Readiness — Potential Made Accountable: 3 Mathematics as Inclination Without Ability

1. Mathematics and the Formal Encoding of Inclination

Mathematics is extraordinarily powerful because it encodes inclination—the systematic tendencies of a system—without always attending to ability:

• Equations, structures, and formal rules specify how a system would evolve if continuation is possible.

• Internal coherence ensures that the system remains consistent along these prescribed tendencies.

• But this formal inclination does not by itself guarantee that relational capacity—the ability to realise these evolutions—actually exists.

In other words, mathematics tells us what should happen, not always what can happen.

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2. Divergence as a Symptom of Collapsed Readiness

When readiness is misaligned:

• Ability has collapsed: the system’s potential space cannot support further actualisation.

• Inclination persists: formal rules continue to demand evolution.

The result is divergence, singularities, or “infinite” solutions:

• Examples include blow-ups in differential equations, divergences in series expansions, and singular points in analytic functions.

• In all these cases, mathematics is correct within its own formal horizon, but the system being modelled has exhausted relational capacity.

Thus, what appears as a formal pathology is in fact a readiness signal.

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3. Hidden Assumptions of Readiness in Mathematics

Many mathematical procedures silently assume aspects of readiness:

1. Continuity – presupposes the ability to interpolate relational potential between points.

2. Differentiability – presupposes the ability to generate infinitesimal distinctions without exhausting potential.

3. Persistence – presupposes that relational axes remain open across successive actualisations.

Where these assumptions fail, formal divergence emerges. Singularities in physics and paradoxes in abstract mathematics are two sides of the same structural pattern: inclination without ability.

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4. Linking Mathematics and Physics

By treating readiness explicitly:

• We see why singularities in physics occur precisely where mathematics is most internally coherent.

• Divergence is not a flaw of mathematics itself; it is a flag indicating over-closure in the relational system being modelled.

• Mathematics becomes a tool for diagnosis, not a claim about ontological boundlessness.

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5. Conclusion

Mathematics is never “ontologically prior” to the world; it is a formal expression of inclination.

• Divergences arise when ability collapses.

• Hidden assumptions like continuity and differentiability are actually readiness assumptions.

• Explicit attention to readiness restores coherence across formal, physical, and conceptual domains.

In the next post, we will explore how modelling practice can integrate readiness checks explicitly, transforming singularities and divergences from metaphysical drama into actionable epistemic signals.