If mathematics sometimes breaks down, this is not because it encounters chaos where it expected order. It is because it quietly presupposes a very specific form of order—one so familiar that it is rarely named.
That form is stability.
Not stability as stasis, nor as permanence, but as the capacity for relations to be held steady across construals. Mathematics does not merely describe relations; it depends on their ability to endure perspectival variation without dissolving.
Stability as a condition of mathematisation
To treat something mathematically is to assume that it can be re-encountered as the same kind of instance. This does not require material sameness, but it does require relational persistence.
At minimum, mathematics presupposes that:
a phenomenon can be cut from structured potential in a repeatable way,
the resulting instance can be re-actualised under comparable construals,
relations instantiated in one cut will not mutate arbitrarily under another.
These assumptions are not conclusions of mathematics. They are entry conditions.
Where they fail, mathematics cannot even begin.
Stability is perspectival, not absolute
It is tempting to imagine stability as a mind-independent feature of reality, something mathematics merely registers. This temptation should be resisted.
Within a relational ontology, stability is never absolute. It is always stability for a perspective. A cut stabilises relations only relative to a construal that holds them in place.
This means that mathematical stability is not discovered but achieved. It is the outcome of a successful alignment between:
the relational dynamics of a system, and
the constraints imposed by a particular mode of construal.
Mathematics works where this alignment can be sustained.
Invariance: stability across transformation
The most powerful expression of mathematical stability is invariance. To say that something is mathematically well-defined is often to say that it remains unchanged under an allowed range of transformations.
Invariance is not a metaphysical guarantee. It is a relational achievement. It presupposes that:
transformations can be specified independently of what they act upon,
the relata survive those transformations as identifiable relata,
relations persist despite perspectival variation.
Where transformation and constitution cannot be separated—where altering perspective alters what there is—invariance collapses. Mathematics does not fail here; it finds no invariants to work with.
Stability, individuation, and mathematical objects
Mathematical objects are not primitive. They are stabilised outcomes of relational cuts.
Numbers, functions, spaces, and structures all presuppose that individuation has already been secured. Something must count as one thing rather than another before it can enter a formal relation.
But individuation itself is not guaranteed. It is a perspectival cline between collective potential and individual actualisation. Where that cline cannot be held—where what counts as an individual shifts with scale or context—mathematical objects lose their footing.
This is why mathematical difficulty often coincides with disputes about what the relevant entities even are. Mathematics presupposes individuation; it cannot supply it.
Stability is not determinacy
It is important to distinguish stability from determinacy.
A system may be highly dynamic, probabilistic, or open-ended and still be mathematically tractable, provided its relations stabilise in lawlike ways. Conversely, a system may be perfectly determinate yet mathematically intractable if its relations cannot be re-instantiated under stable cuts.
Mathematical stability concerns repeatability of relation, not predictability of outcome.
When stability cannot be held
There are situations in which every attempt to stabilise relations alters the relations themselves. The act of construal becomes entangled with what is construed. Cuts slide. Boundaries leak. Instances refuse to recur.
In such regimes, mathematics does not uncover hidden variables or missing equations. It reaches the limits of its presuppositions.
What fails is not rationality, but the assumption that relational becoming can be frozen long enough to be formalised.
What stability tells us about mathematical limits
Seen relationally, stability is not a background condition of reality. It is a fragile achievement that mathematics depends upon but cannot guarantee.
This reframes mathematical limits entirely. They are not signs of ignorance waiting to be overcome, nor of reality withdrawing from reason. They are markers of where relation outruns structure.
In the next post, I will turn to a closely related presupposition: separability. Stability alone is not enough for mathematics to function. Relations must also be decomposable into parts that can be treated as independently variable.
Where separability collapses, mathematics encounters a different—and equally instructive—boundary.
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