Minds & Machines (2008) 18:331–348
DOI 10.1007/s11023-008-9103-9
Explanation in Dynamical Cognitive Science
Joel Walmsley
Received: 28 March 2007 / Accepted: 16 June 2008 / Published online: 2 July 2008
Springer Science+Business Media B.V. 2008
Abstract In this paper, I outline two strands of evidence for the conclusion that
the dynamical approach to cognitive science both seeks and provides covering law
explanations. Two of the most successful dynamical models—Kelso’s model of
rhythmic finger movement and Thelen et al.’s model of infant perseverative
reaching—can be seen to provide explanations which conform to the famous
explanatory scheme first put forward by Hempel and Oppenheim. In addition, many
prominent advocates of the dynamical approach also express the provision of this
kind of explanation as a goal of dynamical cognitive science. I conclude by briefly
outlining two consequences. First, dynamical cognitive science’s explanatory style
may strengthen its links to the so-called ‘‘situated’’ approach to cognition, but,
secondly, it may also undermine the widespread intuition that dynamics is related to
emergentism in the philosophy of mind.
Keywords Covering laws Dynamical cognitive science Emergence Explanation
Introduction
Could he, whose rules the rapid Comet bind,
Describe or fix one movement of his Mind?
Who saw its fires here rise, and there descend,
J. Walmsley (&)
Department of Philosophy, University College Cork, Cork, Ireland
e-mail: [email protected]
123
332
J. Walmsley
Explain his own beginning, or his end?
Alas what wonder!1
With these lines from his Essay on Man, Alexander Pope famously reminds us
that despite Newton’s groundbreaking work in the physical sciences, the mystery of
the human mind remains unsolved (and perhaps unsolvable). Whilst Newton had
shown that the same laws of nature govern both earthly and celestial motion, he
hadn’t discovered, as it were, the laws of thought. Nearly 300 years after his death,
however, the field of mathematics that Newton popularised—dynamics—has
become a fashionable way for man to study himself. The last 15 years have seen a
resurgent interest in a nexus of concepts which has become known as the
‘‘dynamical approach to cognition,’’ and an influential series of papers in
neuroscience2, psychology3, AI and A-life4 and especially philosophy5 have
examined the claim that the branch of mathematics known as dynamical systems
theory (DST) is the appropriate formalism needed to do cognitive science. Most
recently, the concepts of DST have even been extended to apply to questions
concerning embodiment and phenomenology6 as well as intentionality and the
mind-body problem more generally.7
Dynamical cognitive science is, in effect, an attempt to show that the mind is in
fact governed by the same kinds of laws as those which govern the motions of
comets. As Kelso notes:
Dynamics has not only permeated the natural sciences, but the social,
behavioural and neurosciences as well. Cognitive science … whose concepts
and paradigms are historically embedded in the language of the digital
computer, may be the last of Les Sciences Humaines to fall under the spell of
dynamics.8
In this paper, I wish to examine the kind of explanation found in dynamical
cognitive science. My main aim is to present two lines of evidence—one stemming
from the developed dynamical models, and the other stemming from the expressed
explanatory goals of dynamicists—for the conclusion that dynamical cognitive
science offers, and seeks, covering law explanations of the kind made famous by
Hempel and Oppenheim (1948). Along the way, I show how this may help
dynamical cognitive science to overcome one prominent objection, and I conclude
with some thoughts about the implications for the relation between dynamical
cognitive science and ‘‘situated’’ and emergent phenomena.
1
Pope (1903, p. 142), lines 35–29. Quoted in Kelso (2003).
2
Globus (1992); Skarda and Freeman (1987); Foss (1992).
3
Busemeyer and Townsend (1993); Thelen and Smith (1994).
4
Brooks (1991); Beer (1995); Bedau (1997).
5
Van Gelder (1995, 1997a, 1997b, 1998); Bechtel (1998); Clark (1997); Horgan and Tienson (1992,
1994, 1996).
6
Thompson (2007).
7
Tschacher and Haken (2007).
8
Kelso (2003, p. 46).
123
Explanation in Dynamical Cognitive Science
333
Before starting, two caveats are in order. Firstly, this paper is intended as an
exploration of the style of explanation in dynamical cognitive science. Whether this
account will extend to the application of DST in other empirical domains (or indeed,
to DST itself, considered as a branch of mathematics) is an entirely different question,
which I will leave aside. Secondly, this analysis is inspired by, and intended to apply
to, the relatively small existing ‘‘canon’’ of dynamical cognitive scientific models.
I do not wish to suggest that dynamical cognitive scientists are restricted to this
style of explanation. Indeed, some authors (e.g., Jost 2005) have suggested that as
the dynamical approach develops to include what we might call ‘‘non-classical’’
dynamics (for example, addressing issues such as chaos, autopoiesis and far-fromequilibrium dynamics), a different story may have to be told.
The aims of the present discussion are merely to show that the initial
development of dynamical cognitive science seems to have chosen to adopt the
covering-law model of explanation, and to highlight some of the ramifications of
such a choice.
Two Models
A good place to start is by examining some of the (relatively successful) models in
dynamical cognitive science which have piqued the interest of philosophers. One
such example is the ‘‘HKB’’ model of rhythmic finger tapping, named after its
originators, Haken et al. (1985). In what follows, I shall examine Kelso’s (1995)
exegesis of the model.
The HKB model is based upon the simple observation that, when asked to place
their hands palm-down and oscillate both index fingers back and forth with the same
frequency, people are reliably and stably able to reproduce only two basic patterns.
One is where the left index finger and right index finger both move to the left or to
the right at the same time (Kelso calls this ‘‘in phase’’ motion; it is the kind of
motion exhibited by the windscreen wipers on most north American automobiles).
The other is where one finger moves to the left, whilst the other moves to the right,
or vice-versa (‘‘antiphase’’ motion). Quantifying this observation, we can say that
the finger movements of subjects are stable when the relative phase of the finger is
either 180 degrees (for in-phase motion) or 0 degrees (for antiphase motion).
If people are asked to increase the oscillation frequency in time with a
metronome, two interesting results are found. Firstly, people who start working in
antiphase motion will, at a certain frequency of movement (the ‘‘critical region’’),
spontaneously switch to the in-phase mode. Secondly, subjects who start in-phase
would exhibit no such switch—the in phase motion will remain stable through and
beyond the critical region. Thus, Kelso writes: ‘‘…while people can produce two
stable patterns of low frequencies, only one pattern remains stable as frequency is
scaled beyond a critical point.’’9 In the language of dynamical systems theory, there
are two stable attractors at low frequencies and a bifurcation at a critical point,
leading to one stable attractor at high frequencies.
9
Kelso (1995, p. 49).
123
334
J. Walmsley
The specifics of the HKB model of finger movement can be elegantly captured
using the mathematics of dynamical systems theory. The HKB model focuses on the
nature of one collective variable—that of ‘‘relative phase,’’ designated as /—and
how it varies as a result of the control parameter: frequency of oscillation, (inversely
proportional to the ratio b/a indicated in Eq. 1).
Since we are concerned with the way relative phase changes over time, the coordination law may be expressed as a differential equation, in terms of the derivative
of / (d/
dt ), thus:
d/
¼ a sin / 2b sin 2/
ð1Þ
dt
In short, this equation states that the rate of change of relative phase is equal to a
periodic function of current relative phase and the frequency of oscillation. It shows
how, as b/a decreases (i.e., as the frequency of oscillation is increased) the system
goes from having two stable states (corresponding to in-phase and antiphase
motion), to only having one stable state (in-phase motion).
It is worth noting two features of this model that Kelso himself takes as
philosophically significant. Firstly, the model accounts for the data without positing
any kind of inner ‘‘switching mechanism.’’ The HKB model shows how the switch
from two stable states at low frequencies to one stable state at high frequencies
occurs as a natural product of the normal, self-organising evolution of the system
without reference to any kind of ‘‘central executive.’’ Secondly, Kelso notes that the
model was capable of making predictions of behaviour which had not yet been
observed in human subjects at the time the model was developed, but were
nonetheless subsequently confirmed in further experiments. Using the model,
Haken, Kelso and Bunz were able to predict the results of selective interference (if,
for example, they applied a small electrical pulse to the person’s hand so as to
temporarily disrupt the co-ordination). I shall return to these issues below when I
come to evaluate the kind of explanation this model provides.
A second well known (and relatively successful) dynamical model is provided by
Thelen et al.’s (2001) model of the ‘‘A-not-B’’ error—a phenomenon which was
first studied by Piaget. If you take an infant between 7 and 12 months old and hide a
toy or a cookie under one of two boxes in front of her, she will reach for the correct
box in order to get the object. However, if after hiding the object under one box
(‘‘A’’) a few times, you switch to hiding it under a second box (‘‘B’’), the infant
makes the ‘‘A-not-B error’’ and reaches towards box A, even though she saw you
hiding it under B.
Piaget originally argued that the error was something to do with the infant’s
understanding of object permanence. Some have contended that the error is due to
the infant’s impoverished concept of space (ego- versus allocentrically represented).
Others have suggested that the infant’s memory is underdeveloped.10 Investigation
of the matter is substantially complicated by the fact that, although the main effect is
quite reliable, it is enormously sensitive to slight changes in the experimental
conditions, such as the delay between viewing and reaching, the way the scene is
10
See Van Gelder (1999) for a good summary.
123
Explanation in Dynamical Cognitive Science
335
Fig 1 Thelen et al’s ‘‘grand equation’’ (Modified from Van Gelder, 1999)
viewed, the number of trials, the presence of distracting stimuli, and so on. Given
the right configuration of experimental conditions, even adults can succumb to the
error!
Each of the hypotheses manages to explain a limited subset of the data, but
fails to deliver an adequate account of the overall phenomenon. Psychologist
Esther Thelen and her colleagues11 contend that this is because all of the
hypotheses focus on the infant’s internal machinery—the way she thinks about the
world. Such accounts view the error as a cognitive one, which is to be explained
in terms of the infant’s grasp of concepts, and her mental representation of
objects.
Thelen et al. suggest that, in explaining the error, we focus instead on the infant’s
reaching activity, where the important factors are looking, planning, reaching and
remembering. Accordingly, they develop a dynamical model of how the infant
comes to reach in a particular direction, by quantifying these factors and relating
them in a ‘‘grand equation’’ (see Fig. 1).
The equation relates a number of factors, including the current state of the
movement planning field, general and specific aspects of the task (such as the
presence of two boxes and the toy being hidden under one of them), biasing
memories of previous reaches (roughly equivalent to habit) and interactions
between locations in the movement planning field. It specifies how the infant’s
inclination to reach in a certain direction changes over time as a result of changes in
these parameters.
11
Thelen and Smith (1994), Thelen et al. (2001).
123
336
J. Walmsley
Thelen et al. show how the dynamical system thus described reproduces the
classic A-not-B error, including many of the contextual subtleties with which
previous hypotheses were unable to cope. Further, they draw some striking
conclusions concerning the nature and necessity of a ‘‘representational’’ account of
mind (of which they are sceptical) and concerning the relationship between
dynamical and ‘‘situated’’ or ‘‘embodied’’ accounts of cognition. From the point of
view of the present discussion, however, the nature of explanation offered by this
model is of particular interest. Before turning to that, however, I would like to
address one possible objection.
Description vs. Explanation
Before considering the question ‘‘What kind of explanation do we find in dynamical
cognitive science?’’, a prior question needs to be answered: ‘‘Does the dynamical
approach explain cognition?’’ It may seem odd to assess the adequacy of
explanations in dynamical cognitive science in advance of determining what kind
of explanation they are. It is necessary, however, since one possible objection to the
dynamical approach is to deny that dynamical accounts explain anything, and argue
instead that they merely describe cognitive phenomena. I think that such an
objection can be overcome, and moreover, the proper response to such an objection
will leave us in a better position to assess the nature of explanation in dynamical
cognitive science.
Let me start by spelling out the objection in a little more detail. As far as I know,
the only place that this objection appears in print is in Tim van Gelder’s writing,
where he suggests it as a possible general objection to the dynamical hypothesis.
The objection, he says, is that ‘‘dynamical models are at best descriptions of the
data, and do not explain why the data take the form they do.’’12 The thought behind
this objection is that, given any set of data points, one can construct a line to connect
them, and any line can be described (to an arbitrarily close approximation) by some
equation or other. Thus, if we already know the data, we could engage in a ‘‘curve
fitting’’ exercise to connect the dots, calculate the equation which describes this line,
and offer the result as an ‘‘explanation’’ of the phenomenon that generated the data.
Van Gelder offers one possible response to the charge that dynamical models
provide mere descriptions rather than genuine explanations. He admits to the
possibility of such an illegitimate exercise in curve-fitting:
A poor dynamical account may amount to little more than ad hoc ‘curve
fitting,’ and would indeed count as mere description. Its problem, however, is
that it is poor, not that it is dynamical.13
But he goes on to note that ‘‘explanations’’ analogous to those given in dynamical
cognitive science are found in many other areas of scientific discourse, where they
are considered genuine and perfectly acceptable:
12
van Gelder (1998, p. 625). My emphasis.
13
Ibid.
123
Explanation in Dynamical Cognitive Science
337
Dynamical theories of cognitive processes are deeply akin to dynamical
accounts of other natural phenomena such as celestial motion. Those theories
constitute paradigm examples of scientific explanation. Consequently, there is
no reason to regard dynamical accounts of cognition as somehow explanatorily defective.14
At best, this reply to the ‘‘description-not-explanation’’ objection is incomplete,
for it could be taken as a reductio ad absurdum of dynamical ‘‘explanations’’ in
general. Along these lines, one might argue that dynamical accounts of celestial
motion are equally defective, because they too are ‘‘curve fitting’’ descriptions
rather than explanations proper.
What is needed is a specification of what it is about dynamical accounts of
celestial motion (and the like) that makes them into genuine explanations. One
could then see whether this property of dynamical explanations in other sciences
was also a property of dynamical explanations of cognition. So: what is it about the
differential equation in question that makes it more than simply a concise
redescripton of the data?
One immediate point to note is that dynamical accounts are clearly supposed to
go beyond the data, and, as it were, tell us more than we already know. A standard
assumption is that one feature of such explanations is that they support
counterfactuals. A good dynamical explanation will enable us to say how the
dynamical system in question would have behaved in various non-actual circumstances, for example if it suffered specific perturbations, or if its control parameters
were altered. Andy Clark notes that this ability is an important feature of
explanations in dynamical systems theory:
A pure Dynamical Systems account will be one in which the theorist simply
seeks to isolate the parameters, collective variables, and so on that give the
greatest grip on the way the system unfolds in time—including (importantly)
the way it responds to new, not-yet-encountered circumstances.15
A concrete example of this kind of counterfactual-supporting reasoning, as we
saw, comes from the HKB model of finger movement. Recall that the HKB formalism
was capable of predicting the results of selective interference with the system. Kelso
and his colleagues were able to predict such features as the amount of time it would
take for the relative phase to stabilise following the application of a small electrical
pulse to one of the fingers or the situations in which such perturbation would lead to
a transition from anti-phase to in-phase movement.
This observation has an important consequence. Dynamical accounts such as
Kelso’s make for a particularly close tie between explanation and prediction. The
ability to explain phase-transitions following selective interference is intimately
linked to the ability to predict phase-transitions prior to the interference. The
importance of this feature will become apparent in the next section.
14
Ibid.
15
Clark (1997, p. 119).
123
338
J. Walmsley
These two related factors—the ability to support counterfactual reasoning, and
the lack of a logical distinction between explanation and prediction—are both
features of what has come to be known as the ‘‘covering law’’ model of explanation.
It seems appropriate, then, to pursue the line of thought that dynamical explanations
in cognitive science might be covering-law explanations. This point is made by Bill
Bechtel:
DST accounts […] are clearly designed to support counterfactuals. This
suggests that it may be appropriate to construe these DST explanations as being
in the covering law tradition.16
Bechtel and Abrahamsen reiterate this claim elsewhere, writing:
The dynamical equations provide the laws for such covering-law explanations,
and by supplying initial conditions (values of variables and parameters), one
can predict and explain subsequent states of the system.17
To my knowledge, the suggestion that explanations in dynamical cognitive
science are a form of covering-law explanation is not argued for anywhere in the
literature. In fact, these two quotations, together with van Gelder’s (1991) denial of
the suggestion (more of which below), are the only two places I have found in which
the suggestion is even mentioned. In what follows, then, I wish to pursue these
suggestions in more detail, and argue that explanations in dynamical cognitive
science are indeed covering-law explanations. First, however, I would like to give a
very brief overview of covering-law explanations.
Covering Law Explanation
My general goal in this section is to show the background against which dynamical
explanations stand with a view to providing an account of explanation in dynamical
cognitive science itself. Needless to say, there is a large literature concerning the
nature and status of covering law explanations, and a comprehensive survey would be
neither appropriate nor productive here. Instead, I simply want to get the basics of the
view on the table in order to evaluate Bechtel’s suggestion, and to show, in the
following sections, that explanation in dynamical cognitive science fits this account.
The covering law model of explanation finds its most famous expression in Carl
Hempel and Paul Oppenheim’s 1948 paper ‘‘Studies in the Logic of Explanation.’’
As the title of their paper suggests, Hempel and Oppenheim view explanations as
having a structure in the form of a sound deductive argument where a law of nature
occurs as a premise, and the conclusion is the explanandum.18 In addition to
16
Bechtel (1998, p. 311).
17
Bechtel and Abrahamsen (2002, p. 267).
18
Putting it like this makes it clear why such explanations are sometimes called ‘‘deductivenomological’’ explanations. Bill Seager has suggested that ‘‘covering law’’ explanation might be
construed as a broader category than ‘‘deductive-nomological’’ if the former is intended to cover
deduction from statistical and probabilistic laws whereas the latter is not. Since I am more interested in
the deductive form of the explanation than the content of the laws, I shall use the former, broader, phrase.
123
Explanation in Dynamical Cognitive Science
C1, C2 ... Ck
Logical
deduct ion
L1, L2, ... Lr
E
339
St at ement s of
ant ecedent
condit ions
Explanans
General laws
Descript ion of t he
empirical
phenomenon t o be
explained
Explanandum
Fig 2 The form of a covering law explanation (Adapted from Hempel and Oppenheim, 1948, p. 138)
containing a law of nature, the explanans may also contain statements of known
facts, though this is not necessary.19 Accordingly, covering-law explanations, on
Hempel and Oppenheim’s view, have a structure such as that shown in Fig. 2.
Accordingly, the covering-law model views explanation as deduction from a law
in conjunction with statements of the antecedent conditions. As Hempel himself
puts it:
A DN explanation answers the question ‘‘Why did the explanandum
phenomenon occur?’’ by showing that the phenomenon resulted from certain
particular circumstances, specified in C1, C2,…Ck, in accordance with the laws
L1, L2,…Lr. By pointing this out, the argument shows that, given the particular
circumstances and the laws in question, the occurrence of the phenomenon
was to be expected; and it is in this sense that the explanation enables us to
understand why the phenomenon occurred.20
In addition to spelling out its form, Hempel and Oppenheim also outline four
conditions of adequacy for any given explanation to be sound. The first three are
logical conditions of adequacy, whilst the fourth is empirical. Firstly, ‘‘The
explanandum must be a logical consequence of the explanans’’; secondly, ‘‘the
explanans must contain general laws and these must actually be required for the
derivation of the explanandum’’; thirdly, ‘‘the explanans must have empirical
content; i.e., it must be capable, at least in principle, of test by experiment or
observation’’; and finally, ‘‘the sentences constituting the explanans must be true.’’21
The properties of explanations which take this form are well documented,
especially insofar as they constitute problems for the model. I shall mention some in
a little more detail below, but for present purposes, let us note one major feature.
19
Hempel and Oppenheim say that initial conditions aren’t necessary so that cases where one law is
deduced from another count as explanations. So, suppose one asked a question such as ‘‘Why is Boyle’s
law such that p = k(v/t)?’’ An answer would show how p = k(v/t) is derived from a law of statistical
thermodynamics. Even though the answer does not mention any initial conditions, Hempel and
Oppenheim still want this to count as an explanation (i.e., an answer to the explanation-seeking question).
20
Hempel (1965, p. 337). Italics in original. Note how this description assimilates explanation and
prediction.
21
All four quotations are from Hempel and Oppenheim (1948, p. 137).
123
340
J. Walmsley
Covering law explanations, thus construed, allow for no logical distinction between
explanation and prediction. Hempel and Oppenheim themselves admit this:
Let us note here that the same formal analysis … applies to scientific
prediction as well as to explanation … It may be said, therefore, that an
explanation is not fully adequate unless its explanans, if taken account of in
time, could have served as a basis for predicting the phenomenon under
consideration. Consequently, whatever will be said in this article concerning
the logical characteristics of explanation or prediction will be applicable to
either, even if only one of them should be mentioned.22
Accordingly, the only difference between a prediction and an explanation is
whether or not the state of affairs described in the explanandum is known to have
obtained. It is therefore solely a pragmatic difference—prediction and explanation
have an identical logical structure, but differ in terms of what one knows and what
one wants to know.
In many cases, this feature of covering law explanations is an advantage. From
the point of view of the practicing scientist, the ability to generate predictions is the
very condition of possibility of the empirical testing of theories. This is due to the
ability of the laws (which feature essentially in covering-law explanations) to
support counterfactuals. Given that genuine laws are more than mere generalisations, and are supposed to allow reasoning about as-yet-unobserved states of affairs,
a covering-law explanation in which the particular circumstances C1, C2,…Ck refer
to states of affairs which have not yet obtained will generate a prediction about what
would happen if they did. This is the essence of the way in which we normally test
scientific laws. One might define an experiment as an arrangement whereby we
bring it about that C1, C2,…Ck obtain, in order to see whether E also obtains. If it
does not, then we might have good reason to believe that L1, L2,…Lr are not true.23
Explanation by Dynamical Models
The dynamical models outlined above seem to conform very well to Hempel and
Oppenheim’s covering law model of explanation. By emphasizing their ‘‘grand
equation’’ (in Fig. 1), Thelen et al. offer what must be regarded as a covering-law
explanation. To explain the infant’s reaching behaviour (that is, how the
‘‘movement field’’ is changing at a given time), one may cite the equation, together
with values for the parameters and variables on the right hand side of the equation—
i.e., numerical values for (a) the current state of the movement field, (b) the general,
specific and memory inputs to the system and (c) the ‘co-operativity’ function,
which integrates competing inputs. The infant’s behaviour (expressed as changes in
22
Ibid., p. 138.
23
I owe this point to Kukla (2001) who writes: ‘‘an experiment may be defined as an arrangement
whereby the antecedent of a counterfactual is satisfied, whereupon we can observe whether the
consequent falls into line.’’ (p. 48).
123
Explanation in Dynamical Cognitive Science
341
these variables) follows as a mathematical (i.e., deductive) consequence of the
grand equation taken together with these initial states.
In this case, the explanation of a particular reaching-event after its occurrence has
the same logical form as the prediction of that reaching-event in advance of its
occurrence. Similarly, in virtue of the grand equation’s support of counterfactuals,
we could determine what would happen in various circumstances which have yet to
be observed; the model makes predictions beyond the known data (predictions
which were, in Thelen et al.’s work, subsequently confirmed). Thelen et al. note this
formal character of the model, together with its counterfactual-supporting abilities:
‘‘We offer a formal dynamic theory and model based on cognitive embodiment that
both simulates the known A-not-B effects and offers novel predictions that match
new experimental results.’’24 So Thelen et al.’s model can be regarded as offering a
covering-law explanation.
A similar story can be told for the HKB model, which specified how the relative
phase of two co-ordinated moving fingers changes as a result of changes in the
frequency of oscillation. That change can be succinctly captured by Eq. 1. We know
the value of /; in most cases it only takes one of two values, 0 or ±180,
corresponding to stable ‘‘in phase’’ and ‘‘anti phase’’ motion respectively. Further,
the ratio b/a is proportional to frequency of oscillation, which we also know in the
experimental condition, since we control it. The two stable states of the system are
the ones where d/
dt is zero (i.e., where the rate of change of relative phase is zero).
The bifurcation point—the critical region where switch from anti-phase to in-phase
motion takes place as the frequency increases—is the point where d/
dt is at a
maximum (i.e., where the rate of change of relative phase is at a maximum).
Depending on our interests, then, we can insert the values we know into the
equation, and solve the equation in order to find the values we do not know. Finding
d/
the values for a, b, and / when d/
dt is 0 would constitute to an explanation of why dt
takes the value it does, or a prediction that the system would be stable at these
values of a, b and /.
Again, we have a case where the explanandum in any given case is a deductive
consequence of the law (expressed by Eq. 1) combined with the initial conditions
(expressed as values of a, b, and /). The equation is actually required for the
derivation of the explanandum, the explanandum is a deductive consequence of the
explanans, and the explanans has empirical content (it was, after all, discovered on
the basis of observations of rhythmic finger movement in human subjects). So the
logical conditions of adequacy of a covering law explanation, as set out by Hempel
and Oppenheim above, are met. Further, since we have good evidence (or, at least,
the best evidence we could hope for) to suggest that the sentences constituting the
explanans are true, the explanation also fulfils the empirical condition of adequacy.
Finally, there is no logical difference between a prediction of a given state or an
explanation of the same state. In short, in predicting and explaining rhythmic finger
movement using the HKB model, one is using a covering law explanation.
I think that such a story can be told for all of the examples of dynamical models
of cognitive performance which are expressed using the quantitative aspects of DST
24
Thelen et al. (2001, p. 1).
123
342
J. Walmsley
(especially differential equations). In the cases where a dynamical model proceeds
on the basis of the qualitative aspects of dynamical systems theory (such as the
technical language of ‘‘phase spaces’’ and ‘‘attractor basins’’), it may be less clear
that the kind of explanation being sought is a covering law explanation. In these
cases, to determine the kind of explanation being sought, one may examine the
goals of dynamicists. This is the next line of evidence I wish to examine, for it
seems that, even if the laws which govern a dynamical cognitive system are not
known, it remains the goal of dynamicists to discover them, in order that they may
provide a covering law explanation.
The Explanatory Goals of Dynamical Cognitive Science
Thus far, I have shown that some dynamical models provide covering law
explanations. Yet this (empirical) observation alone is not enough to show that
dynamical cognitive scientific explanation is covering law explanation. It might be
that there is nothing essentially covering law about dynamical explanation in
general but rather that this view of explanation in dynamical cognitive science is an
artefact of the current state of the field. Perhaps future dynamical models will
provide different kinds of explanation. In this section, I intend to address such a
thought by examining the theoretical goals of dynamical cognitive scientists.
To this end, I feel it is most productive to let the advocates of dynamical
cognitive science speak for themselves; some quotations should help to illustrate my
point. Van Gelder outlines the aim of dynamical explanations in general, writing;
In studying and explaining the behaviour of dynamical systems one aims at
formulating equations which describe the evolution of the system, and can be
consequently used to explain why the system is in the state it is in, or to predict
what states it will come to be in.25
He later goes on specifically to emphasise the deductive nature of dynamical
explanations:
If we know the current state of the system—i.e., the point in state space it
currently occupies—we can use the equations governing the behavior of the
system to determine what point it will occupy next.26
This view on deducibility is reiterated in a deterministic form by Van Gelder and
Port:
Dynamical modelling … involves finding … a mathematical rule, such that
the phenomena of interest unfold in exactly the way described by the rule.27
The same authors later draw our attention to the fact that knowledge of both the
covering laws and the initial states is necessary for explanation-cum-prediction:
25
Van Gelder (1991, p. 500).
26
Ibid.
27
Van Gelder and Port (1995, p. 14).
123
Explanation in Dynamical Cognitive Science
343
Dynamical models based on differential equations are the pre-eminent
mathematical framework science uses to describe how things happen in time.
Such models specify how change in state variables at any instant depends on the
current values of those variables themselves and other parameters. Solutions to
the governing equations tell you the state that the system will be in at any point in
time, as long as the starting state and the amount of elapsed time are known.28
The same sentiments are expressed even by those with a lower level of
enthusiasm for the dynamical approach. Andy Clark is perhaps better viewed as a
sympathiser with, rather than an advocate of, the dynamical approach. In describing
explanation in DST more generally, he writes:
The mathematics typically specifies a dynamical law that determines how the
values of a set of state variables evolve through time. (Such a law may consist,
for example, in a set of differential equations.) Given an initial state, the
temporal sequence of states determined by the dynamical law constitutes one
trajectory through the space.29
He goes on to relate this style of explanation specifically to cognitive matters. In
describing how explanation in the dynamical approach seems to be radically
different from more familiar forms of explanation in cognitive science (more of
which below), he writes:
Most important in the present context, Dynamical Systems Theory also
provides a new kind of explanatory framework. At the heart of this framework
is the idea of explaining a system’s behaviour by isolating and displaying a set
of variables (collective variables, control parameters and the like) which
underlie the distinctive patterns that emerge as the system unfolds over time
and by describing those patterns of actual and potential unfolding in the
distinctive and mathematically precise terminology of attractors, bifurcation
points, phase portraits and so forth.30
Finally, John Holland, well known for his work in artificial life, considers how
the tools of dynamical systems theory may be applied to those models.
The object in constructing a dynamic model is to find unchanging laws that
generate the changing configurations… the laws of change specify the
succession of states [e.g.] the weather configuration eight hours from now,
twenty-four hours from now, and so on.31
I think that these quotations, coupled with the lack of an alternative metatheoretical stance towards dynamical explanation, are sufficient to establish that the
explanatory goal of dynamical cognitive scientists is to provide covering-law
explanations whereby a cognitive phenomenon is explained by way of citing the
28
Ibid., p. 19.
29
Clark (1997, p. 100).
30
Ibid., p. 115.
31
Holland (1998, p. 45).
123
344
J. Walmsley
laws (qua differential equations) that govern the system that produces it. To be sure,
such covering laws have not been, as yet, provided for all dynamical cognitive
models. Given that the provision of a covering law is one of the logical conditions of
adequacy that Hempel and Oppenheim (1948) outline, then, we must regard such
explanations as incomplete, pending the discovery and formulation of the covering
laws. Nonetheless, it remains the case that the goal is to do just that.
An Objection
Peter Carruthers recently noted ‘‘It is fair to say that a deductive-nomological
approach to explanation is now a minority position.’’32 If this is so for explanation in
general, it is particularly apparent within the cognitive domain. Psychology and
cognitive science have generally sought to provide explanations of a style different
from Hempel and Oppenheim’s covering law model. There are good theoretical
reasons for this; not since behaviourism has psychology tried to provide covering
law explanations, and behaviourism’s failure is at least partially attributable to the
form of explanation peculiar to it.
Most notably, there are a number of widely discussed problems with the covering
law model of explanation concerning the ‘‘direction’’ of explanation33 and the issue
of relevance.34 In fact, for the psychological domain, the problem is arguably even
worse. Psychological laws (or at least law-like, counterfactual supporting,
exceptionless generalizations like E = MC2 or F = MA) have been notoriously hard
to find. Further, where putative psychological laws are on the table, they seem to
contain ineliminable ceteris paribus clauses. In such cases one cannot account for a
psychological explanandum, since one cannot show how it follows from a
psychological (ceteris paribus) law, taken together with some specification of the
(psychological) initial conditions, without knowing if ceteris are paribus and what
those other things that have to be equal are. Suffice to say, there are a number of
historical and theoretical reasons why psychology has avoided the covering law
model of explanation to the point where the provision of covering law explanations
may even be taken as a reductio ad absurdum of a psychological theory.
In seeking to provide covering law explanations, dynamical cognitive science
will ultimately have to address these problems. Nonetheless, such problems do not
undermine my conclusion that explanations in dynamical cognitive science are
covering-law explanations. All such objections show is that explanations in
dynamical cognitive science will be subject to the same set of criticisms, in virtue of
the form they take.
32
Carruthers (2004, p. 159).
33
We might want to say that the height of the flagpole explains the length of the shadow, but we would
not normally want to say that the length of the shadow explains the height of the flagpole. The covering
law model offers no way to distinguish between the two with the former being the genuine explanation.
34
Consider the following (covering law) explanation: All males who take birth control pills regularly do
not get pregnant, John Jones takes birth control pills regularly, therefore John Jones does not get pregnant.
Such an argument conforms to a covering law scheme, but we would not normally want to say that the
premises are a good explanation of why John Jones fails to get pregnant.
123
Explanation in Dynamical Cognitive Science
345
Perhaps with considerations like these in mind, van Gelder has sought to argue
against the claim that dynamical explanations are covering law explanations. He
claims that dynamical explanations are not covering law explanations, because ‘‘we
can use dynamical explanations in situations where the equations governing the
evolution of the system are not themselves general physical laws.’’35 It is unclear
why van Gelder thinks that the laws which feature in covering law explanations
must be laws of physics. Such a requirement appears nowhere in Hempel and
Oppenheim’s discussion of the conditions of adequacy of a covering law
explanation. All that is required is that the laws in question have empirical content;
unless one is an eliminativist, there is more to empirical content than can be found
in the laws of physics. It is true that dynamical systems theory itself, as a branch of
mathematics, has no empirical content, but the application of its formalism to a
specific empirical domain gives rise to a new theory which does;36 the HKB model,
for example, is a hypothesis with empirical consequences about the rhythmic finger
movements that people can actually perform. In short, van Gelder’s counterargument against the thesis of this paper will not work, since it does not show that
dynamical explanations fail to meet the conditions of adequacy for covering law
explanations.
In any case, it should be noted that despite the well-known problems with the
covering law account of explanation, it has not been abandoned wholesale. Kim
(1999), for example, points out that the use of laws (particularly causal laws) is an
important virtue of explanations, since it forces them to ‘‘bring into the open the
explanatorily salient properties of the phenomena involved.’’37 The idea here is that
nomic subsumption, at the very least, may be kept as an essential feature of
explanations—Bird (1999) for example, writes that ‘‘we may be able to retain
Hempel’s guiding intuition that explanation is a matter of subsuming facts under
laws’’ even if other aspects of the covering-law model are inadequate.38 Finally,
Glymour has argued that explanations with a deductive structure are important for
supplying understanding-conveying contrastive explanations (i.e., for saying why
one event occurred, rather than another).39
Further considerations about the extent to which the covering law model of
explanation is worth reviving will have to be set aside for now. For the time being,
however, this goes to show that at least the use of covering-law explanations does
not constitute a reductio of the dynamical approach to cognitive science.
Conclusions and Implications
It seems, then, that a closer examination of both current dynamical cognitive models
and the expressed goals of dynamical cognitive scientists support Bechtel’s
35
Van Gelder (1991, p. 500).
36
Thanks to Christian Lacroix for pointing this out.
37
Kim (1999, p. 17).
38
Bird (1999, p. 17).
39
Glymour (2007).
123
346
J. Walmsley
suggestion that dynamical explanations are covering law explanations. I’d like to
conclude by hinting at two directions in which this observation may lead us.
Firstly, it’s a fact that psychologists have standardly sought modes of explanation
other than covering-law explanations. It seems fair to say that most cognitive
scientific explanations may be categorised as belonging to the covering-law model’s
principal competitor; Salmon’s (1984) causal account of explanation. In brief,
according to this model, to explain something is to specify the causal processes that
brought it about; often an explanans is a description of the causal mechanism that
generates the explanandum. In both classical and connectionist cognitive science,
explanations have standardly proceeded by appealing both to the structure of the
system in question (often including a specification of the interaction or co-operation
between the parts of that structure) and to the specified abilities of whatever it is that
is so structured. Dynamical cognitive science departs from this form of explanation,
since specifying the laws which govern a system does not necessarily specify how it
is physically constituted. In adopting this method of explanation, however,
dynamical cognitive science seems to be strengthened in other respects—in
particular, its explanations are not limited by the constraints of implementation, and
the explanatorily relevant features of a system need not correspond to the system’s
ontologically or causally delineated parts. This is particularly salient given
dynamical cognitive science’s interest in ‘situated’ cognitive phenomena, where
the constraints on a system extend beyond the boundaries of the system itself;
covering law explanations certainly seem better able to cope with this kind of
explanandum, since there is no limit as to what the variables in the law can refer to,
and thus they may track properties that obtain in the relation between an organism
and its environment.40
Finally, the fact that dynamical explanations are covering-law explanations
seems to cast doubt on the widespread intuition that dynamical cognitive science
will vindicate or illuminate emergentism about the mind. Since covering law
explanations require deducibility of the explanandum from the explanans, whilst
most conceptions of emergence (as a non-reductive position) require the absence of
such deducibility, it seems that dynamical cognitive science is in direct conflict with
emergentism about the mind. Given that covering law explanation is often seen as a
good example of reductive explanation, behaviours which can be explained
dynamically cannot be regarded as ‘‘emergent’’ in any non-reductive sense.
The only way to save the intuition that there is a relationship between dynamical
cognitive science and emergentism about the mind would be to reconstrue
emergence such that it is a thesis about laws. On such a view, it may be that the
dynamical laws (qua differential equations) are themselves not deducible from
‘‘lower level’’ laws of physics and/or physiology. The extent to which this idea
could be borne out by the future development of dynamical cognitive science
remains an open question.
Whether these count as problems for the dynamical approach remains to be seen.
Nonetheless, the nature of explanation in dynamical cognitive science causes some
familiar issues from psychology’s past—the search for the ‘‘laws of thought’’ in the
40
See, for example, Chemero (2001).
123
Explanation in Dynamical Cognitive Science
347
form of equations, the nature of explanation offered, and the implicit reductive
tendencies—to resurface. Even though dynamical cognitive science is often seen as
revolutionary (at least in the Kuhnian sense), it may be better, therefore, to regard it
as counter-revolutionary.
References
Bechtel, W. (1998). Representations and cognitive explanations: assessing the dynamicist’s challenge in
cognitive science. Cognitive Science, 22(3), 295–318.
Bechtel, W., & Abrahamsen, A. (2002). Connectionism and the mind (2nd ed.). Oxford: Blackwell.
Bedau, M. (1997). Emergent models of supple dynamics in life and mind. Brain and Cognition, 34, 5–27.
Beer, R. (1995). A dynamical systems perspective on agent-environment interaction. Artificial
Intelligence, 72, 173–215.
Bird, A. (1999). Explanation and laws. Synthese, 120, 1–18.
Brooks, R. (1991). Intelligence without representation. Artificial Intelligence, 47, 139–159.
Busemeyer, J., & Townsend, J. T. (1993). Decision field theory: a dynamic-cognitive approach to
decision making in an uncertain environment. Psychological Review, 100, 432–459
Carruthers, P. (2004). Reductive explanation and the explanatory gap. Canadian Journal of Philosophy,
34(2), 153–173.
Chemero, A. (2001). Dynamical explanation and mental representation. Trends in Cognitive Science,
5(4), 141–142.
Clark, A. (1997). Being there: Putting brain, body and world together again. Cambridge, MA: MIT Press.
Foss, J. (1992). Introduction to the epistemology of the brain: Indeterminacy, micro-specificity, chaos,
and openness. Topoi, 11, 45–57.
Globus, G. (1992). Toward a noncomputational cognitive science. Journal of Cognitive Neuroscience, 4,
299–310.
Glymour, B. (2007). In defence of explanatory deductivism. In J. K. Campbell, M. O’Rourke & H.
Silverstein (Eds.), Causation and explanation. Cambridge, MA: MIT Press.
Haken, H., Kelso, J. A. S., & Bunz, H. (1985). A theoretical model of phase transitions in human hand
movements. Biological Cybernetics, 51, 347–356.
Hempel, C. G. (1965). ‘‘Aspects of Scientific Explanation’’ in his aspects of scientific explanation and
other essays in the philosophy of science. New York: Free Press
Hempel, C. G., & Oppenheim, P. (1948). Studies in the logic of explanation. Philosophy of Science, 15,
135–175.
Holland, J. (1998). Emergence: From chaos to order. Reading, MA: Perseus.
Horgan, T., & Tienson, J. (1992). Cognitive systems as dynamic systems. Topoi, 11, 27–43.
Horgan, T., & Tienson, J. (1994). A nonclassical framework for cognitive science. Synthese, 101, 305–
345.
Horgan, T., & Tienson, J. (1996). Connectionism and the philosophy of psychology. Cambridge, MA:
MIT Press.
Jost, J. (2005). Dynamical systems. Berlin: Springer-Verlag.
Kelso, J. A. S. (1995). Dynamic patterns: The self-organisation of brain and behavior. Cambridge, MA:
MIT Press.
Kelso, J. A. S. (2003). Cognitive coordination dynamics. In W. Tschacher & J.-P. Dauwalder (Eds.), The
dynamical systems approach to cognition (pp. 45–67). Singapore: World Scientific.
Kim, J. (1999). Hempel, explanation, metaphysics. Philosophical Studies, 94, 1–20.
Kukla, A. (2001). Methods of theoretical psychology. Cambridge, MA: MIT Press.
Pope, A. (1903) In H. W. Boynton (Ed.). The complete poetical works of Alexander Pope. Boston:
Houghton Mifflin.
Salmon, W. (1984). Scientific explanation and the causal structure of the world. Princeton: Princeton
University Press.
Skarda, C. A., & Freeman, W. J. (1987). How brains make chaos in order to make sense of the world.
Behavioral and Brain Sciences, 10, 161–195.
Thelen, E., Schöner, G., Scheier, C., & Smith, L. B. (2001). The dynamics of embodiment: A field theory
of infant perseverative reaching. Behavioral and Brain Sciences, 24, 1–86.
123
348
J. Walmsley
Thelen, E., & Smith, L. B. (1994). A dynamic systems approach to the development of cognition and
action. Cambridge, MA: MIT Press.
Thompson, E. (2007). Mind in life: Biology, phenomenology and the cognitive sciences. Harvard:
Harvard University Press.
Tschacher, W., & Haken, H. (2007). Intentionality in non-equilibrium systems? The functional aspects of
self-organized pattern formation. New Ideas in Psychology, 25, 1–15.
van Gelder, T. (1991). Connectionism and dynamical explanation. In Proceedings of the Thirteenth
Annual Conference of the Cognitive Science Society. Hillsdale N.J.: L. Erlbaum Associates (pp.
499–503).
van Gelder, T. (1995). What might cognition be, if not computation? Journal of Philosophy, XCI(7),
345–381.
van Gelder, T. (1997a). Connectionism, dynamics and the philosophy of mind. In M. Carrier & P. K.
Machamer (Eds.), Mindscapes: Philosophy, science and the mind (pp. 245–269). Pittsburgh:
University of Pittsburgh Press.
van Gelder, T. (1997b). Dynamics and cognition. In J. Haugeland (Ed.), Mind design II. Cabridge, MA:
MIT Press.
van Gelder, T. (1998). The Dynamical Hypothesis in Cognitive Science. (with peer commentary).
Behavioural and Brain Sciences, 21, 615–665.
van Gelder, T. (1999). Revisiting the dynamical hypothesis. Preprint No. 2/99, University of Melbourne,
Department of Philosophy. Available at http://www.arts.unimelb.edu.au/*tgelder/papers/Brazil.pdf
. Accessed 13 August 2004.
van Gelder, T., & Port, R. (1995). ‘‘It’s about time: An overview of the dynamical approach to cognition.
In R. Port, T. Van Gelder (Eds.), Mind as motion. Cambridge, MA: MIT Press.
123