# Contextual emergence

Harald Atmanspacher and Peter beim Graben (2009), Scholarpedia, 4(3):7997. | doi:10.4249/scholarpedia.7997 | revision #122848 [link to/cite this article] |

**Contextual emergence** characterizes a specific kind of relationship between different domains of scientific descriptions of particular phenomena. Although these domains are not ordered
strictly hierarchically, one often speaks of lower and higher levels of description, where lower levels are considered as more fundamental in a certain sense. As a rule, phenomena at higher levels of description are more *complex* than phenomena at lower levels. This increasing complexity depends on contingent conditions, so-called contexts, that must be taken into account for an appropriate description.

Moving up or down in the hierarchy of descriptions also decreases or increases the amount of *symmetries* relevant at the respective level. A (hypothetical) description at a most fundamental level would have no broken symmetry, meaning that such a description is invariant under all conceivable transformations. This would amount to a description completely free of contexts: everything is described by one (set of) fundamental law(s). Indeed, this is sometimes called (the dream of) a "theory of everything", but it is equally correct to call it – literally – a "theory of nothing". The consequence of complete symmetry is that there are no distinguishable phenomena. Broken symmetries provide room for contexts and, thus, "create" phenomena.

Contextual emergence utilizes lower level features as necessary (but not sufficient) conditions for the description of higher-level features. As will become clear below, it can be viably combined with the idea of *multiple realization*, a key issue in supervenience (Kim 1992, 1993), which poses sufficient but not necessary conditions at the lower level. Both contextual emergence and supervenience are interlevel relations more specific than a patchwork scenario as in radical emergence and more flexible than a radical reduction where everything is already contained at a lower (or lowest) level.

Contextual emergence is intended as a *structural* relation between different levels of description. As such, it belongs to the class of synchronic types of emergence (Stephan 1999). It does not address questions of diachronic emergence, referring to how new qualities arise dynamically, as a function of time. Contextual emergence also differs from British emergentism from Mill to Broad. An informative discussion of various types of emergence versus reductive interlevel relations is due to Beckermann et al. (1992), see also Gillett (2002).

Finally, it should be emphasized that contextual emergence is conceived as a relation between levels of descriptions, not levels of nature: It addresses questions of *epistemology* rather than ontology. In agreement with Esfeld (2009), who recently advocated that ontology needs to regain more significance in science, it would be desirable to know how ontological interlevel relations can be addressed. Contextual emergence, which decisively depends on epistemic contexts, is not designed for this purpose. (A possible option to relate epistemic and ontic stances along the lines of Quine's (1969) ontological relativity is due to Atmanspacher and Kronz (1999).)

## Contents |

## The conceptual scheme

The basic idea of contextual emergence is that, starting at a particular level \(L\) of description, a two-step procedure can be carried out that leads in a systematic and formal way (1) from an *individual* description \(L_i\) to a *statistical* description \(L_s\) and (2) from \(L_s\) to an individual description \(M_i\) at a higher level \(M\ .\) This scheme can in principle be iterated across any connected set of descriptions, so that it is applicable to any case that can be formulated precisely enough to be a sensible subject of a scientific investigation.

The essential goal of step (1) is the identification of equivalence classes of individual states that are indistinguishable with respect to a particular ensemble property. Insofar as this step implements the multiple realizability of statistical states in \(L_s\) by individual states in \(L_i\ ,\) it is a key feature of a supervenience relation with respect to states. The equivalence classes at \(L\) can be regarded as cells of a partition. Each cell can be regarded as the support of a (probability) distribution representing a statistical state.

The issue of composition or constitution, which is emphasized in alternative types of emergence, is to be treated in the framework of this step (1). In contextual emergence, however, the point is not the composition of large objects from small ones. Rather than size, the point here is that statistical states are formulated as probability distributions over individual states. This way they can at the same time be considered as compositions and as representations of (limited) knowledge about individual states.

The essential goal of step (2) is the assignment of *individual* states at level \(M\) to coextensional *statistical* states at level \(L\ .\) This cannot be done without additional information about the desired level-\(M\) description. In other words, it requires the choice of a *context* setting the framework for the set of *observables* (properties) at level \(M\) that is to be constructed from level \(L\ .\) The chosen context provides conditions to be implemented as a *stability* criterion at level \(L\ .\) It is crucial that this stability condition cannot be specified without knowledge about the context at level \(M\ .\)

The mentioned stability criterion guarantees that the statistical states of \(L_s\) are based on a robust partition so that the *emergent* observables in \(M_i\) are not ill-defined. (For instance, if a partition is not stable under the dynamics of the system at \(L_i\ ,\) the assignment of states in \(M_i\) will change over time and is not well-defined in this sense.) The implementation of a contingent context of \(M_i\) as a stability criterion in \(L_i\) yields a proper partitioning for \(L_s\ .\) In this way, the lower-level state space is endowed with a new, *contextual topology* (see Atmanspacher (2007) and Atmanspacher and Bishop (2007) for more details).

From a slightly different perspective, the context selected at level \(M\) decides which details in \(L_i\) are relevant and which are irrelevant for \(M_i\ .\) Differences among all those individual states at \(L_i\) that fall into the same equivalence class at \(L_s\) are irrelevant for the chosen context. In this sense, the contextually determined partition at \(L_s\) is based on both stability and relevance conditions.

This interplay of context and stability across levels of description is the core of contextual emergence. Its proper implementation requires an appropriate definition of individual and statistical states at these levels. This means in particular that it would not be possible to construct emergent observables in \(M_i\) from \(L_i\) directly, without the intermediate step to \(L_s\ .\) And it would be equally impossible to construct these emergent observables without the downward confinement arising from higher-level contextual constraints.

This way, bottom-up and top-down strategies are interlocked with one another in such a way that the construction of contextually emergent observables is *self-consistent*. Higher-level contexts are required to implement lower-level stability conditions leading to proper lower-level partitions, which in turn are needed to define those lower-level statistical states that are co-extensional with higher-level individual states and associated observables.

## Example: From mechanics to thermodynamics

As a concrete example, consider the transition from classical point mechanics over statistical mechanics to thermodynamics (Bishop and Atmanspacher 2006). Step (1) in the discussion above is here the step *from point mechanics to statistical mechanics*, essentially based on the formation of an ensemble distribution. Particular properties of a many-particle system are defined in terms of a statistical ensemble description (e.g., as moments of a many-particle distribution function) which refers to the statistical state of an ensemble (\(L_s\)) rather than the individual states of single particles (\(L_i\)).

An example for an observable associated with the statistical state of a many-particle system is its mean kinetic energy, which can be calculated from the distribution of the momenta of all N particles. The expectation value of kinetic energy is defined as the limit of its mean value for infinite N.

Step (2) is the step *from statistical mechanics to thermodynamics*. Concerning observables, this is the step from the expectation value of a momentum distribution of a particle ensemble (\(L_s\)) to the temperature of the system as a whole (\(M_i\)). In many standard philosophical discussions this step is mischaracterized by the false claim that the thermodynamic temperature of a gas *is* the mean kinetic energy of the molecules which constitute the gas. In fact, a proper discussion of the details was not available for a long time and has only been achieved by Haag et al. (1974) and Takesaki (1970).

The main conceptual point in step (2) is that thermodynamic observables such as temperature presume thermodynamic equilibrium as a crucial assumption, which we call a *contextual condition*. It is formulated in the zeroth law of thermodynamics and not available at the level of statistical mechanics. The very concept of temperature is thus foreign to statistical mechanics and pertains to the level of thermodynamics alone.
(Needless to say, there are many more thermodynamic observables in addition to temperature. Note that also a feature so fundamental as irreversibility in thermodynamics depends crucially on the context of thermal equilibrium.)

The context of thermal equilibrium (\(M_i\)) can be recast in terms of a class of distinguished statistical states (\(L_s\)), the so-called Kubo-Martin-Schwinger (KMS) states. These states are defined by the KMS condition which characterizes the (structural) stability of a KMS state against local perturbations. Hence, the KMS condition implements the zeroth law of thermodynamics as a *stability criterion* at the level of statistical mechanics. (The second law of thermodynamics expresses this stability in terms of a maximization of entropy for thermal equilibrium states. Equivalently, the free energy of the system is minimal in thermal equilibrium.)

Statistical KMS states induce a contextual topology in the state space of statistical mechanics (\(L_s\)) which is basically a coarse-grained version of the topology of \(L_i\ .\) This means nothing else than a partitioning of the state space into cells, leading to statistical states (\(L_s\)) that represent equivalence classes of individual states (\(L_i\)). They form ensembles of states that are indistinguishable with respect to their mean energy and can be assigned the same temperature (\(M_i\)). Differences between individual states at \(L_i\) falling into the same equivalence class at \(L_s\) are irrelevant with respect to a particular temperature at \(M_i\ .\)

While step (1) formulates statistical states from individual states at the mechanical level of description, step (2) provides individual thermal states from statistical mechanical states. Along with this step goes a definition of new thermal observables. All this is guided by and impossible without the explicit use of the context of thermal equilibrium.

The example of the relation between mechanics and thermodynamics is particularly valuable for the discussion of contextual emergence because it illustrates the two essential construction steps in great detail. In addition to the work quoted, a more recent account of what has been achieved and what is still missing is due to Linden et al. (2008).

There are other examples in physics and chemistry which can be discussed in terms of contextual emergence: emergence of geometric optics from electrodynamics (Primas 1998), emergence of electrical engineering concepts from electrodynamics (Primas 1998), emergence of chirality as a classical observable from quantum mechanics (Bishop 2005, Bishop and Atmanspacher 2006), emergence of diffusion and friction of a quantum particle in a thermal medium (de Roeck and Fröhlich 2011, Fröhlich et al. 2011), emergence of hydrodynamic properties from many-particle theory (Bishop 2008).

## Mental states from neurodynamics

In the example discussed so far, descriptions at \(L\) and \(M\) are usually well developed so that a formally precise interlevel relation can be straightforwardly set up. The situation becomes more difficult in situations where no such established descriptions are available. This is the case in those areas of cognitive neuroscience or consciousness studies, focusing at relations between neural and mental states (e.g., the identification of neural correlates of conscious states). That brain activity provides necessary but not sufficient conditions for mental states, which is a key feature of contextual emergence, becomes increasingly clear even among practicing neuroscientists, see for instance the recent opinion article by Frith (2011).

For the application of contextual emergence, the first desideratum is the specification of proper levels \(L\) and \(M\ .\) With respect to \(L\ ,\) one needs to specify whether states of neurons, of neural assemblies or of the brain as a whole are to be considered; and with respect to \(M\) a class of mental states reflecting the situation under study needs to be defined. In a purely theoretical approach, this can be extremely tedious, but in empirical investigations the experimental setup can often be used for this purpose. For instance, experimental protocols include a task for subjects that defines possible mental states, and they include procedures to record brain states.

The following discussion will first address a general theoretical scenario (developed by Atmanspacher and beim Graben 2007) and then a concrete experimental example (worked out by Allefeld et al. 2009). Both are based on the so-called state space approach to mental and neural systems, see Fell (2004) for a brief introduction.

**Theoretical approach**

The first step is to find a proper assignment of \(L_i\) and \(L_s\) at the neural level. A good candidate for \(L_i\) are the properties of individual neurons. Then the first task is to construct \(L_s\) in such a way that statistical states are based on equivalence classes of those individual states whose differences are irrelevant with respect to a given mental state at level \(M\ .\) This reflects that a neural correlate of a conscious mental state can be multiply realized by "minimally sufficient neural subsystems correlated with states of consciousness" (Chalmers 2000).

In order to identify such a subsystem, we need to select a context at the level of mental states. As one among many possibilities, one may use the concept of "phenomenal families" (Chalmers 2000) for this purpose. A phenomenal family is a set of mutually exclusive phenomenal (mental) states that jointly partition the space of mental states. Starting with something like creature consciousness, that is being conscious versus being not conscious, one can define increasingly refined levels of phenomenal states of background consciousness (awake, dreaming, sleep, ...), sensual consciousness (visual, auditory, tactile, ...), visual consciousness (color, form, location, ...), and so on.

Selecting one of these levels (as an example) provides a context which can then be implemented as a stability criterion at \(L_s\ .\) In cases like the neural system, where complicated dynamics far from thermal equilibrium are involved, a powerful method to do so uses the neurodynamics itself to find proper statistical states. The essential point is to identify a partition of the neural state space whose cells are robust under the dynamics. This guarantees that individual mental states \(M_i\ ,\) defined on the basis of statistical neural states \(L_s\ ,\) remain well-defined as the system develops in time. The reason is that differences between individual neural states \(L_i\) belonging to the same statistical state \(L_s\) remain irrelevant as the system develops in time.

The construction of statistical neural states is strikingly analogous to what leads Butterfield (2012) to the notion of *meshing dynamics*. In his terminology, \(L\)-dynamics and \(M\)-dynamics mesh if coarse graining and time evolution commute. From the perspective of contextual emergence, meshing is guaranteed by the stability criterion induced by the higher-level context. In this picture, meshing translates into the *topological equivalence* of the two dynamics.

For multiple fixed points, their basins of attraction represent proper cells, while chaotic attractors need to be coarse-grained by so-called generating partitions. From experimental data, both can be numerically determined by partitions leading to Markov chains. These partitions yield a rigorous theoretical constraint for the proper definition of stable mental states. The formal tools for the mathematical procedure derive from the fields of ergodic theory (Cornfeld et al. 1982) and symbolic dynamics (Marcus and Lind 1995), and are discussed in some detail in Atmanspacher and beim Graben (2007) and Allefeld et al. (2009).

**Application to experimental data**

A pertinent example for the application of contextual emergence to experimental data is the relation between mental states and EEG dynamics. In a recent study, Allefeld et al. (2009) tested the method using data from the EEG of subjects with sporadic epileptic seizures. This means that the neural level is characterized by brain states recorded via EEG, while the context of normal and epileptic mental states essentially requires a bipartition of that neural state space.

The analytic procedure rests on ideas by Gaveau and Schulman (2005), Froyland (2005), and Deuflhard and Weber (2005). It starts with a (for instance) 20-channel EEG recording, giving rise to a state space of dimension 20, which can be reduced to a lower number by restricting to principal components. On this state space, a homogeneous grid of cells is imposed in order to set up a (Markov) transition matrix reflecting the EEG dynamics on a fine-grained auxiliary partition.

The eigenvalues of this matrix yield time scales for the dynamics which can be ordered by size. Gaps between successive time scales indicate groups of eigenvectors defining partitions of increasing refinement – in simple cases, the first group is already sufficient for the analysis. The corresponding eigenvectors together with the data points belonging to them define the neural state space partition relevant for the identification of mental states (Allefeld and Bialonski 2007).

Finally, the result of the partitioning can be inspected in the originally recorded time series to check whether mental states are reliably assigned to the correct episodes in the EEG dynamics. The study by Allefeld et al. (2009) shows perfect agreement between the distinction of normal and epileptic states and the bipartition resulting from the spectral analysis of the neural transition matrix.

## Macrostates in neural systems

Contextual emergence addresses both the construction of a partition at a lower-level description and the application of a higher-level context to do this in a way adapted to a specific higher-level description. Two alternative strategies have been proposed to contruct \(M_i\)-states ("neural macrostates") from \(L_i\)-states ("neural microstates") previously: one by Amari and collaborators and another one by Crutchfield and collaborators.

Amari and colleagues (Amari 1974, Amari et al. 1977) proposed to identify neural macrostates based on two criteria: (i) the structural stability of microstates as a necessary lower-level condition, and (ii) the decorrelation of microstates as a sufficient higher-level condition. The required macrostate criteria, however, do not exploit the dynamics of the system in the direct way which a Markov partition allows. A detailed discussion of contextual emergence in Amari's approach is due to beim Graben et al. (2009).

Another alternative, which can be applied to neural systems, is the construction of macrostates within an approach called computational mechanics (Shalizi and Crutchfield 2001). A key notion in computational mechanics is the notion of a "causal state". Its definition is based on the equivalence class of histories of a process that are equivalent for predicting the future of the process. Since any prediction method induces a partition of the state space of the system, the choice of an appropriate partition is crucial. If the partition is too fine, too many (irrelevant) details of the process are taken into account; if the partition is too coarse, not enough (relevant) details are considered.

As described in detail by Shalizi and Moore (2003), it is possible to determine partitions leading to causal states. This is achieved by minimizing their statistical complexity, the amount of information which the partition encodes about the past. Thus, the approach uses an information theoretical criterion rather than a stability criterion to construct a proper partition for macrostates.

Causal states depend on the "subjectively" chosen initial partition but are then "objectively" fixed by the underlying dynamics. This has been expressed succinctly by Shalizi and Moore (2003): Nature has no preferred questions, but to any selected question it has a definite answer. Quite similarly, our notion of robust statistical states combines the "subjective" notion of coarse-graining with an "objective" way to determine proper partitions as they are generated by the underlying dynamics of the system.

## Mental Causation

It is a long-standing philosophical puzzle how the mind can be causally relevant in a physical world: the *problem of mental causation* (for reviews see Robb and Heil 2009 and Harbecke 2008, Ch.~1). The question of how mental phenomena can be causes is of high significance for an adequate comprehension of scientific disciplines such as psychology and cognitive neuroscience. Moreover, mental causation is crucial for our everyday understanding of what it means to be an agent in a natural and social environment.
Without the causal efficacy of mental states the notion of agency would be nonsensical.

One of the reasons why the causal efficacy of the mental has appeared questionable is that a horizontal (intralevel, diachronic) determination of a mental state by prior mental states seems to be inconsistent with a vertical (interlevel, synchronic) determination of that mental state by neural states. In a series of influential papers and books, Kim has presented his much discussed *supervenience argument* (also known as *exclusion argument*), which ultimately amounts to the dilemma that mental states either are causally inefficacious or they hold the threat of overdetermining neural states. In other words: either mental events play no horizontally determining causal role at all, or they are causes of the neural bases of their relevant horizontal mental effects (Kim 2003).

The interlevel relation of contextual emergence yields a quite different perspective on mental causation. It dissolves the alleged conflict between horizontal and vertical determination of mental events as ill-conceived (Harbecke and Atmanspacher 2011). The key point is a construction of properly defined mental states from the dynamics of an underlying neural system. This can be done via statistical neural states based on a proper partition, such that these statistical neural states are coextensive (but not necessarily identical) with individual mental states.

This construction *implies* that the mental dynamics and the neural dynamics, related to each other by a so-called intertwiner, are topologically equivalent (Atmanspacher and beim Graben 2007). Given properly defined mental states, the neural dynamics gives rise to a mental dynamics that is independent of those neurodynamical details that are irrelevant for a proper construction of mental states.

As a consequence, (i) mental states can indeed be causally and horizontally related to other mental states, and (ii) they are neither causally related to their vertical neural determiners nor to the neural determiners of their horizontal effects. This makes a strong case against a conflict between a horizontal and a vertical determination of mental events and resolves the problem of mental causation in a deflationary manner. Vertical and horizontal determination do not compete, but complement one another in a cooperative fashion. Both together deflate Kim's dilemma and reflate the causal efficacy of mental states. Our conclusions match with and refine the notion of *proportionate causation* introduced by Yablo (1992).

In this picture, mental causation is a horizontal relation between previous and subsequent mental states, although its efficacy is actually derived from a vertical relation: the downward confinement of (lower-level) neural states originating from (higher-level) mental constraints. This vertical relation is characterized by an intertwiner, a mathematical mapping, which must be distinguished from a causal before-after relation. For this reason, the terms *downward causation* or *top-down causation* (Ellis 2008) are infelicitous choices for addressing a *downward confinement* by contextual constraints.

## Remarks

1. The combination of contextual emergence with supervenience can be seen as a program that comes conspicuously close to plain reduction. However, there is a subtle difference between the ways in which supervenience and emergence are in fact implemented. (In a related sense, Butterfield (2011a,b) has argued that emergence, supervenience and even reduction are not mutually incompatible.) While supervenience refers to states, the argument by emergence refers to observables. The important selection of a higher-level context leads to a stability criterion for states, but it is also crucial for the definition of the set of observables with which lower-level macrostates are to be associated.

2. Statistical neural states are multiply realized by individual neural states, and they are coextensive with individual mental states; see also Bechtel and Mundale (1999) who proposed precisely the same idea. There are a number of reasons to distinguish this coextensivity from an identity relation which are beyond the scope of this article. For details see Harbecke and Atmanspacher (2011).

3. The reference to phenomenal families a la Chalmers must not be misunderstood to mean that contextual emergence provides an option to derive the appearance of phenomenal experience from brain behavior. The approach addresses the emergence of mental states still in the sense of a third-person perspective. "How it is like to be" in a particular mental state, i.e. its qualia character, is not addressed at all.

4. Besides the application of contextual emergence under well-controlled experimental conditions, it may be useful also for investigating spontaneous behavior. If such behavior together with its neural correlates is continuously monitored and recorded, it is possible to construct proper partitions of the neural state space. Mapping the time intervals of these partitions to epochs of corresponding behavior may facilitate the characterization of typical paradigmatic behavioral patterns.

5. It is an interesting consequence of contextual emergence that higher-level descriptions constructed on the basis of proper lower-level partitions are compatible with one another. Conversely, improper partitions yield, in general, incompatible descriptions (beim Graben and Atmanspacher 2006). As ad-hoc partitions usually will not be proper partitions, corresponding higher-level descriptions will generally be incompatible. This argument was proposed (Atmanspacher and beim Graben 2007) for an informed discussion of how to pursue "unity in a fragmented psychology", as Yanchar and Slife (2000) put it.

6. Another application of contextual emergence refers to the symbol grounding problem posed by Harnad (1990). The key issue of symbol grounding is the problem of assigning meaning to symbols on purely syntactic grounds, as proposed by cognitivists such as Fodor and Pylyshin (1988). This entails the question of how conscious mental states can be characterized by their neural correlates, see Atmanspacher and beim Graben (2007). Viewed from a more general perspective, symbol grounding has to do with the relation between analog and digital systems, the way in which syntactic digital symbols are related to the analog behavior of a system they describe symbolically. This might open up a novel way to address the problem of how semantic content arises as a reference relation between symbols and what they symbolize. This problem is not restricted to cognition; it may also be a key to understand the transition from inanimate matter to biological life in information theoretical terms (Roederer 2005).

7. For additional directions of research that utilize ideas pertaining to contextual emergence in cognitive science and psychology see Tabor (2002), beim Graben et al. (2004), Dale and Spivey (2005) and Jordan and Ghin (2006). They are similar in spirit, but differ in their scope and details.

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**Internal references**

- John W. Milnor (2006) Attractor. Scholarpedia, 1(11):1815.

- Edward Ott (2006) Basin of attraction. Scholarpedia, 1(8):1701.

- Valentino Braitenberg (2007) Brain. Scholarpedia, 2(11):2918.

- Olaf Sporns (2007) Complexity. Scholarpedia, 2(10):1623.

- Paul L. Nunez and Ramesh Srinivasan (2007) Electroencephalogram. Scholarpedia, 2(2):1348.

- Tomasz Downarowicz (2007) Entropy. Scholarpedia, 2(11):3901.

- Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.

- Rodolfo Llinas (2008) Neuron. Scholarpedia, 3(8):1490.

- Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.

- David H. Terman and Eugene M. Izhikevich (2008) State space. Scholarpedia, 3(3):1924.

- Brian Marcus and Susan Williams (2008) Symbolic dynamics. Scholarpedia, 3(11):2923.