Mathematics has long served as the privileged language of physics. From Newton’s calculus to the tensor equations of general relativity and the abstract formalism of quantum mechanics, mathematics has not merely described physical reality—it has defined the form in which reality is intelligible.
But what becomes of this formal role when the metaphysical ground shifts? If reality is not a collection of objects in space, but a relational field undergoing constrained transformation, what is the ontological status of mathematics? How do equations retain meaning if there are no particles to move, no background time to evolve through?
This post explores how mathematical formalism functions within a relational ontology, and how the power of mathematics is preserved—but also reinterpreted—when we abandon substance-based metaphysics.
1. The Classical View: Mathematics as Mirror of Structure
In the classical view, mathematics functions as:
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A descriptive tool for capturing regularities in the behaviour of physical entities.
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A predictive engine for forecasting trajectories or outcomes based on laws and initial conditions.
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An ontological blueprint—reality itself is sometimes said to “be mathematical” in structure.
This works well under an object-based metaphysics: equations describe how objects change state over time, interacting under forces, following paths through spacetime.
But in quantum physics, the meaning of the equations becomes obscure:
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What does the wavefunction describe, if not a particle with a trajectory?
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What is time in Schrödinger’s equation, if it doesn’t correspond to physical observables?
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What are quantum operators acting on, if there is no background of determinate state?
And in relativity:
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What does a spacetime metric represent, if spacetime itself is emergent or relational?
These tensions suggest the need to rethink what mathematical structure is about.
2. Mathematics as Map of Constraint
In a relational ontology, mathematics does not describe the motion of entities. It models the structure of constraint within a system of relations.
That is:
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Equations define how possibility is distributed across the field.
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Variables correspond to degrees of freedom—dimensions along which coherence can transform.
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Operators and dynamics encode the rules of modulation—how the system can evolve under internal tension.
The wavefunction, then, is not a probability amplitude of particle location, but a field of potential coherence—a map of how the system might resolve itself across a topological constraint landscape.
In this frame, mathematics retains its role, but its referential target changes: it no longer mirrors object trajectories, but articulates the affordance structure of the system.
3. Interpretation Without Objectification
One of the deepest challenges in quantum mechanics is the so-called “measurement problem”: how does a mathematical structure (the wavefunction) relate to what we actually observe?
From a relational perspective:
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The wavefunction is not a representation of a real entity, but a relational configuration encoding potential transitions.
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Measurement is not a collapse of an object’s state, but a punctualisation—a selection of one coherence path from many, based on a new constraint imposed by an observing context.
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The predictive success of the formalism is not a window onto “hidden variables” but evidence of stable systemic coherence under repeated constraint patterns.
Mathematical formalism is thus not a map of what is, but of how systems become under specific structural tensions.
4. Objectivity Without Absolutes
Some may worry that this leads to relativism or undermines the objectivity of science. On the contrary:
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Mathematics continues to deliver stable, communicable predictions because it encodes how transformations propagate within structured constraint regimes.
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Different observers (or experiments) arrive at the same results not because they share a common background reality, but because their systems instantiate compatible fields of constraint resolution.
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Objectivity, in this model, means invariance of relational structure across transformations—not the existence of a hidden object accessible to all.
This matches well with general relativity, where physical laws are generally covariant (independent of coordinate system), and with quantum theory, where observables depend on the measurement context, but the structure of probabilities is robust.
5. Implications for Theory and Practice
A relational reinterpretation of mathematics offers several practical and philosophical advantages:
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It sidesteps ontological puzzles by decoupling formalism from misplaced object metaphors.
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It reorients interpretation toward systemic coherence, not mechanical cause.
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It opens the door to new kinds of modelling—for instance, using topological, categorical, or constraint-based formalisms that need not presume a background manifold.
More broadly, it allows physics to maintain its mathematical rigour while acknowledging that its true subject is not substance in motion, but relational transformation within dynamic fields of potential.
Closing Thought
Mathematics has always been a language of pattern. The patterns it now describes are not those of billiard balls and forces, but of relational intensities, affordances, and constraints—a deeper structure beneath space, time, and matter.
In the next post, we will consider how this relational view might inform new approaches to unification in physics, where the divide between quantum theory and general relativity may be seen not as a clash of incompatible formalisms, but as a symptom of unresolved ontological assumptions.
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