There is a familiar moment in contemporary discourse where mathematics is said to break down.
We are told that in certain physical situations—at extreme scales, at critical thresholds, at points of apparent indeterminacy—our best mathematical tools no longer apply. Equations diverge. Models lose predictive power. Formal descriptions refuse to close.
The usual responses are well rehearsed. Either reality is declared intrinsically mysterious, or mathematics is defended as merely incomplete, awaiting a deeper formalism still to come. Both positions share a tacit assumption: that mathematics is, in principle, the correct language for describing whatever relations there are.
This series begins from a different question entirely:
What must relation be like for mathematics to succeed at all?
And, by inversion:
What does the breakdown of mathematics actually tell us about relation itself?
Mathematics as relational measure—or something narrower?
It is often said, quite correctly, that mathematics is the science of relations. Numbers relate quantities; functions relate variables; structures relate elements within formal systems. From this, a tempting inference follows: if mathematics fails, perhaps there are no relations left to measure.
From a relational-ontological perspective, this inference is untenable.
Relations are not pre-given features of a world waiting passively to be measured. They are constituted through construal. There is no relation independent of the perspectival cut that brings it forth as a phenomenon. Consequently, mathematical failure cannot signal the absence of relation. At most, it signals the failure of a particular way of construing relations to stabilise.
This distinction matters.
Mathematics does not measure relation as such. It measures relations that meet very specific conditions: stability, repeatability, separability, and invariance. Where those conditions are met, mathematics performs spectacularly. Where they are not, mathematics does not so much fail as lose its footing.
Instantiation, cuts, and mathematical traction
In the framework developed across The Becoming of Possibility, instantiation is not a temporal process by which something vague becomes concrete. It is a perspectival shift: a cut from structured potential to event. An instance is not produced; it is actualised.
Mathematics presupposes the availability of such cuts.
To mathematise a phenomenon is to assume that:
the cut can be held steady,
the resulting instance can be re-instantiated,
the relations involved can be mapped again under comparable construals.
Where this is possible, mathematical objects emerge. Where it is not—where cuts cannot stabilise, where identities smear across scales, where relations mutate with the act of construal itself—mathematics finds nothing it can consistently hold.
This is not because relation disappears, but because relation becomes too dynamically entangled with perspective to be frozen into structure.
Breakdown without absence
The language of “breakdown” is misleading. It suggests collapse, failure, or deficit. What we are often witnessing instead is a mismatch between levels of organisation.
Mathematics operates at a second-order level: it abstracts across phenomena to stabilise patterns. Some physical situations, however, remain stubbornly first-order. They are all event, no reusable form; all occurrence, no invariant structure. In such cases, applying mathematics is not wrong—it is category-mistaken.
Mathematical indeterminacy, on this view, is not a sign that reality has become irrational. It is a sign that we are attempting to extract metaphenomenal structure from phenomena that resist abstraction.
Separability, individuation, and the limits of structure
Many celebrated cases of mathematical difficulty can be re-described without invoking mystery at all. What collapses in these situations is not relation but separability.
Mathematics excels when relata can be individuated prior to their relations. It falters when relata are co-constitutive—when what something is cannot be specified independently of how it relates.
This is not a marginal limitation. It cuts to the heart of mathematical ontology itself. Formal systems presuppose individuation. Where individuation becomes a perspectival cline rather than a fixed condition, mathematics loses its anchoring assumptions.
Coordination is not meaning
Finally, it is crucial to distinguish mathematical success from ontological adequacy.
Mathematics optimises coordination: prediction, compression, control. These are values, not meanings. Mathematical models succeed when systems can be stabilised for coordinated action. When coordination fails—because the system cannot be held still—mathematics falters, even though meaning as construed phenomenon remains fully present.
The error lies in mistaking coordination success for access to reality itself.
What this series will argue
The claim developed across this series is simple but far-reaching:
Mathematics breaks down not where relation disappears, but where relation becomes too rich, too unstable, or too perspectivally entangled to be stabilised as structure.
Seen this way, mathematical limits are not embarrassments to be explained away. They are clues. They tell us what mathematics requires in order to function—and, by contrast, what kinds of relational becoming exceed its reach.
In the posts that follow, I will examine these requirements in detail: stability, separability, individuation, and invariance. Along the way, I will argue that a relational ontology not only explains why mathematics sometimes fails, but predicts precisely where it must.
Mathematics does not describe the world in general.
It describes the world where relation has already been tamed.
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