By this point in the series, much has been taken away.
Language has been stripped of its representational mandate. Grammar has been recast as constraint rather than structure. Reference has been detached from objects, truth from meaning, and ambiguity from defect.
What remains might seem uncomfortably loose.
And yet, mathematics and logic still work.
Not only do they work—they work extraordinarily well.
This post is about why that success is real, why it is local, and why it depends entirely on language rather than standing above it.
The Temptation of Reversal
Having dismantled the representational hierarchy, it is tempting to swing to the opposite extreme: to treat formal systems as mere fictions, arbitrary games played with symbols.
That temptation should be resisted.
Formal systems are not illusions. Their power is not rhetorical. Their successes are not accidents. They enable prediction, coordination, explanation, and control on a scale unmatched by any other symbolic practice.
The question is not whether they succeed.
It is how they succeed.
Formalisation as Deliberate Narrowing
Formalisation is a practice of disciplined reduction.
From the broad, flexible space of linguistic meaning, a narrow corridor is carved. Ambiguity is excluded. Context is regimented. Reference is stabilised. Grammar is frozen into rule sets. Only certain moves are permitted, and only certain distinctions are allowed to matter.
What is gained is invariance.
What is lost is possibility.
This is not a defect. It is the price—and the condition—of formal power.
Why Formal Systems Need Language
Formal systems do not float free of language. They are born within it and sustained by it.
Every formalism relies on linguistic practices to define its symbols, state its axioms, explain its rules, interpret its results, and decide when it applies. Even the most austere symbolic systems presuppose shared linguistic competence to function at all.
Language is not an implementation detail.
It is the ground that makes formalisation intelligible.
Local Success, Not Global Authority
The power of formal systems arises precisely because their scope is limited.
Within carefully delimited domains—where relations can be stabilised, distinctions fixed, and variability controlled—formal reasoning delivers results of astonishing reliability. Outside those domains, its authority evaporates.
This is not because reality becomes irrational.
It is because the conditions that made formalisation possible no longer hold.
Formal success is always conditional. It depends on constraints that must be actively maintained. When those constraints fail, the system does not reveal a hidden truth about the world; it reveals the boundary of its own applicability.
Formal Breakdown as Diagnostic
When mathematics or logic “break down” in application, this is often treated as a failure—either of the formalism or of our understanding.
But breakdowns are informative.
They tell us that relations have become unstable, inseparable, or context-sensitive in ways that the formal system cannot accommodate. They mark transitions where linguistic construal must widen again, allowing ambiguity, perspective, and negotiation back in.
Formal systems do not fail at these points.
They stop being the right tools.
The Ecological View
Seen ecologically, language and formal systems are not competitors but collaborators.
Language provides the expansive space of meaning in which relations can be explored, tested, and reconfigured. Formal systems provide pockets of high stability within that space, enabling calculation and constraint.
Neither replaces the other.
Each does work the other cannot.
Nothing to Settle
The deepest mistake is to assume that one symbolic practice must ground all others.
Language does not need to justify itself to logic. Logic does not need to apologise for its limits. Mathematics does not need to pretend to universality to be powerful.
Each succeeds locally, under the conditions that make its success possible.
Formalisation is not the endpoint of meaning.
It is one of its most disciplined achievements.
Preparing the Close
With this in place, the series can now close without polemic or repair.
Language is not defective because it cannot be fully formalised. Formal systems are not defective because they cannot scale universally. Meaning, relation, and possibility unfold across practices with different constraints and different kinds of success.
In the final post, we will draw these threads together and return to the central claim of the series: that language is the native ecology of meaning, and that nothing has gone wrong—either with language, or with our formal tools—once we stop asking them to do the same work.
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