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.