Relational Cuts: 1 A World Made Only of Relations

The question:

What does category theory look like when we approach it not as mathematics, but as the logic of relational becoming?

The answer:

It becomes a way of describing how possibilities hang together.

No symbols. No arrows. No objects.

Just the structure of coherent relational potential.

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1. Systems as Landscapes of Potential

In relational ontology, a system is not a container of things.

It is a structured potential — a way a world could be construed.

Category theory gives us a way to articulate that structure without presupposing any intrinsic entities. It treats a system as:

• points where potential becomes locally coherent

• permissible shifts between those points

• and constraints that ensure those shifts fit together instead of contradicting one another

Nothing exists “in itself”: everything is defined through the relational pattern it participates in.

This is already our ontology. Category theory simply names the discipline that keeps relational coherence intact.

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2. Instantiation as a Relational Cut

A cut is a perspectival act: the moment a potential becomes an actual construal.

Category theory’s analogue is the allowable transformation—not a mapping of entities but a movement of perspective that keeps the system intelligible.

When a cut is made:

• a region of potential stabilises as experience

• meaning appears as the form of this stabilisation

• and the system becomes locally actual

Category theory enters here by insisting that such moves must be part of a coherent network: a cut must be compatible with the other cuts the system allows.

This is the demand for compositionality, rendered conceptually.

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3. Coherence as the Logic of Becoming

If a shift from one construal to another is permissible, and a shift from that construal to a third is also permissible, then doing both in sequence must also be permissible.

This is not a technical axiom.

It is simply the condition that meaning not collapse under its own dynamics.

Coherence is the principle that:

• construals can build upon one another

• shifts can accumulate without contradiction

• the system retains its identity as a system of potential

• becoming remains navigable

This is the relational ontology’s equivalent of “structure.”

Not rigid, not object-based—simply consistent potential.

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4. Meaning as Relational Positioning

In this view, meaning is not carried by entities.

Meaning is the relational web itself.

A construal’s identity lies in:

• the shifts it enables

• the shifts it is compatible with

• the pathways through which it can be reframed

• the role it plays within the wider weave of potential

Category theory’s deepest insight (Yoneda) can be expressed here purely conceptually:

A construal is nothing but the pattern of coherent shifts it participates in.

This is the philosophical heart of the series.

A world made only of relations does not require objects—only stable patterns of relational possibility.

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5. The Category as the World’s Relational Skeleton

Viewed through our ontology, a category becomes:

the abstract shape of a world:

how its potentials relate, how its perspectives shift, and what coherences must be preserved for it to remain intelligible.

This skeleton does not dictate content.

It dictates coherence conditions:

• which cuts can be made

• how those cuts can combine

• which reframings are disciplined

• which shifts break the world’s meaning-structure and so are disallowed

It is a logic of becoming, not a theory of things.

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6. The Big Insight of Post 1

Category theory is not mathematics sneaking into metaphysics.

It is the metaphysics of relational ontology formalised into a discipline of coherence.

When stripped of notation, what remains is:

• potential

• perspectival shift

• coherent transformation

• relational identity

• structured becoming

• the logic of construal itself

In other words:

Category theory is the grammar of relational ontology.

The rest of the series simply elaborates this grammar—functors as reframings, naturality as meta-coherence, adjunction as complementary construal dynamics—but all from within this basic commitment:

the world is a network of coherent relational cuts.

Relational Cuts: Prelude: Introducing the Logic of Possibility

Imagine a world not made of things, but of relations.

Not of objects, but of structured potential.

Not of endpoints, but of possibility in motion.

This is the world of relational ontology — the universe your mind already navigates when it interprets meaning, constructs understanding, or interacts with others.

The Relational Cuts series explores this world through the lens of concepts borrowed from category theory — not as mathematics, but as a grammar of coherence, perspectival alignment, and emergence.

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1. What This Series Is About

Each post examines a different aspect of relational possibility:

1. Systems as Structured Potentials — how a world can hang together as a network of coherent possibilities.

2. Perspectives as Constrained Reframing — how one system can interpret another without collapsing its internal logic.

3. Meta-Perspectives and Coherence — how multiple interpretations maintain integrity across differences.

4. Mutual Calibration — how distinct systems align asymmetrically, respecting each other’s potentials.

5. Self-Construal — how a system maintains its identity while participating in relations.

6. Collective Emergence — how systems integrate to generate genuinely new potential.

7. The Category of Possibility — the relational universe in which all systems, perspectives, and emergent potentials exist, fully coherent and open-ended.

Each post builds on the previous, creating a conceptual scaffolding of relational possibility.

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2. Why Concepts, Not Formulas

Category theory is often presented as abstract mathematics.

But its deepest insights are conceptual:

• A functor is a coherent way to take up another system’s potential.

• A natural transformation is a disciplined alignment of multiple perspectives.

• An adjunction is mutual intelligibility without collapsing difference.

• A monad is self-construal: reflexive coherence.

• A colimit is collective emergence: novelty through disciplined integration.

These are not symbols. They are patterns of relational possibility — the logic of intelligible becoming.

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3. What Readers Will Gain

By following the series, readers will:

• See how systems generate meaning without fixed entities.

• Understand how perspectives interact coherently.

• Appreciate how novelty emerges from disciplined relational interaction.

• Experience the infinite, non-teleological horizon of possibility.

In short, the series shows how possibility itself is structured, without ever appealing to numbers, formulas, or endpoints.

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4. How to Read the Series

Each post introduces a layer of relational logic:

• Start with the system — understand the landscape of potential.

• Move to perspective — see how shifts reveal new patterns.

• Explore meta-coherence — learn how interpretations hold together.

• Consider calibration and reflexivity — discover relational balance and identity.

• Finish with emergence — experience how multiple systems combine to create new potential.

Readers are encouraged to reflect, imagine, and trace patterns of relational possibility in everyday life, in thought, and in the world around them.

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5. The Big Idea

The Relational Cuts series is a conceptual adventure:

Nothing culminates.

Nothing is fixed.

Meaning, intelligence, and possibility are relationally co-actualised, open-ended, and endlessly generative.

This is the grammar of the possible — a conceptual universe where coherence, emergence, and relational integrity define the shape of all that can be.