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Reinterpreting Memetics in a Multi-Level View of Evolution and Behaviour
Part I: Conceptual Analysis
(Draft Version)

Robert Clewley
Dept. of Engineering Mathematics, University of Bristol, United Kingdom

23 September 1998

Abstract

Part II of this essay is a synthesis -- a discussion of how this situation might be improved, given that there are aspects of the Memetic Stance [Den] which are very important to retain. Concepts already well developed in cognitive and evolutionary psychology, and in the social sciences, are applied to the problem of reinterpreting memetics in those terms. The multi-level aspect is motivated using ideas from dynamical and cybernetic systems theory.

Richard Dawkins effectively defined the field of memetics by introducing the term "meme" as "a unit of cultural transmission or a unit of imitation" [Daw]. Memes are defined to behave as "self-replicators", as part of the process of their transmission between human "hosts" [Ly2]. This mechanism is loosely inspired from epidemiology, and particularly virology. Despite the variations of the definition of memetics in the literature, the original definition is sufficient for the purposes of this paper.

Section 2 motivates the importance of certain theoretical issues in the scientific understanding of "complex" and "emergent" phenomena. These issues are relevant to memetics, which can be viewed as treating memes as emergent features of a neuropsychological process, and culture as emerging from the behaviour of memes. A philosophical framework is briefly introduced, from which an idealised view of scientific methodology is developed. This view allows a clear analysis of the issues described in Section 3 as they concern existing theories, but the methodology cannot be immediately applied to build new theories relating to "complexity" and "emergence". However, Section 2.4 summarises the adjustments that are needed in order to remedy this. In doing so there remain inherent limitations to practical models, which must therefore be identified in memetic theory. This view allows some possible philosophical objections to memetics to be assessed.

The second part of this essay [Cl2] follows up the analysis of this paper, and suggests that the potential problems of memetics might be avoided under a suitable reinterpretation of the role of memetic theory as part of a "multi-level" view of human behaviour. Such "unification" with other theories of culture and cognition could provide a more complete scientific picture. A possible reinterpretation of memetics is outlined, using ideas from the present methodology.

The science of the objects and interactions of normal human experience is not unified in its foundations. There can be significant incompatibility between scientific theories having overlapping domains but which use language based on different representations of natural features. For example, the fact that the mind-body problem remains unresolved means that theories of human behaviour and neural behaviour still use very different elementary objects and interactions. Explaining the diversity of organic life based on an analysis of pre-biotic chemistry has similar difficulties. It is appealing to consider the possibility of an "ontological unity" [Kan] under a "reductionist" assumption, so that, for instance, a theory with one set of basic representations could ultimately underlie all accounts of the observations at the levels of both neurobiology and social behaviour.

There are two main reasons why such problems of unity remain unresolved. Firstly, a natural process is involved which works simultaneously at very different scales of time, or space, or both. Secondly, an observer is able to appreciate the variety of form and behaviour exhibited by that process at the different scales taken individually, but has difficulty relating them precisely in an appreciation of the process as a whole. Thus an analysis is needed of the philosophical inconsistencies between traditional single-scale models that ultimately represent aspects of the same process.

The general advantages of ontological unity are essentially twofold: obtaining logical consistency between theories having non-identical domains of applicability, and obtaining a coherence which can "transcend the narrow borderlines of each [traditional] discipline" [Kan, p. 262]. Together, these goals motivate the idealised scientific methodology developed in the next section. However, a full account of the argument advocating a search for ontological unity in the social sciences can be found in Tooby and Cosmides [Too].

There is continued under-appreciation of the role of methodology in making scientific progress, such as the strive for unity in science. However, "methodological well-foundedness has been constitutive of, rather than tangential to, the most important appraisals of theories." [Lau, p. 59]. In this section, a view of scientific methodology is summarised which is inspired from the issues discussed above.

The methodology that is only outlined in this paper is designed to be general purpose. Its outline has many similarities to the methodologies discussed in the "complex systems" and cybernetics literature (for instance, the work of Crutchfield [Cru], Heylighen [Hey], or Hoppensteadt [Hop]), but its detailed development can be found elsewhere [Cle]. In aspiring to be general purpose it has to deal with the most formal theories of science: the axiomatic mathematical theories commonly found in theoretical physics, for instance.

This paper discusses an application of the methodology, and because models in social science are based much more in natural language than axiomatic mathematics, many of the technical aspects of the methodology are not required here. However, the methodology represents an idealisation of any scientific theory and its relationships with other theories. Many of the definitions in this section that are primarily intended for formal theories nevertheless suggest a useful view of social and psychological theories. To the extent that the definitions can be used to characterise the form of social and psychological theories (Section 5), the analytical methods discussed below can provide new insights.

Central to the proposed methodology is a form of scientific instrumentalism, similar to that used by d'Espagnat [dEs]. In this view, an observer has a certain, limited, perceptual ability, and a certain set of physical instruments (having intrinsic limits of resolution). Theories are seen as syntheses of measurements and the consequent perceptions of form that an observer is capable of. Scientific theories relate to perceptions of structure and function in natural phenomena that can differ greatly according to the scale at which the observer measures. Clearly some uniformity is necessary between the instruments and perceptions of different observers, in order that observations can be corroborated and theories can be generally accepted. However, as a methodology no assumptions of "absolute scientific truth" or "independent physical reality" are required, so that the methodology may be almost universally applicable throughout the natural sciences.

The ideal scientific "explanation" of a natural phenomenon would be a theory that could predict the observations possible under all scientifically valid experimental conditions. This would include pathological cases such as the extremes of the time or space scales of the observer. The view that in principle a "single theory" could achieve this may be termed methodological reductionism (MR) (similar to its use in Ref. [Kup], or the "ontological reductionism" as used in Ref. [Sto]). By a "single theory" it is meant that causal explanations of all observable features of the phenomenon can be provided by a set of formal models which can be precisely related in a unified, multi-scale theory¹. A theory with these qualities will be referred to as a comprehensive theory, and is the basis of MR explanation as used here. (A clearer definition will be made once the rest of the methodology is introduced.) The existence of a comprehensive theory of a natural phenomenon will provide the instrumentalist meaning of "ontological unity" (or simply "unity", for convenience).

Formalism affords logical clarity and power in accounting for the structures and mechanisms in a complex system. In MR, it is common that formalisms are given the primary role in the explanation of a natural phenomenon. Examples of formalisms are differential or integral equations, maps, and symbolic calculi, which have different inbuilt notions of time, space, and state. In this methodology the term "model" will be associated with a formal dynamical system that is given an explicit interpretation. The interpretation gives meaning to the model's elementary atomic components and formal mechanisms, by relating them to the properties of a more concrete model (that reflects a more detailed view of the phenomenon). The interpretation can be seen as an interface between the pair of levels. The higher and lower levels of the pair may then be called the emergent and underlying levels, respectively. Thus the form of a model is a set of representations of some chosen structural and functional aspects of the observed process. The representations are of a more or less abstract (or conversely, concrete) character, which motivates the idea of a model's level of representation (subsuming the intuitive notion of "scale").

An interface defines the domain of applicability of the emergent model relative to the underlying model. Three aspects of this domain are distinguished here: the temporal, spatial, and state sub-domains of the underlying model. These are mathematically precise concepts. For convenience, the three sub-domains together will be called a spatio-temporal-state domain (STS-domain), where it is understood that it is in fact a sub-domain of the underlying level's model.

The logical consistency of the interface is the degree to which the elementary components and mechanisms of the emergent model (seen there as a set of axioms) can be derived from the dynamic properties of the underlying model's restriction to the STS-domain. A (totally) logically consistent pair of models have an interface that consists entirely of theorems relating the underlying level's features to the formal elements of the emergent level model.

It is now possible to define a comprehensive theory of a natural phenomenon as one that can demonstrate a logically consistent hierarchy of mathematical models at all levels of representation.

Some adjustments are necessary in order to make the idealistic methodology practicable. This is because (a) it has no a priori base level of analysis; and (b) it asserts that "true explanations" can be achieved only by comprehensive theories, which potentially involve model hierarchies of arbitrary "height" and "density".

The progress of scientific explanation involves the testing of working hypotheses, leading to their gradual sophistication or replacement. This justifies the temporary use of a root level in a model hierarchy, until developments in instrumentation or mathematical tools permit a more accurate and lower-level model to be put below it. This resolves problem (a), whilst acknowledging the practical sense of the term "explanation" as being relative to the contemporary model of "reality".

Intrinsically dominant scales are evident in the perception of natural processes. Thus it is sufficient to select a small set of appropriate levels of representation above the root level, thereby resolving problem (b). A general view of levels is motivated by Foster [Fos], and an illustration of it is found in Lister and Weingarter [Lis, Ch. 1].

Finally, the form of efficient models in practice is seldom totally determined by an underlying level. If it were, the models would often be cumbersome, losing their descriptive flexibility and conceptual clarity. Thus it is unavoidable that some features will be introduced into the models that are not present in a more concrete view of the phenomenon. These may be called artefacts. The ideal of total logical consistency is therefore not attainable. Nevertheless, a suitable degree of consistency may be agreed upon, when the inconsistencies due to the artefacts are minimised by suitably restricting the domain of applicability of the model.

3. A critique of memetics

According to the definitions of Laudan [Lau], a conceptual problem in a theory is internal if there is logical inconsistency or vagueness of its basic categories of analysis, and external if there is a logical incompatibility with another theory that has an overlapping domain of application. In the context of the present methodology a symmetry between these forms of conceptual problem can be noted, which simplifies the ideas behind a conceptual analysis.

Consider the internal conceptual problems of a theory. In a multi-level, MR view of nature the basic categories of any theory residing above the root level can be analysed as emergent features of some underlying level. As with an "external" problem, therefore, an "internal" problem of a theory also relates to a logical incompatibility with another theory, providing that this other theory is at an underlying level of representation and the two theories are related at an interface. The logical consistency of this interface is the issue in question when considering an "internal" conceptual problem of the emergent theory. In the analysis undertaken below, the emergent level is memetics and the underlying level is usually taken to be cognitive psychology.

Now consider external conceptual problems. By the definitions given in Section 3, there are no interfaces between theories that are not in an emergent-underlying level pair. This feature was chosen to reflect the fact that a comparison between theories having overlapping domains of application requires the existence of "common ground" -- a frame of reference against which the comparison may take place. Naturally, the MR view is that the "common ground" is actually an underlying theory, upon which there are two interfaces that make suitable choices of abstraction from the underlying theory in order to construct the form of two different emergent theories (see Figure 1). External conceptual problems are most easily spotted and understood by comparing these two interfaces. This is because the interfaces explicitly represent the choices about the form and interpretation of emergent-level models (in the technical senses of the terms "form" and "interpretation" developed in Clewley [Cle]), and thus the logical compatibility of these different choices is transparent and unambiguous.

In the present application of these ideas it is the traditional models from the realms of social psychology and evolutionary psychology that are the theories having overlapping domains of application with memetics. That is, in some sense they can all be based in terms of individual psychology and the levels that are implied below that (loosely, biological and ecological levels of representation, that fit the physical individual into an environment of other individuals and objects).

The issue of conceptual problems in current memetic theory can now be discussed. It is important to remember that the analysis will always be referring to the current state of memetics--that is, as a new discipline which has not yet found its feet. The criticisms of some recent attempts to develop memetic theories are therefore not to be taken as suggesting that memetics has no future. It may overcome its problematic areas after suitable adjustments are made. Just such an attempt to address these problems is discussed in the second part of this essay [Cl2].

Consider the following statements.

1. Although the concept of a meme may seem appealing, it is misleading to think of its apparently symbolic and atomic nature as a true feature of a culture in reality. Contemporary memetics cannot justify a meme as any more than a metaphor.
2. Memetics cannot account for all cultural phenomena without oversimplifying a sophisticated multi-scale process.
3. Memetics is sterile and unscientific.

In order to judge their validity, the statements will be recast in the language of the proposed methodology, and each point will be treated separately. The first two require a conceptual analysis, whereas the third concerns an analysis of the empirical methodology of memetics.

This statement suggests a problem similar to the classic "symbol grounding problem" [Har]. Such problems rely on a distinction between the interpretation of an elementary component of a model (a) as a direct representation of a feature present in the root level -- i.e. the instrumentalist interpretation of "physical reality"; or (b) as a simplifying, abstract representation of a complex feature present in a lower level. Under interpretation (a), a meme would have an independent existence of its own, and it's form or function would be not be reducible to features of any lower-level model of reality. Ideas and their interactions are presumed to have a neuropsychological substrate, and acts of communication involve a physical mechanism. Therefore interpretation (a) can be rejected, meaning that the apparently symbolic and atomic nature of memes and their interactions is an artefact. What does in fact constitute the nature of a meme will be discussed later.

3.3.1 An internal conceptual problem

Regarding the issue of grounding, it is important to note that memeticists rarely make claims about the psychological or biological substructure of memes and their dynamics. For instance, consider a memetic formalisation due to Lynch [Ly2]. The basic objects of his model are "mnemons" (memory abstractions) and human "hosts", and the basic interaction between a host and a meme is a "replication event". Lynch aims for a "non-metaphoric memetics" by avoiding explicit analogy between memes and the replicators found in biology (genes, viruses) or computer science (software viruses). To do this a symbolic calculus is used to describe the replication process. This yields a clear description of an abstract mechanism, avoiding reference to the mechanisms found in the analogous domains, and so this aspect of memetics is internally logically consistent. However, the clarity of the symbolic mechanisms in a formalisation hides the vagueness of the interpretation of the symbols themselves. Formalisation alone cannot rid these basic representations of their intrinsic vagueness, and so an internal conceptual problem is present. In justification of this situation in contemporary memetics, Lynch [Ly2] claims:

"[Since] science has achieved no direct observation of the neural encoding of ideas, which might have provided a precise language for discussing ideas ... we must settle for an abstract representation of the information stored."

The admission that the mnemon, the basic concept of the model, has no direct interpretation in a lower-level theory of culture or cognition means that it has no precise meaning, according to the definitions of Section 2. Unfortunately, what meaning remains is largely due to the analogy with a gene, and for memetics to mature this is clearly undesirable. This will now be investigated further.

The conclusion of the previous section is that memes, as they are currently used in the literature, have not been adequately justified as anything more than metaphorical representations. Hope comes from realising that whilst the first part of Lynch's claim is undoubtedly true, it does not imply that science will never develop a "language based on a concrete mechanism of information storage" [Ly2]. In fact, a systematic attempt to develop such a language is vital, so that a purely abstract representation of a meme may only be a temporary necessity. Such an attempt is described in Part II [Cl2]. A direct way to treat the first statement is given in this section. The object of this is to expose potential inconsistencies of memetic theories, in order that they can be addressed explicitly in the synthesis of Part II.

The essence of the argument in this section will be to show that memetics mistakes the syntactic nature of a meme for a semantic nature. Dawkins refers to a meme as a "unit" of culture, and it will be suggested that this should rather be interpreted as referring to a basic "type" of cultural entity -- essentially, a syntactic class.

Memetic theories usually characterise the mechanisms of memes using a symbolic calculus that involves the atomic or composite memes and their simple combination, replication and mutation. The variables that represent memes in formalisms are syntactic entities (symbols). In order to justify their relevance to real culture, a suitable semantic interpretation of them is necessary. In other words, it is desirable that each basic concept in a typical memetic theory--such as a meme (or mnemon), a host, and a transmission mechanism--have a concrete interpretation in terms of entities at a lower level of representation, and have a uniform interpretation in all situations to which memetics is applicable.

The uniform nature of an interpretation is necessary so that all memes have the same underlying form. Effectively, this means that they would all belong to the same type in their calculus. This feature is essential in order that the calculus can apply consistently to all memes. If a concrete and uniform interpretation of a meme is possible, a meme would be a true "unit" of culture. Here, the intended sense of a meme as a "unit" is that "all ideas have the same underlying form that allows them to be compared". This is in the same sense that a kilogram is an objectively definable property of all masses, and so as a unit of mass it provides a precise dimensioned quantity on which to base measurements of one kind of "similarity" between physical objects. It is also in this sense that a gene is an objective unit of heredity².

The issue of the concreteness of a hypothetical interpretation at a lower level of representation has already been discussed (Section 3.3.1). It remains to address the issue of its uniformity.

3.4.1 The "sameness" of private ideas

Normally, definitions of a meme in memetic theories enable a meme to have an internal hierarchical structure made up of memes. This means that it is appropriate to attribute a recursive type structure to a formal meme. For instance, a symphony can be interpreted as a meme, but so can the individual tunes within it. This property of a recursive type structure in the syntactic view of a meme has an appealing sense of symmetry, but there is a limitation to its relevance in real situations. The limitation is that there can be no exhaustive analysis of the depth of an idea's internal structure in the mind. This is now demonstrated for ideas held by a single individual -- so-called "private" ideas.

A private idea is a subjective entity in the mind of an individual, and objectively it has no directly observable properties. In daily life people name, categorise and agree on ideas. These activities may lead to a belief that the concept of a private idea is a tangible, discrete, and decomposable entity. An individual who tries to decompose a private idea into its constituent parts in order to discover its origin in other ideas (related either logically or temporally) will realise that such a belief is fallacious. Although a single, relatively unambiguous, decomposition of an idea is often possible, at every further recursion an unambiguous decomposition becomes more difficult, and the origins of the idea become ever more intangible. An idea is not reducible to a closed hierarchy of constituent ideas because each level of the hierarchy (as an idea itself) suffers the problem of ambiguity in its definition by that individual. Thus the recursive type structure is not an entirely accurate picture.

The lack of uniformity in the "meme as a unit" scenario is now discussed. The problem with any hypothetical uniform grounding of memetics is that it would embrace the misleading analogy of the software-hardware characterisation of the mind and brain. This may, in part, be due to memetic principles drawing some inspiration from the nature of computer viruses. Here, the "software" in question is the mechanism of memes as typically described by a symbolic calculus, and the "hardware" underlying its form would be the brain processes associated with cognition.

A memetic theory is about mental events that represent the parts of the "life-cycle" of memes (infection, induced symptomatic behaviours including those responsible for re-transmission). Typically, the course of these events are assumed only to be influenced by other mental events. For instance, memetic theories typically consider limited characterisations of emotional events as brands of mental event, but their actual biological basis--as brain processes that are not consciously accessible--is assumed irrelevant, and not considered. Mental events are all given the same status in memetic theories, as if they had the same origin in the brain.

It is true that phenomenologically all ideas are mental entities, but their causal origins (and to some extent even the physical origins) in the brain can be argued to vary greatly depending on their content. This is partly due to the lessons of cognitive and evolutionary psychology, and partly due to what is known of the neurological basis for motivation, emotion, intention, inference, and so on. It has been strongly argued in the literature of cognitive neuroscience that there is a richly varied biological basis to the subjective experience of "ideas in general" which should not be dismissed (for instance, see Lister and Weingarter [Lis]). Hence the mental structures underlying different "memes" may work on very different principles and have different sources in the brain. This makes it crucial to know how the abstract forms that these structures appear to generate actually interact at the "hardware" level. Memetic theories tend to homogenise the nature of ideas without a solid justification.

The conclusion is that a real idea cannot be identical to a meme as it is typically described in memetic theories. However, a meme is clearly an approximation to an idea, taken in the abstract and unattainable limit of perfectly discrete and decomposable ideas (which would be true "units").

3.5 The second statement

The second statement demonstrates two examples of the external type of conceptual problem (Sections 3.5.2 and 3.5.4). The first example involves other theories of social cognition that use cultural transmission³ as their primary mechanism for cultural variation, but take into account a wider range of psychological factors in its action. This leads to predictions and posited mechanisms which are incompatible with a naïve approach to memetics. The second example concerns a theory of culture which posits a mechanism underlying cultural variation that is totally different in nature to the mechanisms proposed in memetic theories.

Memetic theories can appeal to an intuitive "sameness of ideas" in order to justify the notion that ideas can be effectively shared in a society of like-minded individuals, and thus take on an objective character. When two people are supposedly having the "same" idea in their minds, "we do not use æsameness' to mean equality in every concrete detail" [Ly2], because as Dawkins [Daw] points out, "almost any statement about two people agreeing with each other would be meaningless." This would appear reasonable, except that one has to be careful about the actual nature of an agreement. For the present purposes, let an idea be characterised as a mental "schema" consisting of a mixture of beliefs and possible behaviours that are relevant and appropriate to some particular situation. An agreement that two people hold the "same" idea requires each party to actively check the contents of the other's schema. As was remarked earlier, no exhaustive or unambiguous definition of an idea is possible, even by the individual who holds it. Furthermore, it is rarely socially appropriate to spend the requisite amount of time necessary to fully discuss the details of each party's version of the idea. Two people may honestly believe that they have reached an agreement on an issue. On the basis of their subsequent actions it can transpire that they actually had quite different ideas in mind at the time. In other words, an agreement guarantees only a degree of similarity between two people's ideas -- depending on the time spent discussing the issue and the ability of those two people to successfully communicate their meaning.

To a greater or lesser degree, the subjective and intangible nature of ideas, in their private and public form, is recognised in introductions to many memetic theories. Despite this recognition it seems not to be considered problematic to the validity of the assumptions of objectivity made subsequently. For instance, it is clear that there will be unmeasurable differences in the connotations and interpretations of a supposed "public" idea held by each individual of a group, for the reasons given above. In time, these differences may result in social behaviour that is inconsistent with a model that presumed the idealised transmission of ideas. An exception to memetic predictions of this kind would have occurred because memetics does not explicitly consider low-variance differences between individuals. This clearly means that the intrinsically subjective nature of ideas can be problematic to memetics.

For memetic transmission mechanisms, such as the process of imitation, there is not even a process of agreement involved between the imitator and the imitated. The detailed mental form and intention of an idea that was merely expressed by some limited physical action is not directly conveyed to an observer: there is no opportunity to verify that an idea formed about the observed actions of a person is anything like the observed person's idea.

The crucial aspect to social interaction is that at all times an observer is inferring a great deal of meaning from the very limited direct perceptions that are available. Such inferential processes are analysed in depth in the literature of social cognition theories, and are discussed briefly in the next subsection.

3.5.2 Social cognition theories

The process of cultural transmission is affected by a huge variety of complex socially- and individually-based psychological factors (which may themselves have some genetic basis -- see Section 3.5.4). If it does not equip itself with a sound psychological substrate, memetics will leave no room for the origin of these factors in its basic definitions. A calculus describing meme combinations and the basic form of transmission mechanisms (a) gives some impression of what is "spreading" from one individual to another, and (b) gives the gross organisation of those events (i.e. a rough impression of how cultural variation occurs). However, a first-order calculus is not able to adequately characterise why particular transmissions occur when they do. This can be attributed largely to the simplistic characterisation of a human mind as a "host" in memetic assumptions. It has been argued that ideas are not "software" that have a form independent of the "host hardware", and are neither discrete nor finitely-decomposable objects. In addition, the memetic "host" concept dangerously casts a human mind (a) as a passive receptacle for memes; and (b) as an oversimplistic vehicle for individual memes to control, and as a meme duplicator.

In contrast to this passive view of the mind, theories of social cognition claim that social interaction involves a reconstructive process rooted deeply in both the prior knowledge of the individuals concerned and the peculiarities of each situation. As Sperber [Spe, p. 31] puts it, "... recall is not storage in reverse, and comprehension is not expression in reverse." As part of the reconstructive process of perception, and as part of the reflective thoughts and actions that follow, a useful first approximation to gross mental structure is the concept of a set of "schemas". In rejecting a passive view of the mind as a host the presence of a mental homunculus is not being implied. The mind may still be characterised as the emergent behaviour of a set of interacting, competing, "evolving" schemas, in a mechanistic fashion. The "active" view of the mind is intended to mean the role that many of these schemas have in determining the creation, maintenance, modification, and use of other schemas. The dynamics occur between the schemas already present in a mind, as they process incoming sensory signals and their own interactions. It is merely a convenient shorthand to talk of a meme's action to "infect a host".

Fiske and Taylor [Fis] define a schema as a cognitive structure containing knowledge, where the knowledge consists of attributes of concepts and the inter-relationships of those attributes. Their purpose is to generalise and to put structure into the perceptions, inferences and actions relating to recurring social situations. They reflect the rich domain-specific cognitive structures present in the mind. Perceptual and inferential schemas are related to the traditional cognitive psychology concept of "expectation", and action schemas to the concept of a "script". In this paper, schema theory is seen as a refined description of the organisation of thoughts, perceptions and actions, compared to the crude concepts of "idea", "belief", or "action". There is an advanced development of the theory of these schemas in the reference, but some basic functional properties of schemas will be discussed in Section 5 as they are relevant to a development of memetic theory. Briefly, Fiske and Taylor distinguish several types of schema: those relating to

1. the physical and psychological traits of other people;
2. the physical and psychological traits of oneself;
3. the appropriate norms and behaviours of people in the context of their social position;
4. the spatio-temporal forms of abstract events (known as "episodes");
5. the abstract, "content-free" form of social relationships (for instance, concerning social hierarchy).

As an example of a formalisation of memetics, Lynch [Ly2] uses a first-order calculus to characterise some of the aspects of memetic transmission. In Artificial Intelligence and the computer sciences classical formal problem solving techniques often involve higher-order logics (especially predicate calculus), which are better able to cope with the sophistication of the real world. As Sperber [Spe, p. 34] remarks, the second-order process of forming mental representations of other mental representations is crucial to the understanding of cognition.

More detail about the proposed mechanism of memetic transmission is provided by Lynch [Ly2] using differential equations of the "population dynamics" type. These form a purely phenomenological model. As with any other formal model of population dynamics in scientific theories, a concrete interpretation of the variables in those equations is needed before one can make a claim that the equations represent any processes actually observable in nature.

It has been shown that memetics points out some apparently atomic structures in social interactions and behaviours (and, as a consequence, human memory), using the notion of a type and some of its observed (grammar-like) functional properties. This characterisation of memes provides the means to show that the "replication event" is a poor account of the mechanism of cultural transmission. Firstly, a meme has been shown not to be a concrete unit. Secondly, it has been shown that the standard "sameness of public ideas" assumption is weakly-founded. As a result, it is difficult to justify any meme transmission mechanism that relies on a unitary property and a public "sameness" property of the memes. The semantic interpretation of a "replication event" relies on both these assumptions. In conclusion, the memetic view that alterations occurring during the replication of an idea are merely accidents (as genetic mutations are -- see Section 3.6) is not consistent with the argument that social interactions are mostly inferential in nature.

Such things as "copy fidelity", "infection", and the encoding of memes into human memory, are clearly notions still tied up in the original inspirations of memetics from genetics, viral epidemiology and computer science (respectively). Even if formal memetic theories avoid explicit reference to concepts from these disciplines, the whole ethos of memetics reflects deep-rooted principles from them. In particular, memetics embodies (respectively) (a) a brand of neo-Darwinism in its account of cultural "evolution" (Section 3.6); (b) a flavour of classical behaviourism, since the role of cognitive and emotional factors in behaviour is trivialised by using the epidemiological analogy that the mind is a passive vehicle that merely responds under the stimuli of memes; (c) a dualistic principle in its treatment of mind as a causally separate domain (i.e. the endorsement of the software-hardware analogy).

3.5.4 Theories of evoked culture

Tooby and Cosmides [Too] provide a strong argument for the existence of cognitive adaptations for social interactions that have arisen by virtue of genetic evolution. As somewhat "innate" abilities they would therefore not be acquired by social transmission. In the same paper, the authors also discuss the concept of "evoked" forms of culture, for which social transmission is again not the mechanism of acquisition. The brief review of their work in this section often follows their line of reasoning closely.

Intuitively, the principle of evoked culture is that a new-born child's mind is not a blank slate on which society chalks up its beliefs and customs. A simple illustration of evoked culture is given in the reference. It involves the types of reciprocity in food sharing that are observed in social groups, where "... a contextual variable--the presence or absence of æluck' in food acquisition--appears to activate different decision rules governing food sharing" [Too, p. 165].

The theory of Social Exchange [Cos] led to psychological experiments [Too] that suggest the existence of a specific ability to spot cheating in stories concerning a violation of a social contract. The evidence suggests that people are not as able to detect pure mistakes in the execution of a similar contract (which consequently benefit neither party), nor to detect altruists who breach a contract to the benefit of others, rather than themselves.

Tooby and Cosmides suggest the existence of highly content-specific evolved mechanisms to account for these social exchange algorithms or, for instance, the observed systems of food sharing. It is apparent from the literature that this degree of genetic determinism is a contentious issue. Fortunately, it is not necessary to justify such a high degree for the present purposes. In the case of their Social Exchange theory, the experimental results of Tooby and Cosmides clearly indicate that there is some non-general process behind a person's reasoning ability in the context of social contracts. It is the degree to which the process has evolved specifically towards cheater detection that is not clear from the results alone. Similarly, with the issue of food-sharing algorithms, it is also not clear the degree to which local circumstances activate entire genetically-programmed behaviours that are highly specific to food sharing, although the differences in sharing behaviour between cultures clearly corresponds to the variance of food acquisition due to the local ecology. The view that there is some content-specific process behind social exchanges and the development of particular food-sharing algorithms is the most simple and plausible explanation for cross-cultural similarities in this cognitive domain.

For many other social situations there is also an argument for content-specific, evolved psychological mechanisms which constitute the building blocks of the relevant cultural behaviour. These mechanisms would have evolved to process information about ancient and important adaptive problems, such as language acquisition, sexual jealousy, kin recognition, emotion recognition, and parenting.

The argument behind this view of evolutionary psychologists runs as follows. The standard "socially constructed" view of culture is that the phenomenon of social exchange is culture-specific and historically contingent. Moreover, the standard model says that whenever social exchange is found to exist, it has to be taught or communicated from scratch. In other words, its every structural feature must be specified by the social environment. However, "it is telling that it is just this explicitness that is usually lacking in social life" [Too, p. 162]. As was discussed in Section 3.5.2, shared assumptions about the world are crucial to the possibility of communication between individuals. This includes the communication responsible for an individual's learning of a culture, either as an immigrant or as a new-born child. Innate domain-specific reasoning procedures supply what is missing in traditional accounts of cultural acquisition.

Consequently, it appears certain that there are at least weakly content-specific cognitive mechanisms that have genetically evolved. These might be seen as "seeds" around which various types of complex, highly content-specific, social behaviour can "crystallise", under the effect of different environmental stimuli. In some sense, therefore, social exchange algorithms might be viewed as a form of evoked culture.

These ideas are developed more rigorously in terms of "attractors" for "socio-physical schemas" [Cl2]. Briefly, they can be illustrated with a simple example (due to Sperber [Spe]). Sperber speculates that humans "have an innate disposition to develop concepts according to certain schemas." His example of why this is an appealing possibility runs roughly as follows. Consider the problems of ostension for a mother who points out and names the species of a bird that is singing in a tree to her infant child. How does the child know what precisely is being given a name: the name could refer to all trees containing birds, or all small, noisy objects, or of that particular bird, or of the underside of its belly? To avoid ambiguity, the child needs some low-level schemas, perhaps reflecting the nature of taxonomy and the economy of expression⁴. These aspects might allow a child to surmise firstly that the mother is referring to the bird in itself, and not as part of its relation to this particular tree, or the fact that the bird happens to be singing. Secondly, these aspects should allow the child to realise that it is the "whole" object of the bird that is being referred to, rather than, say, only its underside. While the child has not yet developed a detailed knowledge of birds and their general relations to other very basic categories in the world, he or she is unlikely to expect the mother to be referring to detailed aspects of a bird. How are these schemas to be acquired socially for such a young infant? The argument is that surely these must be genetically "programmed" in essence.

A summary of the analysis of individual and cultural variation within the context of a universal human nature is now reproduced, with minor modifications, from Cosmides and Tooby [Cos] (my emphases, to highlight the consequences for memetics).

1. By virtue of being members of the human species, all humans are expected to have the same adaptive mechanisms, either at the level of developmental programs, which govern alternative developmental pathways, or (more likely in the case of social exchange) universally developing cognitive specialisations.
2. In consequence, certain fundamental ways of thinking about social exchange will be the same everywhere, without needing to be socially transmitted.
3. As is standardly believed, social transmission may indeed shape social exchange. But it does so not by manufacturing the concept of social exchange de novo. Instead, there are probably specific avenues through which social transmission can act: for example, by influencing the valuation placed on items or actions, by providing information that helps one identify appropriate partners, or by providing information that allows one to identify appropriate contexts.
4. Our social exchange psychology should be highly context-dependent. Consequently, many dimensions along which social exchange varies, both within and between cultures, may be instances of evoked culture. The presence or magnitude of certain cues, conditions, or situations should cause mechanisms within our social exchange psychology to be calibrated; to be activated or inhibited; or to be switched into alternative modes of activation. When local circumstances trigger a particular mode of activation of the social contract algorithms, for example, this may cause a highly structured, multi-individual behavioural outcome. Therefore, when one sees similar patterns of social exchange in widely different parts of the world, one cannot assume that the similarity is determined primarily by social transmission; the similarity may instead be an instance of evoked culture.

In conclusion, cultural evolution clearly cannot be neatly separated from content-specific cognitive faculties having some genetic basis. Memetics must be developed so as to be compatible with this. Some current treatments of memetics interpret occurrences of evoked culture as a transmitted form in memetics, or do not recognise them at all. Whilst it is true in principle that "there is no reason why success in a meme should have any connection whatever with genetic success" [Daw], memetics cannot account for all cultural phenomena on the assumption that memes are entirely "independent replicators" (Cf. Blackmore [Bla]). Memeticists must be clear about the limits on the domain of applicability of their theories, acknowledging exceptions to their basic tenets.

3.6 Memes and "cultural evolution"

Memetic theories use an essentially Darwinian view of "cultural evolution". This view sometimes suffers from strained analogies with genetic evolution⁵. To some degree, the process of memetic transmission is Lamarckian [Bla]. That is, an act of imitation consists of a person A copying the behaviour of another person B, not the instructions which B's brain holds for producing that behaviour. Often, however, memetics goes for a stronger correspondence with genetic evolution: neo-Darwinism. This non-Lamarckian interpretation of simple Darwinism arises because of the dangerous elevation of a meme to an ontological unit which can be used as the object of memetic "heredity". The three principles of neo-Darwinian theory are (a) the source of variation, and (b) the mechanism of selective retention of (c) a particulate heritable entity. Here, "heritable" refers to any kind of conservative mechanism by which the entities are replicated and "passed on".

To begin with, it has already been shown that a meme is not a concrete unit of culture with a uniform interpretation. Since social interaction is an inferential and reconstructive process (Section 3.5.2), the norm is that acts of communication transform ideas rather than simply replicate them. As a result there is no straightforward interpretation of a meme as a "heritable entity", and consequently principles of variation and selection based on the differential fitness of memes become problematic. These two principles are discussed in turn.

Underlying the process of variation is an analogy to the concepts of genetic mutation and recombination. On the whole, genetic mutation is assumed to be a statistically unbiased ("undirected") process that causes only "minor" changes to a gene. Loosely speaking, this "normal" kind of mutation yields similar genes which compete because (a) they interact mostly with the same enzymes as each other, and (b) they interact with the same enzymes in only subtly different ways. Despite the lack of a universal definition of the gene unit [Daw, p. 28], its sound biochemical substrate at least (a) gives an indication of what the dimensions are to the space of possible changes to a gene, and (b) gives the gene a finitely-decomposable property, in that it is composed of base-pairs which are the limit of resolution for the processes of mutation and recombination.

In the case of memetics the lack of a concrete substrate leads to problems. The acceptance that communication events transform ideas to a potentially arbitrarily-large degree, especially in the case of imitation, means that the assumption that only slight alterations occurring during "replication" is only justified when a common mental context can be established between the individuals involved. Also, the transformations may have a large bias according to the schemas used by that individual: in other words, a bias due to the cognitive expectations or emotional dispositions of the individual reconstructing the meaning of a communication. The acceptance of a wide range of factors involved in causing variation to ideas means that there is no clear set of dimensions along which memetic mutation can be defined. Thus the analogy to a mutation breaks down.

Similarly, the lack of (a) a clear set of dimensions along which an idea can be decomposed, and (b) a discrete and concretely-defined notion of an idea, means that the concept of memetic recombination has no precise meaning either.

Underlying the process of selective retention is the assumption of "general" competition between memes to occupy a host (for instance, see Lynch [Ly2]). In light of the analysis of the meme concept, this assumption is weakly-founded because of the variety of possible mental origins of an idea -- both within individuals and between them. "General competition between memes" is an artefactual concept because memetics relies on the software-hardware characterisation of mind and brain (see Section 3.4.1).

An analogy illustrates the significance of the earlier conclusion that a "meme is only an abstract type" to the understanding of the problem of general inter-meme interactions. Consider the concept of a noun in the grammar of a natural language, where two such examples might be "wealth" and "coin". A noun is essentially a broad syntactic type, with some general semantic interpretation provided by the rules of grammar that are applicable to them. Understanding the use and role of these two nouns in language is not elucidated by their identification as nouns, beyond the similarity in their grammatical rules.

The basis for formal models in memetics concerns the representational aspect of ideas, and is often understood in terms of the "functionalism" commonly found in the social sciences. A functional description of a situation may be adequate for some purposes, even perhaps to characterise predictions about measurable properties of a society (see Section 3.8). However, Sperber makes the following point. "As abstract objects, representations have formal properties, and enter into formal relations among themselves. On the other hand, abstract objects do not directly enter into causal relations" [Spe, p. 62]. Thus a population dynamics model of memetics may provide a more concrete description of meme interactions, but it still embodies the assumption that the abstract memes "directly enter into causal relations". On this basis, therefore, no a priori justification for general competition between memes is possible.

Another aspect to general inter-meme competition is the need for ideas to have some kind of universal "fitness" attribute. Such a simple notion as "fitness" of ideas trivialises the psychological motivations behind an individual's (conscious or subconscious) action to accept or reject new ideas. For instance, consider individuals with an emotional susceptibility to certain types of belief, or a tendency to pick up habits or to be lazy. These are properties of those individuals which have a complex personal basis, and are not quantifiable properties of the ideas themselves. The psychological basis of habit formation or laziness is hard to reconcile with the basic memetic model, without simply adding more primitive mechanisms to the list of memetic assumptions -- mechanisms that permit memes to somehow induce habits or laziness. Any intentionality given to memes is a weakly-founded attempt to patch over the exceptional cases to standard memetic predictions. It is counter-productive and ontologically unfounded to view the ideas (in the form of memes) as inherently possessing either inductive abilities or quantifiable "fitness-like" features.

The intentionality of memes that can be implied in loosely composed assumptions of memetics has prompted accusations of mysticism and vitalism. These accusations are motivated partly by arguments against the ontological elevation of a meme to a true "unit", and partly by the dangers in the oversimplified view of the human mind as a "host" that results from this elevation (as was discussed in Section 3.5.2). A declaration of the intended meaning of so-called "short-hand statements" (for instance, see Blackmore [Bla])--which might otherwise be construed as allowing memes intentionality--is necessary to avoid the subconscious traps laid by using imprecise terms.

An example of a social situation in which an attribute of "fitness" seems inappropriate is now examined. Consider a teenager picking up pet sayings and mannerisms from her classmates, and her younger sister (still at junior school) copying them at home. There are several possible motivations for the younger sister imitating her sister. These might include merely wanting to get attention, or to be included with more "grown up" children, or to retaliate for some previous teasing. Obviously, the big sister's school-based idiomatic and vernacular language and behaviours stand out at home, and are a clear target for imitation. However, while the spread of those behavioural patterns in the school playground may be successfully modelled by memetics, their uptake at home by the younger sister may well be a different matter. The "fitness" of a clique's idioms (seen as "memes") within a social group of similar individuals is not a salient factor in its transfer to a younger sister at home who is jealous, attention seeking, or aggrieved. In fact it is unlikely that the form of the memes is relevant at all -- any language unusual to the normal home situation is a potential target. No non-trivial "fitness" attribute is relevant to this type of transfer: the idioms are only "fit" in the trivial sense of standing out from the norm, and being in a particular place and time when their presence is used to the advantage of a child who is jealous, etc. Certainly, an evolutionary selection process does not underlie the transfer. Such situations are modelled with ease in the schema model discussed in Clewley [Cl2], as are situations having an "evolutionary" basis, without explicit talk of "fitness".

A widespread mathematical description of processes that involve "fitness" and "selection" is a "cost-benefit" model. In fact, the clearest case studies in contemporary memetics appear to be restricted to situations where there are intuitively obvious interpretations of "cost" and "benefit" associated with the acceptance or rejection of publicly communicated ideas⁶. Examples of these interpretations in the memetic and social science literature are "utility", "reward", "threat", "emotive hook", "bait" or "exo-toxicity" (e.g. Refs. [Gra, Cos]). In such studies, reasonable idealisations have permitted the statistical analysis of surveys to yield simple evolutionary models of cultural transmission. Essentially, this involves the inference of a causal link between (a) the frequency of supposed memetic transmission events in a population of hosts, and (b) the relative measures of "fitness-like" quantities of the competing memes present in that population. The restriction to these more easily-interpreted situations goes largely unacknowledged, since it is common to find expositions claiming or implying that memetics is universally applicable. The apparently subconscious restriction made on the cases considered is possibly due to the conscious pursuit of soundness in the execution of empirical studies (see Section 3.8).

The idealisations that allow memetic theory to be applied to the results of social studies are now discussed. A cost-benefit account of the success or failure of memetic replication events relies on an assumption that memes have an idealised interaction with a host and with each other. This assumption requires that a meme interacts with only a small number of other memes which seem relevant to the social situation. There must also be a simple and direct cause-effect relationship between the action of a meme and its competing "alleles", and its "inhibiting" or "activating" memes, and a host. Furthermore, this relationship has to be homogeneous in the domain of hosts deemed to be "infected" by the same memes. As discussed above, such idealisations must greatly narrow down the field of cultural examples that are amenable to cost-benefit analyses.

It is instructive to note that the neo-Darwinian theory of biological evolution was in a similar predicament before the molecular basis of genetics was developed. Presumably, Gregor Mendel was only able to conduct experiments involving the cross-breeding of pea plants by assuming the existence of a hitherto-unknown heritable factor, and assuming its direct causal link with easily distinguishable phenotypic differences in the plants. Thus falsifiable hypotheses were only possible for plants having simple genetically-determined phenotypic properties. Those case studies were chosen precisely because there appeared to be a direct causal link between a gene and its expression, without obvious influences from other genes or environmental factors.

Memetics may not have the opportunity to enjoy benefits similar to those brought by the understanding of the molecular basis of genetics. Whilst it is now clear how to study the developmental role of genes even when they don't have such easily-visible phenotypes, no such corresponding technique would appear possible for memes. It is possible for many subtleties in the process of evolution to be revealed. An example is being able to be certain that copies of genes are literally transferred through generations; or to be able to account for the role of activator or inhibitor genes in the study of the relationship between genotypes and phenotypes. Without such a low-level basis for genetics it would not be possible to conduct more complicated studies of genetic adaptation and selection that involve cross-breeding and observed phenotypic expression. The lack of a concrete unitary property of memes means that there is no obvious way to study the "genotype" directly. The phenotypic expression of a set of memes (taken as the genotype) can be defined as the public acts of communication which they "code for". Firstly, the Lamarckian property of memetic evolution means that it is the "phenotype" which is public, and the thing that is "transmitted". Secondly, earlier analysis established that it is only the "phenotype" that can be observed and sometimes "measured" from an individual. Without any way to probe deeper into some lower-level mechanism, there can be no certainty as to other causal influences, and hence no precise "low-level" science of memes.

The simple neo-Darwinism of memetic theories can now be seen as a weak foundation. The existence of interactions "between" particular types of belief and behaviour is a crucial matter underlying the justification of fundamental memetic assumptions. The inability of current memetic theories to explicitly and systematically draw distinctions between types of belief and behavioural schemas and their interactions is therefore a crucial weakness.

There are some memetic terms associated with a genetic analogy which are relatively unproblematic in their interpretation in cognitive psychology. These are the notions of meme combinations and complexes. Grant defines a meme-complex as "a set of mutually assisting memes which have co-evolved a symbiotic relationship" [Gra]. In discussing small sets of (e.g. two or three) functionally-related memes, Lynch uses the term "combination" [Ly2]. Clearly an individual holds several inferential schemas simultaneously and has many developed behavioural schemas. The only misleading feature of the memetic view is that the terms "combination" or "complex" imply an intrinsic and (perhaps) necessary connection between particular schemas. The fact that several otherwise-unrelated memes might play a part together in a particular individual's social development is actually an extrinsic and historically contingent connection between memes. For an individual having some set of disjoint schemas, connections will only be created if the thoughts and perceptions of that individual at some later time happen to relate to several schemas. This type of connection between memes is strongly dependent on that individual's social and ecological context, and memetics fails to emphasise the importance of this factor as an explicit element in its models.

All of the basic components of a typical memetic theory have now been analysed: hosts, memes, and the mechanisms underlying meme dynamics. A brief recap over the major conclusions reached in the analysis of contemporary memetic assumptions is now given.

A person's representations and schemas may differ greatly from those of another person. In a particular social context these divergent sets of representations and schemas may merely outwardly function in a similar way, according to the observations the individuals involved. Therefore it is crucial to realise that in a social group two factors must be present in a particular social exchange in order to justify the "sameness" of public ideas. These are (a) common social and physical experiences, and (b) common psychological motivations.

The intangibility of private and public aspects of ideas means that the concepts of a meme and its neo-Darwinian "cultural" evolution are a poor basis for the fundamental category and mechanism in a widely-applicable theory of culture. It was argued that the resulting idealised models of memetics belie a sophisticated multi-level process.

Sperber [Spe, p.31] neatly sums up the need to involve lower level processes in theories of culture: "Memory and communication transform information. Thus, to treat representations, whether mental or public, as material causes among other material causes implies rooting the study of thought and of communication in cognitive psychology." Arguing for the involvement of cognitive and evolutionary psychology in a sound understanding of culture means that the levels of representation inherent in those disciplines must become part of the multi-level picture of culture, as must all the lower levels which underlie their theories.

3.8 The third statement

In the view of the present methodology, the substantiation of the first statement's claim implies that the level of representation of memetics is not identical to any traditional root level. In addition, it has been argued that this mismatch is not adequately accounted for in the form of an explicit "interface" between memetics and lower-level theories. Purely by the MR definition of explanation it follows that contemporary memetics is not coherently related to other levels of representation. As it stands, therefore, it does not form part of a comprehensive theory of culture. Thus the third statement claims that by itself memetics can no explanatory power -- under the MR view of explanation. In this sense, therefore, it might be branded "sterile".

The issue of whether memetics is "unscientific" involves the notion of falsifiability. Firstly, "the sort of predictions generated by mathematical models and computer simulations are falsifiable" [Ly2]. Memetics is able to falsify these predictions on the same grounds as any other branch of social science: that is, due to the similarity in the empirical methodology used in all these disciplines. To some extent all social sciences rely on idealisations of human psychology which are not always easy to justify. Assuring the accuracy of measurements of the beliefs, attitudes and memory of small samples of individuals from a large population is always difficult. This affects the testing of theories belonging to any of the social sciences. In particular, estimation and extrapolation are required in order to generate data sets that might represent a whole population. Assumptions of homogeneity of social groups are required to do this, and there are many unmeasurable influences on individuals which have to be assumed to average out to a "norm" in the limit of large population size.

Thus memetics seems no less scientific than the other social sciences. However, it will be argued that the conclusions of the preceding conceptual analysis mean that despite having a sound empirical methodology, memetic theories are hard pressed to make any firm interpretation of survey data. As is argued by Lynch [Ly2], data sets can be used to test hypotheses concerning the spread of beliefs, and how that spread correlates to certain types of behaviour. A typical memetic hypothesis might be that a particular sex taboo (presumed to be a meme) increases its hosts' average number of offspring. Lynch uses the example of a national health survey containing data about respondents' attitudes towards various sex acts, which also contains data on reproductive histories: "After making suitable functional definitions based on the questionnaire, one can statistically reanalyze the raw data set to see how much of the variance in reproductive history is attributable to variance in a particular attitude response" [Ly2].

It might have been more accurate to describe the results of a statistical analysis in terms of suggesting a "correlation" rather than an "attribution". This is because the scientific problem for memetics lies in accurately identifying the mechanisms that produce the correlations (or non-correlations) between the measurable properties or quantities of individuals (such as attitudes, or number of offspring). Merely observing statistical correlations between attitudes and number of offspring obviously does not imply a causal connection between them. That process of identification will be hindered by a poor conceptual foundation of memetics. For instance, a memetic mechanism may never be found to explain a correlation if unrecognised factors are involved which may be exceptional to the basic tenets of memetics.

One can make predictions about the results of population studies with memetic models, under restriction to cases with relatively straightforward cause-effect relationships between memes and behavioural/belief outcomes. A set of successful predictions in some domain of application is a kind of explanation, but lacks depth and any generality. For instance, it will not have been established what really is the cause of the observed effects, which -- in the chosen parameter domain -- appear to be predictable according to an abstract model. Taking into account a wider state and parameter space one might find that the models break down, and require a substantially different model in order to make accurate predictions.

1. This is related to the concept of a "metamodel", as defined by Heylighen [Hey].

2. The relation of memetic units to genetic units is developed in Section 3.6.

3. In traditional cognitive psychology, the use of the term "transmission" is standard, and is not meant to imply an active process in which culture literally transmits itself between individuals. However, in memetics, the interpretation of "transmission" is often literal.

4. Of course children often do make some mistakes when inferring meaning from the statements of adults!

5. A deeper analysis of the metaphors inherent in evolutionary theory, and how an MR view can address the associated problems, can be found in a forthcoming paper.

6. Some examples of these studies can be found in Lynch [Lyn].

For draft reviews and helpful discussions I am indebted to Simon Garrett, Alan Champneys, Steve McCoy and Nello Cristianini.

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