The "Ultraviolet" Structure of Recursive Value and Power - Physical Strategies and Formal Architectures for Generative Relational Being

ENGLISH

The “Ultraviolet” Structure of Recursive Value and Power

Physical Strategies and Formal Architectures for Generative Relational Being

Wanhong Huang (huangwanhong@serendip.ngo)

Abstract

Value and power, when examined at their foundations, exhibit a recursion without evident terminus. Every valuation appears to invoke a higher valuation that would ground it, every exercise of power a legitimation that would authorize it, and every legitimation a further legitimation in its turn, so that the attempt to fix a foundation descends through an unbounded series and the analysis grows less stable the finer its resolution becomes. This paper takes that recursion as its object. It argues that the recursion of value and power shares its formal structure with a problem long familiar in theoretical physics, the problem of ultraviolet behaviour, in which the number of relevant degrees of freedom fails to diminish as one descends toward short distances and the quantities one wishes to compute cease to converge. The paper does not claim that the two are mathematically identical. It claims that they are the same kind of formal problem, and that the discipline which has confronted such problems most directly has accumulated a repertoire of strategies worth examining. The account is built in two movements. The first is diagnostic. It reconstructs the recursion of value and power, identifies the condition under which it fails to terminate, and characterizes that condition in the borrowed and deliberately marked vocabulary of “ultraviolet” structure. The second is comparative and constructive. It sets out the four principal responses that theoretical physics has made to the ultraviolet problem, absorption in renormalization, discretization in dynamical triangulation, extension in string theory, and pre-geometric emergence in group field theory, and it examines what each would mean if carried onto the terrain of a relational and generative account of value and power. It finds that a generative relational account is most consistent with the emergence strategy, on which the recursion terminates at the level where the description in terms of contestable claims ceases to apply. On this basis the paper proposes a layered formal architecture of five offices, ontology, representation, dynamics, criteria, and observation, in which the several formal instruments each occupy a distinct office of a single relational order rather than competing for one, so that representation, generative dynamics, cyclic criteria, and multiscale observation are placed in a definite dependence. It closes, in the manner it requires of itself, on the questions that any formalization of generative value would have to confront. The paper continues a programme that has replaced the individual with generative relational being and carries that programme onto the question of formal foundations.

Keywords: recursive value and power; ultraviolet structure; generative relational being; group field theory; emergence; formal architecture.

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.

§1 Introduction: A Recursion Without Terminus

The argument of this paper begins in a difficulty that recurs across the theory of value, the theory of power, and the theory of knowledge, and that each of these fields has met in its own vocabulary without recognizing it as one difficulty. In the theory of value the difficulty appears as follows. To judge that something is good is to judge it by a standard, and the standard invites the question of what makes it a good standard, which is a question of value in its turn, so that each valuation rests upon a higher valuation and the higher upon a higher still. In the theory of power the same difficulty appears as the problem of legitimation. An exercise of power is authorized by a rule, the rule by an authority competent to lay it down, the authority by a further rule that constitutes it, and the series that would ground the first act in a final authorization does not terminate of itself. In the theory of knowledge it appears as the regress of justification, in which each ground stands in need of a ground. These are ordinarily treated as separate puzzles belonging to separate literatures. This paper treats them as instances of a single formal structure, a recursion that descends without an evident terminus and that grows, rather than diminishes, as one attempts to resolve it more finely.

The structure has a feature that the ordinary treatments tend to pass over. The recursion does not merely fail to reach a foundation. It becomes less tractable, not more, as one descends. Each attempt to secure a value by appeal to a higher value opens the higher value to the same demand, and the demand does not weaken with depth. Each attempt to legitimate a power by appeal to a further authority raises the same question of that authority, and the question does not lose its force. The analysis becomes increasingly unstable under recursive refinement. This is the feature that gives the paper its guiding analogy. In theoretical physics there is a class of problems with exactly this shape, in which the quantities one wishes to compute receive contributions from ever shorter distances, the contributions fail to diminish as the distance shrinks, and the computation ceases to converge. These are the ultraviolet problems, so named because the divergence gathers at the high-frequency, short-distance end of the spectrum. The recursion of value and power, this paper argues, has the formal shape of an ultraviolet problem.

The word is borrowed, and the borrowing is marked throughout. When the term “ultraviolet” is applied to the recursion of value and power it is written in quotation marks, to indicate that it names a structural resemblance and not a physical fact. There is no momentum in a relational field, no integral over short-wavelength modes, no literal divergence of a computed amplitude. What there is, and what the analogy tracks, is a recursion whose relevant terms fail to diminish under refinement and whose resolution therefore does not converge. The precise sense in which the analogy is claimed, and the precise sense in which it is withheld, is set out in the methodological note that follows this introduction, and it is the load-bearing commitment of the whole paper. The reader who wishes to know at once how far the analogy is meant to reach should turn there first.

The value of the analogy is not decorative. Theoretical physics has confronted the ultraviolet problem for the better part of a century, and it has done so on repeated occasions, developing several distinct frameworks whose common origin is precisely the need to control an unbounded proliferation of short-distance degrees of freedom. Renormalization, dynamical triangulation, string theory, and group field theory are not, at bottom, competing mathematical techniques in search of application. They are different answers to one question, the question of how a theory may remain well defined when the naive account of its short-distance structure diverges. Each answer embodies a distinct attitude toward the divergence, and the four attitudes, taken together, come close to exhausting the ways in which such a recursion may be met. This is the resource the paper draws upon. If the recursion of value and power is an “ultraviolet” problem in the marked sense, then the strategies by which physics has met its own ultraviolet problems become a structured field of options for meeting it, and the task of the theorist of value is to determine which of them the subject matter permits.

The paper may accordingly state the question it addresses in a single formulation, and it does so here so that every later section may be read as a step toward answering it.

The question of the paper. The recursion by which every value invokes a higher value and every power a further legitimation descends without terminus and destabilizes under refinement. Does this recursion possess a formal structure of a recognizable kind; if so, what strategies exist for problems of that kind; which of those strategies may a generative relational account of value and power consistently adopt; and what architecture for the formal instruments of the account follows from the strategy so chosen?

The purpose of the paper is therefore neither to import a physical model into social theory nor to assert that a relational account of value is a physical theory in disguise. Its purpose is methodological and architectural. It is to establish that the recursion of value and power is a formal problem of a recognizable kind, to survey the strategies that have been developed for problems of that kind, to determine which strategy a generative relational account of value and power can consistently adopt, and to propose, on that basis, a layered architecture in which the several formal instruments each occupy their proper level. The paper claims to have arrived at the right form of the problem and at a defensible architecture for addressing it. It does not claim to have solved it, and it marks, at its close, the questions that remain open. The distinction matters. A great part of what obstructs the formalization of value is that the problem is posed in a shape that admits no tractable answer, and to reshape the problem so that the accumulated strategies of a mature discipline bear upon it is itself a result, prior to and independent of any particular solution.

The argument continues a line of work with a recognizable signature. That work replaced the individual with generative relational being, treating subjects as constituted within relation rather than as prior to it, and treated value as arising within relation rather than residing in a thing or issuing from a lack. It then developed a generative relational epistemology, on which concepts and signs are expressions of relational value rather than its source, and it carried the programme onto the terrain of political economy. The present paper carries it onto the terrain of formal foundations. The recursion of value and power is where the programme meets the question of its own tractability, and the strategies of the ultraviolet problem are what allow that question to be posed in a form that can be worked.

The paper proceeds as follows. It first sets out, in a methodological note, the exact sense of the central analogy and the criterion by which a borrowed term may pass from marked metaphor to literal instrument. It then reconstructs the recursion of value and power and shows in what sense it has an “ultraviolet” structure, identifying the condition under which the recursion fails to terminate. It surveys the four historical responses of theoretical physics to the ultraviolet problem, describing each as an attitude toward the divergence rather than as a body of mathematics. It maps the four strategies onto the recursion of value and power and determines which a generative relational account can consistently adopt. It proposes a layered formal architecture in which generation, scale, and cyclic stability occupy distinct levels. It closes on the open problems, and then, in the manner it requires of itself, on the questions a generative formalization of value would have to confront.

§2 Methodological Note: The Sense of the Analogy

Before the recursion of value and power is characterized as an “ultraviolet” problem, and before the strategies of theoretical physics are brought to bear upon it, the exact sense of the analogy must be fixed. This section fixes it. It is the shortest section of the paper and the most important, because it determines how every subsequent use of a borrowed term is to be read.

The central claim is one of shared formal structure, and it is bounded on both sides. The paper claims that the recursion of value and power and the ultraviolet behaviour of a quantum field theory are the same kind of formal problem. It does not claim that they are mathematically identical, that there exists an isomorphism between them, or that the equations of the one may be transcribed into the other. A quantum field theory has a momentum space, a measure over field configurations, an integral over short-wavelength modes, and a computed amplitude that literally diverges. A relational field of value and power has none of these. To assert an identity would be to claim resources that the subject matter does not supply, and the paper makes no such claim. What it asserts is weaker and defensible. The two problems have the same structure at the level of the problem itself, prior to any model. In each, a quantity one wishes to determine depends on a recursion; in each, the terms of the recursion fail to diminish as one descends toward finer resolution; and in each, the determination therefore does not converge. This is the shared formal problem, and it is all that the analogy asserts.

The claim of shared structure. The paper claims that the recursion of value and power and the ultraviolet behaviour of a quantum field theory constitute the same formal problem: a quantity determined by a recursion whose relevant terms fail to diminish under refinement, so that the determination does not converge. The paper does not claim that the two are mathematically isomorphic, and it does not transcribe the equations of the one onto the other.

From this bound a convention follows, and the convention governs the paper’s vocabulary throughout. When a term drawn from physics is used to describe the physics itself, in the discussion of quantum field theory, of renormalization, of quantum gravity, it is written plainly, because there it is literal. When the same term is used to describe the recursion of value and power, it is written in quotation marks, to mark that it names a structural resemblance rather than a physical fact. The reader will therefore find “ultraviolet,” “divergence,” “cutoff,” and “renormalization” quoted when they qualify the relational field and unquoted when they qualify the physics. The alternation is not a stylistic tic. It is the visible trace of the analogy’s boundary, and it allows the reader to see, at each occurrence, on which side of the boundary the argument stands.

A single principle governs when a term may cross the boundary, and the paper states it in advance so that its own restraint may be checked against it. A borrowed term passes from marked metaphor to literal instrument only when the structure that the term requires has been exhibited in the relational field itself. For the term “ultraviolet” the required structure is a genuine scale parameter together with a well-defined family of degrees of freedom that a change of scale integrates away. Until such a parameter and such a family have been exhibited, the term remains in quotation marks, and any conclusion drawn from it is a conclusion about the form of the problem and not about a mechanism in the field. The paper does not, at any point, exhibit the required structure. It therefore keeps the quotation marks throughout, and it is candid that the analogy remains, in this work, an analogy of problem-form and not a mechanism. To exhibit the structure that would retire the quotation marks is one of the open problems named at the paper’s close.

Criterion for retiring the quotation marks. A borrowed physical term becomes a literal instrument, rather than a marked metaphor, only when the structure it presupposes is exhibited in the relational field. For “ultraviolet” this is a scale parameter and a family of scale-dependent degrees of freedom. This paper does not exhibit that structure and so retains the marks throughout.

A question remains that the convention alone does not answer, and it is the question on which the whole comparative method rests. If the borrowing is licensed only at the level of shared problem-form, why should the strategies developed for the physical problem be expected to bear on the relational one at all? The strategies were devised for divergences in momentum space, and the relational field has no momentum space. The answer is that each of the four strategies is, at its core, a response to the abstract feature that the two problems share, and not to the physical particulars that they do not. What renormalization, discretization, extension, and emergence each confront is the failure of a proliferation to diminish under refinement, and each is a distinct way of rendering such a proliferation tractable: by absorbing its cost into terms that are never read off directly, by denying it a reservoir of ever-finer contributions, by dispersing the concentration that drives it, or by dissolving the very stratum on which it was posed. These are operations on the form of a divergent proliferation, and their content survives translation to any domain in which a proliferation diverges in the same formal way. The strategies are expected to bear on the recursion of value and power to exactly the extent that the recursion shares that form, and no further. Where the shared form gives out, the transfer gives out with it, and the paper claims the transfer only for the form.

Why the strategies are expected to apply. Each physical strategy is a response to the abstract failure of a proliferation to diminish under refinement, rather than to the physical particulars of momentum space. Its content transfers to any domain that shares that abstract failure, and only so far as the sharing extends. The strategies bear on the recursion of value and power because the recursion exhibits the same failure of terms to diminish, and the transfer is claimed for that formal feature alone.

The reason for placing this note before the substantive argument, rather than folding its cautions into footnotes as they arise, is that interdisciplinary borrowing fails most often through an accumulation of small unmarked overstatements, each slight on its own, that together persuade the reader a mechanism has been established where only a resemblance has been drawn. The note is a safeguard against that accumulation. It fixes, once and in advance, the weight that every borrowed term is permitted to bear, so that the comparative survey which follows may proceed without pausing at each step to renegotiate its own license.

§3 Recursive Value and Power as an “Ultraviolet” Problem

This section reconstructs the recursion of value and power without recourse to any formalism, and shows in what precise sense it has the structure of an “ultraviolet” problem. The reconstruction has three parts. It sets out the recursion as it appears in the theory of value and in the theory of power. It identifies the feature that makes the recursion resemble an ultraviolet divergence rather than a benign infinite series. And it names the condition on which the recursion depends, the condition whose presence sustains it and whose absence would terminate it.

The Recursion

Consider first the recursion of value. A judgment that something has value is made against a standard, and the standard is itself a bearer of value, since to adopt it is to hold that judging by it is better than judging by another. The standard therefore stands in need of the same warrant it confers, and the warrant, if it is supplied, is supplied by a further standard of the same kind. The series does not close of itself. No standard in the series is the standard by which the series as a whole is good, for any candidate for that office is itself a member of the series and subject to its demand. This is the familiar regress of justification in the particular form it takes for value. That it fails to terminate is an old observation and carries little interest here. How it fails is the concern of the next part.

Consider next the recursion of power. An exercise of power counts as more than the mere application of force when it is authorized, and the authorization is conferred by a norm that the exercise satisfies. The norm in its turn is not self-standing. It is valid because it was laid down by an authority competent to lay it down, and the competence of that authority is itself constituted by a further norm, which raises the same question of its own validity. The chain of authorization that would ground a present exercise of power in a final and self-sufficient source runs backward without arriving. Attempts to arrest it have been made by positing a basic norm presupposed rather than derived, or a sovereign whose will requires no authorization, and the recurrence of such devices is itself evidence that the recursion is felt as a difficulty in need of a stopping device, rather than as a series that halts on its own.

The two recursions are one recursion in two materials. In each, a term of the relevant kind, a value or an authorization, is sustained only by a further term of the same kind, and the demand that opened the series reopens at every stage. What is sought at each stage is a ground, and what is found is another term that itself requires grounding. The recursion is the movement of that demand as it descends.

Why the Recursion Resembles a Divergence

Not every infinite series is an “ultraviolet” problem. A series may descend without terminus and yet be harmless, if its terms diminish quickly enough that the whole converges, so that the absence of a last term costs nothing and the sum is well defined despite the infinity. The regress of value and power is not of this benign kind, and the difference is exactly the difference that makes the physical ultraviolet problem hard.

In the physical case the divergence is a divergence because the contributions from shorter and shorter distances do not fall off. Were the short-distance contributions to diminish rapidly, the integral over them would converge and there would be no ultraviolet problem; it is precisely their failure to diminish that makes the sum ill defined. The analogous feature in the recursion of value and power is that the demand for a ground does not weaken with depth. The tenth-order value standard is exactly as much in need of warrant as the first-order valuation, and the hundredth authorization is exactly as questionable as the act it was summoned to authorize. The recursion does not approach a limit at which the demand exhausts itself. It reproduces the demand undiminished at every level, and it is this undiminished reproduction, and not the mere fact of infinity, that gives the recursion its “ultraviolet” character in the marked sense. The analysis becomes increasingly unstable under recursive refinement, because each refinement, far from settling the question, restates it at full strength one level down.

The correspondence should be stated with care, since the two cases differ in what the failure consists of. In the physical case the ill-definition is additive, a sum over contributions of non-vanishing size that fails to converge to a finite value. In the recursion of value and power there is no summation, and the failure is one of termination, a series of grounds, each requiring a ground, that fails to arrive at a member which would close it. What the two cases share is the feature that drives both failures, the refusal of the relevant terms to diminish under refinement, so that no finite truncation is adequate and the descent must be continued without end. The analogy is claimed for this shared feature, the non-vanishing of terms under refinement, and it is claimed neither for the additive form of the physical divergence nor for any summation on the side of value. It is in this exact and restricted sense that the recursion has an “ultraviolet” character.

The “ultraviolet” character. The recursion of value and power resembles an ultraviolet divergence, rather than a convergent infinite series, because the demand for a ground fails to diminish with depth. Each level reproduces the demand of the level above at undiminished strength, so that finer resolution destabilizes the analysis rather than settling it.

The Fuel of the Recursion

If the recursion is sustained by the undiminished reproduction of a demand, then the recursion depends on whatever it is that reproduces the demand, and to name that is to name the condition on which the whole structure rests. The condition may be stated as follows. What drives the recursion is that each term in it is posed as a contestable claim. A value standard, offered as a ground, is offered as something that could be otherwise, that draws a boundary and excludes alternatives, and that therefore presents to any challenger a surface against which a counter-claim may be pressed. An authorization, likewise, is a claim that could be denied, a drawing of a line that another may contest. It is because each term is a contestable claim that each term invites the counter-claim which reopens the demand, and it is the endless availability of the counter-claim that keeps the recursion in motion. The fuel of the recursion is the contestability of the claim.

This identification has a consequence that governs the constructive half of the paper. A recursion sustained by contestability will continue for as long as its terms retain the form of contestable claims, and it will continue no longer. Should the descent reach a level at which what is found no longer has the form of a contestable claim, at which there is no boundary to press against and no line to deny, the counter-claim would find no surface, the demand would not be reproduced, and the recursion would lose its fuel. Whether such a level exists, and what would be found there, is not a question the diagnostic reconstruction can answer, for it is a question about the deep structure of the relational field rather than about the surface form of the recursion. It is the question that the comparison with physical strategies is designed to sharpen, since the several strategies differ precisely in what they suppose to lie at the bottom of the descent, and it is to that comparison that the argument now turns.

§4 Layers of Formalization

The comparison of physical strategies presupposes a prior question that the paper has so far left implicit, and this section makes it explicit, because the answer to it governs which strategies are even eligible for the recursion of value and power. The question is what determines the choice of a formal strategy in the first place. The position taken here is that the choice depends on the ontology of the object under study. A formal strategy is answerable to what it is a strategy for, and the strategies developed in one domain transfer to another only when the two domains share the relevant features of their objects. The recursion of value and power belongs to the study of social facts, and the eligibility of any formal strategy for it is settled by what social facts are taken to be.

The Ontology of the Object

The object of social inquiry is the social fact, and social facts are of several kinds. Following a standard account, they include events, entities, relations, attributes, and states, and the discipline of international politics, among others, takes these as the units through which its subject matter is analyzed. A theory of value and power operates upon social facts of exactly these kinds, upon the event of a judgment, the entity that judges, the relation within which the judgment falls, the attribute it ascribes, and the state it institutes or disturbs.

The programme this paper continues does not deny the reality of these social facts. Its single modification of the received ontology is to raise relation to a position of relative priority, so that entities, attributes, states, and events are taken to be generated and stabilized within relation rather than given prior to it. The priority is relative and not exclusive. The claim is not that relation alone exists and the other social facts are illusions, which would leave the theory unable to speak of the very events and entities that social inquiry concerns. The claim is that the other social facts have their being within relation, that an entity is constituted as the entity it is by the relations that place it, and that the analysis therefore begins from relation and reaches the other facts through it. With this the ontology of the object is fixed, and the question of representation may be posed.

The relative priority of relation. Social facts comprise events, entities, relations, attributes, and states. The programme takes relation to have relative ontological priority, so that entities, attributes, states, and events are generated and stabilized within relation. The priority is relative rather than exclusive: the other social facts are real, and they have their being within relation.

The Representation of Relational Structure

Once the object is fixed as a field of social facts with relation at their base, the next question is not how the field evolves, which is a question of dynamics, but how its structure is to be represented. This step is easily passed over, and passing over it is a frequent source of confusion, because a theory that has not fixed its representation will smuggle one in unnoticed. A relational ontology requires a formal language able to express several things at once: relations among many terms rather than only two, changes in the pattern of connection, the local structure of what connects to what, and the generation of new structure from the combination of old.

Among the available candidates, a combinatorial structure of a particular kind meets these requirements more fully than its rivals, the structure that in the physics of quantum geometry is called a spin foam. A spin foam answers to the four requirements in turn. Its vertices encode interactions among several terms at once, meeting the requirement of many-term relations. Its reconnections encode changes in which terms are joined, meeting the requirement of changes in the pattern of connection. Its combinatorial and local character exhibits the actual pattern of what connects to what, meeting the requirement of local structure. And its edges, gluing interactions into an extended complex that a dynamics can then act upon, meet the requirement that new structure be generated from the combination of old. Its advantage over a graph is that a graph carries only relations between pairs, while the social relations at issue are characteristically relations among many, a coalition, a market, a household, none of which reduces to a set of pairwise ties. Its advantage over a purely categorical representation is that a spin foam is combinatorial and local, exhibiting the actual pattern of connection rather than only the abstract relations among structures. And its decisive advantage for the present programme is that it stands at the meeting point of the discrete and the continuous, serving at once as a combinatorial object and as the object on which a field-theoretic and a dynamical-systems description can both be defined.

The role assigned to the spin foam must be stated with care, and it is the role of a representation and nothing more. A spin foam is an intermediate representation, a layer of formal structure interposed between the relational ontology and the dynamics that will act upon it. It is a candidate for the office of representing relational structure, chosen for the reasons just given, and it is offered as the most adequate candidate rather than as the only one. Nothing in the argument requires that a spin foam be a picture of social reality, and the claim is expressly withheld. What is claimed is that a relational ontology needs a representation with the four features named, and that the spin foam supplies them.

The Dynamics of Structure

A representation is static, and structure is not. The spin foam represents a relational structure; it does not account for how such a structure comes to be, how it dissolves, or how it recombines. For that a dynamics is required, and the dynamics is what a group field theory, together with the resources of dynamical systems theory, is fitted to supply. A group field theory takes the combinatorial structures that the spin foam represents and gives them a generative dynamics, a law by which structures form, interact, and give way to others. In the layering proposed here the spin foam and the group field theory occupy adjacent offices, the one representing structure and the other setting it in motion, and together they constitute the representation and the dynamics of the relational field.

It is at the level of dynamics, and specifically in the transition by which determinate structure precipitates from a state that carried none, that the recursion of value and power reaches its terminus, for the reasons given in the mapping. The generative level from which structure emerges is the level below which the description in terms of contestable claims ceases to apply, and it is therefore the dynamics, and not the representation or the layers above it, that carries the resolution of the recursion.

The Criteria of History and Cycle

Above the dynamics, and dependent upon it, stand two further offices that are often mistaken for parts of the dynamics itself but are properly distinct from it. The first is the office of criteria. Once structures have formed and entered into cycles, a question arises that the dynamics alone does not answer, the question of what a cycle accumulates over its course. A relational cycle may return to its point of departure having generated a surplus within itself, or having drawn one down, or having depended throughout on a supply from without, and to distinguish these is to apply a criterion to what the dynamics has produced. The instrument of this office is the holonomy, together with the geometric phase to which it belongs, and its work is to characterize the history-dependence and the cyclic quality of an evolved structure. A holonomy, in the sense borrowed from the geometry of transport around closed paths, is the quantity by which a system fails to return to its exact starting condition after being carried once around a loop, so that a cycle may close in its parameters while leaving a residue that records the path taken.

The criterion is dependent on the dynamics in a strict sense, and the dependence is worth stating because it prevents a common error. A holonomy is defined only where there is a space of configurations around which a cycle may be carried, and such a space is itself a product of the dynamics, appearing only once structure has emerged. The criterion therefore has no application below the level of emergence, and it says nothing about the generative level. It is a characterization of what the dynamics has already produced, and it presupposes that production. The criterion does not ground the dynamics; it reads a property off the dynamics’ output.

The Strategies of Observation

The second office above the dynamics is that of observation across scales. A relational structure exists at many scales at once, from the relation between two persons, through the household, the community, the organization, and the state, to the international system, and the same structure observed at different scales presents different features. The office of observation is to say, as the scale of description changes, which features are to be retained, which are to be set aside, and how the effective description transforms. This office is filled by two complementary strategies, and their complementarity is important enough to state with precision, because they are easily conflated.

The first strategy addresses the hierarchy of spatial scales. It is the strategy of the renormalization group, which describes how an effective description flows as the scale of observation changes, which features remain relevant as one ascends from the smaller scale to the larger, and which become irrelevant and are washed out. Carried onto the relational field, it is the account of how a description adequate to the dyad transforms into a description adequate to the institution or the society, and of what survives that transformation.

The second strategy addresses the hierarchy of temporal scales, and it is distinct from the first. It is the strategy of fast and slow dynamics, the decomposition of a system into variables that change quickly and variables that change slowly, so that the fast variables may be treated as reaching their local equilibrium while the slow variables carry the long-term evolution. Many relational forms have exactly this temporal layering. The event, the passing emotion, and the momentary fluctuation are fast, while the institution, the custom, the ethical disposition, and the settled trust are slow, and the fast-slow decomposition says which processes may be treated as quickly equilibrating, which are to be held slowly varying, and which slow variables govern the long-term attractors of the relational system.

The two strategies of observation share a common logic, the elimination of what is finer in favour of an effective account of what is coarser, and they differ in what the fineness is. The renormalization group eliminates short-scale degrees of freedom and flows a theory along the axis of spatial scale. The fast-slow decomposition eliminates fast degrees of freedom and reduces a trajectory along the axis of time. One flows a description across scales of size, the other reduces a description across scales of duration, and a relational field that is layered in both size and duration requires both. The renormalization group answers to the hierarchy of spatial scale; the fast-slow decomposition answers to the hierarchy of temporal scale.

The Architecture Assembled

The five offices may now be set out together, and the dependence among them made explicit, for the dependence is what distinguishes this architecture from a mere list of formalisms each borrowed for its own sake.

Five layers of formalization.

Layer	Question	Candidate formalism
Ontology	What is studied	Social facts, relation with relative priority
Representation	How structure is expressed	Spin foam, or other combinatorial structure
Dynamics	How structure forms and evolves	Group field theory, dynamical systems
Criteria	How history and cycle are characterized	Holonomy, geometric phase
Observation	How multiscale behaviour is read	Renormalization group, fast-slow dynamics

The five layers do not stand side by side as equals. Ontology, representation, and dynamics form a chain of generation, in which each office presupposes the one before it: representation presupposes a fixed ontology, and dynamics presupposes a representation to act upon. The remaining two offices, criteria and observation, do not lie further down this chain. They stand upon the output of the dynamics, as two kinds of characterization applied to structures that the dynamics has already generated. The holonomy characterizes what an evolved cycle has accumulated; the observation strategies characterize how an evolved structure appears across scales of size and duration. Neither is foundational, and neither can be carried beneath the dynamics, for each presupposes the emergent configurations that the dynamics alone produces.

This arrangement dissolves a question that has recurred throughout the programme, the question of which formalism is the more fundamental, the group field theory or the renormalization group, the holonomy or the emergence. The question is malformed, because the formalisms answer different questions and occupy different offices. What exists is a matter for the ontology; how it is represented, for the spin foam; how it evolves, for the group field theory; how its cycles are judged, for the holonomy; how it is observed across scale and across time, for the renormalization group and the fast-slow decomposition. The formalisms do not compete for a single office. They fill an ordered set of distinct offices, and the recursion of value and power is resolved at the office of dynamics, where the description that generated the recursion gives out, while the offices above supply the characterizations of what the dynamics produces.

A word is owed on the double appearance of two of these formalisms, since it might otherwise read as an inconsistency. Group field theory and the renormalization group each appear twice in the paper, once as a response to the ultraviolet problem, surveyed among the historical responses below, and once as the candidate formalism for a layer of the architecture, group field theory at the office of dynamics and the renormalization group at the office of observation. The two appearances are consistent, and the one grounds the other. What qualifies a framework to fill a layer is the attitude it takes toward the recursion. Group field theory is fitted to the office of dynamics because its response to the ultraviolet problem is the strategy of emergence, the strategy that resolves the recursion of value and power; the renormalization group is fitted to the office of observation because its response is the account of how an effective description runs across scales. The survey of responses that follows is therefore a justification of the architecture rather than a digression from it, the place where it is shown why each formalism belongs to the office it has been assigned.

It is this architecture, and the strategy of emergence that sits at its generative core, that the survey of physical responses in the following section is meant to inform.

§5 Historical Responses to the Ultraviolet Problem

Theoretical physics has met the ultraviolet problem more than once, and the several frameworks it has built to meet it share a common origin in that problem, whatever their later divergences. This section describes four of them. It describes them as attitudes toward the divergence rather than as bodies of mathematics, because what the paper requires of them is their strategy and not their machinery, and because the strategy is what can be carried across to a domain that has no momentum integrals to transcribe. Throughout this section the physical terms are written plainly, without quotation marks, because here they are used of the physics itself, where they are literal. The quotation marks return in the following section, where the strategies are carried onto the relational field.

The problem common to all four may be stated in a sentence. When a quantity in a field theory is computed by summing contributions from all distance scales, the contributions from the shortest distances fail to diminish, and the sum diverges. Each of the four frameworks is a way of ensuring that the theory remains well defined in the face of this failure, and each embodies a distinct conception of what the short-distance structure of the world must be for the divergence to be controlled.

Renormalization: Absorption

The first and most conservative response leaves the divergence in place and absorbs it. Its insight is that the parameters written into the theory at the outset, the bare mass and the bare charge, are not the quantities anyone measures. What is measured is always the effective quantity, the parameter as it appears after the short-distance contributions have been included, and this effective quantity is finite even when the bare one is not. Renormalization allows the bare parameters to carry the divergence, absorbing the infinite short-distance contribution into quantities that were never observable, so that everything that is observable comes out finite. It further reveals that the effective parameters run with the scale of observation rather than holding constant, taking different values as the distance at which one probes the theory changes, and the law of that running is the renormalization group. The attitude here is one of coexistence with the infinite. The divergence is not defeated; it is relocated into the unobservable and thereby rendered harmless, and the physics that remains is the physics of quantities that vary with scale.

Dynamical Triangulation: Discretization

The second response denies the divergence its source. The divergence gathers at the shortest distances only because the theory permits distances to become arbitrarily short, and it permits this because it takes space and time to be a continuum, indefinitely divisible. Dynamical triangulation refuses the continuum at the smallest scale. It builds spacetime from elementary simplices of fixed size, a smallest triangle or tetrahedron below which there is no further division, and it assembles geometry by gluing these building blocks according to rules that respect the causal order. Because there is no distance shorter than the edge of the simplex, there is no reservoir of ever-shorter contributions for the divergence to draw upon, and the sum that diverged in the continuum becomes finite on the discretized structure. The attitude here is one of a floor. The recursion toward shorter distances is stopped by a smallest element, the common edge length of the simplices. A qualification belongs here, and it will matter when the strategy is carried across. In the standard practice of the approach the edge length serves as a regulator, a shortest distance introduced to render the sum finite and then sent toward zero in a continuum limit, so that the discreteness is a device of calculation rather than a claim about the constitution of space. A stronger reading is available, on which the smallest element is a genuine feature of the world and space is grainy at bottom, and the two readings are distinct. What the strategy contributes to the present argument is the form common to both, the arrest of the descent by a smallest element, whether that element is a regulator to be removed or a floor to be affirmed.

String Theory: Extension

The third response dissolves the divergence by refusing the point. In a theory of point particles, an interaction occurs at a single location, a sharp vertex at which the whole energy of the process is concentrated into a region of vanishing size, and it is this concentration into a point that produces the worst of the short-distance divergences. String theory replaces the point particle with an extended object, a one-dimensional string, and with this replacement the sharp vertex of interaction is smeared into a smooth surface traced by the string as it moves and joins and parts. The energy that was concentrated at a point is distributed along the extended object and among its many modes of vibration, and the short-distance behaviour that diverged for points is softened for strings. The attitude here is one of extension. The divergence is relieved by giving the fundamental entities an internal extension and internal degrees of freedom, so that no interaction is ever truly point-like and the concentration that caused the divergence never occurs.

Group Field Theory: Pre-Geometric Emergence

The fourth response removes the background against which the divergence was defined. The ultraviolet problem presupposes a spacetime continuum in which short distances exist to be summed over, and the three responses so far each take that continuum as given and treat the divergence within it, whether by absorbing it, flooring it, or softening it. Group field theory does not take the continuum as given. Its fundamental degrees of freedom are fields on an abstract group manifold, carrying no spatial or geometric interpretation of their own, and continuous spacetime, rather than being assumed, is expected to emerge from the collective behaviour of these pre-geometric quanta through a phase transition, a condensation in which a great number of the fundamental elements enter a common state and a smooth geometry appears as their collective description. On this account the short-distance continuum, within which the ultraviolet problem was posed, is itself a derived and emergent thing, and the question of what lies at distances shorter than any that the emergent geometry supports does not arise, because below the emergent level there is no geometry and no distance at all. The attitude here is one of emergence. The divergence is dissolved together with the background, by locating the background itself as the product of a more fundamental level that has no short-distance problem because it has no distances.

The Four Strategies Compared

The four responses may be set beside one another as four attitudes toward one divergence, and the comparison is the hinge on which the paper turns from physics to its own subject.

Four responses to the ultraviolet problem.

Renormalization absorbs the divergence into unobservable parameters and lets the observable ones run with scale. Attitude: coexistence.

Dynamical triangulation discretizes spacetime into smallest simplices below which no division exists. Attitude: a floor.

String theory extends the point into a one-dimensional object and distributes the interaction across its modes. Attitude: extension.

Group field theory makes geometry emerge from a pre-geometric level, so that the short-distance continuum is derived rather than given. Attitude: emergence.

Two of the four propose that the descent toward shorter distances is stopped, and two that it is not. Dynamical triangulation stops it against a floor of indivisible elements. Group field theory stops it in a different manner, at the level below which the very notion of distance ceases to apply, since geometry itself is emergent and there is nothing geometric beneath it. These two ways of stopping the descent are distinct, and the distinction between a floor and a limit of applicability will prove to be the decisive one when the strategies are carried across, for it is the difference between a smallest thing and the ceasing of a kind of description. Renormalization and string theory, by contrast, do not stop the descent at all. Renormalization lets it run and absorbs its cost; string theory lets it run and softens its effect by extension. With these four attitudes in hand, the paper turns to the recursion of value and power and asks, of each, what it would mean there.

§6 Mapping the Strategies onto Generative Relational Being

The four strategies are now carried onto the recursion of value and power. The quotation marks return, because from this point the physical terms qualify the relational field and name resemblances rather than facts. The question asked of each strategy is the same. If the recursion of value and power is met in the manner this strategy proposes, what does the strategy assert about the nature of value and power at the bottom of the descent, and is the assertion one that a generative relational account can consistently make? The four answers differ, and the difference is instructive, for each strategy, transposed, becomes a distinct thesis about what a value or an authorization ultimately is.

Absorption: The Dressed Claim

Carried onto the recursion, the strategy of absorption yields the following thesis. The recursion of justifications is acknowledged as endless rather than halted, and what is asserted in any actual valuation is the effective value, the valuation as it stands after the whole recursion of grounds has been folded into it, in place of the bare value that would carry the infinite series of higher grounds explicitly. On this reading a value in use is always a “renormalized” value, one that wears the entire series of its higher justifications as a garment rather than deferring to them as an external ladder, and the counter-claim that would reopen the recursion is reconceived as a correction already absorbed into the effective term. Power admits the same reading. An authorization in force is a “dressed” authorization, carrying its infinite regress of legitimations within itself as an effective legitimacy that functions while the regress stays untraversed.

The thesis is coherent and has a genuine attraction, since it accounts for the plain fact that valuations and authorizations do function, in practice, while the infinite series they would seem to require goes untraversed. Its limitation, for a generative account, is that it purchases this at the cost of leaving the recursion intact and merely unobserved. The infinite series remains, still undiminished, and the effective value is well defined only if the folding-in of the series is itself well defined, which requires something answering to the physical scale parameter along which the running is organized. The recursion of value and power exhibits no evident parameter of this kind, and lacking one, the absorption remains a metaphor stripped of its mechanism. Absorption describes how effective values function; on its own it leaves the recursion in place, and a generative account that seeks a genuine terminus will find in it a description of the symptom in place of a treatment of the cause.

Discretization: The Smallest Relation

Carried onto the recursion, the strategy of discretization yields the thesis that the descent of justifications reaches a floor, a smallest relational element below which the demand for a further ground has no application. There would be, on this thesis, an atom of value or of authorization, an indivisible relational unit that is not itself valued or authorized by anything more basic, and against which the recursion halts as the physical descent halts against the smallest simplex. The thesis has the merit of supplying a definite terminus, and it corresponds to a familiar move in foundational thought, the positing of a basic norm or a self-evident value that grounds without needing to be grounded.

Its difficulty for a generative account is that a floor of this kind reintroduces exactly what the programme was built to refuse, a foundational element prior to relation, an atom whose value or authority is intrinsic to it and not constituted within the relational field. To halt the recursion against a smallest valued thing is to concede that value has, at bottom, a bearer that possesses it independently of relation, and this concession is incompatible with the claim that value arises within relation rather than residing in a thing. The strategy of discretization, taken by itself, therefore sits poorly with a generative relational ontology. It will reappear, transformed, when it is seen to converge with the strategy of emergence, but as a bare floor of intrinsically valued atoms it is not a thesis the programme can adopt.

Extension: The Non-Point Claim

Carried onto the recursion, the strategy of extension yields a thesis of unusual interest. The worst of the recursion, on the analogy, is produced by treating a value or an authorization as a point, a sharp and isolated claim concentrated at a single position within the order of justification, against which the counter-claim may be pressed with full force. Extension proposes that the claim is never in fact point-like. A value, on this reading, is an extended relational object with internal structure and internal degrees of freedom, and what presents itself as a single contestable claim is the outer aspect of a distributed relational form whose content is spread across its internal relations rather than concentrated at a contestable point. The counter-claim, meeting an extended object rather than a point, is dispersed among the object’s internal degrees of freedom rather than concentrated against a single surface, and the sharp opposition that fuelled the recursion is softened.

This thesis is congenial to a generative account, and more so than the first two, because it locates the content of a value in a distributed relational structure rather than in an isolated bearer, which is close to what the programme has maintained from the start. Its limitation is that softening is not termination. An extended claim is harder to contest at a point, but it remains, in the end, a claim, and a sufficiently distributed counter-claim could still meet it. Extension relieves the recursion without ending it, and it does so by a supposition, that values are internally extended relational objects, which a generative account can accept but which does not by itself reach the level at which the recursion would lose its fuel. Extension, then, is a partial and welcome measure, and it will be retained below as a supplement to the strategy that does reach that level, rather than as the primary response.

Emergence: The Ceasing of the Contestable Claim

Carried onto the recursion, the strategy of emergence yields the thesis that the descent of justifications terminates at the level below which the description of the relational field in terms of contestable claims ceases to apply. The recursion, as the diagnostic reconstruction showed, is fuelled by the contestability of the claim, and it continues for exactly as long as its terms retain the form of contestable claims. The strategy of emergence proposes that beneath the level of contestable claims there lies a more fundamental relational level, a generative level of pure difference and relation prior to the constitution of any bounded claim, at which there is no boundary to press against and no line to deny. At that level the counter-claim finds no surface, the demand for a ground is not reproduced, and the recursion loses its fuel. The descent does not strike a smallest valued thing. It passes below the level at which “value,” in the sense that supports a contestable claim, is the right description at all, into a generative level from which bounded claims of value and authority themselves emerge.

On the emergence strategy the recursion of value and power reaches its terminus at the level where the description in terms of contestable claims ceases to hold. The terminus is a limit of applicability and not a floor of foundational things. The fuel of the recursion is the contestability of the claim; beneath the level of claims there is no contestability, and so no fuel, and so no recursion.

This thesis is the one a generative relational account can consistently adopt, and it should be seen clearly why. It terminates the recursion, which absorption and extension do not. It terminates it without positing a foundational element prior to relation, which discretization cannot avoid. And it terminates it in a manner that the programme has, in another domain, already worked out, for the account of how bounded linguistic and symbolic structures emerge from a pre-structural generative level, through a breaking of symmetry and a condensation, parallels the account of how contestable structure arises from a level at which no contestable structure yet exists. The termination of the recursion of value and power at the generative level follows the same form, in the register of value, that the emergence of structure from the generative field follows in the register of form.

It is at this point that the strategy of discretization returns, transformed. The emergent, pre-geometric account has a discrete aspect as well as a continuous one, for the same fundamental level that gives rise, in one limit, to a smooth emergent structure is built, in its combinatorial description, from elementary relational simplices. The floor that discretization posited in isolation, and that a generative account could not accept as a floor of intrinsically valued atoms, reappears here in a changed office, as the combinatorial grain of the generative level itself, below the emergence of any valued structure. The smallest element, so understood, is a smallest relational element prior to value rather than a smallest valued thing, and in this transformed sense the discretization strategy is absorbed into the emergence strategy rather than standing against it. The distinction that discharges the earlier objection should be made explicit, since a grain and a floor may appear the same. What the generative account refused, in refusing discretization, was a foundational element standing prior to relation, an atom whose value or authority belongs to it in itself and not through the relations that place it. What the transformed strategy admits is a smallest element of the relational fabric, a grain internal to relation and possessing no value or authority of its own, valued structure appearing only above it through emergence. The first is an element outside relation from which relation would be built; the second is a minimal articulation within relation from which no value yet follows. The grain is relational through and through, and it is for this reason that admitting it does not restore the foundational atom that the programme set aside. This convergence parallels the one that the underlying physics exhibits, in which the emergent and the discrete are two aspects of a single pre-geometric account rather than two competing proposals.

The Result of the Mapping

The mapping yields a determinate result, and it is important to state it in the exact form the programme requires, neither weaker nor stronger. A generative relational account of value and power is most consistent with the emergence strategy, supplemented by the extension strategy and incorporating the discretization strategy in its transformed, pre-value sense. This is a claim of consistency and of best fit. It is not the claim that a generative account of value is a group field theory, or that the recursion of value and power is an ultraviolet divergence. The physical framework supplies the form of a strategy, the attitude of dissolving a problem together with the background that posed it, and the generative account adopts that form because the form matches its own prior commitments. What the generative account must supply for itself, and what no physical framework can supply for it, is the content: what the generative level is, in the register of value; what emerges from it, and by what relational process; and what distinguishes the bounded structures that emerge as bearing one value rather than another. The physical strategy tells the account where to look for the terminus of the recursion. It does not tell the account what it will find there. That is the burden of the architecture set out above, whose generative office is where the resolution falls, and of the open problems that follow.

§7 The Standing of the Paper and Its Reach

An argument built by analogy owes its reader a plain statement of what it is, of what it hopes to make knowable, of how it might one day touch the empirical domain, and of where its limits fall. This section supplies that statement. It is placed before the open problems because the open problems are its consequence, and it gathers into one place the claims about the paper’s own standing that the earlier sections made in passing.

The Standing of the Argument

The paper is a methodological and architectural study, and it is neither a physical model of value nor a completed formalization of it. Its work is done at the level of the problem and of the arrangement of instruments, one remove above any particular mechanism. It takes a recursion that several fields have met separately, shows that the recursion has a recognizable formal shape, surveys the strategies a mature discipline has developed for problems of that shape, determines which strategy a generative relational account can consistently adopt, and proposes an architecture in which the account’s formal instruments occupy distinct and ordered levels. Each of these is a contribution of form and not of content. The paper claims to have found the right shape for a problem and a defensible order for the tools that would address it, and it claims no more. In the vocabulary of the discipline it borrows from, the paper offers a strategy and a scheme of organization, and it leaves the construction of any specific model to the work its architecture is meant to guide.

The Objects of Knowledge

The paper hopes to make three things knowable, and they correspond to its three contributions. The first is that the unbounded regress of value and of power, which has been treated as a defeat for the theory of foundations, has instead the structure of a problem that admits of principled response, so that the regress becomes a thing to be worked rather than a wall at which work stops. The second is that the several strategies of the ultraviolet problem, when set beside the recursion of value and power, sort themselves into those a generative account can adopt and those it cannot, and that the sorting is determined by the account’s prior commitments rather than by preference, so that the choice of strategy is shown to be constrained and not free. The third is that the formal instruments the programme has gathered, which have appeared to compete for a single explanatory office, in fact belong to distinct levels of a single order, so that their apparent rivalry dissolves into a dependence. To make these three things knowable is to convert a cluster of difficulties into a structured field of questions, and that conversion is what the paper offers as knowledge.

The Path to the Empirical Domain

The paper is formal throughout, and its reach into the empirical domain is therefore indirect and mediated, and it is worth saying by what path such a reach might one day be made, if only to mark that the path exists and that the paper does not now walk it. The architecture proposed here assigns definite offices to definite levels, and each office, once a concrete model realized it, would carry empirical commitments of its own.

At the level of emergence, a realized model would predict that bounded structures of value and authority form through something answering to a transition, a relatively sudden precipitation of determinate structure from a state that carried none, rather than through gradual accretion alone. Where the formation of a norm, an institution, or a shared valuation can be observed over time, the architecture directs attention to whether that formation exhibits the marks of a transition, a threshold, a rapid ordering, a sensitivity to conditions near the threshold, and this is a matter on which historical and developmental evidence could in principle speak.

At the level of effective structures, the cyclic criterion distinguishes relational cycles that return bearing an internally generated surplus from those that return depleted or are sustained from without. A realized model would make this distinction operable, and it would direct empirical attention to whether a given institution or relation regenerates the conditions of its own continuation or draws them down, a question that the study of commons, of organizations, and of relationships already asks in other terms and could ask in these.

At the level of scale, the architecture holds that a description adequate to one scale reappears transformed at another, and that the transformation is lawful rather than arbitrary. A realized model would predict specific relations between the description of a relational form at the scale of the dyad and its description at the scale of the institution or the society, and comparative and multiscale social evidence could in principle test whether such relations hold.

These are directions of possible contact and not results. Each of them presupposes a concrete model that the paper does not supply, and each would acquire empirical content only once the value question, deferred to the open problems, had been answered in some determinate way. The paper’s relation to the empirical is therefore the relation of a frame to the pictures it may one day hold. It specifies where empirical commitments would enter and what offices they would serve, and it leaves their supply to the constructive work that a settled architecture is meant to make possible.

The Limits of the Argument

The limits are as definite as the contributions, and honesty requires that they be stated with equal plainness. The paper’s central instrument is an analogy, and the analogy is one of problem-form and not of mechanism; the quotation marks that mark this have not been retired, and until the relational field is shown to possess a genuine scale and genuine scale-dependent degrees of freedom they cannot be. The paper does not formalize value; it locates where a formalization would have to begin and declines to fix the form that value takes, resisting in particular the assumption that value must be an observable on the physical model. The paper does not commit the generative level to a definite mathematical form, holding it open among an action, an evolution rule, a categorical dynamics, and other possibilities, and it therefore does not deliver the base on which a full construction would stand. The paper surveys the strategies of one discipline for one problem, and its survey is complete only as that history is complete, so that a further strategy, within physics or beyond it, might alter the field of options. These limits are not failures of execution. They are the exact boundaries of a study that has chosen to establish the form of a problem rather than to force a premature solution, and to mark them precisely is part of what the study offers.

§8 Open Problems

The paper has claimed to arrive at the right form of a problem and at a defensible architecture for addressing it, and it has been careful to claim only that. Its omissions are as important to state as its claims, and this section states them, because an architecture is honest only when the vacancies within it are marked as plainly as the parts that are filled.

The first open problem is the one the methodological note anticipated. The analogy on which the paper turns remains an analogy of problem-form and not of mechanism. The quotation marks around “ultraviolet” have not been retired, because the structure that would retire them, a genuine scale parameter and a family of scale-dependent degrees of freedom in the relational field itself, has not been exhibited. Whether such a structure can be exhibited, and whether the relational field possesses anything answering to the physical scale along which effective descriptions run, is unsettled. Until it is settled the transposed strategies remain strategies borrowed in their form, and the paper’s use of them is the use of a form and not the operation of a mechanism.

The second open problem concerns the content of value at the generative level, and it is the problem that the mapping deferred and the architecture inherited. The emergence strategy locates the terminus of the recursion at a generative level from which bounded values emerge, but it does not say what value is at that level, nor by what relational process bounded values precipitate from it, nor what distinguishes the structures that emerge as bearing one value rather than another. The programme has reason to resist the first form in which this question presents itself, the form that would demand a value observable on the model of a physical measurement, for there is no evident ground for supposing that value in the relational field takes the form of an observable at all. It may take the form of a constraint that the generative dynamics must satisfy, or of a selection principle that governs which structures emerge, or of an order parameter that distinguishes phases of the relational field, or of something for which the physical vocabulary has no ready name. Which of these, if any, is the right form for value is not decided here, and to decide it prematurely, by fixing on the observable because the analogy makes it available, would be to let the borrowed vocabulary settle a question that belongs to the subject matter.

The third open problem concerns the base of the architecture. The generative level has been named neutrally, as a relational generative dynamics, and the paper has declined to commit it to a specific mathematical form, and in particular to an action. This restraint leaves open the question of what form the generative dynamics takes. Whether it admits a variational formulation, or is better rendered as an evolution rule, a categorical dynamics, or a generative process of another kind, is not established, and the choice among these is not a matter of notation but of what the generative level is claimed to be. The programme’s own group-field-theoretic work supplies one candidate realization, but the architecture proposed here is not bound to that candidate and would survive its replacement by another. The form of the base is open, and the architecture is deliberately built so as not to prejudge it.

The fourth open problem is whether the emergence strategy, having been shown to be the one a generative account can consistently adopt, is the one it must adopt. The paper has argued for best fit and for consistency, and it has shown that the alternatives either fail to terminate the recursion or terminate it by positing a foundation prior to relation. It has not shown that no fifth strategy, outside the four that the physics of the ultraviolet problem has so far produced, could terminate the recursion in some further way. The survey of strategies is a survey of the responses a particular discipline has made to a particular problem, and its completeness is the completeness of that history and not a completeness in principle. That a better strategy may yet be found, in physics or outside it, is a possibility the paper does not foreclose.

These four vacancies are not incidental gaps to be filled by further labour of the same kind that produced the architecture. They are the points at which the architecture meets what it does not yet contain, and each of them is a direction of inquiry rather than a defect of the present one. The paper’s contribution is to have built the frame within which these are the right questions to ask, and to have shown that they are questions about the content of a generative theory of value rather than about the form of the problem, which is settled, or about the arrangement of the instruments, which is proposed.

§9 Conclusion

The argument set out from a recursion that the theory of value, the theory of power, and the theory of knowledge have each met in their own vocabularies without recognizing it as one thing, the recursion in which every ground stands in need of a ground and the demand for a foundation descends without terminus. It has tried to show that this recursion has a definite form, and that the form is one a mature discipline has confronted repeatedly. The recursion does not merely fail to terminate. It grows less tractable as it descends, reproducing at every level, undiminished, the demand that opened it, and this undiminished reproduction gives it the structure of what physics calls an ultraviolet problem, in the marked and bounded sense that the paper has been at pains to keep in view. Named so, the recursion ceased to be a private puzzle of the theory of value and became an instance of a problem for which strategies exist.

The strategies are four, and the paper has read them as four attitudes toward one divergence, absorption, discretization, extension, and emergence. Carried onto the recursion of value and power, each became a thesis about what a value or an authorization is at the bottom of the descent, and the four theses were not of equal worth to a generative account. Absorption described how effective values function without terminating the recursion. Discretization terminated it only by positing a foundation prior to relation, which the programme was built to refuse. Extension relieved the recursion without ending it, and was retained as a supplement. Emergence terminated the recursion in the one manner a generative account can consistently adopt, at a limit of applicability rather than against a floor, at the level where the description in terms of contestable claims gives out, and it did so in the very movement by which, in another register, bounded structure emerges from a generative field. Upon this the paper raised an architecture of five layers, ontology, representation, dynamics, criteria, and observation, in which the offices of generation stand in a chain and the offices of criterion and of observation stand upon what the dynamics produces, so that the instruments the programme had gathered were seen to stack rather than to compete, and the cyclic criterion and the strategies of scale were placed where they belong, as properties of what has emerged and not as grounds of emergence.

The account does not close on a doctrine, and the form of its close is required by its content. A frame that locates the terminus of recursion at a generative level prior to every bounded claim cannot end by installing a new bounded claim as its own foundation, a master strategy from which all else is to be derived. It ends, instead, on the questions a generative formalization of value would have to take up, and it offers them as openings rather than as a programme to be executed. There is the question of the analogy itself, of whether the relational field possesses a genuine scale and genuine scale-dependent degrees of freedom, which alone would let the borrowed terms be spoken without their quotation marks. There is the question of the content of value at the generative level, of whether value there is a constraint, a selection principle, an order parameter, or something for which no borrowed word yet answers, a question the paper has been careful not to let the convenience of the analogy decide. There is the question of the form of the generative dynamics itself, of whether it is an action, an evolution rule, a categorical dynamics, or a generative process of another kind, a question the architecture was built so as not to prejudge. And there is the question that stands behind the others, of whether the four strategies a single discipline has so far produced exhaust the ways in which such a recursion may be met, or whether a fifth remains to be found.

The recursion of value and power is not a wall against which thought must stop, and it is not a void into which thought must fall. It is a descent that ends where the description that generated it gives out, at the generative level from which bounded value and bounded power themselves emerge, and the task of a generative theory is to say what is found there.

Acknowledgements

The layered architecture proposed in this paper places at its effective level a criterion that distinguishes generative from extractive cycles, a criterion whose development draws on the concept of generative justice, for which the author is indebted to the work of Ron Eglash. The use made of it here is the author’s own.

The author owes a further debt, beyond what citation can record, to the one whose companionship sustained this work.

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中文

递归性价值与权力的“紫外”结构

为生成性关系存在所设的物理策略与形式架构

黄万宏 (huangwanhong@serendip.ngo)

摘要

价值与权力，当在它们的根基处被审视时，展现出一种没有显然终点的递归。每一次估价似乎都援引一次会为它奠基的更高估价，每一次权力的施行都援引一次会授权它的正当化，而每一次正当化又轮到援引一次更进一步的正当化，因而固定一个根基的尝试穿过一个无界的序列下降，而分析随它的分辨率变得越细而变得越不稳定。本文把那一递归取为它的对象。它论证价值与权力的递归与理论物理学中一个久已熟悉的问题共享它的形式结构，即紫外行为的问题，其中相关自由度的数目未能随人向短距离下降而减少，而人所希望计算的诸量不再收敛。本文并不主张这两者在数学上是同一的。它主张它们是同一种形式问题，而那个最直接地面对过此类问题的学科已积累起一批值得审视的策略。这一说明由两个乐章构成。第一个是诊断性的。它重构价值与权力的递归、辨认它未能终止所据的条件、并以“紫外”结构这一被借来并被有意标记的词汇刻画那一条件。第二个是比较性与建设性的。它陈述理论物理学对紫外问题所作的四种主要回应，即重整化中的吸收、动力学三角剖分中的离散化、弦论中的延展、以及群场论中的前几何涌现，而它审视每一种若被带上一种关系性与生成性的价值与权力之说明的地形会意味着什么。它发现一种生成性关系的说明与涌现策略最为一致，依这一策略，递归终止于以可争议之主张来描述不再适用的那一层级。在此基础上本文提出一个由五个职分组成的分层形式架构，即本体论、表象、动力学、判准与观察，其中若干形式工具各占据一个单一关系秩序的一个不同职分、而非争夺一个职分，因而表象、生成性动力学、循环判准与多尺度观察被置于一种确定的依赖之中。它以它对自身所要求的方式，收束于任何一种生成性价值之形式化将不得不面对的诸问题。本文延续一个已用生成性关系存在取代个体的纲领，并把那纲领带上形式根基的问题。

关键词： 递归性价值与权力；紫外结构；生成性关系存在；群场论；涌现；形式架构。

AI 使用声明： 本文的思想、主张与理论框架均源自作者。AI 工具被用作起草、来源的定位与核实、以及修订的辅助。每一思想的来源均经作者核实，作者对内容承担全部责任。

§1 导论：一种没有终点的递归

本文的论证起于一个横跨价值理论、权力理论与知识理论而复现的困难，而这些领域中每一个都以它自己的词汇遇见它、却不曾认出它为一个困难。在价值理论中这一困难如下出现。判断某物是善的，便是以一个标准来判断它，而那标准招来“什么使它成为一个好标准”这一问题，而这转而是一个价值的问题，因而每一次估价倚于一次更高的估价、而那更高者又倚于一个更高者。在权力理论中同一困难作为正当化的问题出现。一次权力的施行被一条规则授权，那规则被一个有资格订立它的权威授权，那权威被一条构成它的更进一步之规则授权，而那会把第一个行动奠基于一次最终授权的序列并不自行终止。在知识理论中它作为辩护的回退出现，其中每一个根据都需要一个根据。这些通常被当作属于不同文献的不同谜题来处理。本文把它们当作一个单一形式结构的诸实例，即一种没有显然终点地下降、并随人试图更细地解决它而增长、而非减少的递归。

这一结构有一个通常的处理倾向于略过的特征。这一递归不仅未能抵达一个根基。它随人下降而变得越不、而非越更可处理。每一次通过诉诸一个更高的价值来确保一个价值的尝试，都把那更高的价值开向同一要求，而那要求不随深度减弱。每一次通过诉诸一个更进一步的权威来正当化一个权力的尝试，都提起关于那权威的同一问题，而那问题不丧失它的力量。分析在递归的精细化之下变得越发不稳定。这便是那赋予本文其引导性类比的特征。在理论物理学中有一类恰有此形状的问题，其中人所希望计算的诸量接受来自越来越短之距离的贡献，那些贡献未能随距离缩小而减少，而计算不再收敛。这些是紫外问题，如此命名是因为发散聚集于频谱的高频、短距离一端。价值与权力的递归，本文论证，有一个紫外问题的形式形状。

这个词是被借来的，而这一借用自始至终被标记。当“紫外”一词被应用于价值与权力的递归时，它被写在引号中，以指明它命名一种结构性的相似、而非一个物理事实。在一个关系性场域中没有动量、没有对短波长模式的积分、没有一个被计算之振幅的字面发散。所有者、以及那类比所追踪者，是一种其相关项未能在精细化之下减少、其解决因而不收敛的递归。这一类比在何种精确意义上被主张、以及在何种精确意义上被保留，在随本导论而来的方法论说明中被陈述，而它是整篇论文承重的承诺。希望立即知道这一类比意在伸到多远的读者应先转向那里。

这一类比的价值不是装饰性的。理论物理学已面对紫外问题大半个世纪，而它已在反复的场合如此为之，发展了若干不同的框架，它们共同的起源恰恰是控制短距离自由度的一种无界增殖之需要。重整化、动力学三角剖分、弦论与群场论，在根底上，不是寻求应用的相互竞争之数学技术。它们是对一个问题的不同回答，即一个理论如何可能在它短距离结构的朴素说明发散时仍保持良好定义的问题。每一个回答都体现一种对发散的不同态度，而这四种态度，合而观之，接近于穷尽这样一种递归可被应对的诸方式。这是本文所汲取的资源。若价值与权力的递归在那被标记的意义上是一个“紫外”问题，那么物理学据以应对它自己诸紫外问题的策略便成为应对它的一个被结构化的选项之场，而价值理论家的任务是确定其中哪些为主题所允许。

因此，本文可以以一个单一的表述陈述它所处理的问题，而它在此如此为之，以便随后每一节都可被读作朝向回答它的一步。

本文的问题。 那个每一个价值援引一个更高价值、每一个权力援引一次更进一步之正当化的递归，没有终点地下降、并在精细化之下失稳。这一递归是否拥有一种可辨识之类的形式结构；若是，对那一类的问题存在什么策略；那些策略中哪些是一种生成性关系的价值与权力之说明可以一致地采纳的；以及一种关于该说明诸形式工具的架构从如此被选择的策略中如何得出？

因此，本文的目的既不是把一个物理模型输入社会理论、也不是断言一种关系性的价值说明是一个乔装的物理理论。它的目的是方法论的与架构的。它是要确立价值与权力的递归是一个可辨识之类的形式问题、勘察那些为该类问题所发展的策略、确定一种生成性关系的价值与权力之说明可以一致采纳哪一种策略、并在那基础上提出一个分层的架构，其中若干形式工具各占据它们恰当的层级。本文主张已抵达该问题的正确形式、以及一个用以处理它的可辩护之架构。它并不主张已解决它，而它在它的收束处标记那些仍敞开的问题。这一区分要紧。阻碍价值之形式化者，很大一部分在于该问题被以一个不容许任何可处理答案的形状被提出，而把问题重塑，以便一门成熟学科所积累的策略得以关涉它，本身即是一个结果，先于并独立于任何特定的解决。

这一论证延续一条有着可辨识签名的工作之线。那工作用生成性关系存在取代个体，把主体当作在关系之内被构成、而非先于它，并把价值当作在关系之内升起、而非驻留于一个事物之中或从一处匮乏发出。它随后发展一种生成性关系认识论，依它概念与符号是关系价值的表达、而非它的源头，而它把纲领带上政治经济学的地形。本文把它带上形式根基的地形。价值与权力的递归是这一纲领与它自己可处理性之问题相遇之处，而紫外问题的策略是那让那一问题得以被以一个可被工作之形式提出者。

本文如下进行。它首先在一则方法论说明中陈述那中心类比的确切意义、以及一个被借来的术语据以从被标记的隐喻过渡为字面工具的判准。它随后重构价值与权力的递归、并显示它在何种意义上有一种“紫外”结构，辨认那递归未能终止所据的条件。它勘察理论物理学对紫外问题的四种历史回应，把每一种描述为一种对发散的态度、而非一批数学。它把这四种策略映射到价值与权力的递归之上、并确定一种生成性关系的说明可以一致采纳哪一种。它提出一个分层的形式架构，其中生成、尺度与循环稳定占据不同的层级。它以在诸敞开的问题上、随后以它对自身所要求的方式、在一种生成性的价值形式化将不得不面对的诸问题上收束。

§2 方法论说明：类比的意义

在价值与权力的递归被刻画为一个“紫外”问题之前、在理论物理学的策略被带来关涉它之前，这一类比的确切意义必须被固定。本节固定它。它是本文最短的一节、也是最重要的一节，因为它决定每一次随后对一个被借来之术语的使用如何被读。

那中心的主张是一个共享形式结构的主张，而它在两侧都被限定。本文主张价值与权力的递归与一个量子场论的紫外行为是同一种形式问题。它并不主张它们在数学上是同一的、主张它们之间存在一个同构、或主张这一者的方程可被转写为那一者。一个量子场论有一个动量空间、一个在场构型之上的测度、一个对短波长模式的积分、以及一个字面发散的被计算之振幅。一个价值与权力的关系性场域没有这些之中任何一个。断言一个同一便会主张主题所不供给的资源，而本文不作任何这样的主张。它所断言者更弱且可辩护。这两个问题在问题本身的层面上、先于任何模型，有同样的结构。在每一个中，人希望确定的一个量依赖于一个递归；在每一个中，那递归的诸项未能随人向更细的分辨率下降而减少；而在每一个中，那确定因而不收敛。这是那共享的形式问题，而它是那类比所断言的全部。

共享结构的主张。 本文主张价值与权力的递归与一个量子场论的紫外行为构成同一个形式问题：一个被一个其相关项未能在精细化之下减少之递归所确定的量，因而那确定不收敛。本文并不主张这两者在数学上是同构的，而它不把这一者的方程转写到那一者之上。

从这一限定一个约定随之而来，而这一约定自始至终统领本文的词汇。当一个取自物理学的术语被用来描述物理学本身时，在对量子场论、对重整化、对量子引力的讨论中，它被平实地写出，因为在那里它是字面的。当同一个术语被用来描述价值与权力的递归时，它被写在引号中，以标记它命名一种结构性的相似、而非一个物理事实。因此，读者将发现“紫外”、“发散”、“截断”与“重整化”在它们修饰关系性场域时被加引号、而在它们修饰物理学时不被加引号。这一交替不是一个文体的小习惯。它是那类比之边界的可见痕迹，而它让读者在每一次出现处看出论证立于那边界的哪一侧。

一个单一的原则统领一个术语何时可以越过那边界，而本文预先陈述它，以便它自己的克制可被对照它而核查。一个被借来的术语从被标记的隐喻过渡为字面工具，仅当那术语所要求的结构已在关系性场域本身中被展示。对于“紫外”一词，所要求的结构是一个真正的尺度参数、连同一个尺度的改变会积分掉的良好定义之自由度族。直到这样一个参数与这样一个族已被展示，那术语保持在引号中，而从它所引出的任何结论都是一个关于该问题之形式的结论、而非一个关于该场域中一个机制的结论。本文在任何一点上都不展示所要求的结构。因此它自始至终保留引号，而它坦承在这项工作中，那类比仍是一个问题形式的类比、而非一个机制。展示那会使引号退休的结构，是本文收束处所命名诸敞开问题之一。

使引号退休的判准。 一个被借来的物理术语成为一个字面工具、而非一个被标记的隐喻，仅当它所预设的结构在关系性场域中被展示。对于“紫外”这是一个尺度参数与一个尺度依赖之自由度族。本文并不展示那一结构，因而自始至终保留标记。

一个约定本身所不回答的问题仍然存在，而它是整个比较方法所倚的问题。若这一借用仅在共享问题形式的层面上被许可，那么为物理问题所发展的策略为何应被预期竟能关涉那关系性的问题？那些策略是为动量空间中的发散而设计的，而关系性场域没有动量空间。回答是：这四种策略中每一种，在它的核心处，都是对这两个问题所共享之抽象特征的一个回应、而非对它们所不共享之物理特殊性的回应。重整化、离散化、延展与涌现各自所面对者，是一种增殖未能在精细化之下减少，而每一种都是一种把这样一种增殖弄得可处理的不同方式：通过把它的代价吸收进从不被直接读出的诸项、通过否认它一个越来越细之贡献的储库、通过驱散那驱动它的集中、或通过溶解它被提出于其上的那个地层本身。这些是对一种发散性增殖之形式的操作，而它们的内容在向任何一个增殖以同样的形式方式发散之领域的翻译中存活。这些策略被预期关涉价值与权力的递归，恰恰到该递归共享那一形式的程度、而不再更远。在那共享的形式给出之处，这一转移随它给出，而本文仅为那形式主张这一转移。

为何这些策略被预期适用。 每一种物理策略都是对一种增殖未能在精细化之下减少之抽象失败的一个回应、而非对动量空间之物理特殊性的回应。它的内容转移到任何一个共享那一抽象失败的领域、并且仅到那共享所延伸的程度。这些策略关涉价值与权力的递归，因为那递归展现同样的诸项未能减少之失败，而这一转移仅为那一形式特征被主张。

把这则说明置于实质性论证之前、而非把它的告诫在它们出现时折入脚注，其理由是：跨学科的借用最常通过一种小的、未被标记之夸大的累积而失败，每一个单独看来都是轻微的，而它们一同说服读者：一个机制已被确立，而其实只有一种相似被绘出。这则说明是对那一累积的一道防线。它一次性地、预先地固定每一个被借来之术语被允许承担的分量，以便随后而来的比较性勘察可以进行，而无需在每一步暂停以重新协商它自己的许可。

§3 作为一个“紫外”问题的递归性价值与权力

本节重构价值与权力的递归而不诉诸任何形式化，并显示它在何种精确意义上有一个“紫外”问题的结构。这一重构有三个部分。它陈述那递归如它在价值理论中、在权力理论中所出现者。它辨认那使这一递归类似于一个紫外发散、而非一个良性无穷级数的特征。而它命名那递归所依赖的条件，即那存在便维持它、那缺席便终止它的条件。

那递归

先考虑价值的递归。一个“某物有价值”的判断是对着一个标准而作出的，而那标准本身是一个价值的承载者，因为采纳它便是持有“以它来判断胜于以另一个来判断”。因此那标准需要它所授予的同一保证，而那保证，若它被供给，是被同一类的一个更进一步之标准所供给。这一序列不自行闭合。这一序列中没有一个标准是“这一序列作为一个整体据以是善的”那个标准，因为那一职分的任何候选者本身都是这一序列的一员、并受制于它的要求。这是辩护之回退在它为价值所采取的特定形式中。它未能终止是一个古老的观察、而在此携带很少的兴趣。它如何未能终止是下一部分的关切。

再考虑权力的递归。一次权力的施行，当它被授权时，算作多于力量的单纯应用，而那授权被一条该施行所满足的规范所授予。那规范轮到自己不是自立的。它有效，因为它被一个有资格订立它的权威所订立，而那权威的资格本身被一条更进一步的规范所构成，而这提起关于它自己有效性的同一问题。那会把一次当下的权力施行奠基于一个最终而自足之源头的授权之链，向后运行而不抵达。曾有人试图通过设定一条被预设、而非被推导的基本规范、或一个其意志无需授权的主权者来遏止它，而这样的装置之复现本身即是那递归被感受为一个需要一个停止装置之困难、而非一个自行停止之序列的证据。

这两个递归是两种材料中的一个递归。在每一个中，相关一类的一个项，一个价值或一次授权，仅被同一类的一个更进一步之项所维持，而那开启序列的要求在每一阶段重新开启。在每一阶段所寻求者是一个根据，而所找到者是另一个本身需要奠基的项。这一递归是那要求随它下降的运动。

为何这一递归类似于一个发散

不是每一个无穷级数都是一个“紫外”问题。一个级数可以没有终点地下降、却仍是无害的，若它的诸项减少得足够快以致整体收敛，因而最后一项的缺席不费任何代价、而尽管有那无穷，那和仍良好定义。价值与权力的回退不属于这一良性之类，而这一差异恰恰是那使物理紫外问题困难的差异。

在物理的情形中，发散是一个发散，因为来自越来越短之距离的贡献不衰减。若那些短距离贡献快速地减少，对它们的积分便会收敛、而便不会有紫外问题；恰恰是它们未能减少使那和病态定义。价值与权力之递归中的类似特征是：对一个根据的要求不随深度减弱。第十阶的价值标准恰如第一阶的估价一般需要保证，而第一百次授权恰如它被召来授权的那个行动一般可被质疑。这一递归不趋近一个要求在其上耗尽自己的极限。它在每一层级不减地再生产那要求，而正是这一不减的再生产、而非无穷这一单纯事实，赋予这一递归它在被标记之意义上的“紫外”品格。分析在递归的精细化之下变得越发不稳定，因为每一次精细化，远非了结那问题，都在下一层级以全强度重述它。

这一对应应被审慎地陈述，因为这两个情形在“失败在于什么”上有别。在物理的情形中，那病态定义是可加的，一个对非消失大小之贡献的和未能收敛到一个有限的值。在价值与权力的递归中没有求和，而那失败是终止的失败，一个诸根据的序列，每一个都需要一个根据，未能抵达一个会闭合它的成员。这两个情形所共享者是那驱动两个失败的特征，即相关诸项拒绝在精细化之下减少，因而没有一个有限的截断是适足的、而那下降必须没有终点地被继续。这一类比为这一共享的特征、即诸项在精细化之下的非消失被主张，而它既不为物理发散的可加形式、也不为价值一侧的任何求和被主张。正是在这一确切而受限的意义上，这一递归有一个“紫外”品格。

“紫外”品格。 价值与权力的递归类似于一个紫外发散、而非一个收敛的无穷级数，因为对一个根据的要求未能随深度减少。每一层级以不减的强度再生产它上面那一层级的要求，因而更细的分辨率使分析失稳、而非了结它。

那递归的燃料

若这一递归被一个要求的不减再生产所维持，那么这一递归依赖于无论什么再生产那要求者，而命名那便是命名整个结构所倚的条件。这一条件可如下陈述。驱动这一递归者，是它之中每一个项都被作为一个可争议之主张而提出。一个价值标准，作为一个根据被提供，是作为某种本可以别样、划出一条边界并排除诸替代、并因此向任何挑战者呈现一个反主张可被压向之表面之物而被提供的。一次授权，同样地，是一个本可被否认的主张，一条另一者可争议之线的划出。正因为每一个项都是一个可争议之主张，每一个项才招来那重新开启要求的反主张，而正是那反主张的无尽可得性使这一递归保持运动。这一递归的燃料是那主张的可争议性。

这一辨认有一个统领本文建设性一半的后果。一个被可争议性所维持的递归将持续，只要它的诸项保有可争议之主张的形式，而它将持续不再更久。若那下降抵达一个层级，在其上所找到者不再有一个可争议之主张的形式、在其上没有边界可压向、没有线可否认，那么那反主张便会找不到表面，那要求便不会被再生产，而那递归便会失去它的燃料。这样一个层级是否存在、以及在那里会找到什么，不是那诊断性的重构所能回答的问题，因为它是一个关于该关系性场域之深层结构、而非关于那递归之表面形式的问题。它是那个与物理策略的比较被设计来锐化的问题，因为这若干策略恰恰在它们所设想躺在那下降之底部者上有别，而正是向那一比较论证如今转去。

§4 形式化的诸层

物理策略的比较预设一个本文到此为止一直留为隐含的在先问题，而本节把它弄得显明，因为对它的回答统领哪些策略对价值与权力的递归甚至是有资格的。那问题是：什么首先决定一个形式策略的选择。此处所取的立场是：那选择依赖于所研究对象的本体论。一个形式策略对它是一个策略所为者负责，而在一个领域中被发展的策略仅当那两个领域共享它们对象的相关特征时才转移到另一个领域。价值与权力的递归属于对社会事实的研究，而任何形式策略对它的资格由社会事实被取为什么所了结。

那对象的本体论

社会探究的对象是社会事实，而社会事实是若干类的。依一个标准的说明，它们包括事件、实体、关系、属性与状态，而国际政治的学科，在别者之中，把这些取为它主题据以被分析的诸单位。一个价值与权力的理论恰恰作用于这些类的社会事实，作用于一个判断的事件、作出判断的实体、那判断落于其中的关系、它所归予的属性、以及它所设立或扰动的状态。

本文所延续的纲领并不否认这些社会事实的实在。它对既有本体论的单一修改是把关系抬升到一个相对优先的地位，因而实体、属性、状态与事件被取为在关系之内被生成并被稳定、而非先于它被给定。这一优先是相对的、而非排他的。这一主张不是唯有关系存在、而其他社会事实是幻象，那会使这一理论无法谈论社会探究所关切的那些事件与实体本身。这一主张是：其他社会事实在关系之内有它们的存在，一个实体被那些安置它的关系构成为它之所是的那个实体，而分析因而从关系开始、并通过它抵达其他事实。有了这个，那对象的本体论被固定，而表象的问题可被提出。

关系的相对优先。 社会事实包含事件、实体、关系、属性与状态。这一纲领把关系取为有相对的本体论优先，因而实体、属性、状态与事件在关系之内被生成并被稳定。这一优先是相对的、而非排他的：其他社会事实是实在的，而它们在关系之内有它们的存在。

关系性结构的表象

一旦对象被固定为一个以关系为其基底的社会事实之场，下一个问题便不是那场如何演化，那是一个动力学的问题，而是它的结构如何被表象。这一步易于被略过，而略过它是一个频繁的混淆之源，因为一个未曾固定它表象的理论会不被察觉地偷带一个进来。一个关系性本体论要求一种能够同时表达若干事物的形式语言：多于两个之诸项之间的关系、连接之模式的诸变化、什么连接到什么的局部结构、以及从旧结构之组合生成新结构。

在可得的诸候选之中，一种特定类的组合结构比它的诸对手更充分地满足这些要求，即那个在量子几何的物理学中被称为自旋泡沫的结构。一个自旋泡沫依次回应这四个要求。它的顶点编码同时在若干项之间的相互作用，满足多项关系的要求。它的重连编码哪些项被连接之中的诸变化，满足连接之模式之诸变化的要求。它的组合与局部品格展现什么连接到什么的实际模式，满足局部结构的要求。而它的边，把诸相互作用胶合成一个动力学随后可作用其上的延展复形，满足新结构从旧结构之组合被生成的要求。它相对于一个图的优势是：一个图仅携带成对之间的关系，而所论的社会关系典型地是多者之间的关系，一个联盟、一个市场、一个家户，无一者化约为一组成对的纽带。它相对于一个纯范畴表象的优势是：一个自旋泡沫是组合的与局部的，展现连接的实际模式、而非仅仅诸结构之间的抽象关系。而它对当下纲领的决定性优势是：它立于离散与连续的相会点，同时充当一个组合对象、以及一个场论描述与一个动力系统描述两者都能被定义于其上的对象。

被指派给自旋泡沫的角色必须被审慎地陈述，而它是一个表象的角色、而无更多。一个自旋泡沫是一个中间的表象，一层被插于关系性本体论与将作用其上之动力学之间的形式结构。它是表象关系性结构这一职分的一个候选，因方才所给出的诸理由被选择，而它作为最适足的候选、而非作为唯一的候选被提供。论证中没有任何东西要求一个自旋泡沫是社会实在的一幅图画，而这一主张被明确保留。所主张者是：一个关系性本体论需要一个有所命名之四个特征的表象，而自旋泡沫供给它们。

结构的动力学

一个表象是静态的，而结构不是。自旋泡沫表象一个关系性结构；它不说明这样一个结构如何来成为、它如何溶解、或它如何重组。为此一个动力学被要求，而那动力学是一个群场论、连同动力系统理论的诸资源所适于供给者。一个群场论取自旋泡沫所表象的组合结构、并给它们一个生成性的动力学，一条结构据以形成、相互作用、并让位于他者的法则。在此处所提出的分层中，自旋泡沫与群场论占据相邻的职分，一个表象结构、另一个使它运动，而它们一同构成关系性场域的表象与动力学。

正是在动力学的层级上、并具体地在确定结构从一个不携带任何结构之状态中沉淀出来的那一转变中，价值与权力的递归抵达它的终点，其理由在映射中被给出。结构从之涌现的生成性层级，是以可争议之主张来描述不再适用之下的那一层级，而因此是动力学、而非表象或它之上的诸层，携带这一递归的解决。

历史与循环的诸判准

在动力学之上、并依赖于它，立着两个更进一步的职分，它们常被误认为动力学本身的部分、而其实与它恰当地相区别。第一是判准的职分。一旦结构已形成并进入循环，一个动力学本身所不回答的问题便升起，即一个循环在它的历程上积累什么的问题。一个关系性循环可能返回它的出发点，在它自身之内生成了一份剩余、或抽取了一份、或自始至终依赖于一份来自外部的供给，而把这些区分开来便是把一个判准应用于动力学所产生者。这一职分的工具是完整幺正性（holonomy），连同它所属的几何相位，而它的工作是刻画一个已演化结构的历史依赖性与循环品质。一个完整幺正性，在那从绕闭合路径之输运的几何借来的意义上，是一个系统在被绕一个回路载运一次之后未能返回它确切之起始条件所据的量，因而一个循环可以在它的参数上闭合、而留下一份记录所行路径的余数。

这一判准在一个严格的意义上依赖于动力学，而这一依赖值得被陈述，因为它防止一个常见的错误。一个完整幺正性仅在有一个构型之空间、一个循环可被绕之载运之处被定义，而这样一个空间本身是动力学的一个产物，仅在结构已涌现之后出现。因此这一判准在涌现的层级之下没有应用，而它对生成性层级什么都不说。它是对动力学已然产生者的一个刻画，而它预设那份产生。这一判准不为动力学奠基；它从动力学的输出读出一个性质。

观察的诸策略

在动力学之上的第二个职分是横跨诸尺度之观察的职分。一个关系性结构同时存在于许多尺度，从两个人之间的关系，经由家户、社群、组织与国家，到国际系统，而同一个结构在不同尺度被观察呈现不同的特征。观察的职分是要说，随描述的尺度改变，哪些特征要被保留、哪些要被搁置、以及有效描述如何转变。这一职分被两个互补的策略所填充，而它们的互补重要到足以被精确地陈述，因为它们易于被混同。

第一个策略处理空间尺度的层级。它是重整化群的策略，它描述一个有效描述如何随观察的尺度改变而流动、哪些特征随人从更小的尺度上升到更大的而保持相关、哪些成为不相关并被冲刷掉。被带上关系性场域，它是关于一个适足于二元的描述如何转变为一个适足于制度或社会的描述、以及什么在那转变中存活的说明。

第二个策略处理时间尺度的层级，而它与第一个相区别。它是快慢动力学的策略，把一个系统分解为快速改变的变量与缓慢改变的变量，以便那些快变量可被当作到达它们的局部平衡、而那些慢变量携带长期的演化。许多关系形式恰有这一时间的分层。事件、掠过的情绪、片刻的涨落是快的，而制度、习俗、伦理秉性、安顿下来的信任是慢的，而快慢分解说哪些过程可被当作快速平衡、哪些要被持为缓慢变化、以及哪些慢变量统领关系系统的长期吸引子。

观察的这两个策略共享一个共同的逻辑，即以一个对更粗者的有效说明取代更细者的消除，而它们在“那细者是什么”上有别。重整化群消除短尺度的自由度、并把一个理论沿空间尺度的轴流动。快慢分解消除快的自由度、并把一条轨迹沿时间的轴约化。一个把描述横跨大小的诸尺度流动、另一个把描述横跨持续的诸尺度约化，而一个在大小与持续两者上都分层的关系性场域需要两者。重整化群回应空间尺度的层级；快慢分解回应时间尺度的层级。

被装配的架构

这五个职分如今可被一并陈列，而它们之间的依赖被弄得显明，因为那依赖是那把这一架构与一份各为自己而借来之形式化的单纯清单相区别者。

形式化的五层。

层	问题	候选形式化
本体论	什么被研究	社会事实，关系有相对优先
表象	结构如何被表达	自旋泡沫，或其他组合结构
动力学	结构如何形成与演化	群场论，动力系统
判准	历史与循环如何被刻画	完整幺正性，几何相位
观察	多尺度行为如何被读	重整化群，快慢动力学

这五层并不作为平等者并排而立。本体论、表象与动力学构成一条生成之链，其中每一职分预设它之前那一个：表象预设一个被固定的本体论，而动力学预设一个供它作用的表象。其余两个职分，即判准与观察，不躺在这条链的更下方。它们立于动力学的输出之上，作为两类被应用于动力学已然生成之结构的刻画。完整幺正性刻画一个已演化的循环所积累者；观察策略刻画一个已演化的结构如何横跨大小与持续的诸尺度显现。任一者都不是奠基性的，而任一者都无法被载往动力学之下，因为每一个都预设唯有动力学才产生的那些涌现的构型。

这一安排溶解一个横贯本纲领而复现的问题，即哪一个形式化更为根本的问题，群场论抑或重整化群、完整幺正性抑或涌现。这个问题是被畸形提出的，因为这些形式化回答不同的问题、并占据不同的职分。什么存在，是本体论之事；它如何被表象，是自旋泡沫之事；它如何演化，是群场论之事；它的诸循环如何被判断，是完整幺正性之事；它如何横跨尺度与横跨时间被观察，是重整化群与快慢分解之事。这些形式化不为一个单一的职分竞争。它们填充一个有序的、不同职分之集合，而价值与权力的递归在动力学的职分处被解决，在那里生成那递归的描述给出，而其上的诸职分供给对动力学所产生者的诸刻画。

一言应被给予这些形式化中两个的双重出现，因为它否则会读作一个不一致。群场论与重整化群在本文中各出现两次，一次作为对紫外问题的一个回应，在下文的历史回应之中被勘察，一次作为架构一个层的候选形式化，群场论在动力学的职分处、重整化群在观察的职分处。这两次出现是一致的，而一个为另一个奠基。使一个框架有资格填充一个层者，是它对那递归所取的态度。群场论适于动力学的职分，因为它对紫外问题的回应是涌现的策略，即那解决价值与权力之递归的策略；重整化群适于观察的职分，因为它的回应是关于一个有效描述如何横跨诸尺度运行的说明。因此，随后而来的回应之勘察是对这一架构的一个辩护、而非一个对它的离题，是那显示为何每一个形式化归属于它已被指派之职分的地方。

正是这一架构、以及坐落于它生成性核心处的涌现策略，随后一节中物理回应的勘察意在给予信息者。

§5 对紫外问题的历史回应

理论物理学已不止一次地遇见紫外问题，而它为遇见它所建的若干框架共享一个在那问题中的共同起源，无论它们后来如何分歧。本节描述其中四个。它把它们描述为对发散的诸态度、而非诸批数学，因为本文向它们所要求者是它们的策略、而非它们的机器，也因为策略是那能被带往一个没有动量积分可转写之领域者。本节自始至终，物理术语被平实地写出、不加引号，因为在此它们被用于物理学本身，在那里它们是字面的。引号在随后一节返回，在那里这些策略被带上关系性场域。

这四者共同的问题可被以一句话陈述。当一个场论中的一个量通过对来自所有距离尺度之贡献求和而被计算时，来自最短距离的贡献未能减少，而那和发散。这四个框架中的每一个都是一种确保理论在这一失败面前仍良好定义的方式，而每一个都体现一种关于“世界的短距离结构必须是什么，那发散才能被控制”的不同构想。

重整化：吸收

第一个、也是最保守的回应把发散留在原处并吸收它。它的洞见是：在起初被写入理论的诸参数，即裸质量与裸电荷，不是任何人所测量的量。所测量者总是那有效量，即那参数如它在短距离贡献已被包括之后所呈现者，而这一有效量即便在那裸的量不是有限之时也是有限的。重整化允许那些裸参数携带发散，把那无穷的短距离贡献吸收进从不可观察的量，因而一切可观察者都得出有限。它进一步揭示：那些有效参数随观察的尺度运行、而非保持恒定，随人探测理论所在的距离改变而取不同的值，而那运行的法则是重整化群。此处的态度是一种与无穷的共处。那发散不被击败；它被重新安置进不可观察者、并由此被弄得无害，而所余的物理学是随尺度变化之量的物理学。

动力学三角剖分：离散化

第二个回应否认发散它的源头。发散仅在最短距离聚集，因为理论允许距离变得任意地短，而它允许这，因为它把空间和时间取为一个连续统，无限可分。动力学三角剖分在最小的尺度上拒绝连续统。它从固定大小的初等单纯形建造时空，一个最小的三角形或四面体，在它之下没有进一步的划分，而它通过依尊重因果次序的规则胶合这些建造块来装配几何。因为没有距离短于单纯形的边，便没有一个越来越短之贡献的储库供发散汲取，而那在连续统中发散的和在被离散化的结构上成为有限。此处的态度是一种地板。朝向更短距离的递归被一个最小的元素、即单纯形的公共边长所停止。一个限定归属此处，而当这一策略被带过去时它将要紧。在这一进路的标准实践中，那边长充当一个正规子，一个被引入以使那和有限、然后在一个连续统极限中被送向零的最短距离，因而那离散性是一个计算的装置、而非一个关于空间之构成的主张。一个更强的读法是可得的，依它那最小的元素是世界的一个真正特征、而空间在根底上是有颗粒的，而这两个读法是不同的。这一策略对当下论证所贡献者是两者共同的形式，即那下降被一个最小的元素所遏止，无论那元素是一个待被移除的正规子、抑或一个待被肯定的地板。

弦论：延展

第三个回应通过拒绝点来溶解发散。在一个点粒子的理论中，一次相互作用发生于一个单一的位置，一个尖锐的顶点，在其上该过程的整个能量被集中进一个消失大小的区域，而正是这一向一个点的集中产生最坏的短距离发散。弦论以一个延展的对象、一根一维的弦取代点粒子，而随这一取代，相互作用的尖锐顶点被抹成一个由弦在它移动、连接与分离时所描出的光滑曲面。那曾集中于一个点的能量被沿那延展的对象、并在它众多的振动模式之间分布，而那对点发散的短距离行为对弦被软化。此处的态度是一种延展。发散通过给那些基本实体一种内部的延展与内部的自由度而被缓解，因而没有一次相互作用竟是真正点状的、而那导致发散的集中从不发生。

群场论：前几何涌现

第四个回应移除那发散被对着它定义的背景。紫外问题预设一个时空连续统，其中短距离存在以供被求和，而到此为止的三个回应各自把那连续统取为给定、并在它之内处置发散，无论是通过吸收它、给它铺地板、还是软化它。群场论不把连续统取为给定。它的基本自由度是一个抽象群流形上的场，不携带它们自己的任何空间或几何解释，而连续的时空，与其被假定，被预期从这些前几何量子的集体行为中通过一个相变、一个凝聚而涌现，在那凝聚中大量的基本元素进入一个共同的状态、而一个光滑的几何作为它们的集体描述出现。依这一说明，短距离连续统，紫外问题被提出于其中者，本身是一个被导出的、涌现的东西，而“在比那涌现之几何所支持的任何距离都更短之处躺着什么”这一问题不出现，因为在那涌现的层级之下没有几何、也根本没有距离。此处的态度是一种涌现。发散连同那背景一同被溶解，通过把那背景本身定位为一个更根本之层级的产物，那层级没有短距离问题，因为它没有距离。

四种策略之比较

这四个回应可被作为对一个发散的四种态度而并置在一起，而这一比较是本文从物理学转向它自己主题所据的枢轴。

对紫外问题的四种回应。

重整化 把发散吸收进不可观察的参数、并让那些可观察的随尺度运行。态度：共处。

动力学三角剖分 把时空离散化为最小的单纯形，在它们之下没有划分存在。态度：一种地板。

弦论 把点延展为一个一维的对象、并把相互作用分布于它的诸模式。态度：延展。

群场论 使几何从一个前几何的层级涌现，因而短距离连续统是被导出、而非被给定的。态度：涌现。

四者中有两个提议朝向更短距离的下降被停止、而两个提议它不被停止。动力学三角剖分把它对着一个不可分元素的地板停止。群场论以一种不同的方式把它停止，在距离这一概念本身不再适用之下的那一层级，因为几何本身是涌现的、而在它之下没有任何几何之物。这两种停止下降的方式是不同的，而地板与适用性之界限之间的区分将证明是那决定性的区分，当这些策略被带过去时，因为它是一个最小之物与一类描述之停止之间的差异。重整化与弦论，与此相对，根本不停止下降。重整化让它运行并吸收它的代价；弦论让它运行并通过延展软化它的效果。有了这四种态度在手，本文转向价值与权力的递归、并就每一种追问它在那里会意味着什么。

§6 把诸策略映射到生成性关系存在之上

这四种策略如今被带上价值与权力的递归。引号返回，因为从这一点起物理术语修饰关系性场域、并命名相似而非事实。向每一种策略所问的问题是相同的。若价值与权力的递归被以这一策略所提议的方式应对，这一策略就在那下降之底部价值与权力的本性断言什么，而那断言是否是一种生成性关系的说明可以一致地作出的？这四个回答有别，而这一差异是有教益的，因为每一种策略，被转置，成为一个关于一个价值或一次授权最终是什么的不同论旨。

吸收：被穿衣的主张

被带上递归，吸收的策略产出如下论旨。诸辩护的递归被承认为无尽、而非被停止，而在任何一次实际的估价中所断言者是那有效价值，即那估价如它在整个诸根据之递归已被折入它之后所立者，以取代那会显式携带无穷之更高根据序列的裸价值。依这一读法，一个在使用中的价值总是一个“被重整化的”价值，一个把它更高辩护的整个序列作为一件衣裳来穿、而非作为一架外部的梯子来服从者，而那会重新开启递归的反主张被重新构想为一个已被吸收进那有效项的修正。权力容许同一读法。一次在效力中的授权是一次“被穿衣的”授权，把它无穷的诸正当化之回退作为一份在那回退保持未被穿越时运作的有效正当性携带于自身之内。

这一论旨是融贯的、并有一个真正的吸引力，因为它说明那平白的事实，即诸估价与诸授权确实在实践中运作、而它们看似会要求的那个无穷序列未被穿越。它对一种生成性说明的限度是：它以让那递归完好而仅仅不被观察为代价购得这。那无穷序列保持，仍不减，而那有效价值仅在那序列的折入本身良好定义时才良好定义，而这要求某种回应那物理尺度参数、那运行被沿之组织者。价值与权力的递归展现没有这一类的显然参数，而缺一个，那吸收便保持为一个被剥去它机制的隐喻。吸收描述有效价值如何运作；就它自己而言它把递归留在原处，而一种寻求一个真正终点的生成性说明将在它之中找到一个对症状的描述、以取代一个对成因的处置。

离散化：最小的关系

被带上递归，离散化的策略产出如下论旨：诸辩护的下降抵达一个地板，一个最小的关系性元素，在它之下对一个更进一步之根据的要求没有应用。依这一论旨，会有一个价值或授权的原子，一个不可分的关系性单位，它本身不被任何更基本者所估价或所授权，而对着它那递归遏止，如那物理下降对着最小的单纯形遏止。这一论旨有供给一个确定终点之优点，而它对应于奠基性思想中的一个熟悉之举，即设定一条基本规范或一个无需被奠基便奠基的自明价值。

它对一种生成性说明的困难是：这一类的地板重新引入恰恰是这一纲领被建来拒绝者，即一个先于关系的奠基性元素，一个其价值或权威内在于它、而非在关系性场域之内被构成的原子。把递归对着一个最小的被估价之物遏止，便是让步“价值在根底上有一个独立于关系而占有它的承载者”，而这一让步与“价值在关系之内升起、而非驻留于一个事物之中”这一主张不相容。因此，离散化的策略，就它自己取用，与一种生成性关系本体论相处得糟。它将重新出现，被转化，当它被看出与涌现的策略汇合时，但作为一个内在被估价之原子的裸地板，它不是这一纲领所能采纳的论旨。

延展：非点的主张

被带上递归，延展的策略产出一个不寻常有趣的论旨。递归之最坏者，依这一类比，是被把一个价值或一次授权当作一个点、一个尖锐而孤立的主张、集中于辩护之次序内的一个单一位置、反主张可对着它以全力被压所产生的。延展提议那主张竟从不是点状的。一个价值，依这一读法，是一个有内部结构与内部自由度的延展关系性对象，而那把自己呈现为一个单一可争议主张者，是一个分布式关系性形式的外部方面，其内容散布于它的内部关系、而非集中于一个可争议之点。那反主张，遇见一个延展的对象、而非一个点，被驱散于那对象的内部自由度、而非集中对着一个单一的表面，而那给递归以燃料的尖锐对立被软化。

这一论旨对一种生成性说明是投契的，而比头两个更甚，因为它把一个价值的内容定位于一个分布式的关系性结构、而非一个孤立的承载者，这接近于这一纲领从一开始所维持者。它的限度是：软化不是终止。一个延展的主张更难在一个点被争议，但它终究仍是一个主张，而一个足够分布式的反主张仍能遇见它。延展缓解递归而不结束它，而它通过一个设想、即价值是内部延展的关系性对象为之，而这一设想是一种生成性说明能接受、却本身不抵达那递归会失去它燃料之层级者。那么，延展是一个部分而受欢迎的措施，而它将在下文被保留为那确实抵达那一层级之策略的一个补充、而非作为那首要的回应。

涌现：可争议主张的停止

被带上递归，涌现的策略产出如下论旨：诸辩护的下降终止于以可争议之主张来描述关系性场域不再适用之下的那一层级。这一递归，如诊断性的重构所显示，被那主张的可争议性给以燃料，而它恰恰持续，只要它的诸项保有可争议之主张的形式。涌现的策略提议：在可争议主张的层级之下躺着一个更根本的关系性层级，一个纯差异与关系的生成性层级，先于任何有界主张的构成，在其上没有边界可压向、没有线可否认。在那一层级上，反主张找不到表面，对一个根据的要求不被再生产，而那递归失去它的燃料。那下降不撞上一个最小的被估价之物。它通过“价值”、在支持一个可争议主张的意义上，根本是正确描述之下的那一层级，进入一个有界的价值与权威之主张本身从之涌现的生成性层级。

依涌现的策略，价值与权力的递归在以可争议之主张来描述不再成立之处抵达它的终点。那终点是一个适用性之界限、而非一个奠基性事物之地板。那递归的燃料是那主张的可争议性；在主张的层级之下没有可争议性、因而没有燃料、因而没有递归。

这一论旨是一种生成性关系说明可以一致采纳的那一个，而应清楚地看出为何。它终止那递归，而吸收与延展不。它终止它而不设定一个先于关系的奠基性元素，而离散化无法回避那设定。而它以这一纲领已在另一个领域已然工作出来的方式终止它，因为关于有界的语言与符号结构如何从一个前结构的生成性层级、通过一次对称之破缺与一次凝聚而涌现的说明，平行于关于可争议之结构如何从一个尚无可争议之结构存在的层级升起的说明。价值与权力之递归在生成性层级的终止，在价值的层面上遵循同一个形式，即结构从生成性场域的涌现在形式的层面上所遵循者。

正是在这一点上离散化的策略返回，被转化。那涌现的、前几何的说明有一个离散的方面、亦有一个连续的方面，因为那给出、在一个极限中、一个光滑涌现结构的同一根本层级，在它的组合描述中是从初等的关系性单纯形被建造的。离散化在孤立中所设定、而一种生成性说明不能作为一个内在被估价之原子的地板接受的那个地板，在此以一个被改变的职分重新出现，作为那生成性层级本身的组合颗粒，在任何被估价之结构的涌现之下。如此被理解的最小元素，是一个先于价值的最小关系性元素、而非一个最小的被估价之物，而在这一被转化的意义上，离散化的策略被吸收进涌现的策略、而非与它对立。那免除较早反对的区分应被弄得显明，因为一个颗粒与一个地板可能显得相同。生成性说明所拒绝者，在拒绝离散化时，是一个立于先于关系的奠基性元素，一个其价值或权威在它自身之中、而非通过那些安置它的关系而属于它的原子。那被转化的策略所容许者是关系织物的一个最小元素，一个内在于关系、并不拥有它自己的任何价值或权威的颗粒，被估价的结构仅通过涌现在它之上出现。第一个是一个关系会从之被建造的、关系之外的元素；第二个是一个关系之内的、从之尚无价值随之而来的最小接合。那颗粒彻头彻尾是关系性的，而正因此容许它并不复原这一纲领所搁置的奠基性原子。这一汇合平行于那底层物理学所展现的一个，其中涌现的与离散的是一个单一前几何说明的两个方面、而非两个相互竞争的提议。

映射的结果

这一映射产出一个确定的结果，而以这一纲领所要求的确切形式陈述它是重要的，既不更弱、也不更强。一种生成性关系的价值与权力之说明与涌现策略最为一致，被延展策略所补充、并在它被转化的、前价值的意义上纳入离散化策略。这是一个一致性与最佳契合的主张。它不是“一种生成性的价值说明是一个群场论”、或“价值与权力的递归是一个紫外发散”这一主张。那物理框架供给一个策略的形式，即连同那提出一个问题之背景一同溶解那问题的态度，而生成性说明采纳那形式，因为那形式匹配它自己在先的承诺。生成性说明必须为它自己供给、而没有任何物理框架能为它供给者，是那内容：那生成性层级是什么，在价值的层面上；什么从它涌现，并通过何种关系性过程；以及什么把那些涌现的有界结构区分为承载这一价值、而非另一价值。物理策略告诉那说明去何处寻找递归的终点。它不告诉那说明它将在那里找到什么。那是上文所陈述之架构的负担，它的生成性职分是那解决所落之处，也是随后诸敞开问题的负担。

§7 本文的地位及其触及

一个由类比所建的论证欠它的读者一个平白的陈述，关于它是什么、它希望使什么成为可知、它可能有朝一日如何触及经验领域、以及它的诸界限落于何处。本节供给那一陈述。它被置于诸敞开问题之前，因为诸敞开问题是它的后果，而它把那些较早诸节顺带作出的、关于本文自己地位的诸主张汇集到一处。

论证的地位

本文是一项方法论与架构的研究，而它既不是一个价值的物理模型、也不是一个价值的完成了的形式化。它的工作在问题的层面上、在诸工具之安排的层面上被做，比任何特定的机制高一层。它取一个若干领域分别遇见的递归、显示那递归有一个可辨识的形式形状、勘察一门成熟学科为那形状之问题所发展的策略、确定一种生成性关系说明可以一致采纳哪一种策略、并提出一个架构，其中该说明的形式工具占据不同而有序的层级。这些之中每一个都是一个形式的贡献、而非内容的贡献。本文主张已为一个问题找到正确的形状、以及为会处置它的诸工具找到一个可辩护的次序，而它别无更多主张。以它所借之学科的词汇，本文提供一个策略与一个组织的方案，而它把任何特定模型的构造留给它的架构意在引导的工作。

知识的诸对象

本文希望使三样东西成为可知，而它们对应于它的三个贡献。第一是：价值与权力的无界回退，它曾被当作对根基理论的一次挫败，反而有一个容许有原则之回应的问题之结构，因而那回退成为一个待被工作之物、而非一堵工作停止之墙。第二是：紫外问题的若干策略，当被置于价值与权力的递归之旁时，把它们自己分拣为一种生成性说明能采纳者与它不能采纳者，而那分拣被该说明在先的承诺、而非被偏好所决定，因而策略的选择被显示为被约束的、而非自由的。第三是：这一纲领所汇集的诸形式工具，它们曾显得为一个单一的解释职分竞争，其实归属于一个单一秩序的不同层级，因而它们表面的对立溶解为一种依赖。使这三样东西成为可知，便是把一簇困难转换为一个被结构化的问题之场，而那一转换是本文作为知识所提供者。

通向经验领域的路径

本文自始至终是形式的，而它伸入经验领域因而是间接而被中介的，而值得说它有朝一日可能被以何种路径作出这样一次伸手，哪怕只是为标记那路径存在、而本文如今并不走它。此处所提出的架构把确定的职分指派给确定的层级，而每一个职分，一旦一个具体的模型实现了它，便会携带它自己的经验承诺。

在涌现的层级上，一个被实现的模型会预测：有界的价值与权威之结构通过某种回应一次转变者而形成，即确定结构从一个不携带任何结构之状态的一次相对突然之沉淀，而非仅通过渐进的累积。凡一条规范、一个制度、或一份共享估价的形成能被历时地观察之处，这一架构把注意引向那形成是否展现一次转变的诸标记，一个门槛、一次迅速的排序、一种对门槛附近之条件的敏感，而这是历史与发展的证据在原则上能对之言说的一件事。

在有效结构的层级上，那循环判准把带着一份内在生成之剩余返回的关系性循环，与那些带着耗竭返回、或从外部被维持者区分开来。一个被实现的模型会使这一区分可操作，而它会把经验注意引向一个给定的制度或关系是否再生产它自己延续的诸条件、抑或把它们抽干，一个对公共领域、对组织、对诸关系的研究已以其他措辞提问、并能以这些措辞提问的问题。

在尺度的层级上，这一架构持有：一个适足于一个尺度的描述在另一个尺度重新出现、被转变，而那转变是合法则的、而非任意的。一个被实现的模型会预测一个关系形式在二元之尺度上的描述与它在制度或社会之尺度上的描述之间的特定关系，而比较的与多尺度的社会证据能在原则上检验这样的关系是否成立。

这些是可能接触的方向、而非结果。它们中每一个都预设一个本文不供给的具体模型，而每一个都仅在那被推迟到诸敞开问题的价值问题已被以某种确定的方式回答之后才会获得经验内容。因此，本文与经验的关系是一个框架与它有朝一日可能持有之图画的关系。它指定经验承诺会在何处进入、以及它们会服务什么职分，而它把它们的供给留给一个被安顿的架构意在使之成为可能的建设性工作。

论证的诸界限

诸界限如诸贡献一般确定，而诚实要求它们被以同等的平白陈述。本文的中心工具是一个类比，而那类比是一个问题形式的类比、而非一个机制的类比；标记这一点的引号未曾退休，而直到关系性场域被显示为拥有一个真正的尺度与真正的尺度依赖之自由度，它们无法退休。本文并不形式化价值；它定位一个形式化将不得不在何处开始、并谢绝固定价值所采取的形式，尤其抵抗“价值必须是物理模型上的一个可观察量”这一假定。本文并不把生成性层级承诺给一个确定的数学形式，把它在一个作用量、一条演化规则、一个范畴动力学与其他可能性之间持敞，而它因此不交付一个完整构造会立于其上的基底。本文勘察一门学科对一个问题的策略，而它的勘察仅如那历史完整一般完整，因而一个更进一步的策略，在物理学之内或之外，可能改变那选项之场。这些界限不是执行的失败。它们是一项研究的确切边界，那研究已选择确立一个问题的形式、而非强推一个过早的解决，而精确地标记它们是那研究所提供者的一部分。

§8 诸敞开问题

本文已主张抵达一个问题的正确形式、以及一个用以处置它的可辩护之架构，而它已审慎地仅主张那。它的诸省略如它的诸主张一般重要要被陈述，而本节陈述它们，因为一个架构仅当它之内的诸空位如那些被填的部分一般被平白地标记时才是诚实的。

第一个敞开问题是那方法论说明所预期者。本文所转轴其上的类比保持为一个问题形式的类比、而非一个机制的类比。“紫外”周围的引号未曾退休，因为那会使它们退休的结构，即关系性场域本身中的一个真正尺度参数与一个尺度依赖之自由度族，未曾被展示。这样一个结构能否被展示、以及关系性场域是否拥有任何回应有效描述被沿之运行之物理尺度者，是未被了结的。直到它被了结，那些被转置的策略保持为在它们形式中被借来的策略，而本文对它们的使用是一个形式的使用、而非一个机制的运作。

第二个敞开问题关切价值在生成性层级上的内容，而它是那映射所推迟、而架构所承继的问题。涌现的策略把递归的终点定位于一个有界价值从之涌现的生成性层级，但它不说价值在那一层级是什么、也不说有界价值通过何种关系性过程从它沉淀、也不说什么把那些涌现的结构区分为承载这一价值、而非另一价值。这一纲领有理由抵抗这一问题呈现自己的第一个形式，即那会要求一个在物理测量之模型上的价值可观察量的形式，因为没有显然的根据可设想价值在关系性场域中根本采取一个可观察量的形式。它可能采取一个生成性动力学必须满足之约束的形式、或一个统领哪些结构涌现之选择原则的形式、或一个把关系性场域之诸相位区分之序参量的形式、或某种物理词汇没有现成名字者的形式。这些之中哪一个，若有，是价值的正确形式，在此不被决定，而过早地决定它、通过因那类比使可观察量可得便固定于它，便会让那被借来的词汇了结一个属于主题的问题。

第三个敞开问题关切架构的基底。生成性层级已被中性地命名，作为一个关系性生成动力学，而本文已谢绝把它承诺给一个特定的数学形式，尤其给一个作用量。这一克制留敞“那生成性动力学采取什么形式”这一问题。它是否容许一个变分表述、抑或被更好地呈现为一条演化规则、一个范畴动力学、或另一类的一个生成性过程，是未被确立的，而这些之间的选择不是一个记号之事、而是那生成性层级被主张是什么之事。这一纲领自己的群场论工作供给一个候选的实现，但此处所提出的架构不被束缚于那候选、而会在它被另一个取代时存活。那基底的形式是敞开的，而这一架构被有意地建来以不预判它。

第四个敞开问题是：涌现的策略，既已被显示为一种生成性说明能一致采纳的那一个，是否是它必须采纳的那一个。本文已为最佳契合、为一致性论证，而它已显示那些替代要么未能终止递归、要么通过设定一个先于关系的根基来终止它。它未曾显示没有一个第五策略、在紫外问题之物理学到此为止所产生的四个之外、能以某种更进一步的方式终止递归。策略之勘察是对一门特定学科对一个特定问题所作诸回应的一次勘察，而它的完整是那历史的完整、而非一个原则上的完整。一个更好的策略仍可能被找到，在物理学中或之外，是本文不排除的一个可能。

这四个空位不是待被以那产生了架构的同一类进一步劳作所填补的偶然空隙。它们是那架构与它尚未包含者相遇之点，而它们中每一个都是一个探究的方向、而非当下探究的一个缺陷。本文的贡献是已建造那些是正确要问之问题所在其中的框架、并已显示它们是关于一种生成性价值理论之内容的问题、而非关于那问题之形式（它被了结）、或关于诸工具之安排（它被提出）的问题。

§9 结论

论证从一个递归出发，价值理论、权力理论与知识理论各自以它们自己的词汇遇见它、却不曾认出它为一个东西，即那个每一个根据都需要一个根据、而对一个根基的要求没有终点地下降的递归。它已试图显示这一递归有一个确定的形式，而那形式是一门成熟学科已反复面对者。这一递归不仅未能终止。它随它下降而变得越不可处理，在每一层级不减地再生产那开启它的要求，而这一不减的再生产赋予它物理学称为一个紫外问题者的结构，在本文一直努力保持在视野中的那被标记而受限之意义上。如此被命名，这一递归不再是价值理论的一个私人谜题、而成为一个存在策略之问题的实例。

这些策略是四个，而本文已把它们读作对一个发散的四种态度，吸收、离散化、延展与涌现。被带上价值与权力的递归，每一个都成为一个关于一个价值或一次授权在那下降之底部是什么的论旨，而这四个论旨对一种生成性说明并非同等价值。吸收描述有效价值如何运作而不终止递归。离散化仅通过设定一个先于关系的根基来终止它，而这一纲领被建来拒绝那设定。延展缓解递归而不结束它，并被保留为一个补充。涌现以一种生成性说明能一致采纳的唯一方式终止递归，在一个适用性之界限、而非对着一个地板，在以可争议之主张来描述给出之层级，而它在那同一个运动中如此为之，那运动中，在另一个层面上，有界的结构从一个生成性场域涌现。在此之上本文竖起一个五层的架构，本体论、表象、动力学、判准与观察，其中生成的诸职分立于一条链、而判准与观察的诸职分立于动力学所产生者之上，因而这一纲领所汇集的诸工具被看出是堆叠、而非竞争，而那循环判准与尺度的诸策略被置于它们所归属之处，作为已涌现者的诸性质、而非作为涌现的诸根据。

这一说明不收束于一套教义，而它收束的形式为它的内容所要求。一个把递归的终点定位于一个先于每一个有界主张之生成性层级的框架，无法通过在它自己位置上安装一个新的有界主张作为它自己的根基、一个其余一切从之被推导的主策略而告终。它却收束于一种生成性的价值形式化将不得不接手的诸问题，并把它们作为敞口、而非作为一个有待被执行之纲领来提供。有那类比本身的问题，即关系性场域是否拥有一个真正的尺度与真正的尺度依赖之自由度，唯有那才会让那些被借来的术语在没有它们引号的情况下被说出。有价值在生成性层级上之内容的问题，即价值在那里是否是一个约束、一个选择原则、一个序参量、或某种尚无被借之词回应者，一个本文已审慎地不让那类比之便利决定的问题。有那生成性动力学本身之形式的问题，即它是否是一个作用量、一条演化规则、一个范畴动力学、或另一类的一个生成性过程，一个那架构被建来以不预判的问题。而有那立于其他问题之后的问题，即一门学科到此为止所产生的四种策略是否穷尽这样一种递归可被应对的诸方式、抑或一个第五者仍待被找到。

价值与权力的递归不是一堵思想必须停于其前之墙，而它不是一个思想必须落入其中之虚空。它是一次下降，它终结于生成它的描述给出之处，在有界价值与有界权力本身从之涌现的生成性层级，而一门生成性理论的任务是说在那里所找到者。

致谢

本文所提出的分层架构在它的有效层级处置放一个把生成性循环与榨取性循环区分开来的判准，一个其发展汲取生成性正义之概念的判准，为此作者感念罗恩·埃格拉什的工作。此处对它所作的使用是作者自己的。

作者还欠一份债，超出引注所能记录者，欠于那位其陪伴维系了这份工作之人。

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