If the task of physics is not to discover ontological primitives but to articulate structured potentials and their compatibility relations, then we require a formal language suited to that task.
Most existing physical formalisms remain object-first:
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particles,
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fields,
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manifolds,
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states.
Relations appear secondarily — as interactions between already-given entities.
But relational ontology reverses that order.
There already exists a mathematical framework that takes this reversal seriously: category theory.
Developed in the mid-twentieth century by Samuel Eilenberg and Saunders Mac Lane, category theory was not originally intended as metaphysics. It was a structural tool for relating different areas of mathematics.
Yet its core move is philosophically radical:
It defines systems not by their internal substance, but by the morphisms — the structure-preserving mappings — between them.
1. Objects as Nodes of Relation
In a category:
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Objects are defined only by their position within a network of morphisms.
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Morphisms encode structure-preserving transformations.
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Composition governs how transformations relate.
An object without morphisms is empty.
It has no meaning outside its relational embedding.
This is not merely a technical convenience.
It formalises a relational stance:
Identity is given through structural position, not intrinsic content.
2. Constraint Systems as Categories
Now consider our earlier formulation.
General relativity and quantum field theory were treated as constraint systems generating instance spaces.
We may reinterpret each constraint system as defining a category:
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Objects: allowable configurations under the constraint.
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Morphisms: admissible transformations preserving coherence.
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Composition: sequential application of transformations.
On this view:
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The relativistic constraint system becomes a category of geometrically coherent configurations.
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The quantum constraint system becomes a category of state-space evolutions preserving operator structure.
Neither is reduced to the other.
Each is structurally complete relative to its own internal rules.
3. Compatibility as Functorial Relation
The classical ambition of unification assumes that both theories must embed into a single larger structure.
Category theory offers a subtler alternative: the functor.
A functor maps one category to another while preserving structure.
This is not embedding into a deeper substrate.
It is translation under constraint preservation.
The question of quantum gravity becomes:
Does there exist a functorial relation between the relativistic and quantum constraint categories that preserves essential structure within overlapping domains?
If yes, we have coordination.
If no, we have principled incompatibility.
The focus shifts from ontology to mapping conditions.
4. Failure as Non-Functoriality
In previous posts, we treated paradox and divergence as boundary markers of failed co-actualisation.
Category theory makes this precise.
Failure occurs when no structure-preserving mapping can be defined between two systems under specified constraints.
The infinities of quantum gravity attempts may be read not as hints of deeper substance, but as indicators that a naïve functor cannot exist under the imposed assumptions.
This reframes the technical crisis:
The problem is not that spacetime and quantum states “really” contradict.
It is that our attempted mappings fail to preserve structure.
The failure is formal.
Not ontological.
5. No Final Category
A temptation immediately appears:
If categories can be related by functors, perhaps there exists a “category of all categories” — a final structure containing everything.
Category theory itself warns against this.
There is no totalising category without paradox. Structure is inherently stratified.
Relational ontology converges here:
There is no global frame guaranteeing ultimate unification.
There are only networks of relations between structured potentials.
The dream of a single, final, all-encompassing theory dissolves into a web of coordinated mappings.
6. A New Ambition
If this direction is correct, the deepest physics would not identify the fundamental object.
It would identify:
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the admissible morphisms between constraint systems,
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the invariants preserved across translation,
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and the boundaries where translation necessarily fails.
The “theory of everything” becomes a theory of relation between theories.
Not metaphysical reduction.
Structural articulation.
Closing Reversal
Physics began with substances.
It moved to fields.
It now flirts with information.
But perhaps its next maturation is relational formalism.
Not discovering what the world is made of.
But mapping how structured potentials can coherently transform into one another.
Relation before object.
Constraint before substance.
Compatibility before unification.
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