Emergence in Physics

McGivern, Patrick H. and Rueger, Alexander, 2010, Emergence in physics, in McGivern, P. H. & Rueger, A. 2010, 'Emergence in physics', in A. Corradini & T. O'Connor (eds), Emergence in Science and Philosophy, Routledge, New York. pp. 213, , 213-‐232.

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Emergence in Physics

Patrick McGivern Alexander Rueger

1. Introduction

Discussions of emergence in physics1 usually adopt one of two general approaches. Starting

from intuitive desiderata a concept of emergence should satisfy, one can (i) develop a

philosophical account of emergence that incorporates and makes precise the initial intuitions and

then try to find illustrations for this model in scientific practice; alternatively, one can (ii) study

cases in science where the intuitive desiderata seem to be satisfied and then develop a

philosophical model of emergence that subsumes those cases. Given these different approaches,

disagreements over which phenomena count as emergent are likely, even if everybody were to

agree on the pre-theoretical desiderata. Differing views about the latter, of course, compound the

conflicts. Whether emergent phenomena, for instance, supervene on ‘base level’ phenomena or

whether the ‘completeness’ of physics at the base level should be satisfied are questions which

different philosophical views will answer in different ways.

Here, we will follow the second strategy. We will start with a relatively uncontroversial

set of pre-theoretical features of emergent phenomena and make them more precise by

characterizing them in terms of features that we find in the mathematical treatment of certain

problems in physics. We can then check whether having made our intuitions more precise has

resulted in an account of emergence that allows us to answer philosophically motivated

questions. Unsurprisingly, many of these questions can be answered only if we adopt further 1 ‘Emergence in physics’ is here understood as a discussion of the question whether there are emergent physical phenomena, not an investigation into the possibility of non-physical phenomena emerging out of physical ones.


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assumptions that aren’t directly entailed by the accounts we extract from physics; for instance,

assumptions about the nature of properties and causation will need to be made. Our aim is to

make it clear what these assumptions are, how they can be justified in the context of our

examples, and what their consequences are for several prominent questions about emergence.

Thus our main focus will not be on detailed arguments in favor of certain assumptions and

against others, but instead on the nature of the assumptions needed to interpret our examples in

various possible ways (although we indicate our own preferences).

The examples we choose from classical physics may at first not appear to be striking

enough to count as cases of emergence, especially when compared with the spectacular cases

that are often mentioned in the literature, such as entangled quantum systems, phase transitions,

or certain effects in solid state physics. Our modest choice of examples, however, has its

advantages: the technical details of the cases are less daunting and, although we do not have

space to elaborate the point in detail, some of the basic features we discuss play important roles

in the more complex cases as well.

In assessing our examples, we will focus on three philosophical questions about

emergence:

(i) Is emergence ‘merely epistemic’ or are there examples of ‘ontological’ emergence in

physics?

(ii) Do emergent phenomena supervene on the ‘base’ phenomena from which they

emerge?

(iii) What is the nature of emergent causation?

The rough idea behind the first question is a distinction between situations where the

appearance of emergent phenomena depends on our way of describing a system (e.g., in a fine-


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grained or coarse-grained way) and situations where the occurrence of emergence is independent

of how the system is represented (for instance, by demonstrating that emergent phenomena are

characterized by special causal powers).2 Many emergentists have hoped to show that there are

indeed cases where emergence has to be understood ontologically. Since in general our claims

about ontological issues in physics must be based on features of physical theories, drawing the

distinction between epistemic and ontological forms of emergence is not as straightforward as

simply noting whether a particular feature is a characteristic of theories or not: answering this

first question will require some further assumptions about what features of theories are

ontologically significant, and when.

For question (ii), the assumption that emergent phenomena supervene on their base

phenomena is widespread, although there are dissenting views (e.g., Humphreys 1997). We do

not assume that supervenience is a requirement for emergence, but instead investigate whether it

is found in each of the cases we examine.

Question (iii) is the most elaborate of the three. There are three closely related worries

about causation and emergence. The first concerns whether or not emergent properties have

distinctive causal powers relative to the causal powers of their base properties. Here the key

problem is to distinguish between the sort of causal novelty associated with emergence and the

‘regular’ novelty associated with mere ‘resultants’. The second worry concerns whether or not

emergent causal efficacy is even coherent: for instance, Kim (1999) argues that emergent

causation is unavoidable linked with ‘downward’ causation, and that downward causation is

incoherent. The third worry concerns the idea that whatever causal efficacy is found among

2 Humphreys (2009) draws a tripartite distinction between ‘inferential’, ‘conceptual’, and ‘ontological’ conceptions of emergence. While the inferential and conceptual varieties seem close to what we call the epistemic conception, the fact that Humphreys does not view his categories as mutually exclusive makes this difficult to decide.


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emergent properties, base level properties will always compete with or exclude those emergent

properties as genuine causes. We will examine each of these questions in detail. However, first

we will clarify the pre-theoretic concept of emergence.

2. The concept of emergence

Emergence is a relational concept: emergent phenomena are always emergent relative to some

‘base’ or reference phenomena from which they emerge. Emergence is also a contrastive

concept: emergence involves phenomena that are ‘emergents’ relative to their base phenomena

rather than ‘resultants’. An account of emergence should be able to accommodate both of these

features, for instance by explaining how emergents are to be distinguished from resultants.

We can distinguish between two general types of emergence, diachronic emergence,

where an earlier state of the system, over time, gives rise to a later state which is classified as

emergent with respect to the former, and synchronic emergence, where the reference phenomena

coexist with putatively emergent phenomena (Rueger 2000a). Classical accounts of emergence

typically contain both types, often without distinguishing them. Again, an account of emergence

should be able to accommodate both.

Given that emergence is a relational concept, what are the relata? Does emergence

involve the emergence of entities, properties, laws, behaviour, or some combination of these?

We take it that emergence must involve emergent behaviour of some sort, since it is only

through a system’s behaviour that we ever have reason to suppose that there are any entities,

properties, or laws at all. Hence, for the most part, we will talk of emergent behaviour.

However, since it is also common to speak of emergent properties, at times we will do this as


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well. In these cases, such properties can be understood in the sense of being the property of

having a particular sort of behavior.

With these clarifications in mind, the core criteria we find associated with emergence are

those of non-reducibility and novelty: emergent phenomena are in some way irreducible to and

novel with respect to their base phenomena, whereas ‘resultant’ phenomena are reducible and/or

non-novel. In fact, we see these two criteria as two sides of the same coin: emergent phenomena

are typically taken to be not only novel but in some way ‘qualitatively’ novel, and talk of

irreducibility often seems intended to capture just this distinctive feature. 3 With this basic

concept of emergence in mind, we will now turn to specific examples in order to draw out the

details.

3. Diachronic emergence

Consider a damped oscillating system with an equation of motion of this form:

€

m d2xdt 2

+ kx + c dxdt

= 0

which characterizes the three forces operating in the system: the inertial force (first term), the

restoring force (

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kx ), and the damping force (

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c dxdt

). The solution of the equation (given

sufficient initial conditions) – the integration of the equation over some time interval – describes

the behaviour of this system, that is, the distribution of properties (here: positions) of the system

over time:

3 There are several other criteria that have often been associated with emergence, such as their inexplicability or unpredictability from knowledge of the base phenomena alone. However, criteria of this sort seem to be biased towards a purely epistemic sort of emergence, whereas we want our pre-theoretic account to be neutral insofar as the question of an epistemic or ontological interpretation is concerned.


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xc (t) = A(t)cos(ωt −δ)

a series of oscillations with frequency ω and phase δ and with a gradually decaying amplitude

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A(t). With increasing time, the trajectory of the system in phase space will spiral down into the

origin, the point where the motion comes to a rest. The system has a ‘focal point attractor’, a

final state to which all trajectories lead, irrespective of which initial conditions they started from.

Now imagine we decrease the damping in the system. This will not change the

qualitative nature of the phase space portrait; it will only take longer for the system to arrive at

the point attractor. This is true until we completely eliminate the damping and change the

system into an ideal harmonic oscillator. At this point the attractor disappears from the phase

space portrait and we see a ‘qualitatively different’ behavior of the system, regardless of how

much time passes: the trajectory has turned into a closed curve (an ellipsis, or a series of

concentric ellipses if we consider different initial conditions), also called a ‘center’. The system

never comes to a state of rest and keeps oscillating. We have arrived at an undamped harmonic

oscillator:

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m d2xdt 2

+ kx = 0

with solutions

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x0(t) = A(t)cos(ωt −δ) in which the amplitude doesn’t gradually decay.


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Figure 1 Representative phase space trajectories for damped oscillator (left) and undamped oscillator (right).

This example illustrates a simple phenomenon that satisfies our criteria for emergence: the

behaviour of the undamped oscillator is both irreducible to and qualitatively novel with respect

to that of the damped oscillator. Here’s why.

(i) Non-reducibility

There’s no doubt that there’s a connection between the equations of motion for the damped and

undamped system: the equations for undamped system are the result of setting the damping

parameter in the damped equations to zero. So in some sense, there is a reduction between the

two.

But this isn’t the sense of reduction we’re interested in here. Rather than asking whether

or not the equations of motion for the undamped systems can be derived from those of the

damped system, we want to know whether the damped and the undamped systems exhibit the

same type of behaviour, and, if not, in what sense their behaviours differ. Hence, what we are

interested in are not the equations of the systems but their solutions which describe the

behaviour. For example, as damping decreases – but before it reaches zero – the system’s

behaviour will change: with lower damping, if left untouched, it would take longer to reach its


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equilibrium state than it would with higher damping. But these superficially different behaviours

can still be seen as behaviours of the same type since the systems eventually wind up in the same

state. Once damping reaches zero, though, the system’s long-term behaviour becomes

significantly different: as long as it is undisturbed, such a system will never settle into a

stationary state.

We can characterize this difference between the damped and the undamped systems in

terms of the relation of limit reduction (Nickles 1973, Batterman 1995, Rueger 2000b). For a

successful limit reduction, the solutions of the damped equations would need to go over

smoothly into the solutions of the undamped equations in the limit of vanishing damping,

analogous to the way the solutions of certain equations in Special Relativity Theory go over into

the relevant solutions of Newtonian mechanics in the limit

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v c→ 0. Intuitively, what we are

asking when we ask about reduction in this sense is whether the behaviour of the damped system

becomes more and more like that of the undamped system as damping is continuously reduced.

For limit reduction to be successful, we would need to be able to show that we could make the

behaviour of the damped system arbitrarily close to that of the undamped system by sufficiently

reducing damping. More precisely, if we choose a measure of the distance between the damped

and undamped solutions,

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ε > 0 , then for a successful reduction, the damped solutions would have

to stay within this ε-neighborhood of the undamped solutions, with decreasing damping

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(c→ 0)

and increasing time. However, limit reduction is not successful in this case: the behaviors of the

damped and undamped oscillators are not connected by a continuous limit but rather by what is

called a singular limit. This shows that the transition from the behavior of the damped system to

that of the undamped one is characterized by a discontinuity. This irreducibility is reflected

formally in the fact that the two limit operations relevant to the two systems – the limit of infinite


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time represented in the phase space portrait (focal point attractor vs. center) and the limit of

damping

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→ 0 – do not commute:

Eq. 1

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limt→∞

limc→0

xc (t) ≠ limc→0 limt→∞xc (t)

where

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limc→0

xc (t) = x0(t). The limit on the right hand side is 0 (the focal point) while the limit on

the left is not defined (since the system keeps oscillating for infinite time).

(ii) Novelty

In the damped system, any change in damping will lead to a change in the system’s behavior.

However, there is a formal sense in which the behaviors of any pair of damped systems are

similar: given the phase space portraits of any two damped systems, we can always find a

smooth mapping from the one space into the other that preserves the phase space trajectories.

Such a mapping will deform the trajectories without changing their topological features. In the

case of the undamped system, however, this is not possible: there is no way to deform the

trajectories of any damped system into those of the undamped system without ‘cutting’ the

trajectories. More precisely, there is no homeomorphism, no one-to-one mapping continuous in

both directions, between the phase space of the damped oscillator and the phase space of the

undamped oscillator, that converts the spiral trajectories of the former into the elliptical

trajectories of the latter. Such a mapping would connect the two portraits in a way that always

mapped neighboring points in the one onto neighboring points in the other: the two portraits

would then be said to be topologically equivalent, and the behaviour of the one system could be

seen as merely a quantitative variation on the other. But in the damped/undamped case, the two


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portraits are topologically inequivalent, and in this sense, the behaviour of the undamped system

is qualitatively novel with respect to that of the damped system. This is a standardly applied

notion in dynamical systems theory which characterizes qualitative changes in the behavior of a

system.4 We can illustrate this inequivalence as follows. In diagram 1, the horizontal arrows

represent the evolution of behaviour described by the phase space portraits of the damped and

undamped oscillators, respectively. The vertical arrows represent a mapping h between the two

phase space portraits. Topological inequivalence is indicated by the fact that no such mapping is

homeomorphic on both the left and the right hand sides of the diagram: for instance, any

mapping that succeeds at always mapping neighboring points to neighboring points on the left-

hand side will inevitably map neighboring points to non-neighboring points on the right-hand

side.5

Diagram 1

We propose to adopt this way of understanding novel behavior in the context of emergence:

Novelty of behavior is to be characterized in terms of topological differences between the

representations of a system’s behavior before and after a control parameter reaches or crosses a

4 For a more precise definition see, e.g., Arnold 1983, 89-91. 5 The diagram also shows that the undamped system is structurally unstable: small changes in the parameter (d) turn the system into a topologically inequivalent one. See Rueger 2000b.


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critical value (here: damping = 0). Merely quantitative differences (e.g., a shorter period of

oscillation when damping is diminished) are not sufficient for the sort of novelty associated with

emergence, as has often been pointed out. That a body of 10 kg can behave differently than a

body of 1 g is not reason enough to call the heavier body’s behavior qualitatively different.

Compare diagram 1 with a diagram that represents the non-commutation of the limits in

eq. 1.

Diagram 2

That there is no appropriate map to be found for the ‘closure’ of diagram 1 corresponds to what

we pointed out earlier in the discussion about non-reducibility: the limits

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t→∞ and

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c→ 0 do

not commute. This suggests that there is a connection between our notions of non-reducibility

and novelty. But they are not identical concepts. Though the limit notion of reduction implies a

sense of novelty of behavior that can be characterized topologically, this is not the same concept

that we defined above with diagram 1. In the failure of limit reduction, the transition from the

behavior of the damped system to that of the undamped one is characterized by a discontinuity.

This discontinuity manifests itself formally when we study the topological space in which the

solutions of the equations of motion figure as points. Our notion of reduction as uniform

convergence of one solution to the other in the limit of vanishing damping imposes the ‘topology

of uniform convergence’ on the space of solutions, a fairly ‘fine’ topology in the sense that it

excludes many sequences of functions from the class of converging sequences. The occurrence


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of a singular limit, as in our example, means that we have to change the topology of the space of

solutions so as to be able to deal with cases like the transition from damped to undamped

behavior. We’ll see later, in the case of synchronic emergence, that there are ways of

‘regularizing’ the discontinuous limit behavior so that a uniformly valid approximation relation

between ‘old’ and ‘new’ behavior can be achieved. Since these techniques, however, do not

restore uniform convergence, they effectively introduce a new topology on the space of

solutions. Since changes in the topology of the space in which we represent the behavior of a

system are usually characterized as qualitative changes, the topological perspective on non-

reducibility suggests an explication of ‘novel’ or ‘qualitatively different’ behavior in topological

terms.

How is this topological notion of novelty related to topological inequivalence? The

latter concept was explicated above in terms of topological inequivalence of the families of

trajectories in phase space, a purely topological or qualitative notion. To characterize non-

reducibility, the discontinuous transition of old to new behavior, as breakdown of uniform

convergence in the space of solutions we need a topological space that is also equipped with a

norm or metric – a measure of distance between points. Hence this notion of novelty is not

purely topological. In the case of synchronic emergence below we’ll see a further application of

this concept.

Perhaps this example seems too simple to illustrate an instance of emergent behavior.

Where is all the ‘complexity’ of the system that one traditionally thought of as a condition for

emergence? We seem to have a case in which the system actually becomes simpler. In response

we have to say that the traditional fixation on emergence as associated with complexity is a

mirage. If the spectacular cases of emergence in phase transitions, solid-state physics, and


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quantum theory are bona fide instances of emergence, then they fulfill this role because they

share the relevant features – novelty of behavior and non-reducibility in our sense – with our

simple example (as we’ll indicate in the final section).

Philosophical questions about diachronic emergence

(i) Epistemic vs. ontological emergence

Is our example a case of diachronic emergence in the ontological or the epistemic sense? Recall

that that distinction is supposed to turn on whether the appearance of whatever characteristics are

indicative of emergence can be attributed to a shift in perspective in our description of a system.

Ontological emergence requires a real change in the system, whereas epistemic emergence

requires only a change in how the system appears from a different perspective. Since the

emergent behaviour of the undamped oscillator is brought about by a change in an actual

parameter in the system (the damping), it seems most natural to view this as a case of ontological

emergence. One way to understand this case as an instance of epistemic emergence would be to

claim that the change in damping represents a change in the conditions of idealization, for

instance as our interests shift to time intervals over which the effects of damping can be

considered negligible. Note that though it may seem that undamped oscillatory behaviour must

be regarded as an idealization, the example of the undamped oscillator can be replaced by a more

‘realistic’ (though more complicated) system like the van der Pol oscillator. In these systems,

damping does not have to be reduced to zero for novel behavior to occur: when the damping


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reaches a critical value, the system develops a new attractor, a limit cycle (see Rueger 2000a).

We return to the issue of idealizations briefly below in a more general context.

(ii) Supervenience

Supervenience is commonly understood as a synchronic relation. In the diachronic case it would

therefore seem inappropriate to ask whether the later state of the system supervenes on the earlier

state. We could define a diachronic supervenience relation, and ask whether the state of the

system at one (earlier) time necessitates its state at another (later) time, or whether the later state

could vary without a change in the earlier state. The answer seems to be ‘no’: understood

diachronically, supervenience fails in our example. The earlier, damped behaviour alone doesn’t

necessarily give rise to the later, undamped behaviour: for this to occur, the damping must be

eliminated. And the later, undamped, behaviour could itself vary – for instance, reverting to

damped behaviour if damping is reintroduced – without any change in the earlier, damped

behaviour. But this is not surprising and has no bearing on the question as it is usually asked,

viz., the question whether some property of the system supervenes on other properties of the

system at the same time.

(iii) Causation

Does the diachronically emergent system have distinct causal powers relative to the base system?

In one sense this question seems trivial in the diachronic case: after all, the systems are distinct,

they occur at different times and the difference in causal powers seems no more mysterious than

the difference in causal powers between a chicken and an egg. However, there is another way of

asking about causal powers which leads to more interesting results in this case.


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Recalling the contrast between ‘emergents’ and ‘resultants’, we can investigate the

difference in relationship between diachronic emergents and their base properties and diachronic

resultants and their base properties. In the case of the oscillators, an example of ‘resultant’

behaviour, relative to a base level of damped behaviour, would be some later damped behaviour

with a reduced level of damping. As desired, such behaviour would both reduce to the base level

behaviour and be non-novel with respect to that behaviour, in our senses of these terms. Since

the resultant system will exhibit behaviour that is distinct from (though not qualitatively distinct

from) that of the base system, we expect there to be causal powers associated with this resultant

system that are distinct from those associated with the base system, just as we expect there to be

distinct causal powers associated with the emergent system. However, the non-commutativity

illustrated in diagram 2 shows a way in which those causal powers are themselves of different

types. We can regard the horizontal arrows as indicating causal processes associated with the

temporal evolution of the system, and the vertical arrows as indicating causal processes

associated with the change in damping. Because the diagram does not commute (that is,

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x0( " t )

cannot be reached starting from

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xc (t) and first going to

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xc ( " t ) ) , we see that these two sorts of

causes really are different in the sense that the order in which they are applied is not arbitrary. (If

we considered a case where the diagram did commute, the two causes would be equivalent.).

We can describe this distinction in terms of Dretske’s (1988) distinction between ‘structuring’

and ‘triggering’ causes, where the causal relations represented by the horizontal arrows

correspond to triggering causes and those represented by the vertical arrows correspond to

structuring causes. In the case of non-reducibility we have a real difference between structuring

and triggering causes: the emergent state of the system

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(c = 0) can bring about later states that


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the earlier state

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(c > 0) could not trigger. Only the structuring kind of causation opens up the

possibility of (diachronic) emergence.

4. Synchronic emergence

The classic case of synchronic emergence involves phenomena occurring simultaneously on

macro and micro levels. To illustrate this, consider the treatment of steady state heat conduction

in a one-dimensional rod of length L.6 This system is described in terms of its temperature T(x)

and its thermal conductivity k(x) which both vary in dependence on the spatial variable x. We

assume that at both ends of the rod (

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x = 0 and

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x = L) the temperature is held constant at some

values. Suppose now that we take the rod, at the micro-level, to have a discrete, ‘atomistic’

constitution, that is, we stipulate that the system consists of individual atoms, separated by empty

space – a periodic lattice with a period of length

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P = εL , with

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ε <<1. The microscopic

conductivity k(x) will then be a rapidly oscillating function of position: high around the location

of each atom, low in the interatomic spaces. This behaviour of the conductivity is indicated by

writing k as a function of x and x/ε. The dependence on x/ε manifests itself as rapid variations

because

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ddx (k( xε )) = (1ε ) d

dx k( xε ) that is, the derivative of k is large for small ε. The temperature

distribution at the micro level is thus described by:

Eq. 2

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ddx

k(x, xε )dT(x)dx

#

$ % &

' ( = 0

Note that any solution of (2), that is, an integration of (2) over the length of the rod, will

represent the property of having such-and-such a distribution of micro-level temperature.

To investigate synchronic emergence, we need to compare the solutions for equation (2)

6 See, for instance, Holmes 1995, 224ff., and Rueger 2006. The example can be modified to contain a time variable and so describe a process: see Frisch 1995, 226-228.


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with the solutions describing the rod from the macro-level perspective. At the macro-level, we

assume that the rod’s structure is continuous rather than atomic, and hence that conductivity is

steady rather than rapidly varying. The description of temperature distribution associated with

the macro-level will thus be different from that associated with the micro-level, but this

difference alone isn’t enough to show that the one is irreducible to or emergent with respect to

the other. To test for reducibility, we need to show that the discrete description of the rod on the

micro scale, indicated by the spacing P between individual atoms of the rod, converges to a

continuous description at a larger scale, characterized by the macroscopic length L of the system,

as the ratio

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ε = P L→ 0 . This ‘continuum limit’ reflects the intuitive requirement that the

macroscopic representation smoothes out the details at the micro level. We therefore seek an

expansion of the solutions of eq. 2 in terms of the small parameter

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ε = P L and expect to obtain,

in the limit

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ε → 0 , the solution of the sought-for macroscopic equation, T0(x):

Eq. 3

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T(x) = T0(x) + εT1(x) + ε2T2(x) +…

A successful relation of this sort would show that the macro description reduces to the micro

one: even though the descriptions are different, the macro one could be seen as a direct

consequence of the micro.

It turns out, though, that letting the parameter ε go to zero results in a singular limit. In

general – that is, unless

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k(x, xε ) is chosen in special ways – the solution T(x) will therefore not

converge uniformly to T0(x) in the limit

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ε → 0 . That is, we have

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limT(x) ≠ T0(x) , for some x

T0(x) is not reducible to T(x). In topological terms, the topological space in which the solutions

of the micro equation live (for

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ε → 0) cannot be characterized by the topology of uniform

convergence.


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The singular limit, however, can be tamed if we explicitly introduce two length scales in

the micro description, the macroscopic scale x and a microscopic scale

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y = x ε .7 We first

replace eq. 3 with an expansion of temperature T in terms of both x and y:

Eq. 4

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T(x,y) ≈ T0(x,y) + εT1(x,y) + ε2T2(x,y) +…

We then substitute the right-hand side of this expansion for T in eq. 2 (the micro theory), and

attempt to solve for the various Tis: the result should give us a good approximation of the exact

solution, T(x, y).

The advantage of eq. 4 over eq. 3 is that by explicitly distinguishing between the two

scales, we can impose constraints on the Tis that guarantee that the series is ‘asymptotic’: the

higher order terms don’t become larger than the lower order terms as

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ε → 0 . Note that the series

in eq. 4 is not convergent: adding more terms to the expansion does not necessarily give us a

better approximation of the exact solution to eq. 2 (or, more precisely, the multi-scaled version of

eq. 2), but truncating the series after a few terms will give us a good approximation of T(x, y).

Imposing these constraints has interesting results. The leading term in the approximation

– T0(x, y) – turns out not to depend on the microscopic variable, y. Instead, it depends only the

macroscopic variable, x. Thus, this term represents a purely macroscopic quantity. Furthermore,

it turns out that the constraint that allows us to force the expansion to remain asymptotically

valid -- the ‘solvability condition’ –– is precisely the macroscopic heat conduction equation we

are trying to recover from the micro equation (2):

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( ddx ) K(x) ddx T0[ ] = 0 , where K(x) is the

‘effective’ macro conductivity – a sort of average over the micro conductivity but not the simple

7 An analogous procedure can be applied in the damped oscillator case to handle the singular limit: by introducing two time scales, a ‘fast’ one to describe the oscillations of the system and a ‘slow’ one to characterize the decay of the amplitude, we can produce an asymptotic approximation of the undamped behavior even though this approximation does not converge uniformly.


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arithmetic mean of the k(x, y) over the length of the rod that one might have expected. It’s

important to stress that we don’t simply assume that the leading term in the expansion is

constrained by this equation: that constraint falls out of the effort to keep the later terms in the

expansion from diverging too quickly, and thus ruining the approximation. The macro equation,

thus, arises as a constraint that has to be imposed on the micro-level description of the behavior

of our system.

With these constraints in place, eq. 4 can be used to give a close approximation of the

exact solution of eq. 2. Asymptotic expansions typically give good approximations after only a

few terms: in fact, the leading term alone, T0(x), gives an empirically adequate, and

mathematically justified, approximation of the exact solution. Still, there is still no reduction of

one description to the other: the approximation of the micro solution is well behaved, but it does

not uniformly converge on the macro solution.

As discussed earlier, irreducibility and qualitative novelty can be seen as two closely

related ways of formally characterizing the core intuition that emergence phenomena are

different in kind from their ‘base level’ or ‘resultant’ phenomena. The rod example satisfies this

criterion of emergence. Furthermore, it is an illustration of a fairly general phenomenon in

physics: whenever a problem is characterized by two scales of very different magnitude, relating

the behavior at one scale to that at the other will typically involve a singular limit.

Philosophical questions about synchronic emergence

(i) Epistemic vs. ontological emergence

Since the limit

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ε → 0 does not represent a change in the system (in contrast to the parameter

change in the diachronic case) one might think that only the epistemic interpretation is available:


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the irreducibility of the macro behaviour to the micro is due merely to the irreducibility of one

type of description to another – nothing about the system itself has been shown to be irreducible

or emergent. Furthermore, the multi-scaling method we described does eventually lead to a sort

of derivation of the macro-level description from the micro, and we could interpret this more

sophisticated relationship as a demonstration that the difference between macro and micro is

merely perspectival. However, an ontological interpretation is also available, and in many ways

this interpretation is more natural.

We saw that although the macro behavior of the system is not reducible to the micro

behaviour a quantitative approximation of the macro behavior can be achieved if we explicitly

distinguish a macro scale from a micro scale in the description of the system. What we get,

€

T x,y( ) , is an approximation of the system’s micro scale behavior that includes its macro

behavior as one component – the leading term in the expansion of eq. 3. If we take this result

ontologically seriously8, we are led to the conclusion that macro and micro behavior of the

system are not entirely distinct; their relation is one of part to whole. That’s what a literal

reading of the expansion eq. (3) indicates: T0(x) is part of T(x, y). 9 The behavior of the heated

rod, the distribution of temperature over the length of the system, consists of several

components, one of which is the purely macro behavior T0(x). On this view, the operation of

8 Arguments for why we should do so are given below. 9 It might seem arbitrary to interpret an equation like

T(x) = T0(x) + εT1(x) + … as representing a part-whole relation in which T(x) is the whole. The equation itself is symmetric in the sense that we can just as well write

T0(x) = T(x) - εT1(x) + … , so that it looks as if T0(x) might be the whole and T(x) a component. But there is no arbitrariness here. Although the equations are symmetric, the asymmetry required for our interpretation is introduced by the perturbation approach itself. We are looking for a representation of the system’s (total) behaviour, an appropriate solution of the equations of motion, which is T(x). T0(x), by contrast, solves the equations of motion only approximately, at the lowest order (e.g., ε0) of the perturbation theory; the complete solution is T(x) and therefore we are justified in interpreting T(x) as the whole and T0(x) as a component.


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taking the limit

€

ε → 0 does not represent a process of changing the system, but instead a process

of paring down the system’s detailed micro level behaviour to its core component: the smooth

macro level behaviour that many slightly different systems can share.

The claim that a system’s macro scale behavior could be part of its micro scale behavior

might at first glance seem absurd: if anything, we might think, the parthood claim should be the

other way around, since it is smaller (micro) things that are part of larger (macro) things. When

we consider the parts of entities, we normally expect parts to be smaller than, and spatially

contained within, whatever ‘whole’ they compose. Call this familiar sense of parthood spatial

parthood. Obviously, the macro behaviour is not a spatial part of the system’s behaviour. But

the spatial sense is not the only way to think about parthood. There are a variety of senses of

‘parthood’ that do not imply a particular spatial relation between parts and wholes. One sense

that is appropriate for our case has been frequently used in the philosophy of mind. Shoemaker

and others have suggested that the ‘realization’ relationship between a supervenient property and

its realizer should be understood as a type of parthood relation between properties: realized

properties are (non-spatial) parts of their realizers.10 This idea is then explicated in terms of a

subset relation between causal powers: the causal powers of realized properties form a subset of

those of their realizing properties. In our case, the corresponding claim needs to be that the

causal powers of the macro behavior are a subset of those of the micro (or mixed) behavior.11

10 See Shoemaker 2001, 78ff. (with further references). Compare also Yablo 1992. One of the attractions of the subset view is that, as Lewis (1991) notes, the subset relation satisfies the standard axioms of mereology. 11 The subset view of realization has also been used to characterize the difference between properties that are reducible and those that are not (cf. Wilson1999). In cases of successful reduction, the causal powers of the realized property form an improper subset of the powers associated with the realizing property. In cases where reduction fails, the powers of the realized property are a proper subset of the set of powers of the realizing property.


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(ii) Supervenience

In the case of synchronic emergence, one might expect emergent properties to supervene on their

base properties (though this is controversial). When we discussed the diachronic example, we

noted that the question about supervenience doesn’t seem to be well posed. The question in the

synchronic case is whether there could be any variations in the macro description without any

changes in the micro description. Since the series for T(x, y) does not converge, one might think

that this question is again ill-posed. But this is not the case. Once we have chosen the series of

coefficients of T0(x), T1(x. y), and so on, in eq. 4 (that is, in our case, the series ε0, ε1, …), the

asymptotic expansion of T(x, y) is uniquely determined (cf. Holmes 1995, 11). So, despite the

fact that this expansion does not converge, its leading term (representing the macro property)

cannot vary without variation in the function (representing the micro property) which the series

asymptotically approximates. And while expansions of the same function in terms of other

series of coefficients may well have different leading terms – thus perhaps threatening macro-

micro supervenience – there is an important constraint that is used to select one expansion series

as the most appropriate for a given problem. That constraint is that the series of coefficients

should give us a good asymptotic approximation of the function with as few terms of the

expansion as possible. Given this constraint, the choice of coefficients – and hence the terms of

the expansion – is determined, and supervenience of the macro on the micro is secured.

(iii) Causation

How do the causal powers associated with the macro-level description of the rod relate to those

associated with the micro-level? If the heated rod case is formulated as a dynamic problem (with


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a time variable and introducing two time scales in addition to the spatial scales: see Frisch 1995,

227), we can set up a diagram analogous to diagram 2 in the discussion above.

Diagram 3

In contrast to the diachronic oscillator case, however, it now is inappropriate to interpret the

vertical arrows as causal relations; the change in the parameter in the rod example doesn’t

represent a change the system itself. But from the fact that the diagram does not commute we

can see that the horizontal causal relations, representing micro and macro causation, are not

equivalent in the sense that there is no homeomorphism that would map the micro onto the

macro on both the left and the right hand sides.12

The remaining question about emergent causation concerns whether or not the causal

efficacy of the macro can be distinguished from that of the micro in a way that alleviates any

worries about the two ‘competing’, or the micro ‘excluding’ the macro as a genuine cause. Here

we can avail ourselves of an argument of Yablo’s (1992, 434f.). Yablo defends the causal

efficacy of higher-level properties and rejects the claim that these properties are causally

preempted by the lower level properties on which they supervene. For this purpose, he develops

a notion of causation that imposes two requirements on property instances (or events) so that 12 Lest this be interpreted as a violation of supervenience of the macro on the micro, remember that the vertical arrows do not represent the supervenience relation.


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property instance C causes instance E: (i) C has to be sufficient for the occurrence of E. This

condition, if it is satisfied by C, is also satisfied by any C* such that C is a part of C*. To rule

out C* as causing E, Yablo imposes that (ii) C is not only sufficient but also required for E. If C

is required for E, then any C** that is a part of C will not be sufficient for E. Genuine causes, in

other words, have to be commensurate or proportional to their effects, that is, the cause has to be

sufficient and required for the effect. Some property instances are sufficient to bring about an

effect E but, intuitively, they contain ‘too much’ detail that is not required for causing E; other

property instances, although causally relevant for E, contain ‘not enough’ detail to cause E. The

cause of E strikes the balance and is therefore called proportional to the effect.

Applied to our case of macro and micro behavior, the proportionality requirement tells us

that, depending on the effect we are interested in, T(x, y) will sometimes be the cause but for

other choices of effect we’ll have to select T0(x) in order to satisfy proportionality of cause and

effect. The worry about T0(x) not having a causal role distinguishable from that of T(x, y) can

now be countered by pointing out T0(x)’s causal efficacy: for certain effects, T0(x) cannot be

replaced by T(x, y) as the genuine cause. Thus we see the nonuniform limit which leads from

T(x, y) to T0(x) as a procedure for isolating that part of the microscopic behavior that is causally

efficacious at the macro level, i.e., the part which can satisfy the proportionality requirement for

macro causes. As we have seen, in our example the limit can be taken (approximately)

successfully only if we introduce an independent macro scale besides the micro scale. It is

tempting to characterize this method as stripping away or eliminating causally extraneous (here:

microscopic) detail. But this mustn’t be misunderstood: the continuum limit does not change the

system as given by the microscopic description of the rod. The limit operation rather isolates a

part of the behavior of the given system.


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This perspective is relevant to the question (mentioned earlier) of whether the macro

description is just an ‘idealization’ which mustn’t be taken ontologically seriously. It should be

clear now that there is an ambiguity in the notion of idealization on which this objection relies.

The macro description is idealized compared to the micro description because the former ‘leaves

out’ certain causal powers included in the micro characterization. This is one sense of

idealization. Another sense is that, therefore, the macro description must be false. But this

second meaning of idealization is not equivalent to the first. According to the argument in this

section, we can say that the macro description is true of the macro phenomena whose reality is

secured by their having different causal powers than the micro base.

Does emergent causation necessarily involve ‘downward’ causation? Within the

mathematical framework we used downward causation would be illustrated, presumably, by the

effect of a higher-level constraint on the lower-level. In the heated rod case, imposing the

solvability condition on the solution of the micro equation might count as a relevant higher-level

constraint. Such interpretations have precedents (e.g., Bishop 2008, Sperry 1986). To the

extent that these interpretations can escape the charge of incoherence against synchronic

reflexive downward causation (Kim 1999), they are tenable. If the way in which they diffuse the

incoherence objection relies on a violation of supervenience, however, we don’t see how they

could be compatible with our example.

4. Ramifications

We said earlier that the relatively simple examples with which we illustrated our understanding

of emergence are representative of the crucial features that underlie the more spectacular cases of


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emergence in physics. What all of these cases share is the occurrence of a singular limit which is

responsible for the non-reducible and novel behavior observed.13

■ Phase transitions are often regarded as candidates for emergent behavior, that is,

transitions from an unordered to an ordered state (or vice versa) when a system parameter (like

temperature) is changed. Examples include the transition to ferromagnetism or when a gas

changes its state to a liquid. In all these cases, the behavior of the ordered phase seems

intuitively novel with respect to that of the unordered phase. Since a system parameter is being

changed, the examples are analogous to our discussion of diachronic emergence in the oscillator

case. To represent such phase transitions mathematically, the system has to be studied in the

‘thermodynamic limit’, the limit in which the volume of the system goes to infinity while the

density is kept constant. This limit is singular.

■ The occurrence of classical behavior at a macro level from a quantum mechanical

micro level is another standard candidate for emergence. Again, the differences are intuitively

striking enough to classify the classical behavior as novel with respect to the quantum

mechanical base. In our classification, this would be a case of synchronic emergence and,

indeed, the mathematical treatment of the relation between the levels involves taking the

‘classical limit’

€

h→ 0 of the Schrödinger operator

€

h2 2m (∂ 2 ∂x 2). This limit again turns out to

be singular.

Space does not permit a detailed comparison of our view with other proposals in the

recent literature. Batterman’s (2001) account of emergence is formally closely related to our

view, although he uses the more advanced techniques of the renormalization group to analyze the

13 More examples can be found in Primas 1998.


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extracting procedure that we phrased in terms of singular limits and multi-scale analysis.14

Batterman, however, shies away from giving an interpretation of his results in terms of causal

powers and thus his view should probably be classified as epistemic. Similar approaches can be

found in Primas (1983, 1998) and various work by Atmanspacher (e.g., Bishop/Atmanspacher

2006). A purely ontological account in terms of a ‘fusion’ operation between properties is given

by Humphreys (1997; 2008), a theory that seems to be applicable only to quantum phenomena.

A sophisticated metaphysical model of diachronic emergence that includes downward causation

is presented by O’Connor and Wong (2005). Both, Humphreys and O’Conner, require violations

of supervenience in their proposals. The relation of our view to Wimsatt’s (1997) suggestion of

understanding emergence as ‘violations of aggregativity’ is sketched in Rueger (2006).

References

V.I. Arnold 1983: Geometrical Methods in the Theory of Ordinary Differential Equations. New

York: Springer

R. Batterman 2001: The Devil in the Details. Oxford UP

R. Batterman 1995: ‘Theories between Theories.’ Synthese 103, 171-201

R. Bishop 2008: ‘Downward Causation in Fluid Convection.’ Synthese 160, 229-248

R. Bishop/H. Atmanspacher 2006: ‘Contextual Emergence in the Description of Properties.’

Foundations of Physics 36, 1753-1777

U. Frisch 1995: Turbulence. Cambridge UP

14 These techniques are related: see Goldenfeld 1992, 318-329, with further literature.


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N. Goldenfeld 1992: Lectures on Phase Transitions and the Renormalization Theory. Perseus

Books

M.H. Holmes 1995: Introduction to Perturbation Methods. New York: Springer

P. Humphreys 1997: ‘How Properties Emerge.’ Philosophy of Science 64, 1-17

Humphreys (2008): ‘Computational and Conceptual Emergence.’ Philosophy of Science 75,

584-594

J. Kim 1999: ‘Making Sense of Emergence.’ Philosophical Studies 95, 3-36

D. Lewis 1991: Parts of Classes. Cambridge (Mass.): Blackwell

T. Nickles 1973: ‘Two Concepts of Intertheoretic Reduction.’ Journal of Philosophy 70, 181-201

T. O’Connor/H. Y. Wong 2005: ‘The Metaphysics of Emergence.‘ Nous 39, 658-678

H. Primas 1983: Chemistry, Quantum Mechanics, and Reductionism. New York: Springer

H. Primas 1998: ‘Emergence in Exact Natural Sciences.’ Acta Polytechnica Scandinavia 91, 83-

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A. Rueger 2000a: ‘Physical Emergence, Diachronic and Synchronic.’ Synthese 124, 297-322

A. Rueger 2000b: ‘Robust Supervenience and Emergence.’ Philosophy of Science 67, 466-489

A. Rueger 2006: ‘Functional Reduction and Emergence in Physics.’ Synthese 151 , 335-346

S. Shoemaker 2001: ‘Realization and Mental Causation.’ In: C. Gillett et al. (eds), Physicalism

and Its Discontents. Cambridge: Cambridge UP, 74-98

R. Sperry 1986: ‘Macro- versus Micro-Determinism.’ Philosophy of Science 53, 265-270

W. Wimsatt 1997: ‘Aggregation: Reductive Heuristics for Finding Emergence.’ Philosophy of

Science 64, S372-S384

J. Wilson 1999: ‘How Superduper does a Physicalist Supervenience Need to Be?’ Philosophical

Quarterly 49, 33-52


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S. Yablo 1992: ‘Cause and Essence.’ Synthese 93, 403-449

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