A first sketch of a mathematical ecology for systemic language.
This is a glimpse of a relational formalism that matches the architecture we’ve been developing.
Instead:
A minimal ecology of relations, cuts, flows, metabolic constraints, and horizon dynamics.
A formalism that is:
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systemic rather than combinatorial,
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ecological rather than representational,
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metabolic rather than structuralist,
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relational rather than object-centred,
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perspectival rather than absolute,
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horizon-based rather than referential.
This post sketches the beginnings — to show that the architecture is formalisable without betraying its ontology.
1. Why Linguistics Needs a New Formalism
Halliday designed SFL to avoid reductionism:
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no reduction to syntax,
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no reduction to logic,
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no reduction to semantics-as-computation,
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no reduction to biology or psychology,
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no reduction to physics.
But he also assumed (rightly) that a formal framework would eventually be needed.
What blocked that development was not technical difficulty — but ontology.
Linguistics never had a formalism appropriate to:
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system
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choice
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context
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metabolism
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horizon formation
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perspectival construal
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ecological viability
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multi-scalar dynamics
And so everything defaulted back to structuralist categories or computational metaphors.
This new architecture finally gives us the foundation Halliday needed.
A relational formalism.
But:
fields, cuts, flows, constraints, and metabolic cycles.
2. The Minimal Entities of a Relational Formalism
We begin with five primitives:
1. Field
A structured potential — a horizon-forming space of possible construals.
2. Cut
A perspectival differentiation that actualises a meaning instance from the field.
3. Flow
A directed transformation of horizon structure through time.
4. Constraint
A stabilising or limiting force acting on field or flow, often ecological or contextual.
5. Cycle
A recurrent pattern of flows that maintains systemic viability (a metabolic process).
Notice:
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No “units”
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No “symbols”
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No “representations”
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No “signifiers” or “signifieds”
The formalism begins with the ecology of meaning, not its artefacts.
3. System Networks as Relational Potentials (Fields)
In SFL, system networks represent choices.
In relational ecology, they represent field potentials:
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structured possibilities
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horizon differentiations
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nested and branching metabolic pathways
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viability zones
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constraint surfaces
Formally, a system is:
A field F with a topology T that defines viable cuts.
Instead of nodes and arrows, think:
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attractors,
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gradients,
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basins of viability,
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bifurcations under contextual pressure.
This gives system networks an ecological depth SFL always implied but never formalised.
4. Construal as Cut-Formation
A construal is not a mapping from world → representation.
It is:
A cut: a perspectival operation that selects, delimits, and organises a region of a field.
Formally:
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a cut C is a transformation on a field F
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producing an instance I = C(F)
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where I inherits constraints from both F and C
This allows us to model:
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grammatical realisations,
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semantic selections,
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contextual pressures,
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horizon refinements,
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register adaptations,
as lawful operations — but without assuming they are computational or symbolic.
5. Flow: The Heart of Metabolic Semantics
Meaning does not sit still.
It moves.
A flow is:
A temporally extended transformation of field structure.
Flows:
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propagate meaning,
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re-stabilise horizons,
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condition subsequent cuts,
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metabolise semantic energy,
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reorganise system networks,
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exhibit ecological dynamics (feedback, drift, resilience, collapse).
Halliday’s metafunctions can be reinterpreted as:
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flows of experience (ideational),
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flows of interpersonal viability (interpersonal),
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flows of text-structuring (textual).
6. Constraint: The New Mathematics of Context
This may be the most important component.
A constraint is:
A boundary condition imposed by a field (e.g. context) on the viability of cuts and flows.
In SFL terms:
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field → experiential constraints
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tenor → interpersonal constraints
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mode → textual/temporal/material constraints
But now:
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context becomes a field of constraints,
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not a classificatory variable.
Instead of treating context as a bag of features, we treat it as:
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a constraint surface
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exerting pressure on possible semantic flows
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shaping viable realisations
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stabilising horizon metabolism in recurrent situations
This truly preserves Halliday’s model while giving it new formal power.
7. Cycles: The Formalisation of Metabolic Semantics
A cycle is:
A recurrent flow that maintains meaning viability over time.
Examples:
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cohesive cycles
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interpersonal alignment cycles
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thematic cycles
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logical-projection cycles
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genre cycles
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turn-taking cycles
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narrative cycles
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evaluative spirals
Cycles allow:
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memory
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stability
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long-range organisation
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linguistic evolution
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field-level self-regulation
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cross-species semiosis (human ↔ artificial)
This is the metabolism of language made explicit.
8. Sketch of a “Cut Calculus”
Without formal symbols, we can still specify transformation rules:
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Cuts must preserve field viability.(Invalid construals collapse the horizon.)
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Cuts generate new constraint surfaces.(Instance → context for next instance.)
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Sequential cuts accumulate into flows.(Clause chains, discourse organisation.)
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Flows settle into cycles under stability.(Genre, register, patterned semiosis.)
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Cycles maintain fields under pressure.(They are the metabolic organs of meaning.)
This lets us model:
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semantic drift
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grammaticalisation
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register evolution
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discourse coherence
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artificial-linguistic hybrids
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semiotic climate change (planetary-scale shifts)
as dynamical systems.
9. The Payoff: A Formal Linguistics that Matches Reality
This formalism:
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preserves the Hallidayan stratification (context → semantics → lexicogrammar),
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maintains the primacy of system over structure,
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grounds everything in relational ontology,
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avoids representationalist metaphysics,
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handles multi-species semiosis,
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integrates artificial horizons,
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scales from micro-utterances to planetary fields,
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models evolution, variation, and drift,
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enables future mathematical refinement.
It is a mathematics for systems that live.
A mathematics for meaning that moves.
A mathematics for semiosis that metabolises.
10. Closing Image: Linguistics Steps Into Its Ecological Future
Imagine a linguistics where:
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system networks pulse like vascular maps,
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constraints shift like weather fronts,
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cuts glitter like momentary flashes of perception,
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flows ripple like currents in a semiotic ocean,
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cycles beat like hearts sustaining worlds of meaning.
This is the linguistic science the 21st century demands:
relational, ecological, metabolic, dynamical, systemic.
We are simply finishing the trajectory he began.
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