Mathematics is not transparent; it is oriented, selective, and inclined.
Modern physics inherited a myth about mathematics—a myth so pervasive it often passes as the air we breathe. It is the belief that mathematics gives us an uncoloured view onto the structure of the universe. That the formalism is transparent: a neutral lens, a pure conduit between the world and its representation.
Nothing could be further from the truth.
To forget this is to treat orientation as ontology, inclination as inevitability, and the map as if it were the territory. The resulting metaphysical confusions saturate 20th-century physics, culminating in the singularity pathologies that haunt both GR and quantum theory.
This post synthesises the argument developed so far in the series: that mathematics does not simply “describe” the universe. It carves it. And every carving is an inclination—an asymmetric act that shapes what can appear as meaningful, what can count as a problem, and what is allowed to actualise as an answer.
1. The old dogma: mathematics as neutral medium
The classical image runs like this:
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Physics tells mathematics what to say.
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Mathematics turns that instruction into symbolic form.
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We then “read off” the structure of the universe from the equations.
This is the fantasy of transparency: mathematics as a mirror that introduces no shape of its own.
The idea that mathematics is merely “neutral” is akin to thinking that the grammar of a language adds nothing to thought. But grammatical architecture is what makes thought articulate in the first place. The same is true of the formal architecture mathematics provides to physics.
2. Every mathematical structure is an inclination
A mathematical formalism is not a representational sheet of glass. It is a bias: a normative infrastructure that constrains how a system can be construed.
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A vector space inclines thought towards linear combination.
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A differentiable manifold inclines it towards smoothness, locality, and tangent-based reasoning.
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A Hilbert space inclines it toward orthogonality, projection, and spectral decomposition.
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A category inclines it toward morphisms, compositionality, and relational invariants.
Each of these inclinations constitutes a perspectival cut in the Hallidayan sense of instantiation-as-shift: a commitment to a way of carving the potential into actualisable structure. The formalisms do not “reflect” possibility—they shape it.
This is not a claim about human psychology. It is a structural claim: the formal systems themselves encode dispositions. When physics chooses a formalism, it adopts those dispositions as if they were the universe’s own.
3. The cost of forgetting the cut: metaphysical confusion in physics
When physicists treat mathematical inclination as ontological necessity, the formal constraints appear as natural laws rather than modelling decisions. Three recurring confusions emerge:
(a) Treating mathematical breakdowns as physical catastrophes
The singularities in GR (already analysed in Post 6) are not “in the universe”; they arise when the Riemannian formalism is pushed beyond the regime where its inclinations are coherent. The blow-up is a feature of the mathematical cut, not of the cosmos.
(b) Mistaking representational choices for metaphysical commitments
Whether a field is continuous or quantised, whether spacetime is smooth or discrete, whether the universe “is” a manifold or a category—these questions often presuppose that the formalism reveals being instead of shaping construal.
(c) Smuggling ontology in through the back door
Even the idea that physics should “solve the equations” presupposes that the shape of the equations is already the shape of the universe. This conflates semantic selection with worldly determination.
The deeper lesson: forgetting inclination creates ontological fantasies.
4. Mathematics as semiotic infrastructure: a relational account
In relational ontology, the mathematical formalism functions as a second-order meaning system: a system of construal, not a layer of reality. It positions the phenomenon by making certain relations salient and suppressing others. It is, in Hallidayan terms, a construal system that actualises meaning from the potential of possible descriptions.
Thus:
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Mathematics is not the universe’s blueprint.
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It is the architecture of construal through which physicists carve intelligibility.
5. Physics as the metaphysics of formal inclination
If mathematics is not transparent, then physics cannot claim metaphysical neutrality. The metaphysical claims of physics emerge from the inclinations of the mathematics it employs.
General relativity is not “what spacetime is.” It is what spacetime becomes under the Riemannian inclination. Quantum theory is not “what reality fundamentally is.” It is what reality actualises as under the Hilbert-space inclination.
Every formalism opens some ontological doors and closes others.
Thus the so-called “incompatibility” between GR and quantum theory is not a rift in nature. It is a clash between two incompatible inclinations.
Physics has mistaken a clash of construal for a clash of worlds.
6. Toward a new discipline: the analysis of inclination
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From transparent medium → to orientational structure.
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From mirror → to cut.
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From ontological claim → to semiotic constraint.
Once mathematics is recognised as a system of inclinations, physics becomes the study of how different formal cuts open different vistas of possibility. It becomes a relational discipline, aware of its own constitutive moves.
This enables a different kind of question:
These questions belong to a discipline physics has never formally named, but always implicitly practised.
7. Closing gesture: transparency is a myth; relational clarity is a method
Mathematics never gives us the universe as it is. It gives us the universe as inclined. What we take to be metaphysical necessity is often nothing more than the residual structure of a perspective.
And recognising this is not the end of inquiry—it is the beginning of a more honest, more rigorous, more relational physics.
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