Modelling a Relational World: From Equations to Constraint Topologies

If the world is fundamentally relational—dynamic, contextual, and non-substantial—then the very act of modelling reality must also change. In this post, we ask: How can we construct models that honour relational ontology without smuggling back in the very entities we aim to displace?

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1. The Limits of Traditional Models

Most physical models are built on the assumption that:

• Systems are composed of distinct objects with intrinsic properties,

• Dynamics are governed by rules of interaction among those objects,

• Space and time form a fixed background in which change occurs.

These models succeed in many domains, but become inadequate when:

• Interactions are nonlocal and context-sensitive (e.g. entanglement),

• Boundaries between system and environment are not well-defined,

• Emergence and coherence replace determinism as primary dynamics.

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2. Relational Modelling Principles

To model relationally, we must shift our representational assumptions. Some guiding principles:

a. Systems as Fields of Potential

• The basic unit is not a thing but a configuration space: a set of mutually-constraining possibilities.

• States are actualisations within this space, not carriers of hidden essence.

b. Constraints as Structure

• Structure is not imposed from outside but arises from patterns of constraint among elements.

• Topology replaces geometry as the dominant spatial metaphor: connectivity over metric.

c. Coherence as Dynamics

• Change is tracked not through force or trajectory but through shifts in systemic coherence.

• Processes emerge as realignments of relational balance under changing affordances.

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3. Mathematical and Computational Tools

Relational modelling may draw on:

• Category theory and sheaf theory, which prioritise mappings and transformations over elements;

• Network theory and dynamical systems, reinterpreted in terms of evolving constraint topologies;

• Process calculi and agent-based modelling, where identity is contextual and emergent;

• Quantum information frameworks that foreground relational correlations and contextual encoding.

These tools allow us to describe systems in which what something is depends on where and when it is embedded—not in terms of static labels but dynamic participation.

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4. Dangers of Recapitulating Substance

Even sophisticated models can regress into substance-thinking if:

• Variables are reified as stand-alone entities,

• Initial conditions are treated as ontological givens,

• Interaction rules assume absolute independence prior to relation.

The challenge is to model relation without reifying relata—to capture coherence and transformation without crystallising structure prematurely.

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Closing

Relational modelling does not discard formal tools—it transforms their ontological commitments. To model a relational world is not just to simulate motion or interaction, but to trace how fields of constraint shape what becomes possible, where, and when.

In the next post, we’ll explore how this shift in modelling might reshape foundational questions in cosmology, including the nature of the Big Bang, time’s origin, and the structure of the early universe.