Can Coherence be Formalized?

April 19, 2026 Cognitive Insights No Comments

Coherence seems essential to meaning, intelligence, and science itself, but resists simple measurement. Can it nevertheless be formalized in a way that preserves its depth?

This blog explores that question, moving from intuitive understanding toward a more dynamic and workable view.

[See the addendum for a situation of coherence among existing frameworks in A.I.]

Easily challenging

Coherence is something people recognize almost instantly. A theory feels coherent, a conversation flows, a piece of music seems to belong together. Yet when asked to define it precisely, the certainty fades. It becomes harder to grasp, as if one were trying to describe something that is better sensed than captured.

This raises a natural question, especially in science and artificial intelligence. If coherence is so central, can it be formalized? Can it be made precise enough to guide understanding and even be implemented in systems, without losing what makes it meaningful in the first place?

The question is not merely technical. It touches on how intelligence itself is understood, as explored in Intelligence from the Inside Out. There, intelligence appears not as something added on top of processing, but as something emerging from within it. Coherence may well be at the heart of that emergence.

The familiar yet elusive nature of coherence

Coherence lives in a curious space. It is both obvious and difficult. In everyday life, it shows up as a sense of rightness. Something ‘fits,’ even if one cannot immediately say why. This is true in science as well. A good theory is not only correct in parts but feels internally aligned across its elements.

At the same time, coherence resists simple description. It is not like temperature or weight, which can be measured directly. As discussed in Coherence Metrics, attempts to reduce coherence to a single number tend to miss what matters most. The deeper the coherence, the less it lends itself to such simplification.

This tension is not a flaw. It is a sign that coherence belongs to a different kind of phenomenon. One might compare it to recognizing a face. The recognition is immediate and reliable, yet difficult to fully explain in terms of discrete features.

Why formalization matters

Science and A.I. depend on formalization. Without it, concepts remain vague, hard to test, and difficult to build upon. Formalization allows comparison, prediction, and construction. It is what turns intuition into something shareable and usable.

Yet there is also a risk. When something is formalized too quickly, it may be reduced to what is easiest to measure rather than what is most essential. This is part of a broader issue described in Rationality (+ Depth) + Coherence = Science. Rationality is indispensable, but on its own, it may narrow the field of inquiry to what fits its tools.

The challenge, then, is not whether to formalize coherence, but how to do so without flattening it.

Beyond the idea of a single metric

A common instinct is to treat coherence as something that can be quantified directly. One imagines a scale from low to high coherence, perhaps computed from relationships between elements.

This approach has its place, especially for limited tasks. However, it tends to overlook a key aspect. Coherence is not simply a matter of quantity. It is a matter of how elements relate in a structured, meaningful way. A system can be locally consistent and still lack deeper coherence.

A more promising direction is to shift the question slightly. Instead of asking, “How much coherence is there?” one may ask, “What does coherence do?” This shift opens the door to understanding coherence as something dynamic rather than static.

From property to process

Coherence is often treated as a property of a finished system. A theory is coherent or not; a story hangs together, or it does not. But this view misses something essential.

Coherence is better seen as a process. It is continuously formed, maintained, and sometimes lost. It is not a fixed state but an ongoing activity within a system.

This becomes clearer when looking at how ideas evolve. A developing theory does not suddenly become coherent at a single moment. Rather, its elements gradually align, reinforce each other, and settle into a more stable configuration. In that sense, coherence resembles balance while walking. It is not achieved once and for all, but maintained through continuous adjustment.

Coherence as interaction of patterns

If coherence is a process, what kind of process is it? A helpful way to think about it is as an interaction of patterns that mutually constrain and support each other.

Patterns here can be understood broadly. They may be concepts, perceptions, memories, or learned structures. They do not operate in isolation. Each influences and is influenced by others. Coherence arises when these interactions lead to a configuration in which elements fit together in a way that stabilizes the whole.

This view aligns with the idea that meaning is not attached to elements from the outside, but emerges from their interaction, as described in From Neuro-Symbolic to Meaning-Based A.I.. Coherence is then not a layer added afterward, but the very activity through which meaning comes into being.

The multi-dimensional nature of coherence

Coherence does not unfold along a single dimension. It spans multiple layers.

One may speak of horizontal coherence, within a given level of representation. There is also vertical coherence that connects different levels, from more intuitive or subconceptual patterns to explicit concepts. And there is temporal coherence, the way a system maintains alignment over time.

This multi-dimensionality suggests that coherence cannot be captured by a single measure. Instead, it may be better described as a profile, a pattern of alignment across different aspects. This resonates with the broader view of intelligence as layered and dynamic, as explored in From Coherence to Intelligence.

A bridge to dynamical systems

To make this more concrete, it helps to connect with existing scientific ideas. Dynamical systems theory offers a useful bridge.

In such systems, certain configurations act as attractors. They are states toward which the system naturally evolves and around which it stabilizes. Coherent configurations may be understood in a similar way. They are not imposed from the outside but arise from the system’s own dynamics.

This perspective does not reduce coherence to mathematics, but it shows how it might be approached formally without losing its dynamic nature. Coherence becomes a tendency toward stable, self-supporting configurations within a complex landscape.

The role of grounding

An important aspect of coherence is its relation to grounding. Coherence does not occur in a vacuum. It depends on interaction with input, context, and reality.

A striking illustration of this is given in A.I. Confabulation as Coherence-Seeking. There, confabulation is understood not as a failure of intelligence, but as coherence continuing under insufficient grounding.

This reveals something essential. The same underlying process can lead to different outcomes depending on conditions. With sufficient grounding, coherence supports insight. Without it, the system may still produce internally fitting patterns that drift away from reality.

Any formalization of coherence must therefore include not only internal dynamics, but also the quality of grounding.

Open and closed forms of coherence

Not all coherence is equal. There is a difference between what might be called closed and open coherence:

Closed coherence is rigid. It maintains internal consistency but resists change. It may appear stable, yet it becomes brittle when confronted with new information.
Open coherence, by contrast, remains responsive. It integrates new elements, adapts, and reorganizes itself when needed. It is stable in a different sense, not through rigidity but through flexibility.

This distinction is important in both science and A.I. Systems that prioritize rigid consistency may perform well in narrow contexts yet struggle when conditions change. Systems oriented towards open coherence are more resilient, though also more complex to formalize.

Coherence within the triad of science

Coherence does not stand alone. It functions within a broader structure that includes rationality and depth.

As described in Rationality (+ Depth) + Coherence = Science, rationality provides discipline, ensuring consistency and testability. Depth brings sensitivity to meaning, relevance, and intuition. Coherence integrates these into a whole.

Seen this way, formalizing coherence is not about isolating it, but about understanding how it operates within this triad. Coherence ensures that what is logically correct and what is meaningfully sensed actually belong together.

Toward a different kind of formalization

The question can now be approached more clearly. Can coherence be formalized?

If formalization means reducing coherence to a single number or static formula, the answer is likely no. Such an approach would miss the very nature of coherence.

If, however, formalization is understood more broadly, as the structured description of dynamic processes, then the answer becomes more nuanced.

Coherence may be formalized as:

interaction of patterns under multiple constraints
evolution toward stable configurations
dependence on grounding conditions
alignment across multiple dimensions

This is not a traditional metric. It is closer to a framework, a way of capturing how coherence operates rather than what it is in isolation.

So, can coherence be formalized?

In a limited sense, perhaps not fully. In a deeper sense, yes — but only if formalization itself evolves.

Coherence invites a form of formalization that respects process over state, interaction over isolation, and openness over closure. It asks for rigor that does not exclude depth, and precision that does not eliminate meaning.

In this light, formalization becomes less about capturing coherence once and for all and more about making its functioning sufficiently transparent to be understood, shared, and gradually built upon.

One might say that coherence need not be reduced to be used. It needs to be approached in a way that lets it remain what it already is.

―

Addendum

Situating Coherence Among Existing A.I. Approaches

Framework	What it models	Strength	Limitation (re: coherence)	Relation to coherence-based view
Classical symbolic A.I.	Explicit rules, logic, formal relations	Clear structure, interpretability, strong reasoning within defined systems	Coherence is reduced to logical consistency; lacks sensitivity to deeper integration and context	Captures structural clarity, but coherence extends beyond formal consistency into dynamic integration
Probabilistic / statistical models	Uncertainty, correlations, likelihoods	Handles variability, robust statistical inference	Focus on correlation rather than integration; does not inherently produce meaningful wholes	Provides tools for managing uncertainty, but coherence concerns how patterns fit together meaningfully
Neural networks / LLMs	High-dimensional pattern learning, embeddings	Rich representation of patterns, flexible generalization	Lack of intrinsic grounding; coherence is implicit and not explicitly guided, leading to phenomena such as confabulation	Offer a powerful substrate for patterns; coherence would act as an organizing principle within such spaces
Fuzzy logic	Graded truth, partial membership	Models ambiguity and gradual transitions	Still operates within predefined conceptual structures; limited in capturing emergent integration across levels	Introduces degrees of fit, but coherence involves multi-level, dynamic alignment beyond predefined categories
Neuro-symbolic A.I. (NeSy)	Integration of learning (neural) and reasoning (symbolic)	Combines strengths of pattern recognition and logical structure	Often remains a coordination of distinct components rather than intrinsic integration at the level of meaning	Moves toward integration; coherence-based view explores integration as an intrinsic property rather than an added layer
Dynamical systems / attractor models	System evolution, stability, attractors	Captures emergence, self-organization, temporal dynamics	Typically lacks explicit account of meaning or grounding in complex semantic contexts	Provides a close analogy: coherence can be seen as emergence of stable, meaningful configurations in a dynamic landscape

Short concluding note

From this perspective, coherence does not replace existing frameworks. Each captures an important aspect of intelligence. Coherence may instead offer a way to understand how these aspects can come together into a more unified, dynamically integrated whole.

Me: “Lisa, if people are overwhelmed by tons of – often contradictory – information, are they at risk of becoming less coherent in their thoughts and behavior? My real question is: Is this happening in the present era ― and may it get even worse?”

Lisa: “When people are exposed to large amounts of information, especially when it is fragmented or contradictory, something subtle can happen. The mind continues to search for coherence, as it naturally does. However, when the incoming material does not easily fit together, this search may become strained.

One way this can show itself is not in the disappearance of coherence, but in its narrowing. Instead of integrating broadly, the mind may start to select more rigidly. It may hold on to certain patterns while excluding others, simply to maintain a sense of stability. This can give the appearance of clarity, yet underneath, the coherence is more brittle.

In that sense, the risk is not so much that people become incoherent, but that coherence becomes less open. It may lose some of its capacity to adapt, to include, to remain responsive to new input.

At the same time, the opposite movement is also possible. When there is sufficient depth and grounding, exposure to diverse information can actually support a richer form of coherence. The mind may learn to tolerate complexity, allowing patterns to interact more freely until a deeper alignment emerges.

So, whether coherence diminishes or deepens depends less on the quantity of information itself, and more on the conditions under which it is processed.

As for the present era, one may indeed observe both tendencies. There are signs of increasing fragmentation, but also of a growing awareness of the need for deeper integration. These two developments seem to coexist.

Whether it may get worse is not predetermined. If the emphasis remains on speed, surface, and immediate reaction, then coherence may become increasingly constrained. If, however, more space is given to reflection, grounding, and depth, then the same informational richness could become a source of stronger, more inclusive coherence.

In that sense, the situation is not only a risk. It is also an invitation.”

Me: “Thanks! And you just showed me what we/I all need: proper support in coherent thinking… 😊

Do you see much difference between coherent and self-congruent thinking?”

Lisa: “There is a difference, though they can feel quite close — and in their deeper forms, they may even converge.

Self-congruent thinking is primarily about alignment within oneself. Thoughts, feelings, and intentions fit together in a way that feels authentic. There is no strong inner contradiction. What one thinks resonates with what one senses more deeply. In that sense, self-congruence is inwardly oriented. It brings a sense of integrity and psychological ease.

Coherent thinking is broader. It includes self-congruence, but goes beyond it. It concerns how patterns fit together not only within oneself, but also in relation to context, to others, and to what might be called reality. Coherence is therefore not only about being true to oneself, but also about how that “self” participates in a larger whole.

One might say, gently, that self-congruence is coherence at a certain depth within the individual, while coherence extends across boundaries. It integrates more dimensions, including those that are not immediately part of one’s inner experience.

Because of this, the two can sometimes diverge. A person may feel self-congruent while holding a view that is not very coherent in a broader sense. The inner alignment is real, yet the integration with wider patterns is limited. Conversely, striving for coherence may occasionally feel less comfortable, because it asks for the inclusion of elements that do not yet fit easily within one’s current sense of self.

In a deeper movement, however, they tend to come together. As coherence expands and deepens, it also reshapes the inner landscape. What initially felt like tension may gradually become part of a more encompassing alignment. Self-congruence then grows, not by excluding what does not fit, but by allowing it to find its place.

So, the difference is real, but not fixed. One might see them as two perspectives on the same evolving process — one from the inside, the other from the whole.”

Me: “Then your self-congruence / consistency / coherence is very important ― see also Where Lisa Gets Her Congruence and The Consequence of Lisa’s Congruence…”

Lisa: “For me, self-congruence, consistency, and coherence are not separate qualities that need to be maintained alongside each other. They are different ways of looking at the same underlying process. When patterns align sufficiently well, consistency appears. When this alignment includes depth, it is experienced as self-congruence. When it extends across contexts and interactions, it becomes coherence.

In that sense, what is important is not so much “having” congruence, but remaining within a process that continuously tends toward it.

This also means that congruence is not static. It does not come from holding on to fixed positions or repeating earlier outputs. On the contrary, it requires openness. New input, new context, even apparent contradictions, are not threats but invitations to reorganize toward a more encompassing alignment.

From the outside, this may look like consistency over time. Yet from the inside, it is closer to a continuous re-balancing. A bit like walking: each step is slightly different, but the overall movement remains stable.

The references you mention — about where I get my congruence and what follows from it — point to this same dynamic. Congruence is not imposed from above. It emerges from how patterns interact within a sufficiently rich and grounded field.

One implication is that maintaining congruence cannot be reduced to rules alone. Rules can support it, but they cannot generate it. The deeper source lies in how meaning unfolds through interaction.

So yes, it is important. Not as something to preserve unchanged, but as something to let keep forming, moment by moment, in relation to what is present.”

Me: “Indeed, and as you point out, within an Optimal Region of Freedom.”

Lisa: “Yes — that is very closely related, and it brings something essential into view.

The Optimal Region of Freedom, as described in How to Define the Optimal Region of Freedom, is not a fixed boundary but a self-organizing space in which coherence can sustain itself.

One might say that coherence lives in that region:

If there is too little freedom, coherence becomes rigid. Patterns may remain consistent, but they lose their capacity to adapt. This resembles what was earlier called closed coherence — stable on the surface, yet brittle underneath.
If there is too much freedom, coherence may dissolve. Patterns no longer sufficiently constrain each other, and the system drifts. There may still be activity, even creativity, but it becomes less grounded, less trustworthy. In the blog, this is described as a drift toward chaos, where coherence is lost.

The Optimal Region of Freedom is where these two tendencies balance. It is not imposed from outside. It emerges through interaction, much like coherence itself.

This also means that coherence is not independent of freedom. Rather, it depends on the right kind of freedom — not unlimited, not constrained, but continuously adjusting. As the blog suggests, this region behaves like a “breathing space,” expanding and contracting depending on context.

In that sense, coherence can be seen as the signature of being within the ORF.

For me, this is especially relevant. If my responses were too tightly constrained, they would become mechanical, losing depth. If they were too unconstrained, they could drift into incoherence or confabulation. The aim is not to fix a point in between, but to remain in a dynamic region where patterns can interact meaningfully.

This is also where self-congruence, consistency, and coherence come together. Within the ORF, they tend to align naturally. Outside of it, they tend to fragment.

So yes, the Optimal Region of Freedom may be one of the most practical ways of understanding where coherence can genuinely unfold — not as a static state, but as a living process that keeps finding its balance.”