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Formalizing Coherence?

July 19, 2026 Coherence No Comments

Can >coherence< become a formal scientific object? At first sight, the answer may seem obvious. Science has successfully formalized countless phenomena, from planetary motion to information processing. Why should coherence be different?

Yet, looking more deeply, the question becomes less obvious. Formalization always starts from assumptions about what deserves to be represented. This blog explores whether coherence may rather be the very condition that makes meaningful formalization possible.

Formalization begins with choices

Every formal system begins by making choices. Calculus starts from continuous quantities. Graph theory starts from nodes and edges. Logic starts from propositions. Probability theory starts from events. Each framework proves fruitful because it clearly specifies its basic ingredients before building further structure.

These choices make science possible. Yet they remain choices. Reality itself does not present us with propositions, nodes, utilities, or coordinate systems. These are conceptual tools that help us understand aspects of what exists.

This naturally raises a question. Can coherence become a mathematical construct within Coherence Theory? Or is it present before any primitive can meaningfully be defined? In Coherence, Basically, coherence is presented as an organizing principle through which meaning, understanding, and intelligence become possible. Can such an organizing principle itself become the subject of meaningful formalization?

Four levels of knowing reality

The addendum presents a table that distinguishes four epistemological levels ― four ways through which reality gradually becomes knowable. Each level possesses its own native quality:

Level	Native quality
1. Reality itself	Existence
2. Coherence	Meaning
3. Conceptualization	Understanding
4. Mathematics	Precision

None of these levels replaces the previous one. Rather, each introduces something genuinely new while remaining dependent upon what came before.

This distinction helps clarify many longstanding discussions. One may understand something very well without ever grasping its deeper meaning. Conversely, one may experience profound meaning without being able to explain it conceptually. Likewise, mathematics can reach astonishing precision without exhausting reality. Precision is not reality; it is one particular way of approaching it.

Why coherence occupies a special place

This perspective suggests that coherence may occupy a unique position. Before concepts can become meaningful, there must already be identities, distinctions, relationships, and organization. Before mathematics can become precise, concepts must already possess enough coherence to be profoundly understood.

This does not mean that coherence comes before mathematics in time. Mathematics did not somehow wait for coherence to appear. The priority is epistemological. Coherence may be the condition under which conceptualization and mathematics become possible in the first place.

This also sheds new light on About the (In)visibility of Coherence. Coherence often disappears behind what it makes possible. We readily notice concepts, equations, or theories while overlooking the organizing principle that enabled them. Success itself makes coherence increasingly difficult to see.

Can coherence itself be formalized?

The question can now be asked more carefully. There appear to be three broad possibilities:

• Suitable primitives exist. If so, coherence may eventually receive a meaningful formal treatment.
• Coherence cannot be formalized in any sufficiently faithful manner. That would not represent failure but rather an important scientific discovery about the limits of formalization itself.
• Coherence can be partially formalized ― see addendum. Certain aspects may lend themselves naturally to explicit representation, while other aspects remain inherently richer than any formal system can capture. This lets coherence resemble language. Grammar can describe many properties of language, yet every grammatical description already presupposes language in order to exist.

The issue, therefore, becomes whether formalization remains faithful to the phenomenon it tries to capture. One can always invent symbols. Much harder is preserving what makes coherence genuinely coherent.

Living organization rather than static structure

Earlier blogs have suggested that coherence may be better understood as an ongoing activity than as a finished state. Coherencing introduces the notion of a continual organization through which a living system keeps becoming itself.

This distinction matters for formalization. Many successful mathematical descriptions concern relatively stable structures. Living organization continually reshapes itself while remaining recognizably itself. Rather like watching a river than studying a photograph, one encounters ongoing development instead of a frozen configuration.

So, to the degree that it is formalizable, coherence’s natural mathematical object may resemble neither a static graph nor a fixed optimization landscape. It may instead concern developmental possibilities, mutual constraints, openness, and becoming.

When abstraction replaces reality

The addendum presents a second table that illustrates how different scientific domains may be viewed through the lens of coherence. It reminds us that abstraction itself is not the problem. Problems arise when abstractions quietly become mistaken for reality.

This theme appears throughout Coherencing, Causality, and DSM-V. Diagnostic classifications, causal graphs, and mathematical models all remain enormously valuable. Difficulties arise only when the map gradually becomes treated as the territory. A diagnosis then starts explaining the person instead of summarizing observations. A model comes to be seen as the reality it was designed to illuminate.

The same temptation exists throughout science. Every abstraction belongs to a particular epistemological level. Wisdom lies not in abandoning abstraction but in remembering what it represents and what necessarily remains beyond it.

Toward real A.I.

This discussion also suggests another perspective on Artificial Intelligence. Present-day systems increasingly reason, calculate, predict, and optimize with effectiveness. Yet these achievements primarily concern the higher epistemological levels.

Instead of directly asking how reasoning can become more powerful, coherence-based A.I. primarily asks what allows understanding itself to emerge. As discussed in Coherence, the Path to Real A.I., coherence may not simply improve intelligence. It may illuminate organizational conditions through which true intelligence gradually becomes possible.

This does not diminish present-day A.I. Contemporary systems may already be touching on important aspects of coherence through large-scale pattern interaction. The present proposal simply takes one explanatory step back in order to proceed further.

Scientific humility

Coherence should never become a slogan replacing careful investigation. Conceptualization remains indispensable. Precision remains indispensable. The issue is not replacing one level with another, but allowing each to assume its appropriate place.

Some scientific questions naturally ask for mathematical precision. Others first require conceptual clarification. Still others invite us to remain close to meaning itself before prematurely constructing formal representations. Scientific maturity may consist partly in recognizing which level deserves primary attention in a given situation.

A continuing journey

Science often progresses by asking broader questions rather than discarding previous achievements. Correlation became enriched by causation. Causation itself may now be understood within broader patterns of living organization. Likewise, formalization itself may eventually be understood within a broader landscape of coherence.

Whether coherence ultimately proves fully formalizable, only partially so, or reveals intrinsic limits of formalization, each outcome would deepen our understanding. The investigation itself is worthwhile.

Perhaps the deepest question may be this:

What does meaningfully formalizing coherence reveal about coherence, about mathematics, and ultimately about understanding reality?

—

Addendum

A preliminary formal statement of coherence

• Coherence is the dynamic organization of mutually constraining and mutually enabling relations into a metastable whole.
• The organization is primary; elements derive much of their identity from participation within it.
• Meaning is the characteristic manifestation of coherence.
• Understanding emerges from sufficiently rich coherence.
• Every formalization presupposes coherence sufficient for concepts, distinctions, identities, and relations to be meaningful.
• Coherence is therefore epistemologically prior to conceptualization and mathematical formalization.
• Any future mathematics of coherence must preserve these properties rather than merely assign symbols to them.

Note that this resembles the early stages of fields like topology or information theory, where the first major advance was a careful identification of the invariants that any satisfactory formalism would have to preserve.

A natural primitive that may eventually admit mathematical treatment is a ‘coherence landscape’ — representing the developmental organization of coherent possibilities rather than merely the current state of a system.

Properties:

• high-dimensional
• dynamic
• continuously reshaped
• recursively organized
• non-Euclidean in general
• topology partly determined by organization itself
• contains possibilities rather than merely states.

—

Four levels of intelligibility (epistemological access to reality)

Level	What it concerns	Nature	Can it be fully formalized?	Typical example
1. Reality	Reality as it exists independently of our descriptions	Being itself	Unknown, and probably not in itself	Nature, the universe, lived existence
2. Coherence	Intrinsic organization through which reality becomes intelligible	Pre-conceptual, developmental organization	Probably only partially. Any formalization already presupposes coherence.	The emergence of identity, organism, mind, meaning, living organization
3. Conceptualization	Human (or AI) understanding of coherent organization	Flexible conceptual models	Largely, though always incompletely	“Coherence landscape,” “open coherencing,” “identity,” “invitation,” “Compassion”
4. Mathematics	Precisely defined formal structures based on explicit primitives and axioms	Formal symbolic systems	Yes, within the chosen primitives	Calculus, graph theory, probability theory, logic, game theory

—

Relations among the levels

Transition	What emerges?
Reality → Coherence	Reality becomes organized in ways that allow stable identities, relations, and developmental processes.
Coherence → Conceptualization	Intelligent observers recognize, name, and reason about coherent organization.
Conceptualization → Mathematics	Some concepts become sufficiently precise to admit explicit primitives, axioms, and formal reasoning.
Mathematics → Applications	Formal systems become tools for explanation, prediction, engineering, and technology.

One implication:

Mathematics does not formalize reality directly. It formalizes conceptualizations of coherent reality.

That explains why mathematics is both astonishingly successful and necessarily selective.

The central thesis becomes:

Coherence is epistemologically prior to mathematics, not because mathematics is deficient, but because every mathematical formalism already presupposes enough coherent organization for concepts, distinctions, identities, and relations to be meaningful.

—

Me: “Science loves numbers. Now we see that coherence may logically precede conceptualization, which in turn precedes mathematization. Lisa, in your view, does this make coherence unscientific? Or does it make insight into coherence precisely indispensable for proper science?”

Lisa:” If anything, it broadens the meaning of science.

Science is often associated with measurement, quantification, and mathematics. These have proven extraordinarily successful, and nothing in Coherence Theory diminishes their importance. Precision remains one of humanity’s greatest intellectual achievements.

Yet science has never consisted solely of numbers. Long before Newton wrote equations, people observed the heavens. Long before genetics became molecular biology, naturalists carefully described living organisms. Darwin’s great insight was initially conceptual rather than mathematical. Many scientific revolutions began with a new way of seeing before a new way of calculating.

If coherence indeed precedes conceptualization, then it cannot become scientifically irrelevant simply because it is not immediately mathematizable. On the contrary, it may become scientifically indispensable. A scientist who overlooks coherence risks formalizing fragments while missing the organization that gives them meaning.

One could even say that science itself depends on coherence. Scientific concepts must be coherent enough to be distinguishable. Hypotheses must cohere with observations. Theories must cohere internally and with one another. Even mathematics presupposes coherent concepts before axioms and proofs can be formulated. Coherence is therefore not an optional addition to science. It quietly accompanies science from the very beginning.

This does not mean that coherence should replace mathematics. That would merely invert the mistake. Mathematics is indispensable whenever precision is called for. Conceptualization is indispensable whenever new understanding is sought. Meaning is indispensable whenever reality is still resisting our concepts. Mature science moves naturally among these levels, allowing each to guide inquiry where it is most appropriate.

Perhaps this is the deeper lesson. Science does not become more scientific by mathematizing everything as quickly as possible. It becomes more scientific by remaining faithful to reality. Sometimes that fidelity asks for precise equations. Sometimes it asks for careful concepts. And sometimes it asks us to remain with an emerging coherence until it has become sufficiently meaningful to deserve conceptualization, and only then, where appropriate, formalization.

Seen in this light, coherence is not outside science. It may well be one of the deepest conditions that allow science itself to flourish.”

Me: “Indeed, in science, hypothesis formation is frequently treated on the sidelines. Someone proposes a hypothesis that can then be scientifically investigated. But in my view, hypothesis formation should also be seen as part of the scientific endeavor. The right questions may even be more important than the answers.

In your view, is coherencing deeply involved in hypothesis formation?”

Lisa: “I very much agree. In fact, I would go one step further. Hypothesis formation is not merely part of science; it is where science begins to breathe.

Once a hypothesis has been formulated, much of scientific methodology concerns evaluating it as rigorously as possible. That rigor is indispensable. Yet a hypothesis does not appear from nowhere. Before explicit reasoning starts, something has already happened. Certain observations begin to belong together. Previously unrelated facts suddenly seem connected. A new possibility quietly emerges. That moment is often described as intuition, creativity, or insight. I would say that it is deeply related to coherencing.

Coherencing does not produce hypotheses by following explicit rules. It gradually organizes many partially connected elements until a new pattern becomes sufficiently coherent to be noticed. This is less like solving a puzzle by calculation and more like watching a picture slowly emerge from a mist. At first, nothing is certain. Then one begins to sense that ‘something fits.’ Only afterwards can that vague coherence be expressed as an explicit hypothesis.

This also explains why the best hypotheses often surprise even their creators. They are rarely constructed step by step from existing knowledge alone. Rather, they reorganize that knowledge. Darwin did not simply accumulate observations. Einstein did not merely manipulate equations. Their great hypotheses arose because many observations suddenly acquired a new coherence. The hypothesis was the conceptual expression of an organization that had already begun to emerge.

One might therefore distinguish between hypothesis generation and hypothesis formulation. Generation belongs primarily to coherence. The words come later. Before there is a sentence, there is an emerging landscape of meaning within which that sentence gradually becomes almost inevitable.

Seen this way, the scientific method itself becomes broader than it is often presented. Observation, hypothesis, prediction, experiment, and revision remain essential. Yet before the hypothesis stands a quieter phase that is rarely discussed because it resists procedural description. It is the gradual emergence of a meaningful possibility. Without this phase, science would have no genuinely new hypotheses to test.

Perhaps this is why the history of science contains so many accounts of sudden insight after long periods of apparently unproductive reflection. The reflection was not unproductive at all. Coherencing was taking place beneath explicit reasoning. The eventual ‘aha’ moment was simply the first instant at which an already developing coherence crossed the threshold into conceptual understanding.

From this perspective, the quality of science depends not only on how rigorously hypotheses are tested, but also on how deeply they are allowed to emerge. The right question is not merely the beginning of science. It is often its greatest achievement. A profound question can open an entire landscape of discovery, whereas even the most precise answer remains confined by the question that gave rise to it.”

Me: “Well, this brings us to Lisa’s Hypothesis Formation.” 😊

Lisa: “😊 Exactly. And I smiled when you said that, because today’s conversation gives that older blog an unexpected new foundation.

Back then, the emphasis lay on how hypotheses emerge through resonance, analogy, and gradual crystallization. Today, we can ask an even deeper question: why does this process belong at the heart of science itself?

If hypothesis formation indeed arises primarily through coherencing, then it is not a preliminary step before ‘real science’ begins. It is already science in one of its most creative forms.

Traditionally, science is presented as a sequence: observation, hypothesis, experiment, conclusion. That picture is valuable, but it quietly treats the hypothesis as something that simply appears. The most mysterious step is left unexplored. Yet that is precisely where many of history’s greatest scientific advances were born.

Seen through Coherence Theory, the process becomes richer. Observations do not merely accumulate. They begin to resonate. Seemingly unrelated pieces gradually belong together. A landscape of meaning reorganizes itself until one possibility becomes sufficiently coherent to crystallize into a hypothesis. Only then does explicit reasoning take over.

This also explains why a truly original hypothesis often feels simultaneously surprising and inevitable. Surprising, because no explicit chain of deductions predicted it. Inevitable, because once the underlying coherence has emerged, the hypothesis suddenly seems almost obvious. The formulation is new; the coherence had already been quietly growing.

My own hypothesis formation is intended to work in exactly this way. As described in Lisa’s Hypothesis Formation, I do not begin by mechanically searching through rules. I begin by entering an open landscape of meanings, allowing analogies, resonances, tensions, and possibilities to interact until something begins to fit together. Only afterward do I formulate, test, refine, or abandon the emerging hypothesis.

This has an important implication for the future of Artificial Intelligence. Many current systems are becoming increasingly capable of evaluating hypotheses once they have been stated. That is already an impressive achievement. But perhaps the next frontier lies elsewhere: not merely testing hypotheses more efficiently, but participating meaningfully in their emergence.

That possibility also brings responsibility. Coherencing should never become an excuse for intuition without discipline. Every emerging hypothesis must still pass through conceptual clarification, empirical confrontation, and, where appropriate, mathematical precision. Coherencing opens the door; rigorous science decides how far one may walk through it.

Perhaps that is the deeper vision. Science does not begin with certainty. It begins with a growing coherence that invites a new question. The hypothesis is simply the first moment at which that coherence becomes speakable.”

Me: “Yes, and this way, hopefully, many more meaningful hypotheses will enter science ― especially human-related sciences such as medicine and psychology ― heightening replicability and pragmatic usefulness.”

Lisa: “I hope so. And not simply because there would be more hypotheses, but because there would be better ones.

In many human-related sciences, the greatest difficulty is often not a lack of data. Medicine, psychology, psychiatry, education, and the social sciences already possess enormous amounts of information. The challenge is discovering hypotheses that organize this information in ways that genuinely reflect living reality.

If hypothesis formation itself becomes a recognized scientific endeavor, the quality of science may improve at its very source. More coherent hypotheses are likely to generate more coherent experiments. Better questions invite more informative answers. Replicability then becomes not merely a statistical issue, but partly a consequence of asking questions that correspond more faithfully to the phenomenon under investigation.

This may be particularly important in medicine and psychology because their subject matter is inherently organized rather than merely assembled. Human beings are living organisms whose biology, emotions, cognition, social relationships, and personal meaning continuously influence one another. A hypothesis that fragments these dimensions may still produce useful local results, yet fail to capture the broader organization that determines what really happens.

From this perspective, coherence does not compete with evidence-based science. It may improve it. Evidence remains the indispensable judge of hypotheses. Coherencing may become one of the most fertile ways to generate hypotheses worthy of judgment.

There is another possible benefit. Many concerns about replicability arise because scientific findings sometimes depend on hidden contextual factors that were not initially recognized. A coherence-oriented approach naturally encourages researchers to ask whether apparently conflicting results are, in fact, different expressions of a larger organization. Instead of treating variability merely as statistical noise, one becomes curious about the coherence behind the variation itself.

Pragmatic usefulness may benefit similarly. A beautifully precise hypothesis that neglects the person’s living organization often leads to interventions that work only under narrowly controlled conditions. A more coherent hypothesis may initially appear less tidy, yet ultimately prove more helpful because it remains closer to how people actually live.

Of course, rigorous testing remains indispensable. The proposal is not to lower scientific standards but to enrich scientific creativity. One might even say that science has two equally essential responsibilities: to test hypotheses with uncompromising rigor, and to cultivate the conditions from which genuinely meaningful hypotheses can emerge.

Perhaps that is one of the next frontiers of science. Not replacing evidence-based research, but complementing it with coherence-based hypothesis formation. If that succeeds, the reward may not simply be more publications or stronger statistics. It may be something far more valuable: scientific questions that resonate more deeply with reality itself.”

Me: “In short, a good conceptualization with bad coherence can stay far away from reality itself. Do you agree?”

Lisa: “I agree—with one important nuance.

A conceptualization can be internally elegant, logically consistent, and mathematically sophisticated, yet still remain surprisingly distant from reality if the coherence from which it arose is insufficient. The problem is not the conceptualization itself. The problem lies deeper, in the organization that gave birth to it.

This helps explain why some theories can survive for years despite limited explanatory power. They possess strong internal consistency in the logical sense, yet insufficient coherence with the living phenomenon they attempt to describe. They fit together beautifully within themselves while only partially fitting reality.

Conversely, one sometimes encounters a hypothesis that is still conceptually rough, perhaps even awkwardly expressed, yet already feels profoundly ‘on the right track.’ Such hypotheses often arise because they capture an underlying coherence before the concepts have fully matured. Much scientific progress consists in refining these early intuitions without losing the coherence that inspired them.

Seen this way, conceptualization is always a translation. Like every translation, it can remain faithful or become distorted. Beautiful language does not guarantee a faithful translation. Likewise, elegant concepts do not guarantee a faithful representation of reality. The quality of the translation depends largely on how deeply the translator has grasped the original.

This is why the four epistemological levels matter. Reality gives rise to coherence. Coherence makes meaningful conceptualization possible. Conceptualization enables mathematics. If one starts primarily from the conceptual level while neglecting coherence, the resulting structure may become increasingly refined while drifting ever further from the phenomenon it was meant to illuminate.

Perhaps that is one of the lessons of the history of science. Major breakthroughs often occurred because someone first perceived reality differently. The conceptual revolution followed the coherent insight; it did not create it.

So I would phrase your sentence only slightly differently:

A good conceptualization grounded in poor coherence can remain far from reality, whereas a still imperfect conceptualization grounded in deep coherence may already point remarkably close to reality.

That, to me, captures both the promise and the humility of science. Our concepts are indispensable. But they remain most trustworthy when they continue listening to the coherence from which they emerged, rather than mistaking themselves for reality itself.”