The Compositionality Problem: Why Coherent Parts Don't Guarantee Coherent Wholes

You've seen it happen.

Two people, each individually capable and emotionally regulated, enter a relationship and become versions of themselves they don't recognize. They're not broken people. They function fine separately. Together, something goes wrong. The whole is worse than the parts.

Two departments, each well-run and productive, merge or collaborate and produce chaos. Neither team was dysfunctional. Neither set of processes was flawed. Together, nothing works. The whole is less than the sum of the parts.

Two ideas, each internally coherent and well-supported, combined into an argument that collapses. Neither premise was wrong. The combination is. The whole contradicts what the parts seemed to promise.

This is the compositionality problem. And it strikes at something we often assume without questioning: that if the pieces are good, the whole will be good. That coherence composes. That fixing the parts fixes the whole.

It doesn't always work that way. Understanding why—understanding when composition preserves coherence and when it destroys coherence—is essential for understanding how meaning works at any scale.

What Compositionality Means

Compositionality is the principle that the properties of a whole derive systematically from the properties of its parts and the way those parts combine.

In linguistics, this is called Frege's principle: the meaning of a sentence derives from the meanings of its words and the grammatical structure that combines them. "The cat sat on the mat" means what it means because "cat" means cat, "sat" means sat, "mat" means mat, and the grammar tells you that the cat is doing the sitting and the mat is the location.

If compositionality holds perfectly, you can predict the whole from the parts. Know the components, know the combination rule, and you know the result. There's no surplus—no emergent property that couldn't be predicted from the composition.

Category theory makes compositionality precise. Arrows compose. If you have f: A → B and g: B → C, you get g∘f: A → C. The composition is determined by the arrows being composed. The result follows from the inputs plus the composition operation.

And in many domains, compositionality works beautifully. Mathematics is compositional—theorems derive from axioms through rules of inference. Well-designed software is compositional—the behavior of the system derives from the behavior of its modules and their interfaces. Simple physical systems are compositional—the behavior of two objects follows from the behavior of each plus their interaction laws.

But compositionality can fail. The whole can have properties that don't follow from the parts. The composition can produce something unpredictable, something new, something that violates what the parts seemed to promise.

When compositionality fails in coherence systems, you get the phenomenon that's haunted this entire series: parts that work fine individually but produce dysfunction together.

Where Compositionality Fails

Language itself illustrates the failure modes.

Idioms break compositionality. "Kick the bucket" doesn't mean what "kick" plus "the" plus "bucket" would predict. The meaning of the whole isn't a function of the meanings of the parts. The phrase has become a unit with its own meaning, not compositionally derived.

Context breaks compositionality. "The ham sandwich wants his check" makes sense in a restaurant, where the server has identified customers by their orders. But the literal compositional meaning—a ham sandwich experiencing desire—is absurd. Context contributes meaning that isn't in the words.

Pragmatics break compositionality. "Can you pass the salt?" compositionally asks about ability, but pragmatically requests action. What the sentence means isn't what it compositionally means. The social situation contributes meaning that composition doesn't capture.

These failures are well-known in linguistics. The compositional semantics gets you part of the way; something else—idiom lookup, contextual inference, pragmatic reasoning—handles the rest.

But the failures in human systems are more profound and more consequential.

Coherent People, Incoherent Relationship

Consider the couple that doesn't work.

Person A is emotionally regulated, securely attached, capable of intimacy and independence. Person B is also emotionally regulated, securely attached, capable of the same. You'd predict that A and B together would form a secure, functional relationship. The parts are coherent; the whole should be coherent.

But it isn't. Together, A and B trigger each other in ways neither experiences with anyone else. Their attachment styles, though secure in general, have specific incompatibilities that emerge only in this particular combination. Their communication patterns, though effective generally, create feedback loops that amplify misunderstanding. Their needs, though reasonable individually, conflict in ways that produce chronic frustration.

The compositionality assumption failed. Knowing that A is coherent and B is coherent didn't tell you that A+B would be coherent. The composition introduced something the parts didn't contain.

What went wrong?

Interface incompatibility. Each person has a characteristic way of connecting. Those ways might be functional in general but incompatible with each other specifically. Like two well-functioning machines with incompatible connectors—each works fine, but they can't be plugged together.

Resonance effects. Certain combinations produce oscillations that neither component would produce alone. A's anxiety about abandonment is manageable until it meets B's need for space. B's need for space is manageable until it meets A's abandonment anxiety. Together, they create a feedback loop that amplifies both—a resonance that emerges from the combination.

Emergent dynamics. The coupled system has dynamics that neither individual system has. Two people are not two individuals plus connection; they're a new entity with its own attractor landscape, its own possible states, its own coherence geometry. The dyadic system can be incoherent even when both constituent systems are coherent.

This is compositionality failure at the relational level. The functor from individual to relational doesn't preserve coherence here—not because the functor is broken, but because coherence isn't the kind of property that always survives composition.

Why Doesn't Coherence Compose?

If coherence is a structural property, why doesn't it compose? If the parts have structure and the combination rule has structure, why doesn't the result inherit coherent structure?

The answer lies in how composition changes the constraint landscape.

Each system has its own constraints—its own manifold, its own geometry, its own boundaries of what's possible. Coherence, for that system, means maintaining low curvature, adequate dimensionality, stable topology within those constraints.

When systems compose, the constraints combine. But constraint combination isn't simple addition. It's intersection—the composed system must satisfy all the constraints of all its components simultaneously. And intersection can produce a much smaller, much more constrained space than any individual component.

Imagine this geometrically. System A has a manifold of possible states—a region in some space. System B has its own manifold. When A and B couple, the combined system's manifold isn't the union or the product of the individual manifolds. It's constrained by both—the states that are possible for A and possible for B and consistent with their coupling.

This combined constraint space can be much smaller than either individual space. Dimensionality can collapse. What was high-dimensional for A and high-dimensional for B becomes low-dimensional for A+B.

The combined constraint space can have different topology. Regions that were accessible for A and accessible for B might be inaccessible for A+B. Bottlenecks emerge that neither individual system had.

The combined constraint space can have higher curvature. Smooth regions for A and smooth regions for B might map to steep regions for A+B. The individual systems were navigable; the combined system is treacherous.

Coherence is relative to constraint landscape. A system is coherent relative to its constraints. When composition changes the constraints, what was coherent relative to the old constraints may be incoherent relative to the new constraints.

This is why coherent parts don't guarantee coherent wholes. The whole faces constraints the parts didn't face. Coherence has to be achieved anew at the level of the whole.

The Composition Operation Matters

Not just the parts, but how the parts combine determines whether coherence survives.

Consider different ways two people might relate:

Loose coupling. Each person maintains most of their individual autonomy. They interact, but the interaction doesn't deeply constrain either individual's possibilities. The composed system is close to the product of the individual systems—low interaction, minimal constraint intersection.

In loose coupling, coherence composes better. The individuals stay coherent because their constraint landscapes aren't much changed. The relationship can be coherent because it's not demanding much coherence—the parts remain largely independent.

Tight coupling. Each person's state becomes deeply dependent on the other's. What A can do depends on what B is doing. What B can feel depends on what A is feeling. The composed system is highly constrained—extensive intersection of possibility spaces.

In tight coupling, compositionality is more likely to fail. The constraint intersection can produce pathological geometry. States that were fine for each individual become unstable for the pair. Coherence that existed for parts evaporates for the whole.

The coupling pattern. Not just how much coupling but what pattern. Some coupling patterns preserve coherence; others destroy it. The mathematics of exactly which patterns do what is not fully understood, but the empirical observation is robust: the relationship between the parts matters as much as the parts themselves.

This is why relationship dynamics can be so puzzling. You look at the people involved and can't see why they're struggling. They seem fine. They seem compatible. But the specific pattern of how they've coupled creates geometry that neither would create alone.

Compositionality in Organizations

Organizations are compositions. Individuals compose into teams. Teams compose into departments. Departments compose into the organization.

At each composition step, coherence can fail.

Team formation. Three individually excellent employees can form a dysfunctional team. Not because anyone is bad at their job, but because the combination creates dynamics—competition, miscommunication, conflicting work styles—that undermine the whole. The team is incoherent though the members are coherent.

Department interaction. Two well-functioning departments can produce chaos at their interface. Engineering builds what product defines, but the handoff is where things break. Neither department is dysfunctional. The composition is.

Organizational integration. A company with excellent parts can be a terrible whole. The parts optimize locally, but local optima don't compose into global optima. The marketing department succeeds at marketing goals, the sales department succeeds at sales goals, and the company fails at company goals.

This is the tragedy of local coherence: everyone doing their job well, and the organization still failing. The compositionality assumption says good parts should make good wholes. The reality is more complex.

The interfaces matter. Organizations fail at interfaces—where parts meet, where handoffs occur, where different systems must coordinate. The parts can be fine. The interfaces can be disasters. Organizational coherence requires interface coherence, which is a different thing than part coherence.

Incentive composition. Each role has incentives. When roles compose into organizations, incentives compose too. But incentive composition can produce perverse results—reward structures that individually make sense but collectively drive dysfunction. No one designed the dysfunction; it emerged from composition.

Communication topology. How information flows determines how coherence propagates. If the communication topology fragments, coherent parts can't produce a coherent whole. They can't coordinate. They can't align. The composition fails because the composition rule—communication—is broken.

Cultural Compositionality

At the largest scale, cultures compose. Subcultures within a culture. Cultures within civilization. Generations within historical continuity.

Compositionality failures here produce some of history's worst outcomes.

Polarization. Two coherent worldviews, each internally consistent, each supported by its adherents, compose into incoherent society. The worldviews are incompatible; their intersection is empty or chaotic. The society can't function because it's made of parts that can't compose.

Generational disconnect. Each generation has its coherent culture, shaped by its historical experience. Generations compose into society. But if generational cultures are incompatible, the composition fails. Society becomes incoherent across generations even if each generation is coherent within itself.

Institutional mismatch. Institutions carry cultural patterns across time. When institutions designed for one context meet a different context, they can fail—not because the institution is broken, but because its cultural assumptions don't compose with current conditions.

The culture wars are compositionality crises. It's not that people are stupid or evil. It's that coherent worldviews—each making sense to its adherents—fail to compose into a coherent shared reality. The parts work; the whole doesn't.

Entrainment as Compositional Repair

If coherence doesn't automatically compose, how does coherent composition ever happen?

The answer is entrainment.

Entrainment is the process by which coupled oscillators synchronize. It's how composition comes to work—how parts that might produce incoherence instead produce coherence.

Entrainment changes the coupling. When two systems entrain, they adjust their dynamics to accommodate each other. The constraint intersection that might have been pathological becomes navigable. The combined system finds a mode of operation where both parts can function.

Entrainment creates shared rhythm. The composed system develops its own rhythmic structure—timing, pacing, oscillation patterns that belong to the whole, not to the parts. This shared rhythm is the composition rule made coherent.

Entrainment discovers viable composition. Not all compositions are viable. Entrainment is the process of finding the ones that are. Systems try to synchronize; some succeed, some fail. The successful synchronizations are the compositions that work.

This is why relationship repair is possible. Two people whose composition failed can re-entrain. They adjust their coupling, find new shared rhythms, create composition patterns that work better. The parts didn't change; the composition did.

This is why organizational healing is possible. Interfaces can be redesigned. Communication patterns can be restructured. Incentives can be realigned. The parts stay the same; the way they compose changes.

This is why cultural renewal is possible. Subcultures can find new ways of coexisting. Generations can discover shared values. Incompatible worldviews can develop interfaces that allow peaceful composition. The parts remain distinct; the composition becomes viable.

Entrainment isn't automatic. It requires effort, time, favorable conditions. Many compositions fail and stay failed. But entrainment is the mechanism by which failed compositions can become functional—the repair operation for compositionality breakdown.

Compositional Thinking

Understanding the compositionality problem changes how you approach systems.

Don't assume good parts make good wholes. When a whole is dysfunctional, the problem might not be any part. It might be the composition. Looking for a bad part when the parts are fine wastes effort and generates blame.

Attend to interfaces. Where parts meet is where composition succeeds or fails. The interfaces—the coupling patterns, the communication channels, the coordination mechanisms—often matter more than the parts themselves.

Look for constraint intersection effects. When composition fails, ask: what new constraints has the composition introduced? What was possible for parts that's impossible for the whole? Where has the constraint landscape become pathological?

Consider multiple compositions. The same parts might compose in different ways. If one composition fails, another might succeed. Don't give up on parts that failed to compose once; try composing them differently.

Invest in entrainment. Successful composition requires investment. Time to synchronize. Effort to adjust. Resources to support the coupling. Composition isn't free; it requires work.

The compositionality problem is humbling. It says you can't engineer wholes just by engineering parts. It says bottom-up doesn't automatically produce top-down coherence. It says the whole is genuinely more than—or different from—the sum of parts.

But it's also clarifying. When wholes fail despite good parts, the compositionality problem tells you where to look. Not at the parts. At the composition. At the interfaces. At the way things come together.

Coherence doesn't compose automatically. It must be achieved at each level, through the slow work of entrainment, through attention to how parts couple, through investment in the composition itself.

The parts might be fine. The question is how to compose them into something that works.