Contextual emergence
| Harald Atmanspacher and Peter beim Graben (2009), Scholarpedia, 4(3):7997. | doi:10.4249/scholarpedia.7997 | revision #200061 [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. 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 desicively depends on epistemic contexts, is not designed for this purpose.
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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 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.
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 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).
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.
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 dimention 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 L_s-states ("macrostates") from L_i-states ("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 statistical states L_s based on their decorrelation in the neural state space. The macrostate criterion that they require for the stability of these states, however, does not exploit the dynamics of the system in the direct way which a Markov partition allows. A detailed comparison of macrostate criteria in contextual emergence and 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.
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. While supervenience refers to states (mental and neural), 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. 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.
3. The formal method to identify proper partitions from experimental data allows in principle as many partition cells as there are eigenvalues of the Markov matrix. If its spectrum shows time-scale gaps, they serve to establish a hierarchy of refined partitions. This opens a controlled way to proceed to more refined mental states than addressed in the application example discussed above.
4. 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.
5. For additional directions of research that utilize ideas pertaining to contextual emergence in cognitive science and psychology see Dale and Spivey (2005) and Jordan and Ghin (2006).
6. In mature basic sciences such as physics or chemistry, the interlevel relation of contextual emergence typically substantiates in detail already existing schemes and ideas. Its full power is to expected in further applications to cognitive neuroscience, where levels of description are not yet ultimately specified or even formalized and higher-level properties can be actively constructed. Beyond the emergence of neural macrostates and mental states, another viable area of corresponding research will be the emergence of behavioral states.
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