The Abstractionist’s Papers · Part III · Chapter 8 of 13
A bandit floors it down the highway, cruisers chasing him, sirens blasting. The state line comes up fast.
The whole audience is rooting for him.
He crosses it, and they pull up short and sit there watching him go. Once again, he escapes the reach of the law.
A rule is real only where something enforces it, and no further.
In nature we expect the opposite. We assume laws hold everywhere at once.
Chapter 4 already gave us reason to doubt that. A rule is enforced by a natural process, and that process applies more or less of the rule according to its own impedance. If the process raises its impedance, the same cause produces less effect. Raise it far enough and a different rule governs what the process does.
Causal spaces model where rules reach and where they don’t. The paradoxes in this chapter turn out to be two spaces with different rules treated as a single one. Separate them and the conflict is reconciled.
Contents
8.1 Ideal Causal Spaces
8.1.1 Where Causal Spaces Are
8.2 Propagation and Impedance
8.2.1 Causal Propagation
8.2.2 Causal Impedance
8.3 Where Boundaries Come From
8.4 Independent and Interdependent Spaces
8.4.1 Independent Spaces
8.4.2 Interdependent Spaces
8.4.3 Feedback and Bidirectional Induction
8.5 How Every Paradox Works
8.5.1 Nimbin’s Paradox
8.5.2 The Light Duality Paradox
8.5.3 The Liar’s Paradox
8.5.4 The Chicken and the Egg
8.6 From Causal Spaces to Natural Spaces
8.7 Closing Remarks
8.1 Ideal Causal Spaces
An ideal causal space has a well-defined boundary that determines where its governing rule applies and where it doesn’t.
Inside this boundary, every process follows the rule completely, while anything outside stays unaffected. Within a given space, causation operates under a single, unbroken principle.
The binary nature of ideal causal spaces makes them useful for studying systems where rules apply without exception. In physics, Newtonian mechanics defines how forces produce acceleration within its range of applicability, while relativistic speeds and quantum scales operate under different principles.
Within an ideal causal space, coherence is absolute. A process conforms perfectly to the governing rule, without deviation. Some spaces work this way for real. In grammar or logic, a rule applies by definition, and definitions don’t wear down. In anything we build, from an abacus to a computer, the rule works because we arranged it to. Natural processes are the ones that drift.
8.1.1 Where Causal Spaces Are
A causal space represents an interpretation of how a process operates in the Blue Space. Think of it as a little chunk of process, the part we draw a boundary around so we can say what it does.
Each process carries its own governing rules internally. The consistency lives in how the process enforces these rules when responding to interaction. An electron exists in the Blue Space, but its causal space, the internal rules that determine its responses, belongs to the electron itself.
Fields work the same way. A field is how we interpret many concurrent, consistent behaviors. If processes consistently pull toward a mass, we call that a gravitational field. The field is our interpretive shorthand for collective causal behavior.
Both the causal space and the field are Red Space interpretations of Blue Space processes. Each gives us a way to map what we can’t directly access.
8.2 Propagation and Impedance
Before going further, what is a rule?
We don’t see transformations. We see a state, and then we see another state, and the change between them is not available to us. So we model it. Cause, effect, and a rule in between. The cause is a state, the effect is a state, and the rule is the transformation.
The rule is ours. It’s how we account for a change we can’t watch happen.
A rule holds wherever its causal space does. In nature the space sits inside the process, which is where Chapter 4 put it. For a formal space like grammar, or a built one like a circuit, the question doesn’t come up. The space enforces the rule and that’s the end of it.
That’s what makes rules local. A rule is a good model where it describes what a process does and a bad one where it doesn’t. The process doesn’t need to know the difference.
Every process within a causal space follows a defined rule of causation, and how it unfolds depends on how the rule propagates and whether it meets resistance.
8.2.1 Causal Propagation
Causal propagation describes how a process moves from one state to another within a causal space.
When a cause state transitions into an effect state, it does so under the enforcement of a rule of causation. Within a defined causal space, this rule applies exclusively to the processes inside, and transformations follow its governing principle. The sharper the boundary and the more direct the rule’s application, the more predictable the propagation.
Not all influence propagates with equal ease.
8.2.2 Causal Impedance
Some processes flow freely, adhering closely to the governing rule with little resistance. Others face greater impedance, while some are entirely blocked from transitioning to their expected effect state.
A system with low impedance allows causal rules to apply efficiently, enabling smooth state transitions. A system with high impedance resists change, distorting or preventing the expected outcome.
The relationship can be expressed mathematically.
C = Z × E
Here, the effect state E is influenced by the cause state C, but its transformation is scaled by the impedance Z. When impedance is low, the effect state closely follows from the cause. When impedance is high, movement is slowed or prevented.
Causal impedance shows up across disciplines. In physics, mechanical systems experience impedance as mass, which resists acceleration. In electrical circuits, impedance determines how signals propagate and affects energy transfer. In biology, enzymes lower impedance by making biochemical transformations more efficient, while environmental extremes increase resistance, inhibiting metabolic processes. In social systems, resistance to change acts as impedance, slowing or preventing the adoption of new practices.
Causal impedance explains why transformations sometimes lead nowhere or take unexpected paths.
8.3 Where Boundaries Come From
A space applies its rule and a process responds. If the process raises its impedance, the same cause produces less effect. Keep raising it and the rule no longer applies at all. The way a rule is applied effectively changes that rule into another rule.
Righty tighty. Turn the screwdriver right and the screw goes in. Now wear the head down. Righty still tightens, most of the time, and slips the rest. Wear it further and righty is barely tightening at all. Keep going and righty strips. Your hand never changed what it was doing. Somewhere in there, tightening became stripping, and no new rule ever announced that it had taken over.
Chapter 4 called this Incoherence and described it from the process’s side. Here we’re describing the same thing from the outside, and from out here it looks like a boundary.
We draw that boundary ourselves, after the fact, once a newer rule describes the behavior better than an older one.
8.4 Independent and Interdependent Spaces
Every causal space operates according to its own rule, but how these spaces relate determines the complexity of the systems they form. Some spaces stay entirely self-contained, functioning independently. Others interact dynamically, influencing and responding to one another in ways that direct the processes flowing through them.
8.4.1 Independent Spaces
Independent causal spaces function autonomously, governed exclusively by their own rules. Processes within these spaces transition from cause states to effect states without external influence. Each transformation occurs solely according to the internal rule of causation, without interaction from other spaces.
Figure 33 shows two causal spaces, A and B.
Within Space A, processes follow its rule from a cause state (CA) to an effect state (EA) without interference from outside. The same holds for Space B, where processes move from CB to EB under a different rule.
These spaces operate independently. Each follows its own rule. Keeping them apart lets us study each one on its own, and keeps what happens inside each space predictable.
8.4.2 Interdependent Spaces
Many systems feature interdependent causal spaces where changes in one context induce transformations in another. While independent spaces function autonomously, interdependent spaces are dynamically linked, allowing processes to influence and respond to one another across contexts.
In Figure 34, Spaces A and B interact.
Through induction, an effect at the boundary of one space induces a cause in another.
The induced cause belongs to the receiving space. It follows that space’s rule, not the rule of the space that induced it.
8.4.3 Feedback and Bidirectional Induction
Figure 29, from Chapter 6, shows the same thing with two processes. An effect at the boundary of Process A induces a cause in Process B, which follows its own rule to an effect. A causal space is a little chunk of process, so the picture holds for causal spaces too.
Induction can run both ways. Just as A influences B, B can influence A. The feedback loop keeps each space adjusting to the other.
The interdependence appears everywhere. In taste and nutrition, sensory perception and metabolic processing form interdependent spaces. The perception of sweetness in the sensory space triggers insulin release in the nutritional space, altering how the body processes sugar. Biochemical needs also influence sensory perception; a sodium deficiency can induce a craving for salty foods.
A similar interdependence exists between sense perception and motion control, where vision and motor function influence each other through bidirectional causation. Visual information about an object’s position induces motion adjustments in the motor space, while motor feedback refines sensory perception, improving balance and coordination. Movement actively transforms how perception functions while perception guides motor responses.
The bidirectional dynamic explains what Gibson called affordances. Affordances arise when a process can both construct meaning from environmental signals and project effects back into that environment. Both directions require compatible impedance.
A musician experiences affordances in instruments that others can’t. Their internal rules have developed matched impedance between auditory processing and motor responses. The environment provides the signals, but the affordance only opens up once that match is there.
In economics and individual perception, large-scale economic changes impact personal decision-making. A rise in inflation induces a perception of decreased purchasing power within the psychological space, leading individuals to adjust their behavior by spending less. This change in consumer behavior then feeds back into the economic space, influencing broader market trends.
To understand how a process develops across contexts, you need to know when causal spaces are independent and when they interact.
8.5 How Every Paradox Works
Which came first, the chicken or the egg? Chickens lay eggs, so the chicken came first. Every chicken hatches from an egg, so the egg came first. Both answers sound right, and both are claims about cause and effect.
A cause rarely stops at a single space. The same cause can reach many spaces, each with its own rule and its own effect. In Figure 35, one cause C reaches spaces 1, 2 and on up to n. Each applies its own rule, Ψ1 to Ψn, and gives its own effect, E1 to En.
Usually these effects stay out of each other’s way. Sometimes they seem to work against each other. A gust of wind (C) blows out a candle (EA). The same gust reaches a campfire, and the fire flares up (EB). In Figure 36 the candle is Space A and the campfire is Space B. We see them as two spaces, so nobody calls it a conflict, and the crossed-out triangle says there is no paradox.
Now hide the candle and the campfire, and picture the whole camping trip as one space. All we see is a gust that puts out a flame and feeds a fire, and nothing explains how. In Figure 37 a dark cover hides both spaces and asks one space, one rule? The yellow triangle between the two effects marks the paradox.
A paradox appears, and with the spaces hidden, it won’t go away. Every paradox rests on one assumption, that a cause can follow only one rule. One cause, one rule, one effect. A single rule can’t produce two opposite effects, so one of the answers has to be wrong.
When two answers seem to clash, look for a causal space behind each one. Once you find the spaces, the answers sit side by side, like the candle and the campfire. Both effects are real, and neither cancels the other.
Here are four of them. Nimbin’s sweet tooth happens to him, not just in his head. Light really does behave both as a wave and as a particle. The Liar comes from a sentence someone wrote down. The chicken and the egg is a question about how life goes on. In each figure that follows, the top half covers the spaces, the way we first meet the paradox, and the yellow triangle marks it. The bottom half lifts the cover, and the crossed-out triangle says the paradox is gone.
Traditional approaches treat these as puzzles to solve within a single framework. Natural Reality recognizes them as evidence that reality operates across multiple domains with distinct principles. What they point to doesn’t fit inside any one set of rules.
How do we identify causal spaces? We model them and test until the model works. When contradictions surface, we examine where incompatible rules might apply. We draw boundaries, test predictions, adjust when results don’t match. We model, test, and refine, and then do it again.
The model works when the paradox disappears. The resolution might feel counterintuitive to our native assumptions, but that discomfort reveals exactly which assumptions we’ve been making unconsciously. When wave-particle duality stops being contradictory because we acknowledge measurement and propagation as distinct domains, we’ve exposed our hidden assumption that all properties must exist uniformly. When the Liar’s sentence resolves by separating what it says from how we check it, we’ve exposed our assumption that saying and checking happen in the same space. These contradictions serve as diagnostic tools, showing us the boundaries of assumptions we didn’t know we were making.
Reconciling a paradox means finding a position from which both results make sense. Sugar is a pleasure in one space and a harm in another. Light is a wave in one space and a particle in another. From there, both answers stand.
A causal space is validated by whether it helps us predict and work with what happens.
8.5.1 Nimbin’s Paradox
Nimbin loves sugary treats and dislikes vegetables. Is sugar good for him or bad for him? To a child, it has to be one or the other.
In the space of taste, sugar meets the taste receptors and gives him pleasure right away. In the space of health, years of eating it wear the body down.
In Figure 38, sugar is the cause. Covered, it gives pleasure and harm, and the two look like a contradiction. Uncovered, tasting gives pleasure now and metabolizing gives harm over the years. Nimbin feels the pleasure of sweetness and takes it as a sign that sugar is good for him. The harm happens in a different space, and he never sees the two side by side. Once he learns to tell the space of taste from the space of health, he can see that one food produces both results. Sugar is good for his taste and bad for his health.
8.5.2 The Light Duality Paradox
For more than a century, light has behaved like a wave in some experiments and like a particle in others.
In the space of propagation, light moves as a continuous wave and shows interference wherever nothing absorbs it. In the space of absorption, a detector takes it up in separate events, and those events are the particles.
In Figure 39, light is the cause. Covered, it gives a wave and a particle. Uncovered, propagating gives the wave and absorbing gives the particle. The two spaces follow different rules. Interference needs something continuous, and detection produces discrete, localized clicks. The change is not in light. It’s in how the absorbing process engages with it.
Propagation is how we model light, and it’s the model that lets us work with it. Light appears as a wave while it propagates and as a particle when it is absorbed. Whether we see one or the other depends on which space governs the interaction.
8.5.3 The Liar’s Paradox
“This sentence is false.”
If it’s true, then it’s false, as it says. If it’s false, it’s telling the truth, so it’s true. Logicians have chased that loop for more than two thousand years.
Start with an easier one. “This sentence has five words.” The sentence does two things. First, it makes a claim about itself, in the space of saying. Then we count. This, sentence, has, five, words. Five. Counting happens in the space of checking. What it said matches what we counted, so it’s true.
The Liar does the same two things. First, it says it is false, so in the space of saying, it is false. Then we check. We put what it says next to what it is, and the two match. So in the space of checking, the sentence is true.
In Figure 40, the sentence is the cause. Covered, it gives false and true, and the arrows around the triangle show the answer flipping back and forth. Uncovered, saying gives false and checking gives true. The trouble starts when we take that “true” and carry it back into the space of saying, as if the sentence had said it. Every trip around the loop makes the same mistake.
Keep each answer in its own space and the loop stops. The sentence is false as it is said and true as it is checked, the way sugar is a pleasure to taste and a harm to health. The question “Which is it?” assumed there was only one space.
8.5.4 The Chicken and the Egg
We get stuck on the chicken and the egg because we picture one long chain, chicken, egg, chicken, egg, and ask which came first. We can model the question with two causal spaces instead.
The cause is the chicken and the egg, the plain fact that both exist. In the space of laying, a hen lays an egg, so the chicken came first. In the space of hatching, an egg hatches into a chicken, so the egg came first.
In Figure 41, the chicken and the egg are the cause. Covered, they give two rival answers, chicken first and egg first. Uncovered, laying gives chicken first and hatching gives egg first. Put the two spaces side by side and both answers are true.
In each of these examples, the answers clashed only while we saw one space. Once we found the second, the paradox was reconciled. A paradox lives only inside a mind, where meaning is made. Outside the mind, causes keep producing effects, noticed or not.
8.6 From Causal Spaces to Natural Spaces
A causal space is a concept we use to describe causality. Causal spaces help us understand how transformations occur, defining a space through rules of causation.
How do these spaces turn into Natural Spaces, environments where causation happens as we experience it in nature?
Through size. With enough entities interacting in enough variety, variability arises without anyone introducing it.
The atomic space, the chemical space, the cosmological space are Natural Spaces. These support a vast range of interactions, processes, and variability, allowing them to behave in ways we know as nature.
When we place a system inside a confined environment, we’re approximating a Natural Space. The difference between the approximation and the real one lies in the scale and diversity of interactions.
Chapter 4 said a Natural Space is out there happening and also in here as the version we built. The space applies its rule out there whether anyone models it or not, and the model is what we work with.
Every Natural Space is a causal space. The reverse is true only at scale.
8.7 Closing Remarks
The police stop at the border because that’s where their authority ends. A process is different. It keeps applying the rule, less and less of it, until what it does is something the rule never described. Nothing marks where that happened. We pick the spot afterward, once a different rule tells the story better.
Causal spaces make this workable. Impedance, admittance, cross-impedance, and bidirectional induction model how influence propagates, where it induces a response at a boundary, and what happens when it does. The paradoxes in this chapter share a source, two spaces with different rules treated as one, and they’re reconciled once what was commingled is separated.
A causal space becomes a Natural Space at scale. The atomic space, the chemical space, and the cognitive space all work the same way.
What happens at those boundaries comes down to impedance.
Chapter 8 in the Chapter Manual · Study questions