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### The Metaphysical BasisofNonlinear Emergence

Jessica Wilson

Department of Philosophy

University of Toronto

Metaphysics of Science Group Talk

Intertheoretic Reduction and Degrees of Freedom

Central APA 2010

Motivations for emergence

- Why care about what emergence is, and whether there is any?
- One reason: the “pretheoretic” appearances
- Special science entities (tornados, plants, people) appear to be broadly synchronically dependent on micro-physical entities.
- Yet, as per their governing special science laws, special science entities seem also to be to some extent autonomous from micro-physical entities, even given that the latter stand in lower-level and other aggregative relations.

- Accounts of emergence aim to accommodate these appearances, one way or another.

Core features of emergence

- Correspondingly, two “hallmarks” of emergence:
- (1) Emergent phenomena are dependent on underlying processes
- (2) Emergent phenomena are autonomous from underlying processes

Is emergence compatible with physicalism?

- The notions of dependence and autonomy admit of many interpretations.
- One question of interest to metaphysicians of science concerns whether there is some way of filling in these notions on which emergence is compatible with physicalism:
- Physicalism: all broadly scientific entities are nothing over and above lower-level physical entities.

- Such emergence is sometimes called Weak emergence.
- The intended contrast is with Strong emergence (as per the British emergentists), involving fundamentally novel powers, forces, or features of the sort that would falsify Physicalism.

Weak emergence andnonlinear phenomena

- Nonlinear systems have been seen as involving a sort of emergence that is clearly physically acceptable:
- “An innocent form of emergence---what I call “weak emergence”---is now a commonplace in a thriving interdisciplinary nexus of scientific activity […] that includes connectionist modeling, non-linear dynamics (popularly known as “chaos” theory), and artificial life” (Bedau 1997, 375)

Epistemological accountsof Weak emergence

- Those focusing on nonlinear systems have characterized the associated accounts of emergence (more specifically: emergent autonomy) in broadly epistemological terms.
- Typical here are Newman (1996) and Bedau (1997, 2002, 2008), who take attention to certain nonlinear systems to motivate conceptions of Weak emergence in terms of unpredictability or algorithmic/explanatory incompressibility.

Such accounts gain compatibilitywith reductive Physicalism

- Epistemological accounts establish compatibility with ontologically reductive Physicalism and epistemologically non-reductive Physicalism.
- They also suggest that nonlinearity is not sufficient for Strong emergence (pace certain suggestions of the British Emergentists)
- On the other hand, such accounts seem not to provide a metaphysical ground for genuine higher-level autonomy of higher-level entities.
- As such, they fail to establish compatibility with ontologically non-reductive Physicalism.

Weak emergence and degrees of freedom

- In fact, a robustly metaphysical conception of Weak emergence can be made out, establishing the viability (if not the truth, which remains an open question) of non-reductive Physicalism.
- I have recently argued (BJPS 2010) that the notion of a degree of freedom (DOF) provides the basis for such a metaphysical conception.
- Here I aim to show how a DOF-based account provides a basis for taking certain non-linear phenomena to be metaphysically emergent.

Plan for the talk

- I’ll start by discussing the features of non-linear systems that Newman and Bedau take to be the basis of non-linear emergence
- I’ll say why such epistemological conceptions of non-linear emergence do not, pace Bedau, provide the basis for genuinely metaphysical emergent autonomy
- I’ll then present my DOF-based account Weak emergence, as involving an elimination in degrees of freedom
- I’ll then argue that nonlinear systems of the sort Newman and Bedau consider plausibly have degrees of freedom that are eliminated in the requisite sense, and so are genuinely metaphysically emergent
- As a sub-theme, I’ll argue that there appear to be at least two distinct sources of metaphysical nonlinear emergence

Linear vs. non-linear systems

- Here, composite systems are linear if all of their characteristic law-governed properties and behavior can be seen as additive functions of the properties and behavior of their parts:
- f(x+y) = af(x)+bf(y)

- Composite systems are nonlinear if they are not linear.
- The nonlinear systems at issue are all assumed to be deterministic.
- A nonlinear system may be chaotic or non-chaotic.
- Newman focuses on chaotic systems, Bedau on non-chaotic systems
- I’ll consider each in turn

A chaotic nonlinear system

- Characteristic of a chaotic nonlinear system is extreme sensitivity to initial conditions (the “butterfly effect”)
- Consider the logistic map, describing, e.g., population growth:
- x_n+1 = a x_n – a(x_n)*2

- ‘a’ is a parameter representing, e.g., birth and death rates; is different for different systems
- The evolution of a given system is almost entirely dependent on a, evolving to a fixed point, or to periods subject to increasingly rapid bifurcation as a approaches 4
- For a = 4, the behavior of the resulting system is chaotic: very small differences in initial conditions x_i, associated with distant decimal places, eventually lead to wildly different trajectories

Strange attractors andchaotic systems

- This sort of chaotic behavior is associated with “strange attractors”:
- “A strange attractor is a non-periodic trajectory in the state space that exhibits sensitive dependence on initial conditions. This means that as the state of a chaotic system evolves toward the attractor in its state space, it will never be in exactly the same state twice, and any two nearby points in the state space will diverge exponentially under the dynamical evolution of the system.” (Newman, 254)

- As with the logistic map, for certain parameters of a given non-linear dynamical equation, the resulting system will evolve toward a strange attractor A in its state space, no matter what its initial state
- Such a system is said to be “in the basin” of strange attractor A

Two epistemic features

- Due to the sensitive dependence on initial conditions we are unable to predict future trajectory of a chaotic non-linear system:
- “[F]or any chaotic system, since the measurements that we can make of its state are less than perfect […] the future states of that system are impossible to predict. […] The kind of unpredictability introduced by chaotic systems is a kind of epistemic impossibility rather than a metaphysical impossibility.” (Newman, 254)

- Difficulties in measurement also mean that it is epistemically “impossible to determine exactly which chaotic system we are dealing with”.
- Newman appeals to these features in his account of Weak emergence.

Newman’s Weak emergence

- On Newman’s view emergent properties are properties that, while identical to some lower-level physically acceptable property, but are such that it is epistemically impossible to make the identification.
- A property designated by a predicate P in an ideal theory T is emergent if and only if the following conditions are met:
- T describes a class of systems which are structured aggregates of entities described by T’; T’ is an ideal theory of those entities, and the entities described by T strongly supervene on those described by T’.
- This clause is intended to characterize emergent dependence as physically acceptable

- Occurrences of the property designated by P are epistemically impossible to identify with occurrences of any property finitely describable in T’.
- Each occurrence of the property designated by P is an occurrence of one of a set of properties PC, which is modelled by T’. Each member of PC is epistemically indistinguishable in T’ from some other member of PC.

- T describes a class of systems which are structured aggregates of entities described by T’; T’ is an ideal theory of those entities, and the entities described by T strongly supervene on those described by T’.

A candidate emergent property

- Newman argues that the property of being in the basin of a strange attractor is such an emergent property of nonlinear systems.
- Every nonlinear chaotic system is physically acceptable (the theory of such systems “strongly supervenes” on the theory of lower-level physical theories).
- Every property of such a nonlinear system is identical to some lower-level physical property.
- But we can never know exactly which chaotic system we are dealing with. In particular, we can’t distinguish systems with different strange attractors A and A’.
- Hence though we can know of a given nonlinear system that it has some or other such property, it is epistemically impossible for us to know which of a class of such properties it has

A non-chaoticnonlinear system

- As noted, not all nonlinear systems are chaotic.
- The game of Life is an example of a non-chaotic non-linear map.
- The game consists in a set of simple rules, applied simultaneously and repeatedly to every cell in a lattice of “live” and “dead” cells.
- Here there is no problem of sensitivity to initial conditions, which conditions consist just in the “seeding” of the lattice
- It’s just that evolution of the lattice as per the rules of the game of Life is non-linear.

Non-chaotic nonlinear systemsare also unpredictable

- Although not subject to “the butterfly effect”, nonchaotic nonlinear systems are (usually) also unpredictable, in not admitting of analytic or “closed” solutions.
- This feature is also shared by chaotic nonlinear systems.
- The absence of analytic or otherwise “compressible” means of predicting the evolution of such systems means that the only way to find out what this behavior will be is by “going through the motions”.
- Set up the system, let it roll, and see what happens.

Bedau’s Weak emergence

- Where system S is composed of micro-level entities having associated micro-states, and where microdynamic D governs the time evolution of S’s microstates:
- Macrostate P of S with microdynamic D is weakly emergent iﬀ P can be derived from D and S’s external conditions but only by simulation. (1997, 378)

- Derivation of a system’s macrostate “by simulation” involves iterating the system’s micro-dynamic, taking initial and any relevant external conditions as input.
- The broadly equivalent conception in Bedau’s (2002) takes Weak emergence to involve “explanatory incompressibility”, where there is no “short-cut” explanation of certain macro-features of a system

A candidate emergent property

- Bedau argues that the property of being a glider gun is such an emergent property of nonlinear systems.
- For cellular automata it is crystal clear that the long-term behavior of the system is completely determined by (“derived from”) lower-level goings-on, as (mutatis mutandis) compatibility with Physicalism requires.

- But that a given system will evolve in such a way as to generate a glider gun cannot be predicted from knowledge of initial conditions (seeding) and the rules.

Emergent dependence as reducibility

- Both Newman and Bedau take the dependence at issue in Weak emergence to be ontological and causal reducibility:
- “weakly emergent phenomena are ontologically dependent on and reducible to micro phenomena” (2002, 6)
- “the macro is ontologically and causally reducible to the micro in principle” (2008, 445).

Emergent autonomy as broadly epistemic

- And both Newman and Bedau take emergent autonomy is understood in epistemic terms:
- “weakly emergent phenomena are autonomous in the sense that they can be derived only in a certain non-trivial way” (2002, 6)

- This is unsurprising: if Weakly emergent entities are supposed to be ontologically and causally reducible, then its hard to see how their “autonomy” could be other than epistemic.

An underlying metaphysical autonomy?

- Epistemic reductive conceptions of emergence don’t so much accommodate the pretheoretic appearance of higher-level autonomy as explain such autonomy away.
- Bedau acknowledges this concern, and responds by maintaining that the autonomy in his conception of Weak emergence is metaphysical.
- He offers two reasons for thinking this.

Reason 1:The objectivity of incompressibility

- “The modal terms in this definition are metaphysical, not epistemological. For P to be weakly emergent, what matters is that there is a derivation of P from D and S’s external conditions and any such derivation is a simulation. […] Underivability without simulation is a purely formal notion concerning the existence and nonexistence of certain kinds of derivations of macrostates from a system’s underlying dynamic.” (1997)
- But such facts about explanatory incompressibility, though objective, are not suited to ground the metaphysical autonomy of emergent entities.
- What is needed for such autonomy is not just some or other metaphysical distinction between macro- and micro- goings-on, but moreover one which plausibly serves as a basis for the ontological autonomy of the former vis-à-vis the latter.

Reason 2:Macro-level patterns and laws Either the metaphysical autonomy or the metaphysical reducibility has to go.

- “[T]here is a clear sense in which the behaviors of weak emergent phenomena are autonomous with respect to the underlying processes. The sciences of complexity are discovering simple, general macro-level patterns and laws involving weak emergent phenomena.” (1997, 395)
- This sounds right, but I don’t see how Bedau can maintain that some Weakly emergent goings-on are not “merely epistemological” and rather reflect an “autonomous and irreducible macro-level ontology”, while also maintaining that all Weakly emergent goings-on are metaphysically ontologically and causally reducible to the micro-level goings-on.

Diagnosis: assuming thatPhysicalism requires reducibility

- The tension in Bedau’s position here results from his assuming, like many of those citing nonlinear phenomena as Weakly emergent---that compatibility with Physicalism requires ontological reducibility.
- Can the assumption be rejected?
- Is there a conception of metaphysical emergent autonomy that is compatible with Physicalism?

Weak emergence anddegrees of freedom

- Yes.
- The account is perspicuously motivated by attention to the notion of a degree of freedom (DOF).
- Call states upon which the law-governed (i.e., nomologically possible) properties and behavior of an entity E functionally depend the ‘characteristic states’ of E.
- A DOF is then, roughly, a parameter in a minimal set needed to describe an entity as being in a characteristic state.
- Given an entity and characteristic state, the associated DOF are relativized to choice of coordinates, reflecting that different sets of parameters may be used to describe an entity as being in the state. This relativization won't matter in what follows, since the relations between (sets of) DOF at issue will be in place whatever the choice of coordinates.

DOF (less rough)

- More precisely, then, the operative notion of DOF is as follows:
- Degrees of Freedom (DOF): For an entity E, characteristic state S, and set of coordinates C, the associated DOF are parameters in a minimal set, expressed in coordinates C, needed to characterize E as being in S.

- I’ll sometimes speak for short of “characterizing an entity”, with state and coordinates assumed.
- Let’s get some examples on the table.

Paradigms

- The configuration state: tracks position.
- Specifying this state for a free point particle requires 3 parameters (e.g., x, y, and z; or r, rho, and theta); hence a free point particle has 3 configuration DOF, and a system of N free point particles has 3N configuration DOF.

- The kinematic state: tracks velocities (or momenta).
- Specifying this state for a free point particle requires 6 parameters: one for each configuration coordinate, and one for the velocity along that coordinate; hence a free point particle has 6 kinematic DOF, and a system of N free point particles has 6N kinematic DOF.

- The dynamic state: tracks energies determining the motion.
- Specifying this state typically requires at least one dynamic DOF per configuration coordinate, tracking the kinetic energy associated with each position coordinate; other dynamic DOF may track internal/external contributions to the potential energy.

Why attend to DOF?

- Note that different entities, treated by different sciences, may be functionally dependent on the same characteristic state (e.g., the configuration state).
- For present purposes the main cash value of attention to DOF lies in the fact that DOF track the details of an entity‘s functional dependence on its characteristic states, in a more fine-grained way than states themselves do.
- What I will be exploring here is the idea that these fine-grained details concerning functional dependence serve as a plausible ontological basis for the individuation of broadly scientific entities.

Reductions, restrictions, and eliminations in DOF

- The DOF needed to characterize an entity may be reduced, restricted, or eliminated in certain circumstances, compared to those needed to characterize a (possibly distinct) entity, when such circumstances are not in place.
- Let's get acquainted with these different relations with some examples.
- To prefigure: eliminations in DOF, in particular, will enter into the upcoming metaphysical account of Weak emergence.

Reductions in DOF

- Constraints may reduce the DOF needed to characterize an entity as being in a given state.
- So, for example, a point particle constrained to move in a plane has 2 configuration DOF, rather than the 3 configuration DOF needed to characterize a free point particle.
- In cases where a given DOF is given a fixed value, the laws governing an entity so constrained are still functionally dependent on the (now constant) value of the DOF; hence such constraints do not eliminate the DOF, but rather reduce it to a constant value.
- Special science e.g.:
- Rigid bodies treated in classical mechanics

Restrictions in DOF

- Constraints may also restrict DOF needed to characterize an entity.
- A point particle may be constrained, not to the plane, but to some region including and above the plane. Characterizing such a particle still requires 3 configuration DOF, but the values of one of these DOF will be restricted to only certain of the values available to the unconstrained particle.
- Cases of restriction in DOF are more like cases of reduction than elimination of DOF, in that, again, the entity’s governing laws remain functionally dependent on specific values of the DOF.
- Special science e.g.:
- Molecules
- Molecular bonds are like springs, restricting the DOF of their composing atoms

Eliminations in DOF

- Sometimes the imposition of constraints eliminates DOF.
- So, for example, N free point particles, having 3N configuration DOF, might come to compose an entity whose properties and behavior can be characterized using fewer configuration DOF, not because certain of the DOF needed to characterize the unconstrained system are given a fixed value, but because the properties and behavior of the composed entity are functionally independent of these DOF.
- Special science e.g.:
- Spherical conductors treated in electrostatics
- E-field due to free particles depends on all charged particles, whereas E-field due to spherical conductor depends only on charges of particles on surface.

An interesting case of elimination of DOF:SM (stat-mech) systems

- Consider an isolated gas E composed of large numbers of particles or molecules e_i. Since E is unstructured, shouldn’t it have the same DOF as the system of unconstrained e_i?
- Answer: no, for while not bonded, the e_i are interacting via exchanges of energy, and such interactions may not only restrict or reduce, but eliminate DOF.
- This is indicated by the fact that the renormalization group method is appropriately applied to SM systems (see Batterman 1998, p. 109):
- “In effect, the renormalization group transformation [tracking stability of evolution under perturbations of, e.g., composition, position, and velocity] eliminates those degrees of freedom (those microscopic details) which are inessential or irrelevant for characterizing the system's dominant behavior […] .”

ei-level constraints

- Two features of the previous cases are important for what follows.
- First is that the holding of the constraints is a matter entirely of physical or physically acceptable processes.
- Such processes suffice to explain why sufficiently proximate atoms form certain atomic bonds, why atoms or molecules engage in the energetic interactions associated with SM ensembles, and so on

- More generally, for each of the special science entities E at issue, the constraints on the e_i associated with the latter's composing E are explicable using resources of the theory treating the e_i (or resources of some more fundamental theory, treating the constituents of the e_i).
- Call such constraints “e_i-level constraints”.

ei-level determination

- A second important feature of these special science entities E is that all of their properties and behavior are completely determined by the properties and behavior of their composing e_i, when these stand in the relations relevant to their composing E.
- Note that this feature does not in itself follow from E's being composed by e_i as a result of imposing e_i-level constraints.

- People disagree about the metaphysical ground for this determination, but all parties agree that the determination is in place.

DOF and Weak emergence

- I'll now argue that eliminations in DOF, in particular, provide a basis for weak emergence of the sort vindicating non-reductive Physicalism.
- There is room to resist taking reductions or restrictions in DOF as indicative of metaphysical emergence.

Weak metaphysical emergence

- Weak metaphysical emergence: An entity E is weakly metaphysically emergent from some entities e_i if
- (1) E is composed by the e_i, as a result of imposing some constraint(s) on the e_i,
- (2) For some characteristic state S of E: at least one of the DOF required to characterize the system of unconstrained e_i as being in S is eliminated from the DOF required to characterize E as being in S.
- (3) For every characteristic state S of E: every reduction, restriction, or elimination in the DOF needed to characterize E as being in S is associated with e_i-level constraints;
- (4) The law-governed properties and behavior of E are completely determined by the law-governed properties and behavior of the e_i, when the e_i stand in the relations relevant to their composing E.

The physical acceptability of weakly emergent entities

- The composed special science entities illustrating eliminations in DOF are uncontroversially intuitively physically acceptable.
- Satisfaction of Weak metaphysical emergence accommodates and justifies this intuitive judgement, in conformity to Physicalism.
- In my paper, I argue for this.
- Since all agree that this is the case for nonlinear phenomena that will shortly be our target, I won’t enter into the details here.

The ontological irreducibility of weakly emergent entities

- I’ll now argue that entities satisfying Weak metaphysical emergence plausibly satisfy Non-reduction, by attention to the main objection to this claim.

The objection from theoretical deducibility

- Laws governing Weakly metaphysically emergent entities are (I am granting) deducible consequences of the laws of the more fundamental theory.
- Does such deducibility indicate that Weakly metaphysically emergent entities E are ontologically reducible to their composing e_i? Does theoretical deducibility entail ontological reducibility?
- An affirmative answer seems initially plausible, and has been commonly endorsed (Nagel 1961, Klee 1984).

A dilemma for the NRPist

- We can put the concern in the form of a dilemma.
- As above, the constraints entering into the composition of our target special science entities are plausibly e_i-level constraints, and as indicated, this feature is crucial to establishing the physical acceptability of the target entities.
- But if the constraints are e_i-level, why can't the associated e_i-level relations enter, one way or another, into an ontological as well as a theoretical reduction of E to its composing e_i?

- This line of thought is, I think, at the heart of suspicions that there is no way for entities to be Weakly metaphysically emergent.

The response from different DOF

- First, let’s get clear on what would count as an ontological reduction of E to the e_i.
- The candidate reducing entities include all the lower-level entities, properties and relations, plus any ontologically “lightweight” constructions---lower-level relational, boolean, or mereological---out of these. Hence…
- If E is to be ontologically reducible to the e_i, then E must be identical to:
- (i) a system consisting of the jointly existing e_i
- (ii) a relational entity consisting of the e_i standing in e_i-level relations, or
- (iii) a relational entity consisting in a boolean or mereological combination of the entities at issue in (i) and (ii).

A necessary conditionon ontological reducibility

- Since E is Weakly metaphysically emergent, there is some state S such that characterizing E as being in this state requires fewer DOF than are required to characterize the unconstrained system of E’s composing e_i as being in S.
- Hence a necessary condition on E's being identical with an entity of the type of (i)--(iii) is that the DOF required to characterize the candidate reducing entity as being in S are similarly eliminated, relative to the unconstrained system.
- I’ll now argue that this condition isn’t met for any of the entities of type (i)--(iii).

Jointly existing e_i

- Consider the e_i understood as (merely) jointly existing, as per (i).
- Such a system of e_i is not subject to any constraints; hence for any state, characterizing this system will require the same DOF as are required to characterize the system of unconstrained e_i.

- So E is not identical to the system consisting of (merely) jointly existing e_i.

Relational entities

- Consider the relational entity e_r consisting of the e_i standing in certain e_i-level relations, as per (ii).
- Though e_r realizes a constrained entity (namely, E), e_r is not itself appropriately seen as constrained---even throwing the holding of the constraints into the mix of e_i-level relations at issue in e_r.
- Why not? Because the laws governing entities (like e_r) consisting of the e_i standing in e_i-level relations are, unlike the laws governing E, compatible with the constraints not being in place.
- Hence characterizing e_r as entering into these laws, hence capable of evolving into states where the constraints are not in place, requires all the DOF associated with the unconstrained system of e_i.

- So E is not identical to e_r.

Boolean entities

- Next, consider a relational entity consisting in a disjunctive combination of entities of the sort at issue in (i) or (ii).
- The occurrence of a disjunctive entity consists in one of its disjunct entity's occurring. Hence for any state, characterizing a disjunctive entity as being in that state will require all the DOF required to characterize any of its disjunct entities as being in that state.
- Such disjunct entities, being of type (i) or (ii), will require all the DOF required to characterize the system of unconstrained e_i, for any state.
- So characterizing a disjunctive relational entity will require all the DOF required to characterize the system of unconstrained e_i, for any state.

- Same goes for conjunctive entities.
- So E isn’t identical to a disjunctive or conjunctive relational entity.

Mereological combinations

- Next, consider a relational entity consisting in a mereological combination of entities of the sort at issue in (i) or (ii).
- Mereological wholes are identified with the mere joint holding of their parts; hence characterizing the whole will require all the DOF required to characterize each of the parts.
- Such parts, being of type (i) or (ii); will require all the DOF required to characterize the system of unconstrained e_i, for any state.
- So characterizing a mereological relational entity will require all the DOF required to characterize the system of unconstrained e_i, for any state.

- So E isn’t identical to a mereological relational entity.

Theoretical deducibility is compatible with ontological irreducibility

- That exhausts the available candidates to which Weakly metaphysically emergent entities might be ontologically reduced.
- In each case a Weakly emergent entity has fewer DOF than the candidate reducing entity, so by Leibniz’s law the former is not identical to the latter.

- Attention to the metaphysical implications of the eliminations of DOF at issue in Weak metaphysical emergence indicates that theoretical deducibility is compatible with ontological irreducibility.
- So physical acceptability is compatible with metaphysical autonomy, as the ontologically non-reductive Physicalist supposes.

Back to nonlinear systems irreducibility

- Let’s return now to cases of nonlinear systems, and consider whether they can be understood as Weakly metaphysically emergent.
- I want to start with chaotic nonlinear systems, and by observing a puzzle about such systems:
- Recall that for Newman, chaotic systems are characterized by their sensitivity to initial conditions.
- But if nonlinear systems are so sensitive and seemingly “chaotic”, how is it that they can be, as they are, an interesting object of study?

Interestingly, irreducibilitya similar puzzle about SM systems

- SM systems are unstructured aggregates---e.g., an isolated gas E composed of large numbers of particles or molecules e_i.
- Since E is unstructured, then (apart from restrictions in DOF associated with E’s boundaries) shouldn’t it have the same DOF as the system of unconstrained e_i ?
- So understood, however, the success of SM is mysterious, since obviously we can’t track ~ 10*26 DOF. As Batterman puts it:
- “One wants to know why the method of equilibrium SM-the Gibbs' phase averaging method-is so broadly applicable; why, that is, do systems governed by completely different forces and composed of completely different types of molecules succumb to the same method for the calculation of their equilibrium properties?” (‘Why Equilibrium Statistical Mechanics Works’, p. 185)

SM systems irreducibilityand the renormalization group (RG)

- We’ve already seen the answer to the last puzzle.
- “In effect, the renormalization group transformation [tracking stability of evolution under perturbations of, e.g., composition, position, and velocity] eliminates those degrees of freedom (those microscopic details) which are inessential or irrelevant for characterizing the system's dominant behavior […] .” (Batterman)

- A similar solution is available to explain why chaotic systems are, in spite of their unpredictability, an interesting and to some extent theoretically tractable object of study
- Let’s see why.

The RG method tracks scale-invariance irreducibility

- The general use of the RG method is to investigate the behaviors of systems at different length scales:
- The RG method takes a system’s governing laws (e.g., its Hamiltonian) and iteratively transforms them into laws having a similar form but (reﬂecting moves to increasingly ‘larger’ scales) fewer parameters—that is, fewer DOF (with individual components at a scale being treated as a ‘block’ at the next largest scale, and parameters adjusted accordingly).
- The method is applicable when the laws governing the components and resulting systems take the same form at all distance scales. In the limit, if all goes well, the resulting Hamiltonian will describe the behavior of a single ‘block’, corresponding to the macroscopic system.

Systems near critical points irreducibilityare scale-invariant

- The RG method generally applies to systems near critical points:
- “It turns out that systems undergoing phase transitions typically exhibit the following property: Their behavior at the critical point is characterized by a small set of dimensionless numbers called "critical ex- ponents”. What is truly remarkable about these numbers is their universality […] the critical behavior of systems whose components and interactions are radically different is virtually identical. Hence, such behavior must be largely independent of the details of the microstructures of the various systems. This is known in the literature as the "universality of critical phenomena”. Surely one would like to account for this universality. The so-called "renormalization group" provides the desired explanation."

Nonlinear chaotic systems irreducibilityhave critical points

- As previously, for a nonlinear system to be chaotic is for it to be extremely sensitive to initial conditions
- Consider, e.g., the logistic map used to model population growth and decay
- Whether a nonlinear system governed by the logistic map system is chaotic depends on the values of the parameter a
- The point at which the system becomes chaotic is a critical point.

So… irreducibility

- Systems that can be modeled by the RG group have eliminated DOF
- Systems with critical points are modeled by the RG group
- Chaotic nonlinear systems have critical points
- Therefore, chaotic systems near critical points have eliminated DOF
- Systems with eliminated DOF are Weakly metaphysically emergent
- Therefore, chaotic systems are Weakly metaphysically emergent

Non-chaotic nonlinear systems irreducibility

- A case can also be made that some non-chaotic nonlinear systems are Weakly ontologically emergent.
- Given that ontologically irreducibility of nonlinear phenomena is back on the table, Bedau’s comments that “The sciences of complexity are discovering simple, general macro-level patterns and laws involving weak emergent phenomena” are now apropos.

Counterfactual-sustaining patterns irreducibility

- Bedau’s further remarks concerning glider guns are also useful:
- The glider gun provides an overarching, macro-level explanation for the second glider stream. Furthermore, this same macro explanation holds for any number of other guns that shoot other gliders. The aggregate micro explanation […] omits this information. Furthermore, this information supports counterfactuals about the stream. […] Any such macro gun would have produced the same macro effect. […] There is a macro explanation that is not reducible to that aggregation of micro histories. If those micro histories had been different, the macro explanation could still have been true. The macro explanation is autonomous from the aggregate micro explanation. (19)

- It is plausible (and I argue in my paper) that such patterns of counterfactual independence from micro-level details are indicative of eliminations of micro-level DOF.

Summing up irreducibility

- Previous accounts of Weak non-linear emergence have characterized nonlinear emergence in terms of unpredictability, associated with sensitivity to initial conditions or to algorithmic incompressibility.
- This may reflect the assumption that physical acceptability requires ontological reducibility. As I argued in presenting my DOF-based account of Weak metaphysical emergence, this is incorrect.
- My DOF-based characterizations of nonlinear emergence, chaotic and non-chaotic, appeal to the metaphysical fact that the behavior of nonlinear systems is in certain respects surprisingly insensitive to micro-level details---an insensitivity that supports thinking that such systems have eliminated DOF.
- As such, nonlinear phenomena are appropriately seen as metaphysically as well as epistemologically weakly emergent, and a rich source of entities of the sort that NRPists maintain there can be.

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