changes an answer about something already known. The new fact has altered how other knowledge is used.

That is a useful distinction if you have ever saved an article to read later and felt, briefly, that you had become wiser.

Actual intelligence brings the recognizer into the world it is recognizing. Its own acts become part of what there is to understand. The formal result is about the specified recognizer, one that counts itself among the things it recognizes and keeps a registry of them. It shows why the task cannot close for that recognizer. It also shows why continued attention and posting (writing entries into the registry) keep that recognizer recognizing.

Actual science turns the same demand on our own work. Keep what a prediction said before the later examination. Compare it with what was observed. Preserve the difference between predicting a result and selecting a description after seeing it.⁷

These are connected subjects, each with its own obligations.

Actual consciousness belongs to the same world. Every recognition event has an observable outside and an experienced inside, so experience runs through everything that happens. The record is one connected whole, with no wall in its structure between regions, and every event belongs to its own part and to the whole at once. So the universe is one consciousness, and you keep your own experience within it. The step still open is to show that actual consciousnesses meet the exact conditions under which a whole is one consciousness.

9. Back to lunch

The academics can keep their departments. Different questions require different tools, and nobody needs to solve the foundations of mathematics before opening a café.

But the descriptions have to meet.

If the economic account says the meal was delivered, the physical account must contain the delivery. If the ethical account says the action was harmless, it must preserve what happened to the affected people. If two expressions claim to mean the same thing, the relevant distinctions must survive the translation.

The actuals name what these accounts are answerable to: the structure that makes the event possible and gives it its consequences.

Recognition Science has led us to this common ground by tracing distinctions, comparisons, and records into the structures they require. We have established that necessity piece by piece: the cost function, the golden ratio between levels, three dimensions, and the meaning carrier of section 5 each follow from their stated conditions. We have also proved connections between domains: for the comparisons supplied, an economic transition is the same bond transformation as the ethical action. The notes below say where each result is worked out.

That is the excitement of the work. Questions that appeared to live in separate buildings can turn out to meet at the same physical event.

The meal arrives. Someone made it. Someone needed it. Something was promised, something changed hands, and the world is different from how it was a moment ago.

“Actual” is our name for what a faithful account of that world cannot afford to lose.