Stability, by itself, is not enough for mathematics to function.
Relations may persist across construals and yet remain mathematically resistant. The missing condition is not further precision or deeper formalism, but separability.
Mathematics presupposes not only that relations can be stabilised, but that they can be decomposed—pulled apart into relata that can be treated as independently variable. Where this presupposition fails, mathematics encounters a distinctive and revealing limit.
Separability as an entry condition
To formalise a relation mathematically is to assume that:
relata can be individuated prior to the relation,
relations can be specified between those relata,
variation in one relatum can be treated independently of variation in another.
These assumptions are so deeply built into mathematical practice that they often pass unnoticed. Variables, functions, coordinates, and state spaces all rely on the idea that components of a system can be cleanly distinguished and recombined.
Without separability, there are no variables to vary.
Relations that constitute their relata
From a relational-ontological perspective, separability is never guaranteed.
There are regimes in which relations do not merely connect pre-existing entities but constitute them. What something is cannot be specified independently of how it relates. Relata emerge only within the relation itself.
In such cases, attempting to isolate components destroys the phenomenon one is trying to describe. The cut that would individuate relata simultaneously dissolves the relations that make them what they are.
Mathematics falters here not because it lacks expressive power, but because its presupposition of independently identifiable relata no longer holds.
Co-constitution and the failure of variable-based description
Variable-based formalisms assume that change can be tracked by adjusting values while holding identities fixed. Separability ensures that what changes is a parameter, not the thing itself.
Where co-constitution dominates, this assumption collapses. Altering one aspect of the system alters what counts as an aspect at all. There is no stable background against which variation can be measured.
The difficulty is not complexity. It is ontological entanglement.
Separability is perspectival, not ontological
As with stability, separability is not an intrinsic feature of reality. It is a perspectival achievement.
A construal may succeed in carving a system into parts that behave as if they were independent. Under another construal, the same system may resist decomposition entirely.
Mathematics operates where a perspective can enforce separability strongly enough for formal manipulation to proceed. Where no such perspective can be sustained without distortion, mathematics loses its purchase.
This explains why mathematical tractability often depends less on the system itself than on how it is approached.
Individuation revisited
Separability and individuation are closely linked but not identical.
Individuation concerns whether something can count as one. Separability concerns whether multiple ones can be treated as independent. Mathematics presupposes both, but separability adds an extra constraint: that individuation be robust against relational change.
Where individuation remains fluid—where what counts as an individual shifts with scale, context, or mode of construal—separability cannot be secured. Mathematical objects then appear and disappear with the cut.
This is not a paradox. It is a signal that the system resists being organised into independently manipulable parts.
When separability collapses
There are situations in which every attempt to isolate a component introduces artefacts that dominate the description. The act of separation does more work than the mathematics that follows it.
In such cases, formal models multiply without converging. Each captures a different forced decomposition, none of which can claim priority. Mathematical disagreement here is not empirical but ontological: it reflects incompatible cuts imposed on a non-separable relational field.
Separability and the myth of total description
The expectation that a complete mathematical description of reality must exist rests on an unexamined assumption: that everything is, in principle, separable.
A relational ontology denies this assumption. Some relations are inherently non-decomposable. They can be participated in, navigated, or enacted, but not exhaustively factorised.
Mathematics does not fail in such domains. It reaches the boundary of what separability allows.
What separability tells us about mathematical limits
Seen this way, the limits of mathematics are not accidental. They arise precisely where relations cease to behave like connections between independently specifiable parts.
Mathematical breakdown, here, is not ignorance awaiting better tools. It is a reminder that structure is not the same as relation.
In the next post, I will turn to a further presupposition that often hides behind both stability and separability: invariance. Not all relations that can be stabilised and separated can survive transformation without remainder—and mathematics depends crucially on those that do.
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