Singularities and the Limits of Construal: When Mathematics Meets Its Horizon

1. Singularities as a boundary of construal

In physics, singularities — whether in general relativity (black holes, the Big Bang) or in other extreme models — are points where:

• Mathematical quantities “blow up” (infinities appear).

• Predictive models fail: equations no longer yield finite, meaningful outputs.

• The usual mapping from formalism to physical phenomena breaks down.

From a relational and Hallidayan SFL perspective:

• Mathematics is a construal system: it is a formal, highly abstract tool for modelling relational patterns.

• Singularities reveal the boundary of what this construal can meaningfully encode.

• They are places where the map no longer fits the territory, not necessarily a breakdown of the world itself.

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2. What this says about mathematics

Mathematics remains powerful, precise, and generative:

• It abstracts and encodes regularities across domains.

• It predicts, explains, and enables the engineering of reality.

But singularities reveal:

• Mathematics is not absolute; it is a language with domains of applicability.

• Infinity and undefined behaviour are not flaws of the universe — they are indicators of limits in our construal tools.

• Certain configurations exceed what a given formal system can express meaningfully.

In other words: mathematics does not create reality; it construes aspects of relational patterns within reality — and there are points where that construal is no longer adequate.

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3. Mathematics and physics: a relational dialogue

Physics often treats mathematics as a mirror of reality, implicitly assuming:

“The universe must behave as if the math is correct, even in the extreme.”

Singularities challenge this:

• Our equations lose predictive power, showing that the “mirror” cracks.

• This does not mean the universe ceases to exist, only that our current construals fail.

• Physics must either modify the theory (e.g., quantum gravity attempts) or reframe the questions, acknowledging the limits of the current mathematical apparatus.

So the relation is not: mathematics dictates physics, but rather:

• Mathematics construes relational patterns in the universe.

• Physics tests those construals against phenomena.

• Singularities are stress tests, revealing where the construal system no longer coherently encodes what is possible in the relational space of actualisation.

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4. Lessons from singularities for relational ontology

1. All construals have limits. Even the most abstract, formal ones — mathematics included — cannot fully exhaust the potentialities of relational structures.

2. Failure of construal ≠ failure of reality. A singularity is not a metaphysical breakdown; it is a boundary of expressive power in a given semiotic system.

3. The dialogue matters more than the formalism. Mathematics, physics, and observation co-construct understanding; one alone does not determine the rest.

4. Potential and actualisation remain primary. Even where equations diverge, relational potentials exist; singularities are just places where standard cuts through the possibility space fail to be meaningful.

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5. Punchline

Singularities are a limit-case lesson: mathematics is an incredibly potent construal, but it is never reality itself. Its “failure” at extremes tells us less about the universe than about the horizon of our conceptual and semiotic tools. Physics, in turn, is not just a translation of mathematics; it is a co-construal of relational patterns, tested and iteratively refined against actuality.

In short:

Mathematics maps, but it does not exhaust; physics probes, but it does not dictate. Singularities remind us that construal is always bounded, and that relational potential is the ultimate ground.