The relevance of dynamic systems theory for cognitive linguistics

1
The relevance of dynamic systems theory for
cognitive linguistics
Wolfgang Wildgen

• Interdisciplinary Themes in Cognitive Language
  Research Symposium
• University of Helsinki and FiCLA
• November 25 26, 2005

2
Contents

• 1 Basic oppositions
• 2 Some features of dynamic models for language
• 3 Talmys force- dynamics
• 4 Lakoff and Johnsons metaphorical mappings
• 5 The problem of compositionality (construal) in
  cognitive grammar (Langacker)
• 6 The dynamics of composition in grammar
• 7 Conclusions

3
Basic oppositions

• Thom was guided by his discussion with Waddington
  on biological morphogenesis and expanded this
  thought to linguistics. In his holistic strategy
  Thom preferred a gestaltist, geometrical,
  morphogenetic view on biology and not so much a
  mechanistic one, which takes the brain as the
  central (and finally) only organ responsible
  for thought, language and culture.
• The research line of cognitive linguistics in
  general since the 50s may be situated in an
  interdisciplinary but rather technically minded
  world that of information theory (Shannon), and
  cybernetics (Wiener). It was developed in the
  philosophical atmosphere of logical empiricism
  (Quine) and formal syntax (Carnap). Whereas
  Chomsky elaborated this field and created a
  compact mentalistic theory, Lakoff (since 1975)
  and with him Langacker and Talmy combined
  insights of gestalt-psychology and modern
  computer vision with ideas stemming from issues
  of generative semantics.

4
Some features of dynamic models for language
Name Germ Corang int.variables Codimension extern.variables Type
fold x3 1 1 A2
cusp x4 1 2 A3
swallow tail x5 1 3 A4
butterfly x6 1 4 A5
hyperbolic umbilic xy2y3 2 3 D4
elliptic umbilic xy2 - y3 2 3 D-4
parabolic umbilic x2y y4 2 4 D5
Figure 1 List of elementary catastrophes.
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Figure 2 Two basic types of dynamics stability
(attractor) and instability (repellor).
Figure 3 A hierarchy of dyna-mical conflicts
typical for the compact catastrophes cusp (A3),
butterfly (A5) and star (A7).
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The process type called capture
Figure 4 Derivation of a process-schema (right)
from a path p in the vector-field of the cusp
(A3).
7
Figure 5 Rössler-attractor (in the case a
0,035 b 0,46 c 4.5 cf. Plath and Wildgen,
2005).
Figure 6 Feigenbaum-scenario (the constant k
increases from 2 to 4 horizontally, whereas the
vertical dimension concerns the state x. At k 3
the curve splits at k 3,58 the tree has
infinitely many branches.
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Restrictions on qualitative dynamic models

• The topological nature of elementary catastrophe
  theory asks for a rough modelling by which only
  general features of the field in question can be
  captured. The dynamics of language, must in the
  beginning consider mainly critical transitions,
  bimodal, trimodal oppositions, etc. Specific
  predictions or an exact reproduction of
  descriptive details cannot be the goal of these
  models, because it is by definition a topological
  (and not a geometrical) model and all description
  have to be interpreted modulo smooth deformations
  (diffeomorphisms) i.e., one cannot simply
  transfer them to the level of metrical measures.
  This means that only very general questions may
  be assessed with the help of qualitative dynamics

9
Talmys force- dynamics

• The material on which Talmys analyses is based
  are two sets of examples with a closed class term
  at their centre, either a preposition (a) or a
  connector (b).
• The ball sailed past his head.The ball sailed
  through the hoop.He ran around the house.He
  walked across the field.(Cf. Talmy, 1975
  201-205)
• The ball kept rolling because of the wind blowing
  on it.The shed kept standing despite the gale
  wind blowing against it.(Cf. Talmy, 1988 5)

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• The pictures in Talmys article demonstrate, that
  the examples in (a) use notions of space, border,
  transition, and motion that may be modelled in
  dynamic system theory.
• Fig. 7 sketches such an elaboration. In the
  sentence He walked across the field, the field
  is a topologically coherent surface with a
  boundary, ideally a circle or a regular polygon.
• We have two dynamics slow (stable) and quick
  (transitory) dynamics, the latter corresponds to
  one of the two types in Figure 2 i.e., an
  attractor is found or a repellor is avoided. The
  verb walk focuses on the stable (slow) motion,
  with an implicit ingressive (start) and egressive
  (stop) phase, whereas across focuses on the
  quick dynamics of change, called a catastrophe.

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across
Be captured by the attractor ENTER
Walk (path)slow dynamics
Let free from the attractor LEAVE
Attractor field
Figure 7 Catastrophe theoretical description of
the major dynamic meaning components in the
sentence.
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• The ball kept rolling because of the wind blowing
  on it.
• intrinsic force tendency of the Agonist (right)
  towards rest (?),
• the Antagonist (left) is stronger (?),
• intrinsic force tendency of the Antagonist
  action (?),
• result of the force interaction action ( ).

Figure 8 Schematisation of force-dynamics by
Talmy (1988).
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• roll motion (attracted by a position of rest)
• consequence for motion is the end (death) of
  motion i.e. a fold-catastrophe (A2).
• B. blow energy gain
• C. because link between energy gain and natural
  (diffusive) loss of energy
• D. keep equilibrium between loss of energy and
  (added) gain of energy.

Rest as attractor
End of motion as fold-catastrophe
Figure 9 A dynamic analogue of Talmys
description
Versal unfolding of the attractor
14
Coupled dynamic systems

• The coupling of two dynamic systems has been
  analyzed in the case of physical systems (the
  classical case are coupled oscillators and
  resonance phenomena). The dynamic systems
  approach produced the interdisciplinary field
  called synergetics by Herman Haken.
• It has been applied to cognitive systems. Haken,
  Kelso, a. o. studied the coupling of finger
  movements, of animal gaits etc.
• Haken (1996) applied the methods of synergetics
  also to effects of synchronization and
  desynchronization shown in EEG and MEG patterns.
• Oullier et alii (2004) expanded this paradigm to
  imagined sensorimotor coordination.

15
Semantic coupling in language

• In the case of two sentences coupled by a
  connector or an adjunct (adjective) coupled with
  a head noun one could imagine an application of
  this methodology, if there was a proper dynamic
  model of single word meanings.
• At least the notion of prototype introduced by
  Eleanor Rosch and others in the 70s shows that
  simple concepts like those of color-terms have an
  attractor like shape.
• The major difficulty is that many simplex
  word-concepts are semantically already complex
  (as they involve different sensorimotor
  parameters, abstraction, metonymy, metaphor). The
  syntactic composition of words must therefore
  first consider a kind of frozen complexity at the
  word level and build a syntactic operation of
  meaning composition on this basis.

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• Although the pictorial illusion of simplicity in
  Talmys model is destroyed, the dynamical
  model-sketch pinpoints basic problems hidden in
  Talmys description.
• There is a mapping between physical dynamics (the
  wind, the ball), the perception or the imagined
  enacting of the process, its memory trace (with
  abstraction) and the linguistic expression. The
  first levels are hidden in Talmys description,
  although his terminology and pictures presuppose
  their existence. Thus part of the cognitive
  aspect is veiled by his description.
• The semantics of these complex sentences blend
  different types of dynamics
• spatio-temporal dynamics with attractors and
  catastrophes (cf. a)
• energy functions and the coupling of subsystems
  (cf. b)

17
Lakoff and Johnsons metaphorical mappings

• If one considers the list of all metaphors
  mentioned in Lakoff and Johnson one sees that
  their relational networks are very shallow.
  Almost all metaphors have relational length 1
  examples for a relational net of length 2 are (a)
  and (b)
• a) field war love
• war is a field / love is war
• b) path journey argument
• a journey is a path / an argument is a
  journey
• In some cases the relation is transitive thus in
  (a) one could deduce (by transitivity) love is a
  field and in (b) argument is a path.

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Two basic types

• a. Those metaphors (A, B) where more fillers for
  B are mapped on one filler of A.
• Example
• A ideas are B objects, commodities, organisms
  (people, plants), resources, products, fashions,
  light-sources
• b. Those metaphors (B, C) where one filler of C
  is mapped on more fillers of B
• Example
• B vision, action, event, activity, state
  is C a field

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abstract terms (A) basic level terms (B) spatial categories(C)
idea love argument time building, light-source, organism commodity, limb, patient container, machine, product fashion, madness, resource force, magic, seeing game, money, sending light-medium, object, war field up/down behind/ahead
Figure 9 Scale of metaphorical transitions.
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A more cognitive explanation of metaphor

• The orientational metaphors are rooted in
  non-linguistic cognition (complex perception and
  action programmes).
• The processes of metonymic mapping use part-whole
  separations and are basically relational. They
  presuppose a dual focus. Current research on the
  attentional blink and relevant time lags which
  allow for the two consecutive stimuli may be
  relevant here. Again a rhetorical principle must
  be reduced to a more basic perceptual and
  mnemonic process.
• Metaphoric mappings exploit differences in
  semantic density. The general rule says that
  expressions which are more concrete (have more
  semantic density) may replace less concrete ones
  (with less semantic density) if some basic
  similarities are given.

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Questions (in a dynamic perspective)

• How stable are such mappings?
• What happens, if the mapping is iterated? Does it
  go to chaos?
• How complex (in terms of dimensionality, number
  of components) can a source space be in order to
  be mapped in a stable fashion?
• Do maps preserve a basic structure? Do they
  reduce the dimension of the object mapped?

22
Chaos attractors of iterated mappings

• It is known from chaos theory (cf. Peitgen et
  alii, 1992 277 ff. and for an application to
  language Wildgen, 1998) that even in the case of
  a two-dimensional input, like that on a
  video-screen, an imperfect map to itself produces
  chaos after some steps only. In classical cases
  it has an attractor intrinsic to the system
  itself and totally independent from the input.
  The input information is lost and the iterative
  process is frozen into a standard pattern.

Figure 10 The Sierpinski triangle as a standard
attractor of a chaotic mapping process , which
involves reduced copy, threefold composition.
23
The problem of compositionality (construal) in
cognitive grammar (Langacker)
Figure 11 Langacker's analysis of the verb ENTER.
24
Figure 12 The constituent analysis of the
sentence A man finds a woman (proposition
FIND-WOMAN-MAN)in Langacker's analysis.
25

• The difference to traditional phrase-structure-mod
  els consists in the fact that pictures are
  inserted into lexical positions. At first sight
  one could presume that the cognitive meaning of
  this procedure lies in the fact that meanings may
  be imagined quasi-spatially, but Langacker
  explicitly rejects such an interpretation. He
  says (Langacker, 1990 12-15)
• The symbolic resources of a language generally
  provide an array of alternative images for
  describing a given scene, and we shift from one
  to another with great facility, often within the
  confines of a single sentence. The conventional
  imagery invoked for linguistic expression is a
  fleeting thing that neither defines nor
  constrains the contents of our thoughts.
• The second possibly cognitive notion introduced
  is the distinction between trajectory (figure)
  and landmark (ground), which is more or less
  taken from gestalt-psychology.

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Langackers cognitive programme is
theoretically ambiguous

• If images are only vague and fluctuating
  after-effects observed in the analysis of
  linguistic structures, how can we ever know
  anything specific about these volatile creatures?
  
• What are the empirical techniques which allow for
  the capturing of these phantoms?
• If we look at the large corpus of image analyses
  presented in Langacker (1987 and 1991) the answer
  is almost shocking
• The individual introspective insight of the
  linguist, supported by heuristic techniques taken
  from current linguistic models are the only
  empirical method used.

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Catastrophe theoretical description
Figure 13 The topological schema of enter and
leave.
28
The dynamics of composition in grammar

• In a simple case which avoids the complexity of
  verbal valence, and rather takes nominal syntax
  as a basic example, one may consider a noun
  related to the form of an object, say a square,
  an adjective of colour, say red, and a present
  participle of motion, say moving
• red moving square
• How does the brain compose a head-noun referring
  to form with two satellites referring to colour
  and motion?

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• Andreas Engel (2004) distinguishes three major
  areas for sense related information The visual
  system (subdivided into the areas V1  V5), the
  occipital areas and the parietal ones. The major
  binding process is one of temporal
  synchronization of assemblies, which form wholes
  (gestalts) from parts and desyn-chronization
  which distinguishes figure and ground. The
  syn-chronization of two perceived stimuli can be
  measured in the Gamma-band (60-70 Hz) and the
  Beta-band (15-20 Hz) of an EEG. The
  fronto-parietal centres select features that are
  then passed to working memory and planning. This
  type of analysis concerns only the composition in
  perception, attentiveness and memory, but one may
  conjecture a parallel process for words (related
  to perceptual information) and their composition
  in syntactic constructions involving nouns and
  adjectives.
• The role of determiners, who have an indexical
  function, is probably another story, which needs
  other experiments and measures.

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Verbal valence patterns

• Here the application of catastrophe theory to
  semantics (cf. as a recent summary Wildgen, 2005)
  has its classical field. The complex but
  nevertheless structurally stable valence patterns
  seems to lie beyond the current experimental
  reach of neurological experiments. Therefore the
  plausi-bility of dynamic semantics must still
  rely on a rough iso-morphism between patterns in
  the real world (physical process patterns) and
  linguistic forms (sentences in different
  languages).
• One can presume that the brain as the mediating
  apparatus has the means to map the eco-logically
  relevant aspects of physical processes into
  stable linguistic patterns. For the moment I do
  however not have any evidence how such complex
  achieve-ments can be observed or measured with
  the techniques of brain imaging available to
  date.

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Conclusions

• We need an intermediate level that generalizes
  the specific findings and constitutes a
  neurodynamic model of semantic processing. Such a
  model will build on the topology of the brain,
  synchronization and desynchronization, coupling
  of subnetworks with self-organization (filtering,
  choice of dominant modes), self-reference and
  monitoring in consciousness, etc.
• The class of models emerging in this field will
  certainly belong to dynamic systems theory,
  although such qualita-tive and simple models as
  catastrophe theory will be insuf-ficient, insofar
  as chaos-attractors, transitions between order
  and chaos, and stochastic processes (with
  diffusion) have to be considered.