A Metamodel for Ontologically Heterogeneous Social Dynamics, Part I - The Highest Abstract Layer

Posted on 2026-07-14 In Philosophy , Theory of Justice Views: Disqus: Word count in article: 16k Reading time ≈ 57 mins.

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Part I: The Highest Abstract Layer

Wanhong Huang · huangwanhong@serendip.ngo

Abstract

This is the first paper of a series. It fixes the highest abstract layer, and it goes no further.

Dynamical modeling of social systems ordinarily begins by fixing a state space. This is already a commitment, and across domains it is one that cannot be met. The states of a normative order, a power configuration, a value system, and a network of recognition inhabit no common manifold and admit no common metric, and the obstacle is not that such a space is hard to construct but that these are not objects of a kind that admit a common placement. The heterogeneity is ontological rather than practical, and it is not answered by better instruments. The proposal here is to begin one layer lower, at a level where change, structure, constraint, and difference are defined but not yet realized. Four axioms are given. The carrier of states is left without structure. The mechanisms acting upon it form a semigroup: associative, without inverses, without an identity. A congruence records what any mechanism is obliged to preserve. And difference is defined, neither as a residue nor as a distance, as the set of distinctions a mechanism collapses. From these, four propositions follow that were not assumed: collapse is irreversible; a group of mechanisms has no collapse at all, and hence no history; a mechanism is non-destructive exactly when it is injective modulo the congruence; and collapse accumulates monotonically under composition. The absence of inverses is shown to be the load-bearing commitment. This paper does not instantiate the metamodel, does not apply it, and derives no substantive result about any particular social system. Those belong to subsequent parts. What is established here is only what a social dynamical system must be able to say, and the price of saying it.

Keywords: dynamical systems; social dynamics; semigroup action; congruence; collapse; irreversibility; formal social theory; generative relational being.

AI usage statement: The ideas, claims, and theoretical framework of this paper originate with the author. AI tools were used as an aid in drafting, in the location and verification of sources, and in revision. The origin of each idea has been verified by the author, who takes full responsibility for the content.

Recap

The metamodel in full. $\mathcal{S}$ is a set without further structure; $\mathcal{M}$ is a semigroup, with neither identity nor inverses; $\sim$ is an equivalence relation on $\mathcal{S}$.

$$\begin{aligned} &\textbf{(Change)} && \alpha : \mathcal{M} \times \mathcal{S} \longrightarrow \mathcal{S}, \qquad (m, x) \longmapsto m(x) \[8pt] &\textbf{(Structure)} && \alpha(m_1 m_2,, x) ;=; \alpha!\left(m_1,, \alpha(m_2, x)\right) \[8pt] &\textbf{(Constraint)} && x \sim y ;\Longrightarrow; \alpha(m, x) \sim \alpha(m, y) \[8pt] &\textbf{(Difference)} && \Delta(m) ;=; \bigl{, (x,y) ;:; x \not\sim y ;\wedge; \alpha(m,x) \sim \alpha(m,y) ,\bigr} \end{aligned}$$

Sections 1 to 3 give the reasons for these four lines, §4 their consequences, and §5 what they decline to settle. The consequences, which were not assumed, are that collapse is irreversible, that irreversibility requires the absence of inverses, and that a mechanism destroys nothing when it is injective modulo $\sim$.

§1 Introduction

1.1 The formalism is layered; this paper gives the topmost layer

The programme to which this paper belongs proposes a formalism for social dynamics at several levels of abstraction. At the highest, one says what any model must be able to express, without saying what it expresses. Below it, one says what structures a model may draw upon (an order, a metric, an algebra) and at what cost. Below that, one exhibits models. The levels are not competing accounts of the same thing. They stand in a relation of dependence: what is settled above constrains what may be built beneath, and what is built beneath must answer to it.

This is the first paper of the series, and it gives only the highest layer. It does not instantiate the framework, does not apply it to a domain, and derives no substantive result about any particular social system. Those belong to the parts that follow.

The reason for beginning at the top, and for treating it separately, is that the commitments made at this layer are the ones that later become invisible. A model which ranks has assumed comparability; a model which measures has assumed symmetry; a model which aggregates has assumed an arithmetic. Each of these is a decision, and each, once buried inside a concrete formalism, is difficult to retrieve and impossible to contest. Isolating the highest layer is a way of making the decisions available for inspection before they are made.

1.2 Writing $x$ already stipulates what counts as a difference

A dynamical system, in its received form, is a triple: a state space, a rule of evolution, and a parameter. The rule is written

$$\dot{x} = f(x;\theta),$$

and the reading it invites is that the left side records change and the right side records the structure from which change proceeds. That reading is correct as far as it goes. What it conceals is the cost of the first term. To write $x$ at all is to have fixed a space in which $x$ lives, and with it a stipulation of what counts as a difference between one state and another: whether such differences can be measured, whether they can be compared, whether they vary continuously.

For the systems at issue here, that stipulation cannot be met. The state of a normative order is a configuration of rules and expectations; the state of a power system is a distribution of asymmetries; the state of a value system is an ordering of what matters; the state of a recognition network is a pattern of who is seen. These do not sit in a common space, and they admit no common metric.

The point requires care, because it is stronger than it may appear. The difficulty is not that a common space is hard to construct, which would be a difficulty of instruments and would be answered by better ones. It is that a rule and a dependence are not two values of one variable, and no refinement of measurement will make them so. The heterogeneity of social states is ontological rather than practical, and a heterogeneity of that kind is not overcome; it is either respected or overwritten. It is the claim that these states are commensurable, that some scale $m : X \to \mathbb{R}$ exists, which the theory of value pluralism has spent a century contesting, and a modeling strategy that begins by assuming the space begins by assuming away what is contested.

1.3 Change, structure, constraint, and difference need no state space

The response proposed is not to find a cleverer space. It is to descend below the point at which a space is required.

Four things must be sayable of any social dynamical system. It must be able to say how it changes. It must be able to say how its changes compose. It must be able to say what its changes are constrained to preserve. And it must be able to say what its changes destroy. These are, respectively, change, structure, constraint, and difference.

The claim of this paper is not that these four are novel. It is that they are, taken together, sufficient, and that they can be given without assuming a state space, a metric, an order, an arithmetic, or a zero. This matters, because each of those assumptions, once made, quietly decides a question the theory was meant to leave open. An order on states presumes commensurability. A metric presumes symmetry, and therefore cannot distinguish betraying from being betrayed. A zero presumes that what a transformation leaves behind is a quantity capable of being null, and so decides, in advance of any argument, that loss is a magnitude.

The metamodel given in §3 makes none of these choices. In exchange, it yields results that were not put in by hand: that collapse is irreversible, that irreversibility is possible only because mechanisms lack inverses, and that a mechanism destroys nothing exactly when it is injective modulo the congruence.

1.4 Loss precedes generativity in the order of dependence

Instantiation is deferred. So is application. So, in particular, is the formal treatment of generativity, which is the term the wider programme exists to serve and which the four axioms of this paper do not provide. What they provide is its precondition: an account of what a mechanism can destroy. A framework that cannot say what is lost cannot say what it would mean to generate. The order of exposition follows the order of dependence, and loss comes first.

§2 The State Space Cannot Come First

Consider four systems and the state each would require.

SYSTEM	STATE
normative order	configuration of rules, obligations, expectations
power system	distribution of dependence and authority
value system	ordering of what is held to matter
recognition network	pattern of who is acknowledged as a subject

No embedding of these into a common vector space is available, and the ones that have been proposed are precisely the objects of critique: utility, price, and the composite index each purchase comparability at the cost of the differences they were meant to preserve.

It is worth being exact about the sense in which this heterogeneity is ontological and not merely practical, since the two are easily confused and only the former has consequences for the form of a theory. Practical heterogeneity would mean that a common space exists but is hard to construct, that the data are poor, the scales incommensurate in practice, the measurement instruments crude. Such a difficulty is answered by better instruments, and it licenses the modeler to proceed as though the space were there, since in principle it is.

The heterogeneity at issue is not of that kind. It is not that a common space is hard to find but that the objects to be placed in it are not of a kind that admits a common placement. A rule and a dependence are not two values of one variable measured badly; they are not two values of one variable. A pattern of recognition is not a coarse rendering of a distribution of authority. These are distinct modes of being, and the question of how far one lies from another is not a hard question but a malformed one. It is in this sense that the demand for a common state space is a substantive commitment rather than a technical inconvenience, indeed an ontological one, asserted in the guise of a modeling convention, and it should not be made by default.

2.2 Order presumes commensurability, metric presumes symmetry, zero presumes magnitude

Order. A partial order on $\mathcal{S}$ furnishes suprema, and a supremum of two value states is a common measure of them. To assume a lattice is to have assumed commensurability.

Metric. A metric $d$ on $\mathcal{S}$ satisfies $d(x,y)=d(y,x)$. Symmetry cannot represent a directed act. Under a metric, a mechanism and its reception are the same distance apart, and the asymmetry of the relational act has been erased before any question about it can be posed.

Zero. An additive structure with a zero element makes the residue of a transformation a quantity that may or may not vanish. This is a determinate philosophical position, the position on which what is lost is a magnitude, and so, in principle, restorable. The rival position holds that what is lost has no place in the new structure at all. The metamodel must be able to host both, and therefore may assume neither.

2.3 Fixing the mechanisms leaves the carrier free

The path taken here inverts the usual order of construction. Rather than fixing states and deriving the transformations that act upon them, we fix the transformations and leave the states unstructured. This is not new; it is the one by which the algebraic approach to dynamics proceeds. Its warrant is that the structure a theory requires can be carried by the acting objects, leaving the carrier as a set on which they act.

§3 Change, Structure, Constraint, and Difference Are Definable without a Structured Carrier

3.1 The carrier is a bare set; the mechanisms carry the structure

Definition 1 (Primitives). A social dynamical metamodel is a triple $(\mathcal{S}, \mathcal{M}, \sim)$ where

$\mathcal{S}$ is a set, the carrier. It is assumed to have no further structure: no order, no metric, no algebraic operation, no distinguished element.

$\mathcal{M}$ is a set, the mechanisms.

$\sim$ is an equivalence relation on $\mathcal{S}$.

Elements of $\mathcal{S}$ are states, in the weakest sense the word can bear: whatever a given domain takes a configuration to be. Elements of $\mathcal{M}$ are mechanisms: whatever, in that domain, acts upon a configuration and yields another. Nothing is presumed about what either is made of.

3.2 The mechanisms form a semigroup, and $\Delta$ is derived from $\alpha$ and $\sim$

Axiom 1 (Change). There is a map $\alpha: \mathcal{M} \times \mathcal{S} \longrightarrow \mathcal{S}$, $(m, x) \longmapsto m(x)$.

A mechanism acts on a state and returns a state. This is all that change is required to be.

Axiom 2 (Structure). $\mathcal{M}$ carries an associative composition $\mathcal{M} \times \mathcal{M} \to \mathcal{M}$, $(m_1, m_2) \mapsto m_1 m_2$, with $(m_1 m_2) m_3 = m_1 (m_2 m_3)$, and $\alpha$ is an action of it: $\alpha(m_1 m_2,, x) = \alpha!\left(m_1,, \alpha(m_2, x)\right)$. $\mathcal{M}$ is a semigroup. It is not assumed to have an identity, and it is not assumed to have inverses.

Associativity is free: it says only that doing $m_1$ then $m_2$, and then $m_3$, is the same as doing $m_1$, and then $m_2$ followed by $m_3$. The two omissions are not free, and they are the substantive content of the axiom. Their consequences are drawn in §4.

Axiom 3 (Constraint). $\sim$ is a congruence for the action: $x \sim y ;\Longrightarrow; \alpha(m, x) \sim \alpha(m, y)$ for all $m \in \mathcal{M}$.

Read: whatever the system preserves, no mechanism may destroy. The axiom does not say what $\sim$ is. It says that a mechanism which fails to respect it is not a mechanism of this system at all. $\sim$ is the system’s law of admissibility, and its content is supplied by the domain.

Axiom 4 (Difference). For each $m \in \mathcal{M}$, the collapse of $m$ is $\Delta(m) ;=; \bigl{, (x,y) \in \mathcal{S} \times \mathcal{S} ;:; x \not\sim y ;\wedge; \alpha(m,x) \sim \alpha(m,y) ,\bigr}$.

Read: $\Delta(m)$ is the set of distinctions that $m$ destroys. Two states that the system could tell apart become, after $m$ has acted, states it cannot tell apart.

3.3 Difference is a lost discrimination, not a leftover element

Three familiar formalizations of difference are rejected, each because it requires structure on $\mathcal{S}$ that the metamodel declines to assume.

Not a residuum. The residuated-lattice operator, $x \backslash m(x) = \max{c : x \cdot c \leq m(x)}$, requires that $\mathcal{S}$ carry both a multiplication and an order. It therefore requires that states be comparable, which is what the ontological heterogeneity of social states denies.

Not a distance. The metric-induced form, $d(x, m(x))$, requires a metric, and a metric is symmetric. It cannot represent a directed act, and the acts at issue are directed.

Not a kernel. The form $\ker \pi = {x : \pi(x) = 0}$ requires a zero, hence an algebraic structure on $\mathcal{S}$. In a semigroup the kernel of a morphism is not a subset at all: it is a congruence. Axiom 4 takes that fact as its cue and defines difference as a relation rather than an element.

Difference, at this layer, is not a thing that is left over. It is a discrimination that has been lost.

The three rejections share a form. Each of the discarded operators returns an element, and in doing so commits the metamodel to a determinate view of what a remainder is, a view on which the traditions that have thought longest about remainders do not agree. What is at stake in this is taken up in §5.

$\Delta$ is a derived object. It is not a fifth primitive: it is defined entirely from $\alpha$ and $\sim$, which is to say from Axioms 1 and 3.

§4 Irreversibility Follows from Associativity and the Congruence Alone

The axioms are weak. They nevertheless entail results that were not stipulated. This section gives them.

4.1 Collapse is irreversible

Proposition 1 (Irreversibility of Collapse). Let $(x,y) \in \Delta(m)$. Then for every $m’ \in \mathcal{M}$, $\alpha(m’ m,, x) ;\sim; \alpha(m’ m,, y)$. That is: no mechanism applied after $m$ can restore the distinction $m$ destroyed.

Proof. By definition of $\Delta$, $\alpha(m,x) \sim \alpha(m,y)$. By Axiom 3, applying $m’$ to congruent states yields congruent states, so $\alpha(m’, \alpha(m,x)) \sim \alpha(m’, \alpha(m,y))$. By Axiom 2, the left side is $\alpha(m’m, x)$ and the right side is $\alpha(m’m, y)$. $\square$

The proof uses only Axioms 2 and 3. Irreversibility is therefore not an extra postulate about the social world, appended because we know from experience that some things cannot be undone. It is what the associativity of mechanisms and the preservation of the congruence, taken together, already say.

4.2 Inverses annihilate difference

Proposition 2 (Groups have no history). Suppose $\mathcal{M}$ is a group acting on $\mathcal{S}$ and $\sim$ is a congruence. Then $\Delta(m) = \emptyset$ for every $m \in \mathcal{M}$.

Proof. Suppose $(x,y) \in \Delta(m)$, so $\alpha(m,x) \sim \alpha(m,y)$ and $x \not\sim y$. Applying $m^{-1}$ and using Axiom 3, $\alpha(m^{-1}m, x) \sim \alpha(m^{-1}m, y)$, that is $x \sim y$, a contradiction. $\square$

The absence of inverses is not a technical weakening. It is the condition under which a system can have a history at all. Where every mechanism can be undone, nothing is destroyed, nothing is preserved against destruction, and the question of what a transformation has cost does not arise. A group of mechanisms is a world without loss.

The two propositions stand in a definite relation. Proposition 1 says that what is collapsed stays collapsed. Proposition 2 says that this is possible only because mechanisms are not invertible. Taken together they locate the exact structural commitment on which everything else in this framework depends: Axiom 2’s refusal of inverses.

4.3 A mechanism destroys nothing iff it is injective modulo $\sim$

Proposition 3. $\Delta(m) = \emptyset$ if and only if the induced map $\bar{m}: \mathcal{S}/!\sim ;\to; \mathcal{S}/!\sim$ is injective.

Proof. Axiom 3 guarantees that $\bar{m}$ is well defined on the quotient. $\bar{m}$ fails to be injective exactly when there are classes $[x] \neq [y]$ with $\bar{m}[x] = \bar{m}[y]$, that is, when there are states $x \not\sim y$ with $\alpha(m,x) \sim \alpha(m,y)$, that is, when $\Delta(m) \neq \emptyset$. $\square$

The proposition supplies a criterion. A mechanism destroys nothing if and only if it is injective modulo the congruence, so it is not necessary to inspect a mechanism’s effects case by case in order to know whether it is destructive. It suffices to ask whether it separates what the system could already separate.

4.4 Composition accumulates

Proposition 4 (Monotonicity of Collapse). For all $m, m’ \in \mathcal{M}$, $\Delta(m) ;\subseteq; \Delta(m’ m)$.

Proof. Let $(x,y) \in \Delta(m)$. Then $x \not\sim y$, and by Proposition 1, $\alpha(m’m,x) \sim \alpha(m’m,y)$. Hence $(x,y) \in \Delta(m’m)$. $\square$

Collapse, once incurred, is carried forward by every subsequent mechanism. The history of a system is thus a monotone accumulation of lost discriminations, and no mechanism in the semigroup can reduce it. This is the formal sense in which a social system does not return to a prior condition.

§5 Loss Is a Magnitude Only If the Carrier Has an Arithmetic

Axiom 4 defines difference as a relation. The natural alternative would define it as an element: a residue, a remainder, a something left over when a mechanism has acted. The alternative was declined, and the reason is that the two traditions with the strongest claim on the notion of a remainder, political economy and psychoanalysis, disagree about what a remainder is. An operator that returns an element settles their disagreement, and settles it by stipulation. What follows sets out the disagreement, shows where the notation would decide it, and shows what a relation-valued $\Delta$ leaves open instead.

5.1 Surplus value is restorable; the objet a has no place to be restored to

The economic residue. In the analysis of value, what a mechanism leaves over is a magnitude. Surplus value is the difference between the value a labourer produces and the value returned to him [12]; it is a quantity, it can in principle be computed, and, this is the point on which the whole normative force of the analysis rests, it could have been returned. That it was not is the injustice. The residue is thus an element of the same order as what was given: it has a location in the value system, it is expressible in the system’s terms, and its restitution is intelligible even where it is not forthcoming. A theory of exploitation requires a residue of this kind, for otherwise it could not say what was taken.

The psychoanalytic residue. In Lacan the remainder is of a different kind altogether [14, 15]. The objet petit a is not a quantity of satisfaction withheld; it is what falls out of symbolization itself, the residue produced by the subject’s entry into the symbolic order and precisely not representable within it. It is not that the object is scarce, or that it has been taken and might be given back. It has no place in the order that produced it. Its whole theoretical function depends on this: were it locatable, were it a magnitude, it could in principle be supplied, and the structure of desire that it sustains would collapse [16].

The economic residue is a magnitude that was not returned. The psychoanalytic residue is not a magnitude at all. The first can be restored. The second cannot, and the obstacle is not that restoration is difficult but that there is nothing of the right kind to restore.

5.2 An element-valued operator decides between them by notation

Suppose difference were defined, as it commonly is, by an operator returning an element:

$$\Delta(m, x) ;=; x \backslash m(x) \qquad\text{or}\qquad \Delta(m, x) ;=; d!\left(x, m(x)\right) \qquad\text{or}\qquad \Delta(m) ;=; \ker \pi_m = {x : \pi_m(x) = 0}.$$

Each of these requires structure on $\mathcal{S}$: the first an order and a multiplication, the second a metric, the third a zero. And each, in requiring it, has already chosen. An operator that returns an element of $\mathcal{S}$, or of some $\mathcal{D}$ possessing a zero, has said that what is lost is a something, a magnitude that may be null, that has a place, that could conceivably be supplied. It has decided for the economic residue and against the psychoanalytic one, and it has done so by notation rather than by argument.

This is the kind of commitment the highest layer exists to hold open. Whether the remainder of a social mechanism is a restorable magnitude or an unrepresentable void is a substantive question, and one on which the traditions divide. A metamodel is arguably not the place to answer it, and one that answers it in its choice of symbols has answered it without argument.

5.3 A relation-valued $\Delta$ leaves both readings open

$$\Delta(m) ;=; \bigl{, (x,y) ;:; x \not\sim y ;\wedge; \alpha(m,x) \sim \alpha(m,y) ,\bigr}$$

$\Delta$ returns a relation, not an element. It says which discriminations a mechanism destroyed. It does not say what, if anything, was left behind, and it requires no order, no metric, no arithmetic, and no zero in order to say what it says.

The two traditions are thereby left open, and each is recovered at instantiation rather than assumed at the outset:

Where an instantiation supplies $\mathcal{S}$ with an additive structure, a magnitude can be associated with each collapsed pair, and the residue becomes a quantity, which is the economic reading. Restitution is then well posed, and one may ask whether it has been made.
Where an instantiation supplies no such structure, $\Delta(m)$ remains a set of pairs and nothing more. Two states have become indiscernible, and nothing of the right type exists to be returned, which is the psychoanalytic reading. Restitution is not withheld here; it is not formulable.

The metamodel is not neutral because it is empty. It is neutral because it locates the disagreement: the question of whether loss is a magnitude is the question of whether the carrier has an arithmetic. That is a decision about a particular social order, to be made and defended in the modeling of that order. It is not, so far as this paper can see, a decision about the form of social dynamics as such, and it should not be made by notation alone.

5.4 The two readings license different demands

It should be said, though it exceeds the scope of this part, that the choice is consequential. A theory of justice built on the economic residue can demand restitution: something was taken, it is of a determinate size, and it can be given back. A theory built on the psychoanalytic residue cannot make this demand, and must ask instead for a different order, one whose congruence separates what the present one has conflated.

The metamodel does not adjudicate between them. It does establish that they are different demands, that they follow from different structural assumptions about the carrier, and that a theory which has not said which assumption it makes has not yet said what it is asking for.

§6 Instantiation Supplies the Structure the Axioms Withhold

The metamodel is deliberately too weak to compute with. It has no order, so it cannot rank. It has no metric, so it cannot measure. It has no arithmetic, so it cannot aggregate. Each of these is supplied only at instantiation, and each supply is a commitment. The purpose of separating the layers is to make the commitments visible, so that of any concrete model it may be asked: at which line did it assume comparability, and with what warrant.

6.1 Every instantiation is a semiflow, and its congruence carries the normative weight

To instantiate is to supply four things: a carrier $\mathcal{S}$, a semigroup $\mathcal{M}$, an action $\alpha$, and a congruence $\sim$. The axioms then constrain what the instantiation may do, and the propositions of §4 hold in it automatically, without reproof.

Two constraints on instantiation follow directly from §4 and are worth stating here, since they hold for every instantiation and therefore belong to the highest layer rather than to any particular one.

Semiflows, not flows. A continuous-time flow with $t \in \mathbb{R}$ is a group action, and by Proposition 2 it has $\Delta \equiv \emptyset$. Reversible dynamics has no collapse. This is the classical fact that a Hamiltonian system loses no information, and it is here a corollary rather than an observation. Any instantiation that is to model a system with a history must therefore be a semiflow, $t \geq 0$. The restriction is not a modeling convenience; it is forced.

The congruence carries the normative weight. In any instantiation, $\sim$ is not a technical choice. It is the system’s statement of what it is obliged to preserve, and it is the sole point at which normative content enters the formalism. Two configurations that a theory of justice would not have conflated are configurations that $\sim$ is required to separate. Choosing $\sim$ is therefore a substantive theoretical act, and one to be defended as such. The metamodel supplies the form of the obligation and none of its content.

6.2 Instantiation, application, and generativity are deferred

Three things are outside the scope of Part I, and are named so that their absence is not mistaken for an oversight.

No instantiation. No carrier, semigroup, action, or congruence is exhibited for any concrete domain. The metamodel is presented as a set of conditions on models, not as a model.

No application. The framework is not read against any substantive problem in social or political theory. Where such readings are possible, and the author holds that they are, and that they alter the description of the phenomena, they require a determinate $\sim$, and hence an instantiation, and hence a later part.

No account of generativity. The axioms describe how mechanisms act on a fixed carrier. Nothing in them produces a state that was not already in $\mathcal{S}$, nor a distinction that $\sim$ did not already admit. Collapse has been given a formal home; emergence has not. This is a genuine limitation and is returned to in §6.

Part I establishes what a social dynamical system must be able to say. It does not say it.

§7 The Axioms Cannot Judge a Collapse, Generate the New, or Evolve Their Own Congruence

The framework should be judged by what it cannot do, and four things it cannot do are named here.

It cannot say that a collapse was wrong. $\Delta(m)$ records which distinctions a mechanism destroyed. It does not say they ought to have been kept. Some collapses are innocuous, and some are necessary: a system that could distinguish every state from every other would be a system incapable of generalization. The normative weight falls entirely on $\sim$, that is, on the prior determination of what a system is obliged to preserve. The metamodel supplies the form of that determination and nothing of its content.

It cannot generate the new. The axioms describe how mechanisms act on a fixed carrier. Nothing in them produces a state that was not already in $\mathcal{S}$, and nothing produces a distinction that $\sim$ did not already admit. This is a genuine limitation for a programme whose central term is generativity. Collapse has been given a formal home; emergence has not. A candidate exists, the passage of a system into an inequivalent representation of the same algebra, where the new is an accession to a sector formerly unreachable rather than a rearrangement of the old, but it requires more structure than four axioms provide, and it is not developed here.

The congruence is exogenous. Axiom 3 posits $\sim$ and does not derive it. Yet in any actual system, what counts as a preserved distinction is itself an outcome of the system’s history. A sharper framework would let $\sim$ evolve, so that a mechanism could alter not only the states but the system’s very capacity to discriminate among them. The immediate obstacle is that $\Delta$ is defined in terms of $\sim$, and a moving $\sim$ makes the definition circular. Resolving this is, in the author’s view, the most important open problem in the framework, and it is the problem to which the next part is addressed.

It is not, by itself, a theory. A metamodel of this kind constrains the class of admissible models; it does not select among them, and it makes no prediction. Its use is diagnostic. It permits one to ask of any social model: what does it take a state to be, what does it take a mechanism to be, what does it oblige a mechanism to preserve, and what, therefore, does it permit a mechanism to destroy. A model that cannot answer the fourth question has not thereby avoided destruction. It has only declined to record it.

§8 Conclusion

Four axioms have been given. A carrier without structure; a semigroup of mechanisms acting upon it, associative and without inverses; a congruence recording what the system is obliged to preserve; and a difference operator defined as the set of distinctions a mechanism collapses. From these, four propositions follow: that collapse is irreversible; that it is possible only in the absence of inverses; that a mechanism is non-destructive exactly when it is injective modulo the congruence; and that collapse accumulates monotonically under composition.

The propositions are not deep. They are three lines each. What is worth noticing is that they were not put in. The framework was constructed to be as weak as it could be made, with no order, no metric, no arithmetic, no zero, and by assuming so little, it was able to obtain irreversibility as a theorem rather than posit it.

A system has a history because its mechanisms cannot be undone. It can lose what it was obliged to keep because its mechanisms can collapse distinctions the congruence required it to hold apart. These are the same fact, stated once algebraically and once normatively, and it is the burden of a formal social theory to show that they are the same.

That burden is not discharged here. This paper has established only the highest layer: what a social dynamical system must be able to say, and what it costs to say it. It has exhibited no instantiation, made no application, and given no account of the term, generativity, for the sake of which the wider programme exists.

The order of exposition, however, follows the order of dependence. A framework that cannot say what a mechanism destroys cannot say what it would mean to generate, since generation is intelligible only against the possibility of loss. Loss has now been given a form. It is a relation, not a magnitude: not a quantity that has diminished, but a discrimination the system can no longer make. What follows from that, and in particular whether a congruence can be made to evolve without circularity, so that a system might recover a distinction it had lost, is the subject of the next part.

References

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中文

面向本体论异质的社会动力学的元模型

第一部分：最高抽象层

黄万宏 · huangwanhong@serendip.ngo

摘要

这是本系列的第一篇论文。它确立最高抽象层，并且不再往前一步。

社会系统的动力学建模通常以确定一个状态空间为起点。这本身已经是一项承诺，而且是一项在诸领域之间无法兑现的承诺。规范秩序的状态、权力格局的状态、价值系统的状态、承认网络的状态，并不居于任何共同的流形之中，也不容许任何共同的度量；而障碍并不在于这样的空间难以构造，而在于这些对象根本不属于那种容许共同安置的种类。这种异质性是本体论的，而非实践上的，它不会因更好的工具而得到解决。本文提出的方案是从更低一层出发，在这一层上，变化、结构、约束与差异被定义，却尚未被实现。本文给出四条公理。状态的载体不被赋予任何结构。作用其上的机制构成一个半群：具结合律，无逆元，无单位元。一个同余关系记录任何机制都必须保持的东西。而差异被定义为一个机制所坍缩掉的区分之集合，既非剩余物，亦非距离。由此推出四个并未被预设的命题：坍缩是不可逆的；机制若构成群则根本没有坍缩，因而没有历史；一个机制是非破坏性的，当且仅当它在同余意义下是单射；坍缩在复合之下单调累积。逆元的缺席被表明是承重的那项承诺。本文不实例化该元模型，不加以应用，也不推出关于任何具体社会系统的实质结论。那些属于后续各部分。此处所确立的，仅仅是一个社会动力学系统必须能够言说什么，以及言说它的代价。

关键词： 动力系统；社会动力学；半群作用；同余；坍缩；不可逆性；形式社会理论；生成性关系存在。

AI 使用声明： 本文的思想、主张与理论框架均出自作者。AI 工具被用作起草、文献的查找与核验以及修订过程中的辅助。每一个思想的来源均已由作者核实，作者对内容承担全部责任。

提要

元模型的全貌。$\mathcal{S}$ 是一个不具更多结构的集合；$\mathcal{M}$ 是一个半群，既无单位元亦无逆元；$\sim$ 是 $\mathcal{S}$ 上的一个等价关系。

$$\begin{aligned} &\textbf{(变化)} && \alpha : \mathcal{M} \times \mathcal{S} \longrightarrow \mathcal{S}, \qquad (m, x) \longmapsto m(x) \[8pt] &\textbf{(结构)} && \alpha(m_1 m_2,, x) ;=; \alpha!\left(m_1,, \alpha(m_2, x)\right) \[8pt] &\textbf{(约束)} && x \sim y ;\Longrightarrow; \alpha(m, x) \sim \alpha(m, y) \[8pt] &\textbf{(差异)} && \Delta(m) ;=; \bigl{, (x,y) ;:; x \not\sim y ;\wedge; \alpha(m,x) \sim \alpha(m,y) ,\bigr} \end{aligned}$$

第 1 至 3 节给出这四行的理由，§4 给出它们的后果，§5 给出它们拒绝裁定的东西。这些并未被预设的后果是：坍缩是不可逆的；不可逆性要求逆元的缺席；以及一个机制在同余意义下是单射时，它什么也不破坏。

§1 引论

1.1 该形式体系是分层的；本文给出最顶层

本文所属的研究纲领在若干抽象层级上提出一套社会动力学的形式体系。在最高层，人们陈述任何模型必须能够表达什么，而不陈述它表达了什么。在其下一层，人们陈述一个模型可以援引哪些结构（一个序、一个度量、一个代数），以及代价为何。再往下，人们展示模型。这些层级并不是关于同一事物的相互竞争的解释。它们处于一种依赖关系之中：上层所裁定的东西约束下层可以建造的东西，而下层所建造的东西必须对上层负责。

这是本系列的第一篇论文，它只给出最高层。它不实例化该框架，不将其应用于某个领域，也不推出关于任何具体社会系统的实质结论。那些属于随后的各部分。

从顶层开始并单独处理它，其理由在于：在这一层上所作的承诺，正是日后变得不可见的那些承诺。一个进行排序的模型已经假定了可比性；一个进行测量的模型已经假定了对称性；一个进行加总的模型已经假定了一套算术。其中每一项都是一个决定，而每一项一旦被埋入某个具体的形式体系之内，就难以取回，也无从争辩。将最高层孤立出来，是一种在这些决定被作出之前使其可供检视的办法。

1.2 写下 $x$ 这一举动本身已经规定了什么算作差异

动力系统在其通行的形式中是一个三元组：一个状态空间、一条演化规则、一个参数。规则被写作

$$\dot{x} = f(x;\theta),$$

它所引导的读法是：左边记录变化，右边记录变化由之出发的结构。就其所及而言，这一读法是正确的。它所掩盖的是第一项的代价。只要写下 $x$，就已经确定了 $x$ 所居的空间，并随之规定了什么算作一个状态与另一个状态之间的差异：这类差异是否可被测量，是否可被比较，是否连续地变化。

对于此处所论的系统，这一规定无法兑现。规范秩序的状态是规则与期待的一种格局；权力系统的状态是诸种不对称的一种分布；价值系统的状态是对何者要紧的一种排序；承认网络的状态是谁被看见的一种模式。它们并不坐落于一个共同的空间之中，也不容许一个共同的度量。

这一点需要审慎对待，因为它比表面看上去更强。困难并不在于共同空间难以构造，那将是工具上的困难，并且会由更好的工具来解决。困难在于：一条规则与一种依赖并非同一变量的两个取值，而任何测量的精细化都不会使它们成为如此。社会状态的异质性是本体论的而非实践上的，而这一类异质性不会被克服；它要么被尊重，要么被覆写。价值多元论的理论用一个世纪所争辩的，正是”这些状态是可通约的、存在某个标度 $m : X \to \mathbb{R}$”这一主张；而一种以假定该空间为起点的建模策略，其起点便是把那被争辩的东西假定掉。

1.3 变化、结构、约束与差异无需状态空间

所提出的回应并不是去寻找一个更巧妙的空间。它是下降到空间成为必需之处的下方。

对任何社会动力学系统，有四件事必须是可说的。它必须能够说出它如何变化。它必须能够说出它的诸变化如何复合。它必须能够说出它的诸变化被约束去保持什么。它还必须能够说出它的诸变化破坏了什么。这四者分别是变化、结构、约束与差异。

本文的主张并不是这四者新颖。而是说，它们合在一起是充分的，并且它们可以在不假定状态空间、度量、序、算术或零元的前提下被给出。这一点重要，因为上述每一项假定一旦作出，就悄悄裁定了一个理论本应留待开放的问题。状态上的一个序预设了可比性。一个度量预设了对称性，因而无法区分背叛与被背叛。一个零元预设了：变换所留下的东西是一个能够为空的量，于是在任何论证之前就裁定了损失是一个量值。

§3 所给出的元模型不作上述任何一项选择。作为交换，它产出了并非以手工放入的结果：坍缩是不可逆的；不可逆性之所以可能，仅仅因为机制缺乏逆元；以及一个机制什么也不破坏，当且仅当它在同余意义下是单射。

1.4 在依赖的次序上，损失先于生成性

实例化被推迟。应用亦然。尤其被推迟的，是对生成性的形式处理；生成性正是更广的纲领所要服务的术语，而本文的四条公理并不提供它。四条公理所提供的是它的前提条件：一个关于机制能够破坏什么的说明。一个无法说出什么被丧失的框架，也无法说出”生成”意味着什么。论述的次序遵循依赖的次序，而损失居先。

§2 状态空间不能居先

2.1 社会状态的异质在于种类，而不在于尺度

考虑四个系统，以及各自所需要的状态。

系统	状态
规范秩序	规则、义务、期待的格局
权力系统	依赖与权威的分布
价值系统	对何者被认为要紧的排序
承认网络	谁被承认为主体的模式

不存在将它们嵌入一个共同向量空间的办法，而那些已被提出的办法恰恰正是批判的对象：效用、价格与合成指数，各自都以牺牲它们本应保全的差异为代价，换取了可比性。

有必要精确地说明这种异质性在何种意义上是本体论的而不仅仅是实践上的，因为二者极易混淆，而只有前者对一个理论的形式具有后果。实践上的异质性将意味着：一个共同空间存在，只是难以构造，数据不佳，标度在实践中不可通约，测量工具粗糙。这样一种困难由更好的工具来解决，并且它许可建模者仿佛该空间已然在场那样推进下去，因为在原则上它确实在场。

此处所论的异质性不属于那一类。问题不在于共同空间难以寻得，而在于有待被安置于其中的对象根本不属于那种容许共同安置的种类。一条规则与一种依赖并不是同一变量被测坏了的两个取值；它们根本不是同一变量的两个取值。一种承认的模式并不是权威分布的一种粗略呈现。这些是不同的存在方式，而”其一距其二有多远”这个问题不是一个困难的问题，而是一个构造不良的问题。正是在这一意义上，对共同状态空间的要求是一项实质性的承诺，而不是一项技术上的不便，实则是一项本体论的承诺，却以建模惯例的面目被断言；它不应作为默认而被作出。

2.2 序预设可比性，度量预设对称性，零元预设量值

序。 $\mathcal{S}$ 上的一个偏序提供上确界，而两个价值状态的上确界即是二者的一个共同尺度。假定一个格，就是已经假定了可比性。

度量。 $\mathcal{S}$ 上的度量 $d$ 满足 $d(x,y)=d(y,x)$。对称性无法表示一个有向的行动。在度量之下，一个机制与它的承受相隔同样的距离，而关系性行动的不对称性在任何关于它的问题被提出之前就已被抹除。

零元。 带有零元的加法结构使一个变换的剩余成为一个可能消失也可能不消失的量。这是一个确定的哲学立场，即认为所丧失者是一个量值，因而在原则上是可归还的。与之对立的立场则主张：所丧失者在新的结构中根本没有位置。元模型必须能够容纳二者，因而不可假定其中任何一方。

2.3 固定机制则载体得以自由

此处所走的路径颠倒了通常的构造次序。我们不是先固定状态再导出作用其上的变换，而是固定变换并让状态不具结构。这并不新颖；它正是代数进路处理动力学时所循的路径。其依据在于：一个理论所需要的结构可以由施动的对象来承载，从而使载体保持为一个供它们作用其上的集合。

§3 变化、结构、约束与差异在没有结构化载体的情况下亦可定义

3.1 载体是一个赤裸的集合；结构由机制承载

定义 1（原始概念）。 一个社会动力学元模型是一个三元组 $(\mathcal{S}, \mathcal{M}, \sim)$，其中

$\mathcal{S}$ 是一个集合，称为载体。它被假定不具有更多结构：没有序，没有度量，没有代数运算，没有特异元素。

$\mathcal{M}$ 是一个集合，称为机制。

$\sim$ 是 $\mathcal{S}$ 上的一个等价关系。

$\mathcal{S}$ 的元素是状态，取该词所能承担的最弱含义：即某一给定领域所视为格局的任何东西。$\mathcal{M}$ 的元素是机制：即在该领域中作用于一个格局并产出另一个格局的任何东西。关于二者由什么构成，不作任何预设。

3.2 机制构成半群，而 $\Delta$ 由 $\alpha$ 与 $\sim$ 导出

公理 1（变化）。 存在一个映射 $\alpha: \mathcal{M} \times \mathcal{S} \longrightarrow \mathcal{S}$，$(m, x) \longmapsto m(x)$。

一个机制作用于一个状态并返回一个状态。这就是变化被要求成为的全部。

公理 2（结构）。 $\mathcal{M}$ 带有一个满足结合律的复合 $\mathcal{M} \times \mathcal{M} \to \mathcal{M}$，$(m_1, m_2) \mapsto m_1 m_2$，且 $(m_1 m_2) m_3 = m_1 (m_2 m_3)$，而 $\alpha$ 是它的一个作用：$\alpha(m_1 m_2,, x) = \alpha!\left(m_1,, \alpha(m_2, x)\right)$。$\mathcal{M}$ 是一个半群。它不被假定具有单位元，也不被假定具有逆元。

结合律是免费的：它只是说，先做 $m_1$ 再做 $m_2$，然后做 $m_3$，与先做 $m_1$，然后做 $m_2$ 接着 $m_3$，是同一回事。那两项省略并非免费，它们才是本公理的实质内容。其后果在 §4 中导出。

公理 3（约束）。 $\sim$ 是该作用的一个同余关系：对一切 $m \in \mathcal{M}$，$x \sim y ;\Longrightarrow; \alpha(m, x) \sim \alpha(m, y)$。

读作：凡系统所保持者，任何机制皆不得破坏。该公理并不说出 $\sim$ 是什么。它说的是：一个不尊重 $\sim$ 的机制根本就不是这个系统的机制。$\sim$ 是该系统的容许性法则，其内容由领域供给。

公理 4（差异）。 对每个 $m \in \mathcal{M}$，$m$ 的坍缩是 $\Delta(m) ;=; \bigl{, (x,y) \in \mathcal{S} \times \mathcal{S} ;:; x \not\sim y ;\wedge; \alpha(m,x) \sim \alpha(m,y) ,\bigr}$。

读作：$\Delta(m)$ 是 $m$ 所破坏掉的那些区分之集合。系统本可分辨的两个状态，在 $m$ 作用之后，成为它无法分辨的两个状态。

3.3 差异是一项丧失了的分辨，而不是一个遗留下来的元素

关于差异的三种熟悉的形式化被拒绝，每一种都因为它要求 $\mathcal{S}$ 上元模型拒绝假定的结构。

不是剩余。 剩余格算子 $x \backslash m(x) = \max{c : x \cdot c \leq m(x)}$ 要求 $\mathcal{S}$ 同时带有一个乘法与一个序。因此它要求状态是可比的，而这正是社会状态的本体论异质性所否认的。

不是距离。 由度量诱导的形式 $d(x, m(x))$ 要求一个度量，而度量是对称的。它无法表示一个有向的行动，而此处所论的行动是有向的。

不是核。 形式 $\ker \pi = {x : \pi(x) = 0}$ 要求一个零元，因而要求 $\mathcal{S}$ 上的一个代数结构。在半群中，一个态射的核根本不是一个子集：它是一个同余关系。公理 4 以这一事实为引导，将差异定义为一个关系而非一个元素。

在这一层上，差异不是一个被剩下的东西。它是一项已经丧失了的分辨。

这三项拒绝共有一个形式。被弃置的每一个算子都返回一个元素，并且在这样做的时候，就使元模型承诺了关于剩余物为何的一个确定看法，而对这个看法，那些思考剩余物最久的传统并不一致。其中的利害关系在 §5 中处理。

$\Delta$ 是一个导出的对象。它不是第五个原始概念：它完全由 $\alpha$ 与 $\sim$ 所定义，也就是说，由公理 1 与公理 3 所定义。

§4 不可逆性仅由结合律与同余关系即可推出

这些公理是弱的。它们却蕴涵了未被规定的结果。本节给出这些结果。

4.1 坍缩是不可逆的

命题 1（坍缩的不可逆性）。 设 $(x,y) \in \Delta(m)$。则对每个 $m’ \in \mathcal{M}$，有 $\alpha(m’ m,, x) ;\sim; \alpha(m’ m,, y)$。即：在 $m$ 之后施加的任何机制都无法恢复 $m$ 所破坏的那一区分。

证明。 由 $\Delta$ 的定义，$\alpha(m,x) \sim \alpha(m,y)$。由公理 3，把 $m’$ 施于同余的状态得到同余的状态，故 $\alpha(m’, \alpha(m,x)) \sim \alpha(m’, \alpha(m,y))$。由公理 2，左端为 $\alpha(m’m, x)$，右端为 $\alpha(m’m, y)$。$\square$

该证明只用到公理 2 与公理 3。因此，不可逆性并不是关于社会世界的一条额外公设，并非因为我们凭经验知道有些事无法撤销而被追加上去。它是机制的结合律与同余关系的保持二者合在一起已经说出的东西。

4.2 逆元消灭差异

命题 2（群没有历史）。 设 $\mathcal{M}$ 是作用于 $\mathcal{S}$ 上的一个群，且 $\sim$ 是一个同余关系。则对每个 $m \in \mathcal{M}$，$\Delta(m) = \emptyset$。

证明。 设 $(x,y) \in \Delta(m)$，于是 $\alpha(m,x) \sim \alpha(m,y)$ 且 $x \not\sim y$。施加 $m^{-1}$ 并使用公理 3，得 $\alpha(m^{-1}m, x) \sim \alpha(m^{-1}m, y)$，即 $x \sim y$，矛盾。$\square$

逆元的缺席不是一项技术上的削弱。它是一个系统得以拥有历史的条件本身。凡每个机制都可被撤销之处，没有什么被破坏，没有什么被守护以免于破坏，而一个变换耗费了什么这一问题根本不会出现。一个由机制构成的群，是一个没有损失的世界。

这两个命题处于一种确定的关系之中。命题 1 说：被坍缩者保持被坍缩。命题 2 说：这之所以可能，仅仅因为机制不可逆。二者合起来，定位了这个框架中其余一切所依赖的那项精确的结构性承诺：公理 2 对逆元的拒绝。

4.3 一个机制什么也不破坏，当且仅当它在 $\sim$ 意义下是单射

命题 3。 $\Delta(m) = \emptyset$ 当且仅当诱导映射 $\bar{m}: \mathcal{S}/!\sim ;\to; \mathcal{S}/!\sim$ 是单射。

证明。 公理 3 保证 $\bar{m}$ 在商上是良定义的。$\bar{m}$ 不是单射，恰恰当存在类 $[x] \neq [y]$ 使 $\bar{m}[x] = \bar{m}[y]$，也就是当存在状态 $x \not\sim y$ 使 $\alpha(m,x) \sim \alpha(m,y)$，也就是当 $\Delta(m) \neq \emptyset$。$\square$

该命题提供了一个判据。一个机制什么也不破坏，当且仅当它在同余意义下是单射；因此，为了知道一个机制是否具破坏性，无须逐例检查它的效果。只需追问它是否分开了系统本已能够分开的东西。

4.4 复合会累积

命题 4（坍缩的单调性）。 对一切 $m, m’ \in \mathcal{M}$，$\Delta(m) ;\subseteq; \Delta(m’ m)$。

证明。 设 $(x,y) \in \Delta(m)$。则 $x \not\sim y$，且由命题 1，$\alpha(m’m,x) \sim \alpha(m’m,y)$。故 $(x,y) \in \Delta(m’m)$。$\square$

坍缩一旦发生，就被其后的每一个机制携带向前。因此，一个系统的历史是丧失了的诸分辨的单调累积，而半群中没有任何机制能够减少它。这就是”一个社会系统不会回到先前状况”这一说法的形式含义。

§5 唯当载体拥有算术，损失才是一个量值

公理 4 把差异定义为一个关系。自然的替代方案会把它定义为一个元素：一个剩余、一个余项、机制作用之后遗留下来的某个东西。该替代方案被拒绝了，理由在于：对”余项”这一概念拥有最强主张权的两个传统，即政治经济学与精神分析，对余项是什么并不一致。一个返回元素的算子裁定了它们的分歧，并且是以规定的方式裁定的。以下陈述这一分歧，指出记法会在何处裁定它，并指出一个取值为关系的 $\Delta$ 转而留下了什么开放。

5.1 剩余价值是可归还的；对象小 a 没有可供归还之处

经济学的剩余。 在价值分析中，一个机制所遗留的是一个量值。剩余价值是劳动者所生产的价值与返还给他的价值之间的差额 [12]；它是一个量，原则上可以计算，并且，这正是整个分析的规范力量所依托之点，它本可以被归还。它没有被归还，这就是不义。因此该剩余是与所给出者同一序列中的一个元素：它在价值系统中有其位置，可以用该系统的术语表达，而其归还即便并未到来，也是可理解的。一个剥削理论需要这一类剩余，否则它无法说出被拿走的是什么。

精神分析的剩余。 在拉康那里，余项完全是另一种类型 [14, 15]。对象小 a 并不是被扣留的某个满足的量；它是从象征化本身当中掉落出来的东西，是主体进入象征秩序所产生的剩余，并且恰恰不可在该秩序之内被表征。问题不在于该对象稀缺，也不在于它被拿走了、或许可以归还。它在产生它的那个秩序中没有位置。它的全部理论功能都依赖于此：倘若它可被定位，倘若它是一个量值，它在原则上就可以被供给，而它所维系的欲望结构便会瓦解 [16]。

经济学的剩余是一个未被归还的量值。精神分析的剩余根本不是一个量值。前者可以被归还。后者不能，而障碍并不在于归还很困难，而在于根本没有恰当类型的东西可供归还。

5.2 取值为元素的算子以记法裁定二者

假设差异如通常那样，由一个返回元素的算子来定义：

$$\Delta(m, x) ;=; x \backslash m(x) \qquad\text{或}\qquad \Delta(m, x) ;=; d!\left(x, m(x)\right) \qquad\text{或}\qquad \Delta(m) ;=; \ker \pi_m = {x : \pi_m(x) = 0}.$$

这些形式各自都要求 $\mathcal{S}$ 上的结构：第一个要求一个序与一个乘法，第二个要求一个度量，第三个要求一个零元。而每一个在提出这一要求时，就已经作出了选择。一个返回 $\mathcal{S}$ 之元素、或返回某个拥有零元的 $\mathcal{D}$ 之元素的算子，已经说出：所丧失者是一个某物，一个可能为空的、拥有位置的、可以设想被供给的量值。它已经裁定支持经济学的剩余而反对精神分析的剩余，而且它是以记法而非以论证做到这一点的。

这正是最高层之所以存在、要为之保留开放的那一类承诺。一个社会机制的余项究竟是一个可归还的量值，还是一个不可表征的虚空，这是一个实质性的问题，而诸传统在其上分歧。可以说，元模型并不是回答它的地方；而一个以其符号的选择来回答它的元模型，是在没有论证的情况下回答了它。

5.3 取值为关系的 $\Delta$ 使两种读法皆保持开放

$$\Delta(m) ;=; \bigl{, (x,y) ;:; x \not\sim y ;\wedge; \alpha(m,x) \sim \alpha(m,y) ,\bigr}$$

$\Delta$ 返回一个关系，而不是一个元素。它说出一个机制破坏了哪些分辨。它不说出有什么（如果有的话）被遗留下来，而且它为了说出它所说的东西，无需序，无需度量，无需算术，无需零元。

两个传统由此皆被保持开放，而每一个都在实例化处被复得，而不是在起点处被假定：

凡实例化为 $\mathcal{S}$ 供给了加法结构之处，一个量值便可被关联到每一个被坍缩的对上，而剩余就成为一个量，这就是经济学的读法。归还于是被良好地提出，而人们可以追问它是否已被作出。
凡实例化不供给这样的结构之处，$\Delta(m)$ 仍旧只是一个对的集合，仅此而已。两个状态已成为不可分辨的，而不存在恰当类型的东西可供归还，这就是精神分析的读法。此处归还并非被扣留；它是不可表述的。

元模型之所以中立，并不是因为它空洞。它之所以中立，是因为它定位了那一分歧：损失是否为一个量值这一问题，就是载体是否拥有算术这一问题。那是一个关于某个特定社会秩序的决定，应当在对该秩序的建模中作出并加以辩护。就本文所能见及者而言，那并不是一个关于社会动力学之形式本身的决定，而且它不应仅仅由记法来作出。

5.4 两种读法许可不同的要求

应当说明，尽管这超出了本部分的范围：这一选择是有后果的。一个建立在经济学剩余之上的正义理论可以要求归还：某物被拿走了，它有确定的大小，并且它可以被交还。一个建立在精神分析剩余之上的理论无法提出这一要求，它必须转而要求一个不同的秩序，一个其同余关系分开了当前秩序所混同之物的秩序。

元模型并不在二者之间作出裁决。它确实确立了：它们是不同的要求，它们出自关于载体的不同结构性假定，而一个尚未说出自己作了哪一项假定的理论，也就尚未说出自己在要求什么。

§6 实例化供给公理所扣留的结构

元模型被有意造得太弱而无法用于计算。它没有序，所以不能排序。它没有度量，所以不能测量。它没有算术，所以不能加总。这些都只在实例化处被供给，而每一次供给都是一项承诺。分层的目的在于使这些承诺可见，从而对任何具体模型都可以追问：它在哪一行假定了可比性，依据何在。

6.1 每个实例化都是半流，其同余关系承载规范性的分量

实例化就是供给四样东西：一个载体 $\mathcal{S}$、一个半群 $\mathcal{M}$、一个作用 $\alpha$ 以及一个同余关系 $\sim$。于是诸公理约束该实例化可以做什么，而 §4 的诸命题在其中自动成立，无需重新证明。

关于实例化的两项约束直接由 §4 推出，值得在此陈述，因为它们对每一个实例化都成立，因而属于最高层而非任何特定的层。

是半流，而不是流。 一个以 $t \in \mathbb{R}$ 为时间的连续时间流是一个群作用，而由命题 2，它有 $\Delta \equiv \emptyset$。可逆的动力学没有坍缩。这就是”哈密顿系统不丧失信息”这一经典事实，而它在此处是一个推论而非一个观察。因此，任何要为一个具有历史的系统建模的实例化，都必须是一个半流，$t \geq 0$。这一限制并不是建模上的便利；它是被迫的。

同余关系承载规范性的分量。 在任何实例化中，$\sim$ 都不是一个技术性的选择。它是该系统关于自身有义务保持什么的陈述，并且它是规范性内容进入形式体系的唯一入口。一个正义理论本不会混同的两个格局，正是 $\sim$ 被要求加以分开的格局。因此，选择 $\sim$ 是一项实质性的理论行动，并且应当作为这样一项行动而被辩护。元模型供给该义务的形式，而不供给它的任何内容。

6.2 实例化、应用与生成性被推迟

有三样东西在第一部分的范围之外，此处予以指明，以免它们的缺席被误认作疏漏。

没有实例化。 未针对任何具体领域展示载体、半群、作用或同余关系。元模型是作为对模型的一组条件而被提出的，而不是作为一个模型。

没有应用。 该框架未曾对照社会理论或政治理论中的任何实质问题来阅读。凡此类阅读为可能之处，而作者认为它们是可能的，并且认为它们改变了对现象的描述，它们都要求一个确定的 $\sim$，因而要求一个实例化，因而要求后续的部分。

没有关于生成性的说明。 诸公理描述机制如何作用于一个固定的载体。其中没有任何东西产生一个尚未在 $\mathcal{S}$ 之中的状态，也没有任何东西产生一项 $\sim$ 尚未容许的区分。坍缩已被给予一个形式上的居所；涌现则尚未。这是一项真实的限制，并将在 §6 中被再次提起。

第一部分确立了一个社会动力学系统必须能够言说什么。它并不言说它。

§7 诸公理无法评判一次坍缩、无法生成新者、也无法演化其自身的同余关系

一个框架应当以它不能做什么来评判，此处指明它不能做的四件事。

它无法说一次坍缩是错的。 $\Delta(m)$ 记录一个机制破坏了哪些区分。它并不说这些区分本应被保留。有些坍缩是无害的，有些则是必要的：一个能够把每个状态与其他每个状态都区分开来的系统，将是一个无力进行概括的系统。规范性的分量完全落在 $\sim$ 之上，也就是落在关于一个系统有义务保持什么的先行裁定之上。元模型供给该裁定的形式，而不供给它的任何内容。

它无法生成新者。 诸公理描述机制如何作用于一个固定的载体。其中没有任何东西产生一个尚未在 $\mathcal{S}$ 之中的状态，也没有任何东西产生一项 $\sim$ 尚未容许的区分。对于一个以生成性为核心术语的纲领而言，这是一项真实的限制。坍缩已被给予一个形式上的居所；涌现则尚未。一个候选方案是存在的，即一个系统过渡到同一代数的一个不等价表示，在其中新者是对一个先前不可抵达之扇区的进入，而不是对旧者的重新排列；但它所要求的结构多于四条公理所提供的，并且此处不予展开。

同余关系是外生的。 公理 3 设定 $\sim$ 而不导出它。然而在任何实际的系统中，什么算作一项被保持的区分，其本身就是该系统历史的一个结果。一个更锐利的框架会让 $\sim$ 演化，从而使一个机制不仅能够改变诸状态，还能够改变系统在它们之间进行分辨的能力本身。当下的障碍在于：$\Delta$ 是依 $\sim$ 而定义的，而一个移动着的 $\sim$ 会使该定义循环。在作者看来，解决这一点是该框架中最重要的开放问题，而这正是下一部分所要处理的问题。

它本身并不是一个理论。 这一类元模型约束可容许模型的类；它并不在其中作出选择，也不作任何预测。它的用处是诊断性的。它使人们得以对任何社会模型追问：它把状态当作什么，它把机制当作什么，它责成机制保持什么，因而它容许机制破坏什么。一个无法回答第四个问题的模型，并未因此避免破坏。它只是拒绝去记录破坏。

§8 结论

四条公理已被给出。一个不具结构的载体；一个作用其上的机制半群，具结合律且无逆元；一个记录系统有义务保持什么的同余关系；以及一个被定义为机制所坍缩掉的诸区分之集合的差异算子。由此推出四个命题：坍缩是不可逆的；坍缩唯有在逆元缺席时才可能；一个机制是非破坏性的，当且仅当它在同余意义下是单射；以及坍缩在复合之下单调累积。

这些命题并不深刻。它们各自只有三行。值得注意的是：它们并非被放进去的。该框架被构造得尽可能地弱，没有序，没有度量，没有算术，没有零元；而正因为假定得如此之少，它才得以把不可逆性作为一个定理获得，而不是把它设定为一个公设。

一个系统之所以拥有历史，是因为它的机制无法被撤销。它之所以能够丧失它本有义务保持的东西，是因为它的机制能够坍缩掉同余关系要求它分开保持的诸区分。这是同一个事实，一次以代数的方式陈述，一次以规范的方式陈述；而表明二者是同一个事实，正是一个形式社会理论的担负。

这一担负在此处并未被履行。本文只确立了最高层：一个社会动力学系统必须能够言说什么，以及言说它的代价是什么。它未曾展示任何实例化，未曾作出任何应用，也未曾就更广的纲领为之而存在的那个术语，即生成性，给出任何说明。

然而，论述的次序遵循依赖的次序。一个无法说出机制破坏了什么的框架，也无法说出”生成”意味着什么，因为生成唯有对照着损失的可能性才是可理解的。损失如今已被给予一个形式。它是一个关系，而不是一个量值：它不是一个已经减少了的量，而是一项系统不再能够作出的分辨。由此而来的东西，尤其是一个同余关系能否在不循环的情况下被造成可演化的，从而使一个系统或许能够复得它曾丧失的一项区分，这正是下一部分的主题。

参考文献