Of computations and dynamic systems - An overview of the dynamicist
controversy in cognitive science1
Sylvain Pronovost
Institute of Cognitive Science, Carleton University
Carleton University Cognitive Science Technical Report 2006-05.
- Abstract This paper constitutes an overview of two competing conceptual frameworks in the
study of cognition, the now standard computational approach and the more recent,
and controversial, dynamical hypothesis in cognitive science, championed by T. van
Gelder et al. Through such conceptual and methodological disputes about the nature
of cognition, a debate about the adequacy of their respective models has been the
main ground for disagreements. I propose to explore each framework, or paradigm,
in turn, by focusing on their definition and use of a number of critical characteristics
of intelligent behavior, namely that of representations, computation, and exactly
what is a cognitive feature or process. The conclusions that I have reached are
twofold: firstly, the dynamicist view of the computational approach to cognition in
no way discredits its relevance to cognitive modeling, since dynamicists are not
concerned with the same features of mental processes in their models, and their
evaluation of what counts as computational is based on a common misconception,
namely a confusion between the abstract and formal concept of computation with
that of physical symbol systems. Secondly, the type of explanation used by the
dynamicist view is quite different, for it concerns nomological explanations (i.e.
explanations through covering laws), whereas the computational view frames its
explanations in a mechanistic manner.
- Table of contents -
Abstract..………………………………………………………...………… ii
Table of contents……………….…….………………………………….... iii
Index of figures……….……………….…………………………………..
Index of acronyms…………………….………………………………….. vii
Introduction: Of models and minds………………………………………….....
- I - Computational and dynamical systems……………………………………
- I.I - Computation and the computational hypothesis in cognitive science…
………………………………………………………………………...… 7
- I.I.I – Origins………………………………………………....….
- I.I.II - Queering up the concept of computation………………... 10
- I.I.III - Cognition as computation………………………………. 12
- I.II - Dynamics and the dynamical hypothesis in cognitive science……. 16
- I.II.I – Origins ………………………………………………….. 17
- I.II.II - Varieties of dynamical systems…………………………. 19
- I.II.III - Cognition as a real dynamical system…………………. 24
- II - Case study: computational and dynamical accounts of sensorimotor
cognition……….………………………………………………………………… 30
- II.I - The evidence………...…………………………………………….. 32
- II.II - The argument…………………………………………………….. 39
- II.III - The model…………………………………………………..…… 44
- III - Issues, controversies, and answers concerning the framing of cognition in
a computational or a dynamical model………………………………………... 55
- III.I - On the nature of cognition vis-à-vis computation and dynamical
systems……………………………………………………………………………. 57
- III.I.I - Giunti and van Gelder on the mathematical properties
required to properly model cognitive systems and processes……………………. 58
- III.I.II - Piccinini on symbols, strings, and neural spikes………. 68
- III.I.III - Bechtel and Eliasmith on the issue of representations in
dynamical systems…………………………………………………………..……
- III.II - On the type of explanation involved in computational and dynamical
models..................................................................................................................... 75
- III.II.I - Piccinini on functionalism and computationalism as
independent characterizations in cognitive science……………………………… 75
- III.II.II - Bechtel, on mechanistic explanations versus nomological
explanations in cognitive science………………………………………………… 80
- IV - Close encounters of the third kind: connectionism..……………………. 85
- IV.I - Misunderstandings so far: on representation and computation, types
of explanation, and the special case of connectionism…………………………..
- IV.II - Types of connectionist models, and what makes them more or less
dynamical………………………………………………………………………… 92
Conclusion: Strange bedfellows? Computational and dynamical models in
cognitive science……………………………………………………………….. 103
Finale…..……………………………………………………………………….. 110
Appendices………………………………………………………………………. 111
Appendix I - Definitions of computation………………………………... 112
Appendix II - Computational and dynamical models of low-level cognitive
processes………………………………………………………………………… 118
Bibliography……………………………………………………………………… 122
- Index of figures Figure 1
A vector field defined over the reals, along a single solution
trajectory…………………………………………………………………………. 22
Figure 2
Two trajectories for a vector field F(x,y) = (y,-x). These trajectories
are periodic cycles……………….……………………………………………….. 22
Figure 3
A toroidal manifold, shown along with the phase portraits of
solutions on, and approaching, the manifold…………………………………….
Figure 4
A geometrical representation of the attractor shape of a frictionless
mass-and-spring system, given three sets of initial parameters………………….. 24
Figure 5
A mass-and-spring system influenced by a friction parameter, here a
constant in the ordinary differential equations…………………………………… 24
Figure 6
The state space of the gradient descent learning function (defined
over weight and error) for a given number of connections in an artificial neural
network…………………………………………………………………………… 29
Figure 7
The geometrical representation of the learning partition, through
gradient descent, of a given number of hidden units in an artificial neural network…
………………………………………………………………………… 29
Figures 8, 9, 10, and 11
Geometrical representations of some parameters of the
inverse kinematics problem, such as it is studied by computer scientists, engineers
and roboticists……………………………………………………………………. 36
Figure 12
A still frame of a 3D rendered simulation of inverse kinematics,
where a “tail’ tries to reach and touch a small green cube, illustrating the complexity
of devising such an algorithm……………………………………………………. 36
Figure 13
A geometrical MDS representation of the performance of an infant in
an object-hiding task named the ‘A-not-B error’, with regards to the development of
object permanence……………………………………………………………….. 49
Figure 14
A geometrical MDS representation of a general ontogenetic
landscape, where development is seen as a series of evolving and dissolving
attractors over time……………………………………………………………….. 50
Figure 15
A geometrical MDS representation of a more sophisticated depiction
of the ‘A-not-B error’ task. It shows some properties of a movement field without
specific input……………………………………………………………………... 51
Figure 16
An artificial neural network, an interconnected groups of nodes, akin
to the vast network of neurons in the human brain………………………………. 92
Figure 17
Neurosynaptic pathways, illustrated through immunofluorescence…
…………………………………………………………... 92
- Index of acronyms -
ANN: artificial neural network
CH: computational hypothesis
CHCS: computational hypothesis in cognitive science
CNS: central nervous system
CS: cognitive system
DH: dynamical hypothesis
DHCS: dynamical hypothesis in cognitive science
MCS: mathematical computational system
MDS: mathematical dynamical system
ODE: ordinary differential equations
PDE: partial differential equations
RDS: real dynamical system
Of computations and dynamic systems - An overview of the dynamicist
controversy in cognitive science
- Introduction - Of models and minds
Model (abstract)
From Wikipedia, the free encyclopedia.
An abstract model (or conceptual model) is a theoretical construct that represents physical,
biological or social processes, with a set of variables and a set of logical and quantitative
relationships between them. Models in this sense are constructed to enable reasoning within an
idealized logical framework about these processes and are an important component of scientific
theories. Idealized here means that the model may make explicit assumptions that are known to be
false in some detail, but by their simplification of the model allow the production of acceptably
accurate solutions […]
What is a conceptual framework? Epistemologists and philosophers of
science may not agree on the minutiae of the concepts of concept, knowledge and
science, but we can roughly sketch an uncontroversial canvas: it is a set of concepts
and methods through which people generate conjectures and theses, and strive to
produce descriptions, explanations, and predictions about entities, events, and
phenomena. Since we would rather conceptualize and explain phenomena in an
interesting manner, that is, with teleological considerations such as accuracy,
efficiency, and consistency, many constraints have to be taken into account to
establish what constitutes a successful conceptual framework. Science, unarguably
the most demanding conceptual and methodological endeavour in the pursuit of
knowledge, has numerous constraints through which are filtered what are considered
acceptable theses, methods, models, and what counts as evidence. Among these
constraints, modern philosophy of science commits us to two universal tenets, (i) an
ontological commitment dubbed materialism, which states that science’s domain is
the material world and that it should not bother itself with spiritual or religious
phenomena, and (ii) an epistemological claim named naturalism, the view that valid
explanations or theories ought to make use of, and only of, entities accessible to
natural science. Much is to be said about and within epistemology and philosophy of
science, such as whether or not paradigms are commensurable and continuous, or if
there exist radical shifts in conceptual frameworks, on the value of reductionism in
view of scientific claims of different levels of description, and on what counts as
criteria of justification or proof for such scientific claims.
Allowing myself the luxury of a metaphor, conceptual frameworks can be
seen as universes of discourse, following in that the semantic theories in the
philosophy of language, spawned from the works of Frege, Peirce, and their
successors of the analytic tradition. One semantic view of analytic philosophy,
notably, attributes meaning and truth value to propositions validated by models, viz.
a larger set of propositions mirroring entities and relations between them, that
Here I use the term of conceptual framework in a broader sense than a simple matrix of concepts
and relations, so as to include models and methods, akin to Kuhn’s concept of paradigm.
represents a ‘possible world’, or complete state of things. Thus we could view the
scientific discourse concerning biology, for example, as a universe or domain of
discourse in terms of semantics, and different theories and models of evolutionary
biology, such as phyletic gradualism, punctuated equilibria, and creationist models
(if there are any that can achieve a reasonable degree of rigor) would validate or
invalidate propositions made about the entities, and relations between such entities,
of the biotic realm . Philosophy of language or otherwise, the emphasis is that
models involve a set of entities and relations by which they purport to accurately
describe, explain (and make predictions about) the phenomena under enquiry.
From a somewhat direct lineage of ancestors such as cybernetics,
information theory and the study of algorithms in mathematics, computationalism
has established itself as a predominant conceptual framework to deal with enquiries
concerned with what we understand by intelligence. Computationalism is our more
recent conception of intelligence, the view that cognition can be understood as
information processing, and has spun models of intelligence inspired by information
processing technologies. It reaches as far as the study of biological cognition and
even the whole of life sciences altogether, ubiquitous in a way that finds its way into
the labelling of our era, the Information Age.
Numerous models of cognition as information processing under the guise of
computationalism have been suggested, from the already classic seminal works of
Turing (1936) on formal, discrete and machine-like computation, and Rumelhart,
McClelland et al (1986) on parallel and distributed, brain-like computation, to their
philosophical critics and promoters like Fodor and Pylyshyn (1988), and P.M.
Churchland (1989, among many other references). Yet, for some skeptics,
computationalism is mainly concerned with simulations of informational processes,
and while it doesn’t seem to be controversial for the purpose of developing
‘intelligent’ devices and technologies, it’s being considered as the basis for such
I chose the semantic conception of a model for its simplicity and scope, but the issue is not
unproblematic in the details, and much of this paper revolves around the very minutiae of models in
cognitive science.
models of cognition does not appeal to everyone. Thus, computationalism has been
challenged on nearly all possible grounds, with regards to its structure, its
constituents and foundations, and its ability to stand for as a qualitative and/or
quantitative model of what is meant by cognition (among others, Brooks 1991, Clark
1992, 1998, Dreyfus 1992, Elman 1998, Freeman & Nuñez 1999, Giunti 1995,
Piccinini 2003, Stufflebeam 1998, Thelen 1995, van Gelder & Port 1995, Wertsch
Criticizing is one thing, proposing solutions is another. Has anyone come
forth with an alternative framework that might deal with the shortcomings of
computationalism and yet bear as much, if more, explanatory and predictive power
as required of a rigorous scientific endeavor? Some believe so, and the answer might
come from a rather physicalist perspective (it certainly doesn’t get any more natural,
as in what we mean by naturalism and the naturalization of cognition), that is, the
theory of systems dynamics. From the conceptual framework of dynamical systems
theory and with the help of its formal and quantitative counterpart, namely
dynamical modeling, I will try to assess their position by confronting what as been
dubbed the dynamical hypothesis (DH hereafter) about cognition, with the dominant
yet quite problematic computational hypothesis (CH hereafter) about cognition.
To this end, the present dissertation is divided in four chapters. The first
presents a quick overview of what the computational and dynamical frameworks are,
what kind of characteristics and ambiguities define and populate such frameworks,
as well as what is entailed by adopting a computational hypothesis in cognitive
science (CHCS hereafter), or the dynamical hypothesis in cognitive science (DHCS
hereafter). The second chapter is an incursion into the cognitive science of
sensorimotor processes, which exposes the application of the previously defined
frameworks to empirical research. Through an attempt to link the dynamicists’
allegedly revolutionary point of view with neuroscientic findings on the workings of
a certain class of low-level cognitive processes, namely sensorimotor control and
T. van Gelder (1995, and subsequent work).
learning, we will then be able to understand the full extent of the claims of both
frameworks on such evidence. This will also make possible a clear, concise ground
on which to position ourselves in further characterization of the issues at hand. The
third chapter presents formal shortcomings and technical issues of both the CHCS
and the DHCS, from areas as different as mathematics and neuroscience. The
definitions of computation established in the first chapter, as well as the
dynamicist’s conceptual repertoire, will be confronted with formal considerations
and empirical evidence. To this end, the chapter exposes criticisms, objections, and
answers from protagonists of both frameworks, concerning the advantages and
shortcomings of their respective stance in cognitive science. The fourth and final
chapter presents the controversial class of models that is connectionism. Since
proponents of both frameworks insist on claiming connectionism as part of their own
view of cognition, the entire chapter is devoted to the clarification of what is at stake
in connectionist models, both formally and empirically. In the conclusion, I will
attempt to synthesize critical issues of, and possible answers to, the clash of such
conceptual frameworks, driven by a scrupulous desire for univocal concepts. I
advocate the adoption of a rigorous vocabulary concerning cognitive, computational,
and dynamical themes, a point that unfortunately needs to be emphasized
notwithstanding its ubiquity in the requirements of a sound academic enterprise. The
fact of the matter is that throughout this thesis, I aim to expose a number of
incorrectly defended positions criticizing computationalism and promoting
dynamics, and such misconceptions undermine the authority of the supported
arguments in a way that requires us to redefine the relative advantages, limitations,
relevance, and scope of both conceptual schemes.
The debate on whether a dynamical framework is preferable to a
computational one can be developed on many avenues, and I have chosen to
emphasize epistemological and semantic issues. This decision is based on two
observations, namely (i) since, as it is exposed in chapter III, a solution to the
disagreements between these two frameworks may be found in the type of
explanation held dearest by their respective protagonists, we may assume that a
discussion of ontological and formal issues would not focus on the essential
divergences between the CHCS and the DHCS that this dissertation aims to disclose,
and (ii) such ontological considerations about the nature of cognition, and formal
issues in the mathematical treatment of cognitive modeling, would be sufficient
grounds to motivate the writing of two other extensive dissertations altogether.
Therefore, while this dissertation indeed exploits qualitative and quantitative
mathematical issues, as well as essential ontological topics about the mind, all such
considerations are secondary to the main line of argumentation. The question of
which of the CHCS or the DHCS constitutes the best possible explanatory
framework may not be entirely independent of formal and ontological issues, it is
nevertheless in a noncommittal stance on such questions that I intend to conduct my
It is worth noting that while I have undertaken an assessment of the
conflicting views of the computational and dynamical frameworks, I do not pretend
that this schism is ‘the’ most fundamental issue at hand in cognitive science, with
respects to models. For the sake of discussion, I will subsume the biophysical
models of neuroscience to the dynamical view (for it is indeed concerned with
systems dynamics, and written in the language of calculus), and such models will be
a central issue in many parts of the following discussion. The particular status of
connectionism will also be addressed along the way, and as it turns out, it will be a
critical element in the assessment of the two paradigms’ claims, but I won’t commit
myself to its characterization yet. It is less a matter of introducing some element of
suspense for the reader, than a preoccupation with mathematical issues that do not
lend themselves to a casual overview.
- I - Computational and dynamical systems
- I.I - Computation and the computational hypothesis in cognitive science
From Wikipedia, the free encyclopedia.
A calculation is a deliberate process for transforming one or more inputs into one or more results.
The term is used in a variety of senses, from the very definite arithmetical calculation using an
algorithm to the vague heuristics of calculating a strategy in a competition or calculating the chance
of a successful relationship between two people […]
From Wikipedia, the free encyclopedia.
Computation can be defined as finding a solution to a problem from given inputs by means of an
algorithm. This is what the theory of computation, a subfield of computer science and mathematics,
deals with. For thousands of years, computing was done with pen and paper, or chalk and slate, or
mentally, sometimes with the aid of tables […]
In order to give a fair treatment to the debate between the computational
framework in cognitive science and its more recent contender, the dynamical
framework, we firstly have to present the two positions in some detail, concerning
their technicalities and their history. We thus begin with the dominant view, that of
computation. Exactly what is it that the computational hypothesis entails in the
realm of cognitive science? In order to answer that question, we have to firstly
characterize the theory of computation, secondly, distinguish two complementary
yet different conceptions of computation, and thirdly, link this theory to the
exploration of cognition. It will also become evident that the term ‘computational’
refers to a great many things, and perhaps unsurprisingly as such, since its mere
formal origins portrayed the concept in a vague, abstract sense.
- I.I.I - Origins
Mathematics spawned the concept of computation. The issue at hand, at the
dawn of the twentieth century, was the question of which formal problems could be
solved, and which couldn’t be. Thus a formal method of analysis had to be
developed to this end. The issue wasn’t trivial at all: defining which set of relations
could be solved was quintessential to formal analysis, and to all of the quantitative
sciences using such formalisms. Science being dependent on mathematics, defining
the class of problems that could in principle be effectively and quantitatively
formalized was no mere undertaking. But just what is a formally solvable problem?
Fregean logic (and its successors) and most if not all of mathematics model
interesting relations as functional relations between arguments and values. As odd as
it may appear, the very definition of a function is not that old, it was coined by
Leibniz in the late seventeenth century in his development of calculus. Euler later
(middle of the eighteenth century) extended the concept to all expressions composed
of arguments. When Weierstrass suggested the adoption of arithmetic as a basis for
calculus rather than geometry, in the late nineteenth century, Euler’s conception of a
function took over the entire field of mathematics. Thus, functions are a special
subset of relations, linking each element of a set to a unique element of another (or
the same) set. Such relations permit effective quantitative analysis and, by
extension, effective and workable science.
As we mentioned at the beginning of this chapter, the most abstract
definition of computation involves a procedure by which one finds a solution to a
problem, given one or more input values (or initial conditions in a broader sense).
This procedure is commonly named algorithm, and further characterized as a finite
and well-defined set of instructions that will produce an equally well-defined result.
Mathematicians thus devised models to meet the challenge of computable problems,
of which the effective characterization is to be treated as functions and arguments. It
was eventually assessed by Alonzo Church (1936ab) and Alan Turing (1936) that
the class of computable functions is equivalent to the class of functions defined by
the following models:
- recursive functions
- lambda calculus
and that class of computable functions is also definable as algorithms calculable by:
- Markov algorithms
- register machines
- Post systems
- Turing machines
In terms of computation, the preceding formalisms, and algorithms operating
over such formal languages, were shown to be equivalent in computational power. In
other words, any and all computations that can be ‘performed’ through one
formalism, can in principle also be performed through any other. For the sake of
mathematical enquiry, that means that many classes of problems are effectively
computable, spanning from partial functions to computable complex numbers, and
Formal definition of a function: a function f from a set X of input values to a set Y of possible output
values (written as f : X → Y) is a relation between X and Y which satisfies:
1. f is total, or entire: for all x in X, there exists a y in Y such that x f y (x is f-related to y), i.e.
for each input value, there is at least one output value in Y.
2. f is many-to-one, or functional: if x f y and x f z, then y = z. i.e., many input values can be
related to one output value, but one input value cannot be related to many output values.
6 Appendix I describes in details the class of computable functions and its equivalents.
find applications even into chaos and quantum related problems. On the other hand,
many more magnitudes of formal problems involved in the construction of
mathematical proofs and mind-boggling numbers are said to be uncomputable, for a
variety of reasons, such as uncountability, computational complexity classes, or the
apparent impossibility of subsumption of interesting phenomena under a
deterministic formalism (for certain areas of applied mathematics), et caetera. This
is a very important issue in computability theory with regard to the rest of this essay,
for part of the controversy at hand between computational and dynamical enthusiasts
is the relevance and scope of the formal tools of computability theory with respect to
cognitive science.
- I.I.II - Queering up the concept of computation
It is worth mentioning, if not essential to underline the computational
equivalence in power of computable functions, and the algorithms defined over
them, to the familiar digital computer, with the relevant yet secondary criterion of
requiring infinite memory in the definition of an abstract computer, by opposition to
the finite constraints of implemented computational devices. The Turing machine is
an abstract model of an algorithm which can calculate any and all of the computable
functions. But the concept of computation itself is very large, and while a recursive
function is calculable by a Turing machine, these two concepts are not identical.
Recursive functions are the class of functions, from natural numbers to natural
numbers, that are computable, but they are a matter of discrete mathematics almost
exclusively, namely number theory and combinatorics (the issue of computable reals
and complex numbers will be raised in chapter III). The study of algorithms that are
Turing Machines is an extension of applied mathematics and computer science, and
as such concern both empirical and formal matters. Thus we can draw a first
distinction between the concepts of computable functions (a strictly formal,
mathematical concept), their computational equivalent classes (formal systems such
as programming languages in computer science, generative grammars in linguistics,
etc), and the algorithms defined over Turing Machines, which concern
implementation issues and are as such beyond the scope of an exclusively formal
account. Yet another contrast worth mentioning, related to the aforementioned
distinction between abstract and material computers, is between universal Turing
machines or UTMs, a definition of an algorithm given infinite resources of
calculation, and digital computers, the actual implementation of such abstract
devices. For the purpose of clarification, further references to systems based on
computation will either call upon a MCS (a mathematical computational system,
pertaining to formal models), or a RCS (a real computational system, viz. a system
which actually performs computations).
We thus draw an elementary distinction between a first, permissive but
trivial, formal definition of computability (large computation, viz. anything
effectively represented as recursive), versus the narrow, and additionally empirical,
concept of Turing-computation (symbolic, discrete, and serial computation, both
abstract and implemented). Of particular interest to us, then, the Church-Turing
thesis as it is commonly named, thus concerns the nature of mechanical devices,
beyond mathematical problems. Following Turing, “Every function which would
naturally be regarded as computable can be computed by a Turing machine.”
(Turing, 1936, p. 230) As fundamental as it is, this thesis can not be proven or
disproven by formal means, since the concept of computable function used in the
formulation is too vague. Some view the Church-Turing thesis as a physical law
(viz. a nomological statement), since it can’t be mathematically or logically proven.
Perhaps the problem lies in the fact that the concept of computation, in
Turing’s sense, relies on another equally vague concept, that of algorithm. A coarse
characterization of an algorithm states that: (i) the algorithm consists of a finite set
of simple and precise instructions that are described with a finite number of symbols,
(ii) the algorithm will always produce the result in a finite number of steps, (iii) the
algorithm can in principle be carried out by a human being with only paper and
Hereafter, I shall use the term of (symbolic) Turing-computation to refer to symbolic computation,
and computation or computability to refer to the class of computable functions, as is understood by
Turing computability.
pencil, and (iv) the execution of the algorithm requires no intelligence of the human
being except that which is needed to understand and execute the instructions. Now,
as intuitive as it may be, the concept of algorithm is not formally defined, since there
is no means to so characterize what is meant by ‘simple and precise instructions’ and
‘required intelligence for the execution of the instructions’.
Those conceptual vagaries, as harmless as they may seem for matters of
mathematics and information sciences, are in my opinion not only the source of
much of the confusion in the clashes of the many proponents of computationalism in
areas such as cybernetics, early cognitive psychology and classic artificial
intelligence , but also one critical point of dissension between computationalists and
dynamicists. Such distinctions in matters of computation will thus be essential in the
following discussion.
- I.I.III - Cognition as computation
What does all of this have to do with cognition? The computational
hypothesis in cognitive science (CHCS), the dominant conceptual framework in
cognitive science, is based on the complementary theses of (i) functionalism,
roughly, the philosophical idea that mental states are functional states (the
ontological commitment of cognitive functionalism), and can thus be accounted for
without taking into account the underlying physical substrate, but instead by
attending to (here, representation-laden) functional states (the epistemological part
of functionalism), and (ii) cognitivism, the epistemological position in the
philosophy of mind which argues that mental functions can be understood by
quantitative, positivist and scientific methods (for instance, that such functions can
be described through information processing models for the sake of psychological
A. Markov (1960).
Also coined ‘GOFAI’, for good old fashioned artificial intelligence, by John Haugeland (1985,
10 Most important in the case of the CHCS, a notion that we will explore further in chapter 3.
Intelligent behavior had been, from the early years of the 20th century until
the 1950s, studied under the dominating paradigm of behaviorism, a strict empirical
approach mainly concerned with the naturalization of human activity through
external observation. Heralds and leaders of this paradigm indeed dominated the
North-American academic scene during the first half of the 20th century, with figures
such as Watson (1913) and Skinner (1938) in psychology, Bloomfield (1933) in
linguistics, and the associated endeavors of Carnap (1966) and Hempel (1966), in
the form of logical positivism in philosophy. But the shortcomings of its
methodology and concepts, along with the evolution of ideas in novel research areas,
gave way to the rise of cognitivism. Daring and far reaching projects shifted the
obsession with externalism to inward and mechanistic exploration, spawning
cybernetics , information science , cognitive linguistics , and the foundations of
artificial intelligence , to name a few (This list is by no means exhaustive). Further
works along interdisciplinary boundaries, concerning conceptual and methodological
issues, have since then both enriched and plagued the computational view.
The functionalist thesis stated above is foundational in cognitive science, as
mental events are to be distinguished from the physical substrate on some ground
(be it properties, if not in terms of substance), albeit to some degree of sophistication
that has evolved beyond the traditional philosophical divide imposed by Descartes.
Indeed, 20th century sciences of mind and behavior thrived to come to terms with
what we call the mind-body problem, but could not escape the boundaries imposed
by our intuitions on the matter. That is precisely what led to the adoption of
cognitivism: the adoption of a scientifically rooted view of the mind, founded on a
functionalist stance, and drawing upon information science to model language,
championed by Rosenblueth, Wiener, and Bigelow (1943, also Wiener, 1948), von Neumann
(Aspray & Burks, 1987, for a collection of papers), McCulloch and Pitts (1943)
12 Turing (1950) and Shannon (1948)
13 Chomsky (1957, 1968, among others)
14 pioneered by Newell and Simon (1956, also Newell, 1980), Minsky and Papert (1969, also
Minksy, 1968)
memory, perception, sensorimotor processes, etc., or in fewer words (but in a coarse
way), everything relating to intelligent behavior.
Most endeavors traded on higher levels of cognitive processes, such as
semantics, deliberation, decision-making, et caetera, like the works of Chomsky and
his generative grammars (1968), Fodor and his language of thought (1975), and
some of the abovementioned thinkers in artificial intelligence, too name a few. Such
proponents are usually dubbed ‘symbolicists’, since they championed philosophical
and scientific views of cognition in the raw and formal way of Turing, namely the
view of cognition as the manipulation of symbols, in the likeness of a universal or
implemented Turing machine, exhibiting characteristics such as discreteness,
seriality, and intentional (representational) contents individuated through semantic
Thus did computationalism provide a framework for cognitive science that
could account for mental phenomena in many advantageous avenues:
the age-old problem of the separation between mind and body, which had
been made quite popular in philosophy through the works of Descartes, was
tossed aside through a functionalist conceptualization, abandoning a
substantiative conception of everything mental to the benefit of a ‘mind is to
the brain as what the software is to the computer’ stance, in an effort to
naturalize cognition,
a formal account of cognition was able to link such mental-related faculties
like language use, logico-mathematical abilities, memory, categorization, et
cetera, with the machine-like conception of a Turing machine, viz. the
implementation of an effective, formal and generalizable procedure meant to
carry out operations on functions and arguments,
the formal and technical properties of computational models were meant to
reflect cognitive ones, including
o the representational nature of mental tokens, which exhibited
intentional, content-bearing states, much like language. The symbolic
aspect of Turing-like computation embraced by symbolicists was
seen as an essential property of high-level cognition, although the
case of lower-level cognition would eventually challenge such a
restrictive take on computation,
o the discreteness and seriality of a Turing machine-inspired conception
of cognition also seemed to fit well the aforementioned mental
faculties of language and logico-mathematical performance. Turing
machines (and common digital computers) process information in a
serial way (successive operations), over discrete (distinct, non
continuous) values, the content of which is individuated by a
representational relation, from symbol to object. This relation is thus
conventional, arbitrary.
Other thinkers slowly but surely championed alternative views, of which
connectionism is the most popular inheritor. On grounds of psychological
plausibility, the parallel and distributed nature of information processing in the
brain, and the implausibility of content individuation through discrete, symbolic
tokens in a significant manner even for simulated cognition, some theorists
resurrected the low profile heritage in cognitive neuroscience of individuals such as
the abovementioned McCulloch and Pitts (1943), Hebb (1949), and Rosenblatt
(1962), to name a few. This would lead to a radical turn in computational modeling,
and the pretences of artificial intelligence would thereafter be severely modified.
The issue of whether some sophisticated models of connectionism have more to do
with computation or dynamical models will be examined throughout this paper, for
it has been raised as an argument to support the claims of protagonists of both
frameworks. Chapter IV examines the issue of connectionism comprehensively, and
with more minutiae.
like Churchland (1986) and Churchland (1989), Rumelhart and McClelland (1986, also Rumelhart,
1989), and Smolensky (1988, 1989)
- I.II - Dynamics and the dynamical hypothesis in cognitive science
Dynamical system
From Wikipedia, the free encyclopedia.
In engineering and mathematics, a dynamical system is a deterministic process in which a function's
value changes over time according to a rule that is defined in terms of the function's current value
Dynamics (mechanics)
From Wikipedia, the free encyclopedia.
In mathematics and physics, dynamics is the branch of mechanics that is concerned with the effects
of forces on the motion of objects […]
The proponents of the dynamical approach to cognitive science are
dissatisfied with the dominant view of cognition as computation. Some suggest a
radical paradigm shift, pretending that dynamical systems theory and dynamical
modeling, inconsistent with the computational view, bear all of the necessary and
sufficient concepts and methods for the study of cognition, while others adopt a
moderate position, suggesting a number of prescriptions to compensate for the
shortcomings of the CHCS, drawing from both the mathematical minutiae and
qualitative resources of dynamics. We will firstly characterize what dynamics stand
for, secondly, observe two varieties of dynamical systems that account for formal
and empirical types of systems, and thirdly, expose the dynamical hypothesis
concerning cognitive science.
- I.II.I - Origins
From Merriam-Webster Online Dictionary.
Function:noun Inflected Form(s): plural cal·cu·li also -lus·es Etymology: Latin, stone (used in
reckoning) 1 a : a method of computation or calculation in a special notation (as of logic or symbolic
logic) b : the mathematical methods comprising differential and integral calculus […] 4 : a system or
arrangement of intricate or interrelated parts.
Whereas computability is the domain of applied discrete mathematics and
computer science (although it also draws on information science), dynamics are a
subset of applied mathematical analysis and the branch of physics concerned with
machines or machine-like objects, in the broad sense of the area of study of
mechanics, but more specifically within the branch of dynamics, the study of the
effects of forces on the motion of objects. Thus on one hand, dynamics are derived
from empirical studies in the physics of motion and forces, with their most
significant lineage tracing back to Newton, in the late seventeenth century, when he
proposed his three laws of motion (the law of inertia, the fundamental law of
dynamics, and the law of reciprocal actions). On the other hand, dynamics have
much to owe to the mathematical formalism that Newton and Leibniz developed
Some evidence suggests that calculus-related methods and concepts were known by Egyptian and
Hellenistic thinkers, notably Eudoxus and Archimedes.
concurrently, but independently: calculus. For Newton, calculus was the necessary
means to quantify and express his findings in classical mechanics. Although calculus
is thus connected to the advent of Newtonian mechanics, it has thereafter evolved
somewhat independently, along the lines of abstract, fundamental mathematics.
Calculus is built on studies in algebra and geometry, and relies on the notions of
functions and limits. It basically involves the study of two concepts that are
indissociable, essentially complementary: that of rates of change and accumulation
of quantities. These two concepts are formally expressed by differential and integral
calculi, respectively.
The developments of infinitesimal calculus, as it is commonly called, were
expanded to all of physics’ domains in the following centuries, from particle physics
to astrophysics, but even into the life sciences, humanities, and social sciences. On
the other hand, mathematicians like Laplace and Lagrange brought the concepts and
methods of dynamics to a full bearing into the study of mathematical analysis, the
study of real and complex numbers, and of the functions defined over them. Thus the
interactions between the empirical applications of dynamics and its formal
counterpart, calculus, have been mutually enriching, contributing to great
developments in physics and mathematics, while inspiring other disciplines to make
use of such conceptual and methodological tools. Following Giunti (1995) and van
Gelder and Port (1995), we can conceive of dynamics’ contribution to other areas of
science as twofold: through the use of (i) dynamical systems theory, we have
concepts, qualitative methods and models from which to draw parallels with other
phenomena, develop explanations, and make predictions, and (ii) dynamical
modeling is the formal means by which we express the relevant features of the
phenomena under study, both qualitatively and quantitatively, through infinitesimal
calculus, ergo differential and integral equations.
- I.II.II - Varieties of dynamical systems
Areas and extensions of calculus include: differential equations, vector calculus, calculus of
variations, complex analysis, time scale calculus, infinitesimal calculus, and differential topology.
18 All references for this section, A. Norton (1995).
The following characterizations are meant to form a simple introduction and
overview of the relevant features of dynamics. We must first distinguish between a
real dynamical system (RDS) and a mathematical dynamical system (MDS), namely
the distinction between the phenomenon under observation (a system in which
features or elements change over time interdependently, like the weather, an ant
colony, the cardiovascular system, etc.), and the mathematical model used to
represent the system’s qualitative changes through variations in the features’ or
elements’ magnitudes. A broad definition of dynamical systems is that they are
deterministic processes in which a function’s value changes over time, according to
a rule that is defined in terms of the function’s current value. More precisely, in the
words of Norton (1995, p. 45),
A mathematical dynamical system consists of the space of all possible states of
the system together with a rule called the dynamic for determining the state
which corresponds at a given future time to a given present state.
The algebraic or geometrical representation of the collection of all possible/relevant
values is called the state space of the system.
Of major hindrance to the study of dynamics are the sensitivity to initial
conditions and the nonlinearity of most systems. As Poincaré pointed out in the late
19th century, most systems, even composed of a few variables, do not allow for the
simple calculation of a solution. Indeed, most nonlinear, and even some piecewise
linear systems, exhibit chaotic behavior, viz. apparently random, unpredictable
behavior from deterministic systems. Poincaré thus suggested that dynamical
systems theory could be the basis for a serious qualitative method of analysis, with
regards to the intractability of most systems. The concepts thereafter developed
involve trajectories, stability, recurrence, attractors and bifurcations, and generic
behavior, all of which provide us with useful methodological means of studying the
Source: Wikipedia, under ‘dynamical system’.
A piecewise linear system is a system whose mathematical characterizations allow certain areas,
but not its overall state space, to be calculated through simple algebraic functions.
overall behavior of simple and complex systems, and generate explanations and
predictions by observing their state space. In Norton’s words again (id., p. 47):
[…] the state of mathematical art dictates that any tractable mathematical
model should not have too many variables, and that the variables it does have
must be very clearly defined. As a result, conceptually understandable models
are sure to be greatly simplified in comparison with real systems. The goal is
then to look for simplified models that are nevertheless useful.
The calculus-based formalism of dynamics allows for two specific types of
systems, if interpreted in terms of continuity and enumerability of time-dependent
variable evolution, namely continuous dynamics and discrete dynamics. The formal
descriptions of the former are the (algebraic) differential equations and (geometrical)
flows, and for the latter, the (algebraic) difference equations and (geometrical)
diffeomorphisms. While continuous dynamics are essential to both fundamental
mathematics (analysis) and applied mathematics (mechanics), discrete dynamics are
a useful tool to predict the qualitative changes of both linear and nonlinear systems.
Discrete dynamics also have similarities with the large field of discrete mathematics
and computability theory, but are interested in describing and predicting timedependent changes within the state space of the concerned system. Such similarities
and differences will play an important role in the following discussions.
A dynamical system is said to be discrete if its time parameter is measured in
discrete steps, i.e. that its state space on the time parameter is a metric of evenly
spaced discrete jumps. Such systems are modeled through recursive relations .
Discrete time dynamics use difference equations, equations defined over integers
(for time values) as well as reals (for values of other parameters), by means of
recursive functions being iterated on chosen initial values. When a system is
modeled has having a continuous time parameter, i.e. its metric for time is a
continuous progression over the real numbers, it is expressed through ordinary
For reference, the logistic map is a simple nonlinear second-degree polynomial, which can be
expressed discretely: xt+1=axt(1-xt).
differential equations (ODE) or partial differential equations (PDE) . A differential
equation in one variable, or one dimension, is an equation composed of a function x
and one or more of its derivatives. Partial differential equations are much more
complex, involving partial derivatives of functions of more than one variable. The
distinction between linear and nonlinear systems is also very important: linear
systems have solutions that form a vector space, and allows the reduction of the
problem from calculus to linear algebra. Indeed, one can solve a continuous linear
differential equation by reducing it to an algebraic equation, through an algorithm
called the Laplace transform method. But as mentioned above, most nonlinear, and
some piecewise linear systems challenge our means of calculation: they can’t be
solved explicitly. The vast majority of natural phenomena being nonlinear under
mathematical formalization, we have to rely on sophisticated qualitative and
quantitative means of analysis.
Geometrical and topological considerations help greatly in the understanding
of such systems. Given a vector field F, one can find the solution trajectories that
pass through the field in the proper way (i.e. given some initial parameters and the
geometrical progression of the relevant differential equations). Each trajectory then
corresponds to a set of input parameter values and their solution to the equations (see
figures 1, 2 and 3 below for examples of vector fields and solution trajectories). The
full solution of an equation is called a flow, using the notation φ(t,x), and describes
the position of a point x on its solution trajectory for a time t. Note that not all
solution trajectories are necessary or relevant, and can be defined over a restricted
surface, or a manifold for higher dimensional state spaces.
For reference, the logistic map can be defined over the reals through the following ordinary
differential equation: dx/dt=ax(1-x).
Two interesting concepts emerging from geometrical and topological
characterizations of dynamical systems are attractors and bifurcations. According to
Norton (id., p. 56), “attractors are important because they represent the long-term
states of systems.” Roughly, attractors can be defined in the following way:
Let F be a vector field on Rn, with flow φ. A closed set A ⊂ Rn is an attractor
for this flow if (i) all initial conditions sufficiently close to A have trajectories
that tend to A as time progresses, (ii) all trajectories that start in A remain
there, and (iii) A contains no smaller closed subsets with properties (i) and (ii).
An interesting subset of attractors is that of the strange or chaotic attractors, which
exhibit diverging nearby trajectories following similar overall directions, and
generally possess a fractal structure, where large scale variations are also found on
smaller scales. Bifurcations reflect states of transitions in a system, “when a
parameter value is reached at which a sudden change in the qualitative type of the
attractor occurs.” (id., p. 57) Bifurcations can be seen as thresholds where certain
parameter values generate different dynamical behaviors. Thus, the system’s overall
behavior is dependent on the conjunction of the dynamic rule(s) and input
parameters. Attractors and bifurcations are of great importance in even the simplest
systems. To clarify this point, let us briefly consider Norton’s example of a
frictionless mass-and-spring system, versus a system which takes friction under
consideration. The passage from a MDS of a frictionless mass-and-spring system,
whose geometrical representation exhibits a periodic attractor shaped in a circle (for
it depends on initial values of position and velocity only), to a more complex system
involving the drag force sliding friction, clearly shows that not only is qualitative
behavior dependent on the parameters involved, but that the simple addition of a
significant real world feature like friction into the dynamics of a system greatly
complicates both qualitative and quantitative analysis. (see figures 4 and 5 below)
Figure 4 (left) is a geometrical representation of the attractor shape of the frictionless mass-andspring system, given three sets of initial parameters. The abscissa x represents the distance from the
rest position, and ordinate u is the velocity. Figure 5 (right) is the mass-and-spring system influenced
by a friction parameter, here a constant in the ordinary differential equations.
- I.II.III - Cognition as a real dynamical system
Dynamical systems theory
From Wikipedia, the free encyclopedia.
Dynamical systems theory is an area of mathematics used to describe the behavior of complex
systems by employing differential equations.
Proponents of the dynamical systems theory approach to cognition […] believe that differential
equations are the most appropriate tool for modeling human behavior. These equations are
interpreted to represent an agent's cognitive trajectory through state space. In other words,
dynamicists argue that psychology should be (or is) the description (via differential equations) of the
cognitions and behaviors of an agent under certain environmental and internal pressures. The
language of chaos theory is also frequently adopted.
What does it mean to have a dynamicist’s view of cognition? The DHCS, or
dynamical hypothesis in cognitive science , is the view that cognitive processes and
related states are best described and explained through the conceptual language and
models of dynamic systems theory and dynamical modeling. While not incompatible
with one of the main tenets of computationalism, viz. the thesis of functionalism (by
taking into account the essentially embodied nature of cognition), it does clash with
the strong claim of cognitivism, in the matter of what type of model best explains
cognitive processes. Against the information processing models championed by
classic computation and classic artificial intelligence (Haugeland’s GOFAI view
mentioned above), dynamicists suggest that the mathematical models of dynamics
The DH, or dynamic hypothesis, was coined by van Gelder (1998b). The DHCS acronym is used
here to contrast with Piccinini’s CHCS.
offer a more accurate depiction of cognitive processes, and allows formal and
empirical coherence at all levels of cognitive modeling. This section exposes the
concepts championed by the core assumptions of the majority of dynamicists,
namely the interdependence of context, corporeality, and cognitive processes
(embeddedness/situatedness and embodiment), the simultaneity and time-dependent
evolution of processes, the emergence of structure and behavior from cognitive
processes’ interactivity, and the heterogeneity of cognitive time scales.
According to van Gelder (van Gelder and Port, 1995, p. 2) , “the heart of the
problem is time. Cognitive processes and their context unfold continuously and
simultaneously in real time.” Now, intuitions and conceptual issues about cognitive
processes are one thing, but dynamicists insist that this deeper a priori problem is
the source of much of the misconceived models of cognition, a legacy of
computationally framed cognitivism that favored Turing’s metaphor of calculation
to generalize it as a theory of mental processes. But computation lacks many features
that seem essential to frame cognition properly, according to dynamicists. van
Gelder et al hold that we already have many reasons to hold on to the DHCS:
We know, at least, these very basic facts: that cognitive processes always
unfold in real time; that their behaviors are pervaded by both continuities and
discretenesses; that they are composed of multiple subsystems which are
simultaneously active and interacting; that their distinctive kinds of structure
and complexity are not present from the very first moment, but emerge over
time; that cognitive processes operate over many time scales, and events at
different time scales interact; and that they are embedded in a real body and
environment. (id., p. 18)
The following issues are meant to illustrate what dynamics have to offer in view of
the shortcomings of the computational theory of mind.
computationalism models cognitive processes in sequences of discrete steps,
dynamics help model processes in real time, specifying not only the states of the
All references for this section are taken from van Gelder and Port (1995).
system but also their time evolution. Time is continuous in dynamical models, it is
also a quantity, or magnitude, on which other cognitive related magnitudes are
dependent, thus providing analyses rich in resolution and detail. Dynamicists stress
the issue of cognition occurring in time, not simply over time like computational
models frame cognition. To say that cognition unfolds in time is to hold that
cognitive processes are time dependent, and that considerations of simultaneity,
embeddedness, and interdependence of a multiplicity of time scales (like neural
processes time, perceptual time, decision making time, learning time, and maturation
time) are fundamental to cognition itself. Dynamics are precisely the kind of
mathematical means to formalize processes that occur over time, with differential
calculus pertaining to rates of change, and integral calculus concerning the
accumulation of quantities. The interdependence of multiple time scales is
formalized by using multiple variables within those equations, which stand for
relevant cognitive magnitudes. State variables and parameters can both be seen as
changing, thus representing the coevolving nature of processes on different scales.
State continuity, and the multiplicity and simultaneity of interactions.
Dynamics provide formal tools to model both the continuity and discreteness of
processes and states, whatever best suits the phenomenon under observation.
Dynamicists are concerned with continuity not only in time, but also in state, of the
processes underlying cognition. While many cognitive tasks are modeled through
discrete dynamics, such as language-related performances and logical and
mathematical calculations, a much larger spectrum of processes unfold continuously
in time and state, such as sensorimotor processes and related procedural tasks. As
pointed out by van Gelder, discreteness of states is also quite often a matter of
perceiving seemingly discrete qualitative changes in continuous processes, such as it
is conceptualized in dynamics by the use of the term catastrophe, viz. sudden and
dramatic changes in the behavior of a system, when a parameter’s change in
magnitude causes a bifurcation, as seen in the previous section. Similarly, dynamics
offers incomparable advantages for the formalization of simultaneity and
interactivity, and at all levels of cognitive processes, such as in the modeling of
interactive agents in interpersonal tasks, sensorimotor processes, and neurobiological
modeling, to name a few. The simultaneity and interactivity of component
subsystems, or of cognitive agents, is essential to carrying on individual and
collective tasks, all of which fall under the domain of calculus and dynamics in the
framing of overall and local behavior, and allows predictions based on both
quantitative and qualitative methods.
Self-organization and emergence. The organization of cognitive systems and
processes exhibits complex and intricate design, or structure. Dynamicists propose to
not only describe existing cognitive structures, but also to provide a framework able
to explain how such structures came to be in the first place, namely a means to
explain the emergence of such design. The conceptual and formal tools of dynamics,
because they involve the modeling of spatial and temporal structures, offer the
possibility of analyzing the time and state evolutions, or morphogenesis, of complex
structures in physics, chemistry, and biology. There are therefore good reasons to
believe that using dynamical systems theory and dynamical modeling for the
purpose of studying the morphogenesis of cognitive processes and structures is a
heuristic avenue. Many physical, chemical, and biotic systems are also studied under
the stance of self-organization principles, which holds that some structures come
into existence with neither a plan, nor an independent, external builder, but through
simple formal and empirical principles governing the organization of elements into
complex and heterogeneous wholes. Here again, dynamics are of great use to model
such phenomena and to elaborate explanations. Dynamicists interested in the study
of cognition propose that we have evidence towards such a view of mental processes
(chapter II will present an application of such theses in the study of ontogenetic
dynamics), and even purport to link cognition and evolution as emergent structures
on a shared spectrum, only pertaining to different time scales dynamics.
Embeddedness and embodiment. Dynamicists refuse to hold on a
conservative definition of a cognitive system as a strictly internal structure. In order
to account for cognitive processes, we must integrate considerations about their
neural correlate, their corporeality, and their embeddedness or situatedness in a
context, an environment. But neural processes, behaviors, and the whole of
environment are already quite efficiently and heuristically modeled through
dynamics! It thus appears that dynamics, beyond their being desirable in the study of
cognition, might even be unavoidable as such. As van Gelder points out, it is also a
matter of advantage that is put forward here, since accounts in terms of dynamics for
internal matters of cognition find themselves in continuity with the dynamics of
behavior and context, thus facilitating the integration of a variety of systems for
explanatory and predictive purposes. Such an integration of component systems into
what constitutes cognition reflects the interdisciplinary endeavors found in cognitive
science, since neither neuroscientific, behavioral or ethological, psychological, or
social systems can be exhaustive by themselves of what counts as cognitive. The
supporters of the DHCS propose to study precisely the interactions between internal
cognitive processes, the body, and different contexts or environments. Dynamicists
accuse computationalism of having simply avoided the problems posed by the
discontinuity and heterogeneity of systems. By tacitly positing the autonomy of the
study of cognition, computationalists have thus avoided, and failed to account for,
most, if not all of the abovementioned issues raised by the DHCS. In van Gelder’s
[…] whenever confronted with the problem of explaining how a natural
cognitive system might interact with another that is essentially temporal, one
finds that the relevant aspect of the cognitive system itself must be given a
dynamical account. It then becomes a problem how this dynamical component
of the cognitive system interacts with even more ‘central’ processes. The
situation repeats itself, and dynamics is driven further inward […] (id., p. 30)
Worth mentioning: Churchland (1989) used dynamical representations (figures 6 and 7) of the
behavior of formal neurons while promoting connectionism against classical (symbolic) Turingcomputation models (the symbolic view of cognition). Why bother, a dynamicist might ask, to
employ a computational framework, when we have everything we need in biophysics and
neuroscience to characterize the (still very mechanistic) functional decomposition of informational
processes in nonlinear and differential equations, rightful domain of dynamics? (see Giunti, Piccinini,
and van Gelder’s arguments in chapter III about that position)
- II - Case study: computational and dynamical accounts of sensorimotor
Cognitive Science
From Wikipedia, the free encyclopedia.
The term "cognitive" in "cognitive science" is "used for any kind of mental operation or structure that
can be studied in precise terms." (Lakoff and Johnson 1999) This conceptualization is very broad,
and should not be confused with how "cognitive" is used in some traditions of analytic philosophy,
where "cognitive" has to do only with formal rules and truth conditional semantics. (Nonetheless,
that interpretation would bring one close to the historically dominant school of thought within
cognitive science on the nature of cognition - that it is essentially symbolic, propositional, and
The earliest entries for the word "cognitive" in the OED take it to mean roughly pertaining to "to the
action or process of knowing". The first entry, from 1586, shows the word was at one time used in the
context of discussions of Platonic theories of knowledge. Most in Cognitive science, however,
presumably do not believe their field is the study of anything as certain as the knowledge sought by
The conceptual frameworks of computation and dynamics having been
summarily exposed, we have then observed their bearing on the study of cognition.
This chapter presents an application of the previously defined frameworks to
empirical research on a particular kind of cognitive processes, sensorimotor control
and learning. This contextualisation of the aforementioned concepts and models of
computational theory and dynamics is essential to the following discussion,
presented in chapter III, since it will provide us with a clear picture of the claims and
allegations upon which are based most of the misunderstandings and quarrels
between supporters of both frameworks. Both the CHCS’ and the DHCS’ views on
such cognitive processes are presented, and the quantitative and qualitative
properties of the suggested models will be exposed so as to understand their full
extent, and evaluate their respective claims. Since computationalism is rather
ubiquitous even in the realm of cognitive neuroscience, I will therefore begin with
the exposition of the aforementioned evidence dealing with sensorimotor processes.
In a second section, I will then argue about the alleged advantages of switching
frameworks for a dynamical account of cognitive processes, by opposition to the
traditional computational view. In the last section, I will present a correlated
dynamical account of sensorimotor processes which bears explanatory and
predictive significance to a higher level of description, that of developmental
psychophysiology (more specifically, psychophysics) , with the help of Thelen’s
(Thelen 1995, Thelen, Schöner, Scheier and Smith 2001, and Smith and Thelen
2003) findings and subsequent model. I therefore aim to evaluate the conjectured
benefits of the adoption of a dynamical perspective on cognition, relative to the (still
controversial) shortcomings of the traditional computational framework, while
Psychophysiology is vaguely defined as the science of understanding the link between psychology
and physiology. Psychophysics appears to be more specific, defined as the branch of psychology
dealing with the relationship between physical stimuli and their perception. All references online,
showing the relevance of said dynamical account in the rather different studies of
sensorimotor cognition from neuroscience and psychophysics.
- II.I - The evidence
From Wikipedia, the free encyclopedia.
In physics, kinematics is the branch of mechanics concerned with the motions of objects without
being concerned with the forces that cause the motion. In this latter respect it differs from dynamics,
which is concerned with the forces that affect motion […]
Inverse kinematics
From Wikipedia, the free encyclopedia.
Inverse kinematics is the process of determining the movement of interconnected segments of a body
or model. For example, with a 3D model of a human body, if the hand is moved from a resting
position to a waving position, how do the connected fingers, forearm, upper arm and main body
move in response? It is a subject of programming and animating. It is approached often in game
programming and 3D modeling […]
A standard way to describe neuroscientific evidence is through
neurobiological modeling (Montague & Dayan, 1998), which in turn relies heavily
on formal characterizations of more or less sophisticated computational design,
when imported into the arena of cognitive science. By contrasting the conceptual
language of neuroscientific studies of sensorimotor processes involved in control
and learning with that of the CHCS, it becomes apparent that neuroscience draws
upon a mathematical language that extends beyond that of computation, towards
dynamics. The case study concerns the problem of inverse kinematics, addressed
from both computational and dynamical perspectives.
Whether we are concerned with what is traditionally considered a high level
(language and decision-making, to name just a few) or low level (emotional
responses, sensorimotor activity) cognitive process, neurobiological modeling is
computation-laden. As we have seen in chapter I, both the original forms and
contemporary offshoots of cognitive science - including some cognitive
neuroscience - (i) view mental states as functional states, and (ii) conceive these
functional states as computational, viz. to be modeled and explained through
information processing concepts and schemes. Computational functionalism is the
conjunction of the two, but they are logically independent theses, according to
Piccinini (2003, 2004b, 2004d). Part of the ambition of this chapter is indeed
concerned with showing that while a functionalist account of cognition can be
appreciated from within a computational framework, it does not entail that
computationalism is the only functionalist model of mental processes, and there are
different aspects of cognition that are worth scrutinizing.
According to Albright (1993, p. 178), “motion processing serves a number of
behavioral goals, from which it is possible to infer a hierarchy of computational
steps.” (My emphasis) The initial step for motion perception is said to be motion
detection, more precisely, the perception of motion direction. What purposes might
motion perception serve? Albright lists a few, such as establishing the volumetric
structure of a scene, posture and balance control, the appraisal of one’s own
trajectories and possible collisions, segregating visual inputs into objects and
background, and identifying and predicting the motion of objects to respond
accordingly, among others. Each of these sensorimotor functions can be in turn
described computationally and neurophysiologically in detailed steps. Research on
motion bestows significance to sensorimotor behavior in a causal and mechanistic
way. For example, parietal cortical stream (areas MT and MST) activity and motor
control of ocular globes activation (by means of dorsolateral pons) suggests a causal
relationship from the former to the latter. Functional decomposition is an essential
part of a mechanistic explanation, shared by a plethora of sciences, both
computational and noncomputational. Thus, following Albright, we can say that the
main function of motion perception for our purpose is the affordance of motor
control. Yet, Albright also attempts to characterize sensorimotor processes in the
A more precise, and therefore different, commitment of cognitivism, which conflates the CHCS
and the possibility evoked by cognitivism of a sound empirical, scientific account of mental
processes. This is not unproblematic, since dynamicists claim to the naturalization of cognition on
different grounds. It is therefore important to distinguish computationalism as part, but also a stricter
form, of cognitivism.
larger perspective of (implicitly) an agent or organism and (explicitly) an
environment (this will turn out to be less trivial than it might sound at first):
Detection and interpretation of these motions are not only crucial for
predicting the future state of one’s dynamic world […] but also provide a
wealth of information about the 3-D structure of the environment. (id., p. 179)
Can a computational account of cognition, based on functional decomposition, be
exhaustive of the actual inner workings of sensorimotor processes? What counts as a
good computational explanation, if not the effectiveness of a simulation inspired by
neurobiological modeling, with the aim of matching inputs and outputs to and from
the cognitive unit under scrutiny? Another question that immediately follows, then,
is: is there any more, or any less computation actually going on in this collection of
cognitive processes? Perhaps some reversal of perspective is needed to accurately
characterize these processes.
According to Bizzi, Mussa-Ivaldi and Giszter (1991)27, some neurons must
calculate the relative positions of body/limbs and objects in order to achieve an
adequate sensorimotor activity (based on egocentric sensorimotor perception and
producing a behavioral output). The CNS also has neurons involved in the
calculation of body/limbs- independent perception, or allocentric perception, in the
representation or signal emission of the concerned objects. In their words:
Recent psychophysical evidence supports the hypothesis that the planning of
limbs’ movements constitutes an early and separate stage of information
processing. [...] during planning the brain is mainly concerned with
establishing movement kinematics, a sequence of positions that the hand is
expected to occupy at different times within the extrapersonal space. […] The
analysis of arm movements has revealed kinematic invariances. […] The data
derived from straight and curved movements indicate that the kinematic
See also Bizzi and Mussa-Ivaldi 1998, Bizzi, Tresch, Saltiel, and d’Avella 2000, Mussa-Ivaldi and
Bizzi 2000, and Gandolfo, Li, Benda, Padoa Schioppa, and Bizzi 2000 for further references on Bizzi
and colleagues’ research on sensorimotor processes. It should be noted that at no point does Bizzi
advocate a dynamical stance over a computational one. These orthogonal considerations are the
author’s designs.
invariances could be derived from a single organizing principle based on
optimizing endpoint smoothness. (id., p. 287. My emphases)
Notwithstanding its computation-laden imagery, this evidence is hardly
consequent of functional decomposition through a strict computational commitment.
It translates to computations and interactions between the CNS, afferent visual and
kinaesthetic inputs, and musculoskeletal outputs – have them calculate on analog or
discrete quantities, whichever is more appealing – the overall picture suddenly
appeals to a different mathematical characterization, that of the changes within and
outside a system according to thresholds, invariants, nonlinearity and obviously,
motion. Such is the mathematical language of dynamics. We witness the emergence
of concepts of the likes of optimization gradients, thresholds and invariances (known
as attractors in systems dynamics), and complex behavioral plasticity resulting from
simple nonlinear ‘organizing principles’ (which are nevertheless computationally
mind-boggling for our commonly linear reductivism).
One central theme cherished by proponents of the application of dynamics to
cognition is the interactivity between cognitive agents and their environment.
Remarkably, following again Bizzi and his colleagues, the CNS is not the source of
coordinates in space; it relies on extrinsic information, as in:
[…] actions are planned in spatial or extrinsic coordinates, [then] for the
execution of movements, the CNS must convert the desired direction and
velocity of the limbs into signals that control muscles. (id., my emphasis)
This has rather interesting consequences. Given extrinsic coordinates- and kinematic
invariances- reliance for the CNS to actually ‘do’ something sensorimotorwise, (i)
sensorimotor cognition is better studied in specific contexts, supporting the claims of
proponents of embodied/situated cognition, and (ii) the world provides enough
‘affordances’ in the language of J. J. Gibson (1966, 1979), and ‘structure’ for a
cognitive agent to navigate without having to build a new world from scratch. In the
equations of dynamicists, the world operates as a whole system itself, albeit not a
cognitive one, and influences continuously and inexorably the cognitive agent within
it, and it works the other way around too, in feedback loops. Granted, discharging
some of the weight of information and information processing (because the world
has structural invariants at all levels) does not make cognition any easier to model,
nor to understand. There is evidence that the CNS calculates inverse kinematic and
inverse dynamic problems in the generation of motion, so much for the dignity of
empirical sensorimotor enquiry and even more of a burden for cognitive modeling
and simulations. Is the account of inverse dynamics computation by the CNS
satisfactory or, to put the issue at hand in other words: do our brains and
computational models of cognitive processes deal the same way with such
informational complexities?
Figures 8, 9, 10 and 11 are geometrical representations of some parameters of inverse kinematics
problems such as they are studied by computer scientists, engineers and roboticists. Figure 12 is a
still frame of a 3D rendered simulation of inverse kinematics where a “tail’ tries to reach and touch a
small green cube, illustrating the complexity of devising such an algorithm.
No. Just as you thought it was over with conceptual and methodological
pitfalls, it’s not that simple. In Bizzi’s words:
One way to compute inverse dynamics is based on carrying out explicitly the
algebraic operations after representing variables such as position, velocity
acceleration, torque, and inertia. This hypothesis, however, is unsatisfactory
because there is no allowance for the inevitable mechanical vagaries
associated with any interaction with the environment. Alternative proposals
have been made that do not depend on the solution of the complicated inversedynamic problem. Specifically, it has been proposed that the CNS may
transform the desired hand motion into a series of equilibrium positions. […]
According to the equilibrium-point hypothesis […] (Feldman, 1974) limb
movements result from a shift in the neurally specified equilibrium point. (id.,
p. 289. My emphases)
Computation is again constrained by structural invariants of the body, as well as
invariances in the world.
[…] With respect to control, the elastic properties of the muscles provide
instantaneous correcting forces when a limb is moved away from the intended
trajectory by some external perturbation. With respect to computation, the
[same] elastic properties offer the brain an opportunity to deal with the
inverse-dynamics problem. (id. My emphases)
Well! There is more to meat than first transpires! Conclusions? (i) Our cognitive
processes are constrained and mediated by some useful designs and useful physical
properties that afford them – much of sensorimotor cognition does not require
symbolic processing, and (ii) there is less to be paranoid about the amount of
computation and information processing that the CNS must carry out for the
information- and computation- obsessed theorist. Things you can suddenly do
without if you are a central nervous system, vis-à-vis sensorimotor limb control:
parameters and variables of inertial forces, gravity, viscosity, required effort
expenditures, et cetera… How does that fit the simulation model? Worth noting is
the stubbornness and resilience of computational schemes to be dealt away with,
even for such cognitive neuroscientists, in saying: “[…] in this context, a
representation in the CNS [of the previous variables] contained in the equations of
motion is no longer necessary.” (id.) A radical dynamicist’s reply could be: “In this
context, a representation in the CNS of the previously discussed parameters and
variables in the equations of motion is nowhere to be found, there is no such thing,
bottom line, the brain does not need it, get over it and change your model…”
It is fascinating that psychophysical research models its observations from an
ambiguous middle ground in the midst of computational and dynamical stances,
such as in the case of du Lac and colleagues (1995, p. 411): “The iterative process of
improving motor performance by executing movements, identifying errors, and
correcting those errors in subsequent movements is called motor learning.” (Note
the vernacular of intentionally and informationally phrased definitions here, my
emphasis) Yet, elsewhere and about the same topic:
The simplest form of motor learning is adaptation, in which muscular force
generation changes to compensate for altered mechanical loads or sensory
inputs. Adaptation can involve movements across either a single joint or
multiple joints, and can occur in both reflexive and voluntary movements. (id.,
p. 418)
It can clearly be seen that whereas the first definition involves a functional
decomposition that is essentially computational in flavour, the second one is quite
recoverable by an effective and quantitative dynamical characterization. As we have
seen with Bizzi and colleagues, the danger of following a model to its deeper logical
conclusions is falling prey to over-characterization and bearing little into matters of
empirical correspondence. Consider the following:
For complex movements, motor learning is required to select and coordinate
the appropriate muscular contractions, to link together motor subroutines, and
to create new motor synergies by combining forces generated across multiple
joints in novel spatial and temporal patterns. (id., p. 416)
Well, the illusion here is that there is an ongoing calculation at every single step of
the described process, and it contradicts the previous precisions on motor control
intricacies; what matters is that the functional decomposition of sensorimotor
processes, as mechanistic as it gets, does not entail what I will dub an
omniparametrism, or more specifically, an omnicomputationalism of processes
under functional scrutiny.
- II.II - The argument
In this second section, we are confronted with the arguments of a proponent
of the dynamical view of sensorimotor control and learning, Esther Thelen . The
broad range of arguments from the DHCS will be exposed against the treatment of
sensorimotor learning from a computational perspective , as Thelen tries to answer a
most important question: how do we relate sensorimotor ontogenetic dynamics with
cognition? Dynamicists aim to provide a more biologically plausible framework for
cognitive science while also aiming for a gain in explanatory and predictive strength
through their models. Thelen’s work is primarily concerned with setting up
empirical psychophysical experimentations to provide enough support for the
following claim: that ontogenetic dynamics are the very source of cognition. Her
premises: (i) embodiment is a necessary condition for cognition, as we have seen
time and again, and (ii) as is conceived through the study of infant psychophysical
development, the major developmental task that tops them all is to gain control of
the body.
Many attempts at modeling sensorimotor processes from a dynamical
framework have been made, such as those of Bingham on visual event recognition,
Grossberg on the neurodynamics of motion perception, recognition learning, and
spatial attention, Saltzman on sensorimotor coordination, Thelen on the development
of sensorimotor (embodied) cognition, and Turvey and Carello on haptic perception
and coordinated movement30. For the sake of a workable basis of reference, we will
only explore Thelen’s work on developmental dynamics, an arbitrary choice, albeit a
perfect exemplar of the relevance of dynamics to cognition.
E. Thelen (1995), Thelen, Schöner, Scheier and Smith (2001), and Smith and Thelen (2003).
Computational solutions of the inverse kinematics problem, and the dynamical equations of Thelen
et al concerning sensorimotor dynamics, can be found in appendix II.
30 All of the preceding authors, 1995, found in Port, R. F., and van Gelder, T., Eds., Mind as Motion.
Cambridge, MA: MIT Press.
Port and van Gelder (1995, p. 69), say that “[…] Thelen argues that taking up
the dynamic perspective leads to dramatic reconceptualization of the general nature
of cognitive development, and indeed of the product of development, mind itself.”
Thelen’s original contribution to developmental psychology has been of integrating
the theoretical tools of dynamics (that is, dynamical systems theory coupled with
quantitative dynamical modeling) within a research program and setting up
experiments to gather empirical support in order to confirm or refute the relevance
of what van Gelder properly named the dynamical hypothesis in cognitive science,
as exposed in chapter I. As we have seen before, the dynamical hypothesis is the
conjecture that cognition and its related processes might better be described in the
conceptual framework of dynamics, rather than, say, a computational one.
Obviously, discontent with computational endeavors motivated this departure, and
as mentioned in the introduction, was spawned by the many shortcomings of such
models. On Thelen’s take on developmental dynamics again:
Changes in behavior come to be understood in terms of attractors, stability,
potential wells, parameter adjustment and so forth. Taking over this
vocabulary facilitates a whole new way of seeing how sophisticated capacities
emerge. New abilities take shape in a process of gradual adjustment of the
dynamics governing the range of movements currently available; this
adjustment is effected by exploratory activity itself. (id.)
Thus, the vocabulary of dynamic systems theory might be understood as a bridge
between outward, externalist descriptions of behavioral changes on one hand, and
inward, internalist descriptions of informational processes on the other hand.
Dynamics is a language of systematicity, above all, it groups, joins, couples
homogeneous (similar entities or
processes understood
systems, or
heterogeneous (different hierarchical levels of interactions between entities or
processes understood as) systems.
Let’s develop further on the arguments supporting a dynamical account of
cognition where developmental psychophysics and psychology are concerned.
Thelen argues that her ambition is to take on not only a noncomputational stance on
the ontogenesis of cognition, but to go as far as deny the traditional view of genetic
determination of such development; a view supported by Piagetian objectivism and
the maturationist account of development. Again, in van Gelder’s words:
Since infants can begin this process of adjustment from very different starting
points, it is highly unlikely that there is any predetermined, genetically coded
program for development. It is rather a self-organizing process in which
solutions emerge to problems defined by the particular constraints of the
infants’ immediate situation. (id.)
The question that begs to be answered is: how do we relate sensorimotor ontogenetic
dynamics with cognition, though? Thelen does recuperate a thesis of Piaget’s
according to which “thought grows from action, and that activity is the engine of
change.” (id., p. 73) As such, Piaget’s fundamental thesis is what we nowadays call
that of embodiment. Piaget’s mistake, though, was to admit a fundamentally
Cartesian separation to the end-state of development in the characterization of a
mature objective mind, thus recreating a discontinuity essential of what is meant by
the Cartesian dualism of mind and body. Cognitive dynamics avoid exactly such a
problem, or it could be said that it is an answer to the very re-enactment of mindbody dualism that permeates the standard model of cognitive science. So, an
embodied cognition appears to be the only way to solve the mind-body
discontinuities, found even in the weakest form that is positing ‘mental properties’.
Thelen’s is a radical position: beyond a noncomputational account of
cognition, she also promotes an antirepresentational stance of cognition. Since this
does not turn out to be a common ground to all or even most dynamicists’ views on
cognition, we will proceed without making this assumption. It is also not a necessary
criterion to promote the dynamical hypothesis as such. A better depiction would be
to say that dynamicists, up to and including Thelen, are against symbolic
representations when it comes to characterizing biological cognition. Another
consequence of Thelen’s arguments is that it trivializes such concepts like the
modularity of knowledge in cognitive processes, the consequences of which are up
to interpretation and will not be discussed here. On the other hand, it certainly does
well in dealing away with the annoyance of the archaic introduction of a distinction
between semantic and episodic knowledge (know-that knowledge) versus procedural
knowledge (know-how knowledge). Seemingly pointless (excluding matters of
neurological localization of relevant faculties and processes) or irksome distinctions
of such nature are always a good test to the relevance of a paradigmatic shift from
one conceptual framework to another. Other problems that might be better solved
through a dynamical framework are ones concerning individual differences in
cognitive processes, context sensitivity of cognition, cognitive tasks such as
categorization, and the integration and seamlessness of cognition and behavior
(internal and external states), or further, that of behavioral changes, ontogenetic
‘learning’, and ontogenetic physiological changes. The methodology? Correlating
continuities in time between ‘physical’ and ‘mental’ events or processes. Thus, it can
be said that dynamicists aim to provide a more biologically plausible framework for
cognitive sciences while also aiming for a gain in explanatory and predictive
strength through their models.
But the meat of Thelen’s work in setting up empirical psychophysical
experimentations is that it provides enough support for the following claim: that
ontogenetic dynamics are the very source of cognition! The premises are indeed
compelling, if not intuitive: (i) embodiment is a necessary condition for cognition, as
we have seen time and again, and (ii) as is conceived through the study of infant
psychophysical development, the major developmental task that tops them all is to
gain control of the body. To prove such claims, it is in turn necessary to
demonstrate the origins of certain mental processes, and Thelen argues that an
analysis of the various time-scales’ dynamics of psychophysical processes shows
them to be interdependent and profoundly embedded structures. There is no place
for discontinuities, in her own words. It should be noted that this is a departure from
Worth noting is that both points (i) and (ii) are not uncontroversial, and only lightly elaborated on
by dynamicists, including Thelen.
earlier positions concerning development in psychophysically related research:
motor development was mistakenly considered as a strictly biological phenomenon
and secondary to the brain or CNS development. This thesis, conveniently dubbed
the maturationist account of development, thus viewed motor control as a byproduct of autonomous brain development, in a rather Piagetian stage-like process of
emergence. It is pretty obvious that the spectre of dualism was ubiquitous even in
the mostly empirical of ontogenetic accounts of cognitive developments. Piaget’s
alternative introduced the idea that mental life is built upon those sensorimotor
processes, but maintained the undesirable dualism by also claiming that mental
processes are distinct, separate phenomena characterized as the end-state
development of abstract and objective mental structures. The whole of cognitive
science, it seems, has a bad tendency of getting drawn back to (or should I say
drowned into) Descartes’ legacy.
A dynamical account of cognition necessarily calls onto a deep commitment
to the thesis of embodied cognition. The legacy of Cartesian dualism is found in
some assumptions of classic and contemporary forms of cognitive science, like
The denial of the relevance of the physical body in all its instantiations through
movement, feeling, and emotion […] [and] the separation of intelligent
behavior from the subjective self, from consciousness, imagination, and from
commonsense understanding. (id., p. 74)
This legacy has long denied embodiment as a necessary condition for cognition, and
is a consequence of its sharing continuity with the methodology of a strictly formal
and functional characterization that is the (classic) computationalist view.
- II.III - The model
“1. We cast the mental events involved in perception, planning, deciding, and remembering in the
analogic language of dynamics. This situates cognition within the same continuous, time-based, and
nonlinear processes as those involved in bodily movement, and in the large-scale processes in the
nervous system […] Finding a common language for behavior, body, and brain is a first step for
banishing the specter of dualism once and for all.
2. Because perception, action, decision, execution, and memory are cast in compatible task
dynamics, the processes can be continuously meshed together. This changes the informationprocessing flow from the traditional input-transduction-output stream to one of time-based and often
shifting patterns of cooperative and competitive interactions. The advantage is the ability to capture
the subtle contextual and temporal influences that are the hallmarks of real life behavior in the
3. We address specifically the developmental origins of cognition. Since Piaget […], it has been
widely acknowledged that all forms of human thought must somehow arise from the purely
sensorimotor activities of infants. But it is also generally assumed that the goal of development is to
rise above the "mere sensorimotor" into symbolic and conceptual modes of functioning. The task of
the developmental researcher, in this view, has been to unearth the "real" cognitive competence of
the child unfettered by performance deficits from immature perception, attention, or motor skills.
This division between what children really "know" and what they can demonstrate they know has
been a persistent theme in developmental psychology [...] We argue here that these discontinuities
are untenable. Our message is: if we can understand this particular infant task and its myriad
contextual variations in terms of coupled dynamic processes, then the same kind of analysis can be
applied to any task at any age. If we can show that "knowing" cannot be separated from perceiving,
acting, and remembering, then these processes are always linked. There is no time and no task when
such dynamics cease and some other mode of processing kicks in. Body and world remain ceaselessly
melded together.”
- 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)
This section develops a correlated dynamical account of sensorimotor
processes which bears explanatory and predictive significance to a higher level of
description, that of developmental psychophysics, with the help of Thelen’s findings
on infants sensorimotor cognition and the subsequent model developed to deal with
the empirical findings. Exposed here are the benefits of adopting of a dynamical
perspective on cognition, relative to some shortcomings of a computational
framework, while showing the relevance of said dynamical account in the rather
psychophysiology. The formal and qualitative characterizations of the dynamical
hypothesis in cognitive science are thus applied directly to the studies of
developmental psychophysics.
Thelen gives three examples of embodied cognition and the relevance of
dynamics to characterize cognition: the containment example, one on symmetry, and
another one on forces. Following Johnson (1987), she points towards psychological
studies from an embodiment perspective, to show how obviously pervading are the
recurrent features and constraints of our physical world not only in our actions, but
even in our language and thoughts as well. Indeed, Johnson puts forward the idea of
what I call ‘embodiment semantics’, such as when we use prepositions like ‘in’,
‘out’, ‘over’, ‘near’, ‘under’, etc. To see the extent of such implications, consider the
following: “I don’t want to leave any relevant data out of my argument.” Metaphor?
Yes, but there’s no chicken-and-egg problem here, the world came first, and then
this embodiment-laden cognition… that’s containment for you, right there. Similarly
with physical and bodily symmetry and polarization, we can see the extent of such
categories as far as in our cultural artifacts, beyond our actual cognitive processes.
Just take a hint and think about literature and poetry, cinema and music, and if you
don’t see schematic and spatial cognition in that, you can keep hoping that
disembodied AI will come up with emergent poetry-writing software... Last but not
least, consider the concept of force embodiment. Forces reach into cognition as
essentially as they involve our every physiological interaction: what about the
semantics of verbs? Language and thought are dynamically-laden, think again of the
following sentence: “I’m attracted to the ideas of Tim van Gelder.” Johnson calls
this prelinguistic meaning, semantics drawn from experience.
It should be noted that in Thelen’s view, a dynamical account of cognition is
still compatible with a functionalist account from a mechanistic perspective.
Indeed, attributing characteristics to an entity as being a real dynamical system is an
ontological commitment as much as a model of such entity; it makes claims about
the nonlinear, emergent, and embedded properties of such an entity, its intrinsic
dynamics. Dynamics are the mathematical study of patterns of flow, expressed in
nonlinear calculus equations. It is concerned with motions and forces, quite a
physicalist level of description and explanation. Yet, from the design of dynamical
systems, complex structures and emergent properties arise, and a nontrivial
qualitative characterization is possible on top of the quantitative bearing of
dynamical modeling. Thelen notes that in order to bear any scientific adequacy and
explanatory power, the dynamical equations must fit the observed behavioral data.
The dynamical hypothesis in cognitive science not only posits that cognition is
essentially a dynamical phenomenon, but that dynamics is the best explanatory
framework so far for the scientific study of cognition. Thus, (real) complex
nonlinear systems can be studied through mathematical dynamical systems (MDSs),
and such MDSs can explain changes as the result of coupled magnitudes fluctuating
interdependently. One fundamental assumption from the adoption of the dynamical
hypothesis is that pattern emerges only in process: it thus rejects symbols,
computational structures , and developmental stages in the programmatic view of
computationalism as ontologically unacceptable in the study of the brain and of
cognitive processes.
But the problem lies in that dynamicists such as Thelen would have their formal account replace
any other formal model of cognition, i.e. they actually believe their dynamical account to be a good
candidate for a functionalist account. Such a claim is hardly supported, and perhaps ill-fated, as we
will see in chapter III with the help of Bechtel’s arguments.
33 Or does it? We will explore the claim of compatibility under arguments about the types of
explanation involved in both frameworks in chapter III, section II.II.
There can be no description of a purely ‘inner life’: every mental and
behavioral act is always emergent in context […] Perception, action, and
cognition form a single process, with no distinction between what people
really ‘know’ and what they perform. (van Gelder and Port, 1995, p. 72)
How exactly do cognitive systems translate into dynamical ones?
A fundamental assumption in a dynamical approach to development is that
behavior and cognition, and their changes during ontogeny, are not represented
anywhere in the system beforehand either as dedicated structures or symbols in
the brain or as codes in the genes. (id., p. 76)
Cognitive processes and behavioral outputs are better thought of as dynamical
patterns of activity, function of the context at hand coupled with the intrinsic
dynamics of an agent. These intrinsic dynamics are in turn the product of the current
architecture of the system and its history of prior activity. Thus, the “behavior
represents a reduction of the degrees of freedom of the contributing subsystems into
a pattern that has form over time.” (id.) Every fascinating aspect of cognitive
processes finds its place in a compelling dynamical characterization: the stability of
an action or thought is considered as the intrinsically preferred states, or attractors,
in the behavioral state space of a system. Strong attractors represent patterns of
cognitive activity that are more likely to be manifested, the more consistent
behaviors. Weaker attractors express instability and perturbation, as well as the
variability and unreliability of such patterns. Development itself is the changing
landscape of preferred behavioral states. Thelen argues that some of these preferred
behavioral states are so ubiquitous within and across individuals of our species that
they are interpreted as discrete developmental stages, such as the ones traditionally
described by orthodox developmental psychophysics and psychology. These stages
are merely high probability states in the behavioral space of cognitive processes. But
attractors in a state space cannot be too rigid and stable, otherwise change wouldn’t
be possible: the combination of moderate attractors and pattern instability could be a
good dynamical translation of the concept of behavioral and neural plasticity, in my
opinion. Behavior is
The product of the confluence of components within a specific problem
context […] [and] development is likewise a series of both gains and losses as
old ways of solving problems are replaced by more functional forms. (id., p.
Figures 13, 14, and 15 (following pages) are geometrical MDS representations of different
sensorimotor cognitive tasks. Figure 13 (page 49) represents the performance of an infant in an
object-hiding task named the ‘A-not-B error’, with regards to the development of object permanence
(Smith and Thelen 2003). Figure 14 (page 50) is a general ontogenetic landscape, where
development is seen as a series of evolving and dissolving attractors over time (Thelen 1995). Figure
15 (page 51) is a more sophisticated depiction of the ‘A-not-B error’ task. It shows some properties of
a movement field without specific input (following that there were no cues or training over the
decision to make) (Thelen, Schöner, Scheier and Smith 2001).
(a) Motor field dynamics in non-cooperative regime with task input only (no specific input) at the
first reach to A (A1). Parameters: Sspec = 0, Stask = 1, Smem = 3 (a). Motor planning field evolution (b)
Corresponding evolution of memory field. In this figure and subsequent figures, x axis denotes field
location, y is time, z is activation. On the y axis a letter code indicates the input present at different
moments in time: T, task input, S, specific input (none added here), and D, the delay where no
specific input is added. (c). Histograms of decisions to A or B from an ensemble of 500 simulations
per trial showing the read-out of the field as a function of time. The decision to reach to A or B is
probabilistic; in this case, A or B is equally likely at any point in time.
Thelen also aims to show the essential chronometric properties of cognitive
processes, that is, that cognition and behavior are best explained through a set of
interdependent variables changing at a number of quantitatively different, embedded
time-scales. She distinguishes (but to further reintegrate) what I dub ontogenetic
time (learning and development time scale) and cognitive time (task resolution time
scale, i.e. real time), time scales that are continuous in her model, for they share the
same dynamics. The embeddedness of time scales is fundamental to cognition and
overall development, and is made available to our quantitative proclivity through
and only through dynamical modeling. Thus is expressed not only the confluence of
behavior and cognition in a given context, conceptualized as ‘local dynamics’, but
this confluence also shapes, affects the overall internal-external pattern
configuration, which we have already named ‘intrinsic dynamics’. Shorter version:
the cognitive time scale (local dynamics) shapes ontogenetic time (intrinsic
dynamics), which feeds back on every subsequent local, real time cognitive
processes. Thelen draws an example from the modeling of a simple damped massspring34 to model early spontaneous limb movements in infants, an activity that leads
to coordinated sensorimotor control through exploration and selection of values
matching the affordances (allowing ourselves a Gibsonian analogy) of the
environment coupled with the goal of the task at hand. See figure 14 for an idea of
an ontogenetic landscape generated through exploratory and selective experience,
where coordination is learnt by exploring “the many different values of the spring
parameters generated by [the infant’s] spontaneous movements and movements
produced in the presence of a goal.” (id., p. 80)
The values ‘selected’ from exploration become attractors in a given class of
actions, the clearest depiction of the causal relationships between local and
ontogenetic dynamics. The mathematical dynamical system (MDS) fits of course the
real dynamical system (RDS) that is the infant’s psychophysiological ‘substrate’:
The mathematical expression of which is mx + kx + sx = f(t), x being the displacement of the spring
and its derivatives, m is the mass, k is the damping coefficient (friction), s is the stiffness of the
spring, and f(t) is the time-dependent quantity of energy produced by the contraction of the muscle.
Activity changes the biochemistry and the anatomy of muscles and bones […]
These changes occur over a more prolonged time scale than do changes in
behavior, but they are part and parcel of the same dynamic. (id., p. 81)
This model can even explain phase shifts and discontinuities of the dynamic
specifications of the sensorimotor system of the infant through the simple damped
mass-spring analogy: Thelen elaborates on the example where newborns are held
upright and make step-like movements. These motions then disappear over the next
few months, which can be explained by an increase in leg mass at a faster rate than
muscle strength. In terms of the damped mass-spring, the parameter m (mass) is
increasing faster than parameter f (energy burst from muscle contraction).
Parameters m and k are constant in local time, but they change over ontogenetic
time, whereas s and f change over both time scales. The consequent behavioral shift
(disappearance of step-like movements when held upright) in ontogenetic dynamics
caused by the faster increase in leg mass than muscle strength is again subject to a
phase shift at a later age (latter part of the first year), when the gain in muscle
strength relative to leg mass is reversed, enabling the child not only to lift their legs
when held upright, but eventually to support their own weight. To summarize,
cognitive time scales are continuous in such a way that doesn’t allow for a clear
distinction between them, hence the thesis of embeddedness.
Sensorimotor activity urges us to conceive behavior, real time cognition, and
ontogenetic dynamics as seamless and indissociable even in their conceptualization.
Isolating one for the sake of analysis is risking a considerable loss of explanatory
resolution. But Thelen’s plan is to assimilate higher cognitive processes in the same
way, to show that beyond biomechanics, developmental processes of sensorimotor
coordination are (i) the same for all psychophysical and cognitive levels of
processing, and (ii) that sensorimotor coordination itself is the foundation of all
mental activities. These serious conjectures indeed provide a ground for further
empirical enquiry, and for the analysis of the relevance and scope of the dynamical
hypothesis concerning cognitive science. But what is under examination here is the
commensurability of two conceptual frameworks concerning the study of cognition,
a type of ‘comparative theoretical cognitive science’, if you would allow such an
exotic epithet, and I will now address the very controversies sustained by proponents
of both stances towards cognition. While this chapter was meant to expose the
adoption of a contending conceptual framework in the exploitation of cognitive
phenomena, we have but barely undertaken a rigorous comparison between the
foundational arguments supporting both frameworks. The arguments of Thelen
concerning embodiment, embeddedness and the precedence and determination of
sensorimotor control over higher-level cognitive processes are most certainly
compelling, but we will see that they are only secondary to the epistemological and
semantic issues opposing computationalism and dynamicism in cognitive science.
- III - Issues, controversies, and answers concerning the framing of cognition in a
computational or a dynamical model
The CHCS and DHCS still raise many debates in cognitive science, and this
chapter presents some of the formal and empirical issues raised for and against them
from areas such as mathematics, neuroscience, and philosophy. The definitions of
computation established in the first chapter, as well as the dynamicist’s conceptual
repertoire, will be brought into play to assess the significance of such arguments.
In chapter II, I have exposed a certain number of controversial and not-so
controversial ideas concerning cognitive science that are related to the issue of
conceptual clashes between computationalism and dynamicism, namely: (i) the
extrinsic nature of many variables related to cognitive processes of sensorimotor
design, versus the interior processing and mapping of everything, leading to (ii) the
idea of environmentally- and physiologically- constrained cognitive processes,
versus an omniparametrism, or rather an omnicomputationalism of such processes,
(iii) some empirical evidence in neuroscience and psychophysics points toward the
adoption of dynamical concepts and models to further our understanding of
cognition, and also (iv) toward the integration, in the study of cognition, of context,
corporeality, and systematicity.
But dynamicism is also silent on many things, namely, (i) it does not provide
a theoretical framework for implementation: the dynamical hypothesis is not an
implementation theory, much like the debate from Fodor and Pylyshyn (1988) on
matters of computationalism versus connectionism. Indeed, connectionism can be
conceived as an attempt to model the implementation of informational processes into
a biologically inspired design. It does not follow that all connectionist models are or
should be computational, even if we invoke the polysemous concept of computation
as it is exposed in the first section of chapter I. Some supporters of the DHCS claim
that some nontrivial types of artificial neural networks have more to do with
nonlinear differential equations and dynamics than with digital computation
(Grossberg 1995, van Gelder 1990, 1998abc, 1999abc), and that will play an
important part in both the following discussion and chapter IV.
Also, (ii) dynamicism, despite having put forward strong arguments for the
relevance of dynamics to cognition, does not constitute in itself a definitive rebuttal
of computationalism or a vindication of the dynamical hypothesis above all other
frameworks. Much more work is needed to this end, through an assessment of the
accuracy and explanatory power of the framework, and the rebuttal of a possible
compatibility, or coexistence, of both stances towards cognition. This will also turn
out to be a fundamental issue in the resolution of this comparative analysis.
Finally, (iii) although advocates of dynamicism claim to avoid restricting
their framework to mechanistic explanations of cognitive processes by observing
mathematical correlations between systems, internal and external, there are
accusations from computationalists of dynamics being a sophisticated new avatar of
behaviorism. This issue will also be addressed in this section, drawing on arguments
about the types of explanation involved in both frameworks.
- III.I - On the nature of cognition vis-à-vis computation and dynamical systems
This first section deals with the arguments concerning the adequacy of
mathematical formalisms with regards to cognition and cognitive processes. The
main concepts and subject matters under scrutiny are those of cognitive processes
and continuous or discrete time dynamics, symbolic representations and neural
computation, and the role of representations in cognitive dynamics. Here will be
argued (i) that the dynamicists’ conceptions of computation and representation are
inadequate, on conceptual and methodological grounds, (ii) that computational
cognitive science needs not be rejected on grounds that its symbolic avatar, spawned
from the early days of artificial intelligence, cannot account adequately for
biological cognition, and (iii) that while representations in models of cognition may
be different in format, they are still required to account for cognitive processes, even
in a dynamical view.
- III.I.I - Giunti and van Gelder on the mathematical properties required to
properly model cognitive systems and processes
Giunti has presented many arguments to promote the study of cognitive
systems as dynamical systems. While his position has become more moderated with
time with regards to the relevance of computational models of cognition, we propose
to expose his earlier arguments on cognition as best studied through dynamics. This
account will help understand the mathematical issues at stake in modeling cognitive
processes, as well as constituting a preliminary acknowledgment of flaws and
confusions concerning the concepts of computation and cognition. Giunti exposes
two sufficient conditions for a system not to be computational, and suggests that
both the time and state space values of computational models lack analytical
resolution (continuity, density, viz. properties exclusive to real numbers). The
remainder of the section follows van Gelder’s discussion with his antagonists on
objections to the definitions of dynamical systems and digital computers that he
champions. Topics of interest concern the scope of such definitions (too narrow or
too broad definitions), and the minute distinctions between computational and
dynamical models concerning the temporality of cognition, their state space, and
considerations on quantification.
Two sufficient conditions for a system not to be computational. Giunti’s
(1995) earlier endeavors in the promotion of the dynamical approach of cognition
was very clever: in order to support dynamics, he proposed a formal, comparative
analysis, both qualitative and quantitative, of the two frameworks with respect to
cognition. His principal thesis, that all cognitive systems are dynamical systems, is
uncontentious. It is his secondary thesis that poses a problem: that computational
systems are a subclass of restricted dynamical systems that would only gain in
explanatory power, if they were ‘released’ from the shortcomings of the
computational framework. While there is no problem in principle with the analysis
of mathematical models from other areas within mathematics, it does not follow that
cognitive science would benefit from the analysis of computational systems in terms
of dynamics. Nevertheless, the exercise is original and enlightening, and here is a
summary of the argument.
As distinguished in the first chapter, there are RCSs and RDSs in the world,
and the mathematical models are said to ‘realize’ the regularities observed in real
dynamical systems. Giunti elaborates on the mathematical characterization of
dynamical systems: they have three elements, namely a time set T, a state space M,
and a set of functions {gt}. Now, both a MDS and a MCS instantiate one or more
aspects of a real system, for purposes of simplification and tractability. Thus, it can
be said that different MDSs and MCSs can describe the same real world system
independently, depending on the parameters and variables of interest. A discrete
MDS, also called a cascade, is thus a MDS <T M {gt}> where functions are
expressed in the form gt+1(x)=g(gt(x)), and the time set is a set defined on the (nonnegative or complete) integers. From a dynamical perspective, thus, a Turing
machine (the foremost exemplar in the computational theory of mind, according to
Giunti and van Gelder) ‘is’ a cascade, viz. a discrete, mathematical, dynamical
system. But dynamicists hold that in order to better understand cognitive processes,
we have to have access to the time evolution of the state space of such processes.
Now, according to the tools made available by dynamical systems theory,
discrete dynamical systems, or cascades, only appeal to a limited part of dynamics,
such as the qualitative concepts of state space, time evolution in terms of periodic,
eventually periodic, or aperiodic ‘orbits’, and attractors. But that’s it, since Turing
machines and symbolic processors lack any interesting topological and metric
properties, according to Giunti. But computational systems have an additional
essential characteristic of being effectively describable. In Giunti’s words:
“Intuitively, this means that the constitution and operations of the system are purely
mechanical or that the system can always be identified with an idealized machine.”
(Giunti, id., p. 559) Thus, a computational system can be more specifically defined
A mathematical computational system. The acronym will be used hereafter. Note that the
‘mathematical’ part of the expression MCS is debatable, since dynamicists specifically appeal to
(symbolic) Turing-computation, which conflates symbolic logic, mathematical computation, and
essential properties of algorithms.
as an effective cascade, or effective discrete dynamical system. This requires, with
regards to fundamental issues in the mathematics of computation, that the state space
M must be a decidable set, and that each state transition function gt is
effective/computable. The two sufficient conditions for a (dynamical) system not to
be computational, then, concern whether its time set or state space is continuous or
not: a system is not computational if (i) its time set is defined over real numbers,
and/or (ii) its state space is not effectively denumerable. Apparently, we should be
satisfied with such scarce formalities.
On the definitions of dynamical and computational systems. Before we move
to some criticism, I want to complement Giunti’s arguments with van Gelder’s on
similar grounds. van Gelder is struggling with his critics on the topic of the proper
treatment of dynamical models and computational ones. He argues that dynamical
systems are significantly different from digital computers, the implementation
exemplar championed by proponents of the CHCS, in that the state space of a
computational system is quite different from a metric space, such as the integers. For
a Turing machine, the relevant set of variables is
[…] head state, head position, and locations on the tape. These are the things
which change over time in the operation of the machine. The state space of the
Turing machine is the set of all possible combinations of values of this set of
variables. Ontologically, this is wholly different than the integers. (van Gelder
1998a, p. 2)
So to speak, van Gelder claims that the state space of a computational model cannot
be equivalent to a metric space except in a trivial sense, and consequently does not
constitute a quantitative system. Further, a system’s metric should be independent of
its behavior, otherwise “we can’t know what the distances are in the state space until
we know how the system behaves.” (van Gelder, id., p. 3) van Gelder thus accuses
Turing machines of having entirely post hoc and uninteresting metric properties.
Therefore, the criterion of a dynamical system pertaining to its being quantitative in
van Gelder, 1998abc, 1999abc. For an overall perspective, 1998a.
state should require the additional condition of having a behavior- independent
Another issue of concern is that of the confusion over dynamical systems
being continuous or discrete. van Gelder claims that it is not the issue at hand while
one attempts to discriminate between computational and dynamical systems, but
rather a matter of having quantitative systems, which is a property of dynamical
systems ‘not’ shared with computational ones. He is perfectly aware that continuity
or discreteness of states are significant in dynamics, but also claims, much like
Giunti, that dynamics allows the study of both types of systems, whereas
computability theory is only interested in discrete systems, and moreover,
“interpreted formal systems” (van Gelder, id., p. 4) on top of that. On the other
hand, there are discrete dynamical systems that have been proposed as models in
cognitive science (van Geert, 1995, for an example), and discreteness alone does not
make a Turing machine, or a digital computer. The essential temporality of cognition
is directly dependent on such matters of quantitative modeling, in van Gelder’s
words again:
The fundamental point is that in systems exhibiting quantitative state-time
interdependence, the time set is not merely an ordered set used to specify the
order of change in which system states are occupied. Rather, it is a metric
space, such that amounts of change in state are systematically related to
amounts of change in time as measured by that metric. (van Gelder, id., p. 14)
While Giunti and van Gelder’s efforts in promoting the dynamical hypothesis in
cognitive science are bold and appealing, on grounds of what dynamics have to offer
to cognitive science, there are quite a few foibles in the arguments above, which I
have split in three categories: (i) arguments in the observation of computational
systems as dynamical systems, or in the comparative advantages of dynamics and
computability for the study of cognition, (ii) matters of state space, time sets,
metrics, and on continuity and discreteness, and (iii) methodological problems
related to the interpretation and use of concepts such as cognition and computation.
The following considerations address such issues.
On the relative advantage of dynamics in comparison with computability
theory, on empirical and pragmatical grounds. There are prima facie two problems
with Giunti’s argumentation on the promotion of quantitative dynamical systems
‘over’ simulation
models of cognition: (i) observing correlations between
magnitudes, and the interdependent time evolution of features of such processes
does not make it any more fundamental to cognition in any way, and (ii) dynamics
still does not answer what counts as cognition in the first place, begging the
question of which cognitive magnitudes we are supposed to care about. He admits to
point (ii) in his conclusions, while not providing a rigorous argument to waive the
issue raised by the first point. Also, Giunti and van Gelder seem to suggest that the
relative advantage of dynamics over computation is up for grabs on empirical and
pragmatical grounds, beyond formal and conceptual matters. Indeed, they concede
that it may turn out that cognition, or a subset of cognitive processes, might best be
accounted for in terms of computation, and then argue that it’s not a big deal, since
computability can also be explained through features and models of dynamics, a
more powerful and resourceful mathematical language. Roughly, we could make the
following syllogistic inference to sketch this vague and unconvincing point of view:
- All MCSs are MDSs (not really an issue),
- Some MDSs are MCSs (again, not controversial),
- All cognitive systems (CSs hereafter) are RDSs (we have but to agree with
that too),
- BUT, I ask, what if most, or all CSs turned out to be possibly modeled as
MCSs, in a way that is both necessary and sufficient for our concerns?
Remember, the DHCS is appealing for its formal and conceptual resources, but it
does not entail that cognitive phenomena might be relevantly modeled through
dynamics, as empirical and pragmatical concerns may waive dynamics in favor of
Giunti calls quantitative, continuous dynamical systems ‘Galilean models’, by opposition with the
limited qualitative and discrete character of symbolic models, which he calls ‘simulation models’.
computability. Thus, one very critical epistemological argument supporting the
supremacy of dynamics over computability, as far as cognition is concerned, is the
- CSs are best described through models that are MDSs, but are not also
Now, postponing criticism on conceptual and formal matters concerning the proper
treatment of such mathematical models, here are unanswered questions about the
aforementioned empirical and pragmatical issues: if both computational models and
dynamical models can, in principle, account for a given cognitive feature, or set of
features, which one is preferable, and on what ground? Aren’t mathematical models
defined with an arbitrary degree of resolution, or precision, and chosen on grounds
of the type of features, and results, that we are interested in?
On conceptual and formal issues concerning the divergences in explanatory
power of both mathematical models. Piccinini disagrees with the temporal
constraints of computational models, by comparison with the alleged advantage of
mathematical dynamical systems. Piccinini claims that
This objection trades on an ambiguity between the mathematical
representation of time and real time. Computations are temporally
unconstrained in the sense that they can be defined and individuated in terms
of computational steps, independently of how much time it takes to complete a
step. But this is not due to the fact that the process being defined is
computational. The same is true of any mathematically described process,
whether computational or not. Differential equations contain time variables,
but per se these do not correspond to real time any more than the time steps of
a Turing machine correspond to any particular real time interval. In order for
the time variables of differential equations to correspond to any particular real
time, a temporal scale must be specified (e.g., whether time is being measured
in seconds, nanoseconds, light years, or what-have-you). By the same token,
the time steps of a digital computing mechanism can be made to correspond to
real time by specifying an appropriate time scale. (Piccinini 2004e, p. 10)
Thus, the temporality of cognition need not be exclusive to dynamics, since the
allegedly uninteresting and post hoc properties of the metric of Turing machines can
be made far more interesting by incorporating a relevant time scale into the
computational model. This time metric need not be specified in terms of the actual
computational steps of the process, and can be made to match the content of what is
being computed.
Furthermore, computational models can be made to perform over continuous
values, either by design (analog computation), or in the specification of algorithms
to this end (computation over continuous values by a digital mechanism, such as
neural networks). Real computation, that is, a hypothetical mechanism computing
over real numbers, is simply not possible outside of its abstract formulation, since
the implementation of such a mechanism defies many levels of physical phenomena,
from macrophysical noise to quantum uncertainty effects. Giunti claims that
An immediate consequence [of the state space not being denumerable] is that
any finite neural network whose units have continuous activation levels is not
a computational system […] A computational system can, of course, be used
to approximate the transitions of a network [with continuous activation levels].
Nevertheless, if the real numbers involved are not computable, we cannot
conclude that this approximation can be carried out to an arbitrary degree of
precision. (Giunti, 1995, p. 561)
But the interdependent evolution of variables defined over reals is itself computable
for a large enough class of numbers and functions. Indeed, Glymour summarizes:
Suppose we consider a dynamical system as a function f(w,t), where t is the
real variable representing time, w is some n-tuple of numerical quantities,
including possibly integers, real or complex numbers, taking values in a space
of k-tuples, u, of similar objects […] Computable complex numbers are
defined in terms of computable reals; a computable real number r can be
defined in various ways—as a computable sequence of rationals converging to
r with a computable bound on the error at any stage in the sequence; as a
number whose digits in some base – say 2 – can be computed by an infinitary
generalization of a Turing machine (essentially a multi-tape Turing machine
that need never stop reading input or printing output), and in other ways.
(Glymour 1997, p. 6)
Glymour stresses that not all definitions of computability are equivalent, for they
depend on the representations involved, “and for computation on the reals, to the
measure of approximation.” (id., pp. 6-7) For example, the simple operation of
multiplication by 3 is not computable in the decimal notation of reals, but actually is
in binary notation!
There are uncomputable systems, including dynamical ones. But such
characterizations depend on formal factors that need not concern us here, and in fact
may not concern cognition at all. For one thing, even many chaotic systems are
computable, and so are some quantum phenomena. What about cognitive systems?
Glymour holds that the functions proposed to model cognitive processes are
expected to be computable, if only for the fact that we may tend to postulate
computable systems, “or because natural dynamical systems, including people, are
mostly computable.” (id., p. 7) Thus, although we can obviously postulate
uncomputable systems in the world, one has yet to come forward with an empirically
grounded observation of a cognitive process which can be modeled only through an
uncomputable dynamical system. The burden of proof should be on the dynamicists,
as this seems like a logically dubious relation, a variation on a faulty generalization,
or inductive fallacy: the possibility of uncomputable dynamical systems suspiciously
supporting the claim that cognitive systems are uncomputable. Note that van Gelder
seems to be rather unfair to computability when it comes to observing similar claims
on the relationship between mathematical model and real world phenomena:
The fact that sequences of discretized states of continuous dynamical systems
can be given (digital) computational descriptions is certainly interesting, but
all it really shows is that we can set up complicated mappings between the
A real number is said to be computable if it can be approximated by some algorithm in the
following sense: given any integer n ≥ 1, the algorithm produces an integer k such that: (k-1)/n ≤ a ≤
(k+1)/n. Another way is for an algorithm to produce a rational number r, given any real error bound ε
> 0, such that | r-a ≤ ε |.
realms of dynamics and digital computation. It doesn’t show that the
dynamical system is a digital computer, any more than the fact that we can
simulate the solar system on a digital computer shows that the solar system is a
digital computer. (van Gelder, id., p. 5)
So, according to van Gelder, it is fair to say that a RDS ‘realizes’ a MDS that stands
as a model of the RDS, but the solar system could not be said to ‘realize’ a MCS? It
seems that MDSs can realize MCSs, but not the opposite. It’s too bad, I guess, that I
can’t appreciate why real world phenomena, dynamical models, and computational
ones, coexist in such an irreconcilable asymmetry…
On the proper treatment of cognition and computation. One thing really
suspicious about the discussion on the comparative advantages of computational and
dynamical models so far is an apparent lack of consistency in the use of the concepts
of cognition and computation. Indeed, authors on both sides of the divide move back
and forth along different intensions and extensions of such concepts, perhaps
unknowingly, out of carelessness, or by an outright commitment to the reduction of
the many senses of the concepts to some core definition shared by all of its
subspecimens. Nevertheless, would the latter case be the actual motivation to do so,
their lack of explicitness should be proof enough to the contrary. Through all the
literature on computation and dynamics, for example, connectionist models are
usually claimed by both sides on grounds of characteristics that they share in
common, somewhat exclusively. But as it is becoming obvious through the
arguments of Giunti, Thelen, van Gelder, and other dynamicists, their arguments
against computability rest on a somewhat archaic intension of the concept of
computation, that is, Turing computation, or a symbolic view of both functionalist
and cognitivist commitments to the CHCS. But as we have seen in chapter I,
computation need not be Turing’s thesis on the implementation of decidable
functions through an abstract mechanism, operating over symbols! That is just one
of many interesting properties of the theory interested in computability, and does not
constitute a strictly formal account, but also empirical criteria on realization and
instantiation considerations. Such considerations will be discussed in section I.III of
this chapter, and the controversial issue of connectionism will be covered in the last
On the issue of cognition, it seems that different authors switch back and
forth between what counts as cognitive, be it internal processes, from neurological
processes to higher level cognition such as decision making or language use, or
behavioral and social processes, involving other agents and an environment which
must be inescapably included to study the relevant cognitive features. It can be said,
thus, that arguments about, and drawn from, the study of cognition, are very
sensitive to the level of description with which they are concerned. For one thing,
arguments about the proper treatment of cognition in matters of modeling may not
turn out to cover all levels of what counts as cognitive: developing a symbolic
information processing model of sensorimotor processes, all things considered, does
sound superfluous, and so does observing the continuous correlations of external
cognitive features, whatever they might be, when studying the processes by which
one performs long division in mathematical problems. Sadly, it seems that many
computationalists and dynamicists think that explanations framed in their respective
concepts and models can deal with any sort of evidence or phenomena. So far, the
strategy of both sides has been to find some cognitive features that can heuristically
be explained through their respective framework, and poorly dealt with from the
‘adverse’ perspective. Thelen, in championing a radical antirepresentational account
of cognition, would have all of cognition reduced to basic organizational principles
of systems dynamics in a largely behavioral viewpoint, focusing on the coevolving
correlations of internal and external magnitudes. Can all of cognition be reduced to
dynamical principles? We will see in section II of this chapter that epistemological
and methodological issues determine, far beyond this type of semantic warfare, an
accurate account of the relevance of both frameworks. Piccinini is no exception, on
the matter of computational explanations of neurological processes, and although he
is not a supporter of the DHCS explicitly, he does favor a departure from
computation-laden models to a more favorable mathematical model. We will see, in
the next section, how such an account constitutes a clever empirical support to the
aforementioned formal considerations of Giunti and van Gelder, but that it
ultimately avoids the problem altogether by refusing to integrate the full extent of
computation as a mathematical tool that reaches far beyond symbolicism (in section
I.III), and constitutes a different type of explanation altogether (in section II.II).
- III.I.II - Piccinini on symbols, strings, and neural spikes
This section deals with Piccinini’s (Piccinini 2004e) account of neuroscientic
models, which contrasts the biophysical models of Rashevsky et al with McCulloch
and Pitts’ computational endeavors. Piccinini considers that the computational
models in neuroscience are inadequate, based on the definitions of the concepts
drawn from computation theory. His reasoning, leading to the conclusion that
neurons do not compute, is roughly as follows: (i) computation is the manipulation
of strings or symbols (ii) neural spikes aren’t symbols, spike trains (or sets) aren’t
strings (iii) the manipulation of spike trains is therefore not computational. He also
suggests that we have no reason to believe that other aspects of neural activity are
computational, and that we therefore have no reason to believe that neural activity is
computation. Some objections will be raised against his conception of what counts
as computation and cognition, much as in the section about Giunti’s take on the
same concepts above.
According to van Gelder et al, the dynamical hypothesis in cognitive science
is not contrived to a mere externalist and observational characterization of cognition,
it is in fact a powerful qualitative and quantitative framework that allows the
coupling of various systems, and such systems can be internal informational
processes much in the same way that functional decomposition from a
computational perspective would have it.
Our own foray into sensorimotor
cognition and behavior, in chapter II, clearly hints towards the possibility of a
dynamical outlook. Piccinini elaborates on how computationalism was absent of
This issue is actually quite controversial, and is the subject matter of section II.II below.
pioneering work in biophysics (Rashevsky 1938, Householder and Landahl 1945)
that provided a framework for neuroscientific modeling. Mathematical biophysics is
the formal means to model the change in behavior of biological phenomena, inspired
by the concepts and methods applied in physical sciences. Rashevsky and his
colleagues used such means to complement a full account of neural mechanisms in
the neuroscience of the 1940s, and of the psychological phenomena that supervene
on them. Neither the concept of computation, nor any considerations derived from
computability theory, were involved in such an undertaking. Rather, ordinary
differential and partial differential equations, along with integral calculus, were the
formal tools constitutive of biophysical accounts of neuromechanics.
The adoption of computation-laden models was the original contribution of
Pitts and McCulloch in neuroscience, drawing from their research and interests in
cybernetics. Since computability theory already meant to a considerable extent the
modeling of informational processes through Turing’s view of computation, i.e.
through operations on symbols, Pitts and McCulloch purported to explain neural and
mental processes in a coherent framework, and thus viewed neural activity as
informational processes in much the same way. Neural spikes, namely the activation
peaks of electrochemical processes in the nervous system, were thus considered as
mathematical symbols, and spikes sets (commonly dubbed spikes trains) were
equivalent to strings of symbols. Piccinini grants that similarities between
mathematical symbols and neural spikes were easily found, such as their
discreteness, and unambiguous individuation relative to the processes in which they
take part. But this account is not satisfactory according to him: we need to look at
their inherent differences too, which are significant enough to undermine a
computational view of neural processing. This observation needs not be surprising at
all, since contemporary neuroscientific modeling is much more similar to
Rashevsky’s mathematical biophysics than to McCulloch and Pitts’ computational
neuroscience. The problem lies in the fact that while today’s neuroscientists seldom
treat neural spikes as symbols, it has ‘contaminated’ the rest of mainstream cognitive
science into adopting such a view according to Piccinini, as the CHCS is exactly the
view that mental processes are computational processes realized by the brain.
Piccinini claims that neuroscience is noncomputational, and gives a detailed account
of the shortcomings of identifying neural spikes as symbols, and spikes sets as
strings of symbols.
He starts by elaborating an account of computationalism as the manipulation
of symbols, and strings of symbols, and emphasizes two properties of (symbolic)
Turing-computation relevant to his endeavors: (i) a symbol’s content or role is
unambiguous, relative to the behavior of the system, and (ii) an output of a
computational process depends solely on the following combination: the internal
state of the system, the input symbols, and the way in which those symbols are
concatenated in a string, for a specific step of the process or a particular time
interval. Piccinini then draws on two features of neural mechanisms that will pave
the way to support his argument on the ‘noncomputationality’ of neural pathways:
neural spikes are all-or-none events (that is, neither ‘simply’ symbolic and discrete,
nor analog, i.e. time-dependent, continuous variables), and neural processes include
a large amount of spontaneous activity, which doesn’t allow for the simple
individuation of input-output matching processes, or the identification of
functionally relevant media to carry them out. What follows is that points (i) and (ii)
above are simply not found in neural processes, since it is nigh impossible to
individuate either functional units in neural signals, or a concatenation relation that
would establish sets (strings) of such units. Time-dependence (the absence of clear
boundaries for the beginning and end of a signal, of consistency of intervals, of
synchronicity), allegedly nondeterministic processes, and the unwarranted
significance of the presence or absence of a token or a string, in view of spontaneous
activity , are all disincentives for a computational account of neural processes.
Piccinini distinguishes between a semantic view and a functional view of computationalism, an
issue which is briefly discussed in section II.I.
41 Not to be confused with ‘noise’, as spontaneous activity might in fact be functionally relevant, by
opposition to the principled irrelevance of noise for functional purposes, in signal processing.
The worse part of the story, according to Piccinini, is not that neuroscientific
phenomena has been wrongly given a computational account, but that this account
has led, through the formulation of the CHCS, to a number of conclusions about
cognition that are ill-founded. Among others, (i) that we posses an explanatory
framework that can accurately account for mental processes through the Turing
conception of cognition, (ii) that the Church-Turing thesis puts neural processes on
the same ground as digital computation for the sake of an explanation of cognition,
and (iii) consequently, it is in principle possible that digital computation might
eventually realize identical cognitive prowesses. I could not agree more with such
arguments. The problem is, as it has been hinted in the previous section on
mathematics and cognition, every single (contemporary) computationalist knows so.
Such is the subject matter of the following section.
- III.I.III - Bechtel and Eliasmith on the issue of representations in dynamical
What transpires so far about the dynamicists’ rebuttal of computation as an
adequate framework for cognition is its apparent lack of distinction between
(symbolic) Turing-computation, championed by the symbolicists quite a while ago
(the era of GOFAI, so to speak), and subsymbolic or nonsymbolic models of
cognition that are nevertheless computational, in the much larger (to the extent of
being somewhat trivial) sense debated on in the first chapter. Piccinini, much like
Giunti and van Gelder, makes an unarguably good case against symbolic, or Turing
computation, at the expense of being a bit behind schedule. More a case of a straw
man argument, then, or as I shall call it specifically, the Don Quixote case against
The flaw in this line of argumentation is thus a matter of conflating digital
computers and the mathematical conception of computability. Piccinini makes a
good case about neural spikes and spikes sets being non- [Turing] computational,
but does he make a case against computability in its largest sense? Again, like in the
case of Giunti and van Gelder, we should differentiate between what constitutes a
computational account of something, from the level of explanation at which we
study something. Symbolic computation may account for logico-mathematical skills,
the use of language, and almost all of the inner workings and behaviors of a digital
computer, it does not appear to fit most of the rest of biological cognition.
Nevertheless, it can be said that we do have computational models of cognition in
the large, albeit more trivial, sense of the word. Indeed, we have subsymbolic models
of cognition, realized on digital computers, for one thing. The digital computer, in
such cases, is not the model, just a platform from which we design the relevant
models, at the relevant level of enquiry, thus symbolic computation is nothing but a
canvas on which are painted appropriate textures and colors mirroring our
conception of cognition, if you allow me the use of such a metaphor.
Bechtel (Bechtel 1998) and Eliasmith (Eliasmith 1997) accuse some
proponents of the DHCS of being antirepresentationalists, much to the demise of
cognitive science, and based on a misconception of the very concept of
representation. While it is incorrect that van Gelder et al are against representations
in the modeling of cognitive processes, much has to be said concerning the role and
format of representation to clear up this fundamental issue. van Gelder does confuse
computation and symbol manipulation, and as he unsuccessfully tries to deal away
with the wrong concept of representation (a strict, symbolic conception of
representations not necessary at all for computational cognitive science), he (as are
most dynamicists) is left with a vague, non-operational definition (one where
anything can count as a representation, such as attractors and trajectories in the state
space of the dynamics of a system). While the DHCS integrates representations in its
mathematical characterizations (by means of interpretation of concepts such as
attractors and trajectories), it further requires an implementation theory (in Fodor’s
and Pylyshyn’s 1988 sense of a cognitive architecture), like connectionism or other
possible implementation models in cognitive science.
Bechtel insists on two essential characteristics of representations in
information processing systems, independent of the nature of the concerned system:
(i) the aspect of representations that we usually express as ‘standing-for’ something
else, and (ii) the format of such representations. While we can adopt different views
towards what counts as representations, he argues that the former point is necessary
as such, and that dynamics do not deal away with representations at all. But
supporting the claim of informational systems requiring representations as Bechtel
does is unnecessary, since van Gelder does not, in fact repudiate their relevance.
Only a number of radical dynamicists, such as Thelen, repudiate the use of
representation-laden cognitive systems, and such an ambition trades on a misreading
of computational models being conflated with symbolicism, as stated above, more
than a sound account against the very concept of representation. Even van Gelder’s
landmark example of a dynamical system, Watt’s centrifugal governor, which is
used to counter the computational explanation of an allegedly inherent dynamical
nature of cognition, can thus be said to be representational. Indeed, the various
components, and interactions, of this type of mechanism nevertheless indicate (stand
for) magnitudes of physical phenomena, and determine its operation (are meant to
operate on, or produce, a spectrum of outcomes depending on the relevant
magnitudes determined by the system).
On the latter point, concerning the format of representations, Bechtel agrees
that dynamicists are innovative in promoting non-symbolic, quantitative values to
stand for informationally driven systems. Granted, this non-symbolic acceptation of
the concept of representation makes it otherwise ubiquitous, and can be said to be
some sort of ‘minimal (as in low-level) representation’. It is nevertheless operational
for the purpose of the framing explanations, relating to “any organized system which
has evolved or been designed to coordinate its behavior with features of its
environment.” (Bechtel, id., p. 16) So representations need not be strictly a matter of
propositional format, and moreover, static ones. The proposition that trajectories and
attractors might stand for representations, to the benefit of the overall behavior of a
system, thus constituting dynamical representations (i.e. representations that change
over time, influenced by other features of the processes involved) marks this original
contribution from dynamics to the study of cognition. Bechtel additionally points out
van Gelder and Port also stress that in DST systems the processes within the
system are not defined over representations. […] DST, like connectionist
modeling as well as much work in neuroscience is concerned with
representations that figure in processes. (Bechtel, id., p. 9. My emphases)
Eliasmith’s criticism is even less reverent towards van Gelder, whom he
accuses of being completely beside the point, on his characterization of
connectionism, for one thing. As van Gelder first confuses computation with
symbolic and digital processing, he then wrongly claims that connectionism has
more to do with dynamics than computation. Apparently, van Gelder overlooked
Newell’s (Newell, 1980, 1990) distinction between the type of computer postulated
to realize cognitive processes, from the family of universal computers. Newell, as
did all symbolicists, postulated just that kind of representational systems, i.e. symbol
systems, not computational systems. On the other hand, to say that connectionist
models are noncomputational on that (misconceived) ground is preposterous: every
and all connectionists have always considered their models to be computational, for
it is indeed the very point of connectionism to model information processing in a
biologically plausible way, in order to better understand cognition. Connectionists
are committed to complex dynamical analysis, as a means to account for essential
features of information processing in neural networks, but the ultimate goal is to
address computational problems (see Churchland and Sejnowski, 1992).
Bottom line is, connectionists, and even symbolicists, have generally had a
much broader conception of computation than dynamicists, such as van Gelder. As
such, there is no computational versus noncomputational division between
symbolicism and connectionism, it is a mistaken characterization on the part of
certain dynamicists. On the issue of representations, Eliasmith also sharply disagrees
with van Gelder’s view of dynamical systems having representations in the loose
sense of trajectories and attractors, for if any kind of pattern or element of a system
might be said to be representational, then the very meaning and use of
representations become patently trivial. Bechtel’s account of representations, above,
should be the preferred view, since it constitutes a minimalist and deflationist
account that is nevertheless operational, and allows for representations in both
frameworks, their only dissimilarities pertaining to format.
- III.II - On the type of explanation involved in computational and dynamical
The second section deals with methodological arguments concerning the
motivation and scope of both frameworks, as many writers have tried to dissociate,
negate, or complement their models in the study of cognitive science. The topics of
discussion concern distinctions between functionalism and computationalism,
mechanistic and covering laws explanations, and the complementary value of
computational and dynamical models. We will firstly consider Piccinini’s account of
functionalism in cognitive science, in light of his conflation of computation and
symbolic processing. In a second section, I will draw on Bechtel’s clever
characterization of the types of explanation involved in the two conceptual
frameworks, a crucial step in the development of this thesis, if not its main grounds
for argumentation.
- III.II.I - Piccinini on functionalism and computationalism as independent
characterizations in cognitive science
Piccinini is concerned with what he calls the semantic view of computation
(wrongly carried into philosophy, the view that computational states should be
individuated by their semantic properties) leads him to propose a strictly
functionalist account of computation. However, functionalism and computationalism
have traditionally been conflated as one concept, and computation turns out to be
only one type of functional explanation, according to the author. The important
distinction between computational explanation and computational modeling is also
introduced, but I will argue that here again, and similarly to van Gelder and most
dynamicists, Piccinini in fact still conflates the empirical thesis of implementation of
computer science’s computation (i.e. it’s much stricter symbolic, serial, and discrete
account of what constitutes computation), with the mathematical class of
computable functions, along with the related algebraic and geometrical properties of
its analysis (not to be confused with mathematical analysis’ sense of the study of real
and complex numbers, and related functions). This will turn out to be of direct
consequence with respect to Piccinini’s thesis, concerning the CHCS, that (i) (if) any
nontrivial computational theory of mind is committed to the existence of appropriate
mechanisms that realize the computations, and (ii) (if) the manipulation of spike
trains, according to neuroscience, is not computational (section I.II), (iii) (then) there
is no nontrivial computational theory that survives the empirical test (according to
his definition of the functional account of computationalism).
Having criticized above, in section I.II, Piccinini’s account of neural
processes with respect to the concepts of computability theory, the section is then
concerned with Piccinini’s (Piccinini 2003, 2004bcd) position on the role of
computational explanations in cognitive science. As seen before, Piccinini disagrees
with the type of consequences that can be drawn from the foundational theses of the
CHCS, for he refuses a computational account of neurological mechanisms that
would support cognition. We have already commented on his twofold shortcomings,
one being his unwarranted conflation of (symbolic) Turing-computation with the
class of formal definitions of computable functions and numbers, the other
pertaining to his reluctance to address anything above neural mechanisms as
possibly being both cognitive and computational. Indeed, he sticks to neural
mechanisms, while debating over the formulation of the CHCS:
According to the [CHCS] , neural mechanisms perform computations, and
neural computations explain mental capacities more or less in the way that the
computations performed by calculators and computers explain the capacities
that are peculiar to them.” (Piccinini 2004d, p. 2)
Piccinini actually uses the acronym CTMB, for a ‘computational theory of mind and brain’, which
carries ontological commitments that are not under evaluation here. I therefore consider, as
mentioned in the introduction, only the epistemological and semantic issues raised by what is meant
by the CHCS, to be on par with the DHCS, in this dissertation.
Since I have already commented on the matter of whether computation has any
bearing on neurological processes, my endeavors here are to address what
Piccinini’s insights, concerning the type of explanation involved by the CHCS,
entail for the debate over the comparative advantages and flaws of the computational
and dynamical frameworks.
Computational explanations are usually defined as postulating mechanisms
operating over representations. A first formulation of computationalism, which
Piccinini names the semantic view of computational explanation, holds that
“computations are individuated at least in part by their semantic properties.” (id., p.
3) But such an account in untenable, as symbolic ascriptions to representational
processes is arbitrary and observer-relative. This first definition of computationalism
needs not concern us, since (i) Piccinini conflates formal computation and
(symbolic) Turing-computation, and (ii) to argue that neural mechanisms aren’t
computational on grounds of not being interpretative mechanisms operating on
arbitrary symbols is patently evident to us anyways. How about computational
explanations being warranted in virtue of a system possibly being modeled as
computational? The problem is that too many things would turn out to be
computational! That would thus trivialize computational explanations, claims
Piccinini. Also, it would establish the phenomena of concern, here being cognitive
processes, as computational a priori, in a dogmatic way. Again, I feel obligated to
reply that some criteria on pragmatical grounds can be invoked, such as
Stufflebeam’s (Stufflebeam 1998) take on what constitutes intrinsic, versus extrinsic
computation: Stufflebeam argues that since anything can be modeled as computable,
that is, given an interpretation as a computational system, we should be concerned
only with what makes a system computational, viz. intrinsically to such a system. It
turns out that whatever can be said to perform computations, either by design or by
an apparently inexorable tendency to be considered as such (thus, on pragmatical
grounds), should be considered as such. Information processing models should be
involved in, and only in, explanations pertaining to entities that process information
in a relevant sense. In short, yes, it’s up to us to decide what is computing, but not
everything need, or should, be considered as a computational process.
So, which processes do deserve to be called computations in a relevant
sense? Piccinini feels compelled to peek into physiology and engineering, to parallel
his study of neural mechanisms and computers, and claims that we must observe the
type of explanation involved in such scientific endeavors to better understand
computation. Now, a general explanatory strategy in applied sciences is to appeal to
functional explanations:
A functional analysis involves the partition of a mechanism into components,
the assignment of functions to those components, and the identification of
organizational relations between the functioning components. For any capacity
of a mechanism, a functional explanation invokes appropriate functions of
appropriate components of the mechanism, which, when appropriately
organized under normal conditions, generate the capacity to be explained. The
components’ capacities to fulfill their functions may be explained by the same
strategy, namely in terms of the components’ components, functions, and
organization. The process of functional analysis bottoms out in components
whose capacities are no longer functionally analyzable; they are to be
explained by other explanatory strategies (e.g., subsumption under physical
laws). (id., p. 8)
How then, are computational explanations related to functional explanations?
Mechanistic and nomological explanations co-occur quite abundantly in
science: “More generally, mathematical descriptions can be employed in conjunction
with functional analyses to yield theories and models of functionally analyzed
systems.” (id., p. 9) The relevance of mathematical descriptions, according to
Piccinini, is threefold: (i) to specify the time evolution of mechanisms, or their
features, (ii) to observe relations of dependence or interdependence between
variables (expressing features of the mechanism), such as input-output matching,
and (iii) to observe how the state space of a mechanism changes, or develops, viz. to
observe trajectories, attractors, and bifurcations. So Piccinini agrees that
mathematical descriptions are complementary to functional analysis (id.), by
providing a means to observe the behavior of a mechanism, or the relations between
its components, both qualitatively and quantitatively. He then proceeds to trying to
convince us that (i) computational and functional explanations are traditionally
conflated (in the literature of cognitive science, he claims), and that (ii)
computational explanations are but a particular type of functional explanations.
While I find the latter to be intuitively sound and uncontroversial, I disagree with the
former point, part of which obviously pertains to Piccinini’s own flawed, conflated
account of the concept of computation. So, functional explanations appeal to the
internal states, processes, and inputs of a system. The problem is thus a matter of
choosing the appropriate type of functional analysis, for refrigeration, digestion, or
photosynthesis have little to do with computation, to name a few. But, avoiding
again the debate on matters of formal computation and symbolic processing, is it not
just stating the obvious, much to the advantage of computational explanations, their
being concerned only with information processing mechanisms in general? By
avoiding Piccinini’s flawed concept of computation, we thus have little left to argue
about, since such derived conclusions were spun by untenable premises.
The fact that Piccinini constantly appeals to (symbolic) Turing-computation
to prove his point about our functional explanations having to shift towards another
explanatory framework, when neural mechanisms are concerned, makes it hard to
disagree with, for symbolicism (the GOFAI era) has faded in popularity quite a
while ago. What is not fair, however, is Piccinini’s treatment of connectionism as
being doomed on the same grounds: connectionism is different enough from
(symbolic) Turing-computation, or symbolicism, to deserve an analysis of its own,
as we shall see in chapter IV. On the bright side, Piccinini has opened up a very
important issue about the complementary value of mathematical descriptions and
functional analysis, which is the subject of the following section.
- III.II.II - Bechtel, on mechanistic explanations versus nomological explanations
in cognitive science
The clash between computationalism and dynamicism may not be about the
alleged ‘nature’ of cognition after all. As we have seen in the previous section,
methodological concerns turn out to be a preeminent issue when comparing the two
frameworks with respect to the study of cognition. For one thing, is the choice
between computability theory and dynamics necessarily one of mutual exclusivity,
or aren’t such concepts and models more a matter of phenomena of concern? Let’s
rewind a bit and think about the data on which Thelen builds her arguments
supporting dynamics, in chapter II: the evidence of concern is psychophysical
phenomena. Now, if we pay attention to the very definition of psychophysics,
From Wikipedia, the free encyclopedia.
Psychophysics is the branch of psychology dealing with the relationship between physical stimuli and
their perception. […]
Psychophysics studies psychological scales for physical stimuli. Hot and cold, for example, are
psychological scalings of temperature stimuli for which such physical measures as degrees Celsius
provide only physical units.
Areas of investigation include sensory thresholds, methods of measurement of sensitivity, and signal
detection theory. (My emphases)
Is it not now clearer that such a modus operandi, and the concepts relevant to the
dynamical framework in cognitive science, have more to do with behavioral
observations, and the mathematical description of patterns and regularities of the
parameters of a system? Psychophysics, for one thing, differs greatly from cognitive
neuropsychology, or task-specific enquiries into psychological faculties, for
example, which are areas of research purporting to identify mechanisms relevant to
the observed cognitive processes. Psychophysics also incorporate cognitive features,
of course, but such cognitive features are already given, the concern is to observe
correlations of physiological and cognitive magnitudes already arbitrarily defined
and chosen. One might jut say that in observing sensory thresholds, sensitivity
measurement, and signal detection, we are observing the dynamics of a cognitive
and sensorimotor system, but in no way is such an endeavor able to produce a
constitutive account of the design of a system. As Glymour states about van Gelder’s
(representative of the DHCS in general) view:
One way of abiding by some of van Gelder’s prohibitions is to adopt a kind of
neo-behaviorism […]. Skinner’s version of behaviorism tried to confine
scientific inquiry and conjecture to functional--indeed dynamical--descriptions
of how human and animal action depends on the environmental history to
which the creature has been exposed. Doubtless van Gelder’s behaviorism
would differ considerably from Skinner’s in what it lets in, but van Gelder
appears to agree with Skinner in wishing to prohibit any inquiry into the
internal mechanisms by which the creature does what it does or thinks what it
thinks. (Glymour 1997, p. 10)
Dynamicism, thus, in its most radical form (such as Thelen’s), would be some sort
of sophisticated behaviorism, albeit integrative of some internal considerations
(which are also to be modeled through correlational observations, not through design
and components).
Bechtel (Bechtel 1998) also presents strong arguments about the conception
of explanation championed by the dynamicists. He holds that while it is indeed
compelling and useful to adopt the concepts and methodologies of dynamics in
cognitive science, it is in no way a refutation of the CHCS and the computational
approach in general, since their respective type of explanation are orthogonal ways
of conducting research. Indeed, while computational models of cognition are
interested in mechanistic characterizations of the processes involved, through
localization and functional decomposition, dynamical models are of another type of
explanation, namely the explanation of cognition through what Bechtel calls
covering laws (following the type of explanation championed by the neopositivists).
Indeed, a more traditional view of science (Hempel 1966) has been to conceptualize
scientific laws as universally true statements, also called nomological statements.
But covering law explanations pose a problem when we depart from physics, such as
See also Craver (forthcoming) for a similar argumentation in favor of mechanistic explanations in
in the domains of life and cognitive sciences, where subsuming phenomena under
universal laws is not as much the goal as is the discovery of particular processes at
work in a given system. The main difficulty, while observing the behavior of
complex phenomena, lies in that we have no way to distinguish statements that
might be universally true from accidentally true statements, and that low-level
physical laws are too simple to be constitutive of nontrivial accounts of such
complex phenomena. So, by appealing to mechanistic explanations, we can analyze
the processes of a system through component processes, “described either physically
or functionally.” (Bechtel, id., p. 10)
Two underlying assumptions of mechanistic explanations are that of
decomposition and localization, that is, (i) “the assumption that the overall activity
results from the execution of component tasks”, and (ii) “the assumption that there
are components in the system that perform these tasks.” (id., pp. 10-11) Complex
phenomena such as biological and cognitive systems have been studied for quite a
while through mechanistic explanations, for it narrows down many conjectures in
testing them through empirical enquiry, and helps formulate sound conclusions
concerning the role of component processes into the overall behavior of a system. In
Bechtel’s words:
This explanatory strategy is common not just in information processing
psychology but in much of contemporary neuroscience; researchers try to
decompose the tasks performed by the brain into component tasks and then
seek evidence that these tasks are actually performed by neural components.
[…] These studies accordingly are seeking to identify hypothesized component
psychological processes with specific brain regions.” (id., p. 11)
Dynamicists hold a somewhat holistic view of cognitive systems and processes,
which they claim is incompatible with mechanistic decomposition and localization
on conceptual grounds. But mechanistic models need not be simple linear and serial
processes, they too can (and ultimately do, in both life and cognitive sciences) be
sophisticated accounts of integrated and nonlinear systems! The information
processing metaphor is important to mechanistic explanations in that it models
“particular components in the system as carrying information about processes
elsewhere in the system.” (id., p. 13)
Perhaps one of the most enlightening developments in Bechtel’s comment is
about a contrasting feature of dynamical models, in comparison with computational
ones, as found in his discussion between connectionist and dynamical models:
The difference and differential equations in [Townsend and Busemeyer’s ]
models are intended to describe patterns of linked change in the values of
specified parameters in the course of the system’s evolution over time. The
parameters do not correspond to components of the system which interact
causally. They are, rather, features in the phenomenon itself. (id., p. 14)
In other words, the parameters of dynamics refer to magnitudes, themselves drawn
from features of behavioral concern in a system’s process, but do not pertain to the
mechanisms’ componential characterization! One links an infant’s capacity to grasp
objects with regards to coordination factors such as perception, motor control, and
physiological features (mass, strength, etc.), as Thelen does, or the arbitrary
valuation of motivational features relative to particular consequences or expected
outcomes, such as in Townsend and Busemeyer’s model. But none of these
parameters appeal to the nature or role of the underlying component processes! In
short, dynamics does well at describing correlations and overall tendencies of
arbitrarily chosen magnitudes relevant to a system’s behavior, but do not answer
‘why’ questions about such processes (since dynamics are not interested in what
does what, to what else, and for what reason), only partly ‘how’ questions (since
causal and organizational issues are not addressed either).
But Bechtel favors, even supports as essential, the complementarity of
mechanistic and nomological endeavors:
In reference to Townsend and Busemeyer’s (1995) study of decision-making using dynamical
Assume that we have a correct [dynamical] account of motor behavior […], of
motor development […], of perception […], or of decision-making […]. Each
of these invites a further question: how is the underlying system able to
instantiate the laws identified in these [dynamical] accounts? One way to
answer this question is to pursue a mechanistic explanation by trying to
decompose the overall behavior and localize subtasks. (id., p. 15)
Accordingly, such methodologies are essentially complementary, and one does not
have any kind of ‘priority’ over the other:
If a [dynamical] account provides an account at this level [of description of
processes], its legitimacy is not undercut by learning how the various
components in the system operate and perform their individual roles. (id.)
Bechtel even adds a further value, or advantage, of co-opting dynamical
(nomological) explanations in the study of cognition: some research conjectures may
be doomed to modeling cognitive processes in a way that is inaccurate, and having a
fair account of the behavior of a system may warrant a good mechanistic explanation
of it a fortiori. This argument is equivalent to that of proponents of the need for the
ecological validity of models in cognitive science (a good example of which is the
characterization of the inverse kinematics problem as seen in chapter II).
Are the CHCS and DHCS incompatible, mutually translatable, or simply
orthogonal characterizations of the same phenomena? Bechtel provides us with good
reasons to believe not only in their compatibility, but even to an essential
complementarity of both frameworks.
Most cognitive science research has been devoted to determining the nature of
the mechanisms underlying cognitive performance, whereas some DST
(dynamical systems theory) accounts are rather directed toward identifying
laws that relate different parameters in a system. But while there is a
difference here between DST accounts and other cognitive accounts, this does
not render the two approaches incompatible. Indeed, they are complementary.
We want to know both what the regularities are in the phenomena, and what
mechanisms underlie them. (id., p. 16)
- IV - Close encounters of the third kind: connectionism
This fourth and final chapter presents the controversial class of connectionist
models. Since proponents from both frameworks insist on claiming connectionism
as part of their own view of cognition, the entire chapter is devoted to the
clarification of what is at stake in connectionist models, both formally and
empirically, and in what way it has anything to do with the comparative analysis of
computational and dynamical models.
- IV.I - Misunderstandings so far: on representation and computation, types of
explanation, and the special case of connectionism
Now that we have spent a lot of ink on conceptual issues concerning the
nature, format, role, and explanations involving computation, representation, and
dynamics, we have to assess how all of this might claim lineage with yet another
type of applied mathematical models, that of connectionism. Connectionism, as we
have seen throughout this paper, has pretenders on both sides of the computationaldynamical divide. This chapter’s aim is to show that connectionism can be seen as
an exemplar model of a particular type of cognitive processes, namely neural
mechanisms, and that such a model combines elements of both conceptual
frameworks in an essentially complementary way, in its aim to provide us with an
accurate account of the biological substrate of psychological processes. But before
we even start characterizing connectionist models, let us summarize what we have
gathered from our earlier reflections on computation and cognition. Firstly, we have
established that computational models need not be (but can indeed be) symbolic
models, as the latter are but a special type of the former, more generic family of
formal models. Secondly, representations are not only necessary features of any
mechanistic explanation involving information processing models, but we can
conceive them in an operational and minimalist, albeit ubiquitous, way that avoids
the aforementioned symbolic characterization and is quite compatible with a
dynamical explanation. Thirdly, cognition is meant to refer to many things, since
cognitive processes aren’t circumscribed to brain processes, or task performances,
and can pertain to environmental and social features too. While connectionism
constitutes a computational view in a nonsymbolic sense, and involves
representations much like in Bechtel’s discussion above, it concerns only internal,
low-level cognitive processes, namely that of neural mechanisms, and how they
exhibit features that can be informative about higher level psychological features.
Piccinini’s (Piccinini 2004e) comments on the adequacy of computational
models to represent neural processes have left us in doubt, since his account of what
counts as computational is severely biased. But, like everyone else, he still holds that
a functional account of neural mechanisms should be, in principle, possible, albeit
not a (symbolicist) Turing-computational one. So, if it is in principle possible to
model neural mechanisms through an adequate enough mathematical model, the
question that remains is: which one? Connectionist models, much like the ones from
computational and dynamical theories of cognition, also have a dual commitment to
a formal thesis (which involves both computable functions and mathematical
analysis) and an empirical thesis (that the realization of computation in biologically
inspired information processing involves parallelism and large-scale distribution,
among other things). Glymour sketches the outlines of the motivations and
pretensions pertaining to connectionism, with regard to cognition:
The most obvious and most important fact about cognitive psychology is that
on almost every dimension this aim [to figure out cognition], and even more
specialized pieces of it, are radically underdetermined by this sort of evidence
[traditional sorts of evidence available to psychologists] […] Psychologists
have given three sorts of responses to [various sources of] evidences of
underdetemination. One is to ignore alternatives and treat speculation as nearly
established fact; another is to try to establish only more modest, but still
relevant, claims about mental processes and their development; and a third is
to try to connect models of mind, in so far as possible, with biology, in the
hope that biology will eventually so constrain such mechanisms that together
with psychological experiments many of the big questions about how
cognition works can be answered. Connectionism is the oldest and most
influential instance of the third strategy. (Glymour 1997, p. 2)
Connectionism is built on the assumption that the brain, in fact the whole
nervous system, is the substrate of our cognitive, here psychological, processes, and
that such processes are carried out in a way that is best captured through a
sophisticated computational model, thus appealing to information processing
models. Connectionism is defined as
From Wikipedia, the free encyclopedia.
Connectionism […] refers to an approach in the fields of cognitive psychology, cognitive science and
philosophy of mind which models mental or behavioral phenomena with neural networks […]
The main assumptions are that (i) a mental state can be represented by a ndimensional vector of numeric activation values, over neural units interconnected in
a network, and (ii) psychological processes commonly referred to as learning and
memory are represented by the modification of the strengths (or weights), or the
architecture , of the connections between such units. Connection weights are
themselves described as NxN-dimensional matrices. The state of a neural unit
is a function of the weighted sums of states of its parents, the function roughly
approximating how changes in cell potentials depend on inputs. Learning takes
place by any of several essentially local algorithms that adjust weights.
Memory resides in the weight values. (id.)
The motivation for connectionist modeling came from the application of
information processing concepts and methods to the physiology of the nervous
system, as we have seen in chapter III, section I.II. But we must also be aware of the
limitations of connectionist models. For one thing, not all of the local learning
algorithms are representative of real neural processes, and artificial neural networks
(ANNs hereafter) are simplified, coarse grained versions of such processes.
Nevertheless, as stated above, the aim is to constrain the explanations about
psychological phenomena through our knowledge of neurobiological processes, an
endeavor that is still in progress, and has already shown considerable success over
By the creation of new connections (representing synaptogenesis) or new neurons (neurogenesis),
paralleling actual neurobiological processes. For references on the biological bases of connectionism,
see Cline (Cline 2001) on neurogenesis and synaptogenesis, Kandel (Kandel, Jessell, and Sanes 2000)
on the neural mechanisms underlying behavior and cognition, Shultz’s models of cognitive
development (Shultz 2003), and Stein, Wallace, and Stanford (1998) on single neuron
the last decades. For example, some neurobiological evidence suggests that local
neurosynaptic learning might be one of two mechanisms, the other one being global
processes, which should be featured in our explanations of neural mechanisms. But
as global processes such as hormonal transmission (Glymour 1997) and interactions
neurophysiological explanations, they can also find corresponding features in
connectionist models. Such sciences, and their models, inform each other and evolve
slowly but surely. Global processes, Glymour argues, might just turn out to make
connectionist modeling easier, rather than more difficult.
The essential assumption of computation throughout such models is
supported not only through formal and conceptual considerations, as we have stated
on so many occasions, but also on the sheer success of such a perspective, judging
by the ubiquity of computational explanations in cognitive science, and beyond,
even in life sciences. In Glymour’s words again:
I think no one with scientific experience can read the papers reviewed by
Churchland and Sejnowski, or many other sources, and doubt that it is real
science, or that computational and representational ideas are essential to it […]
In almost all of [the research in cognitive neuroscience], an essential
assumption is that cognition depends on computable biological processes.
(Glymour 1997, p. 4)
But connectionism need not be pulled into the allegedly opposite directions of
computationalism and dynamicism, it does actually constitute a field rich in
mathematical enquiries that is determined by its one motivation: to frame a
biologically adequate explanation of cognition. The end justifies the means, and
such means are drawn from many branches of mathematics, even statistical
analysis! Bechtel illustrates this through the example of Elman’s (Elman 1995)
A remark from Glymour concerning the mathematical inspirations and requirements of
connectionism: “[…] the two directions scarcely exhaust the methods by which people try to
understand why connectionist systems behave the way they do – at least as important, perhaps more
so, is the application of rather conventional statistical techniques to try to gain a qualitative
understanding of the causal relations among features of a complex connectionist system.” (Glymour
1997, pp. 4-5, note 2)
connectionist model of a language related tasks, the prediction of successive words
in a sentence.
Elman uses a recurrent neural network which has both computational and
dynamical features. There is no doubt that such a model is mechanistic, for one
thing, since it appeals to neural mechanisms and the functional role of their
components. The model is also obviously computational for the same reasons stated
throughout this paper. But dynamics are here used to analyze the behavior of such
The question motivating this research is whether recurrent connections provide
sufficient information for the network to predict words of grammatically
appropriate categories. Elman demonstrated that when an appropriate training
regime was used the network’s predictions would respect even fairly long
range grammatical dependency relations. (Bechtel 1998, p. 15)
Elman is curious about the way this is achieved. How does the network manage to
do so? Since his network is significantly complex (involving many neural units, and
many more connections between them), the ‘information’ stored by the network is
bound to be massively distributed, and single unit investigation is therefore
pointless. Elman consequently uses formal strategies issued from dynamics and
statistics, such as cluster analysis and principal components analysis, to observe a
reduced state space of its behavior (by observing the qualitative features of the
dynamics of a certain number of variables). By comparing the behavior of the
network on nearly identical tasks, it is then possible to pinpoint relevant differences
in processing, and thus give a satisfactory account of the performance of such a
complex mechanism. Elman’s combination of mechanistic assumptions, viz. that of
decomposition and localization, with dynamics’ heuristics of cluster and components
analyses, provides him with compelling information from which he can then give a
detailed account of the phenomena under observation. Note that such dynamical
features need not be strictly methodological, and external, characteristics. For one
The next section develops on such specifications.
thing, the recurrence of the network itself is a dynamical feature, and so are many of
the features of component learning algorithms, as we will see in the next section.
To summarize in a more concise, disambiguated vocabulary, connectionist
models (i) involve subsymbolic (a contrasting feature with GOFAI computation)
computational models, (ii) are simplified models of real neural networks, but
nevertheless exhibit many of their interesting features (otherwise, there wouldn’t be
any point to pursue such venture), (iii) are traditionally simulated (here,
implemented) on digital computers to make use of their computational power, (iv)
realize a mathematical model that involves complex, nonlinear algebraic
calculations, and exhibit parallelism and massively distributed representations, and
(v) may, or may not (at the risk of excessive simplicity, or triviality) involve
essential dynamical features, and/or appeal to dynamics for the purpose of framing
adequate explanations. We need to put enough emphasis on that last point, as many
researchers, such as Smolensky (one of the founders of parallel and distributed
processing), have argued that the direction connectionist models will take is towards
fully continuous, high-dimensional, nonlinear dynamic systems approaches.
- IV.II - Types of connectionist models, and what makes them more or less
Neural network
On that topic, see Elman, Bates, Johnson, Karmiloff-Smith, Parisi, and Plunkett, 1997, and Elman
From Wikipedia, the free encyclopedia.
A neural network is an interconnected group of neurons. The prime examples are biological neural
networks, especially the human brain. In modern usage the term most often refers to artificial neural
networks (ANN) […]
An artificial neural network is a mathematical or computational model for information processing
based on a connectionist approach to computation.[…] It involves a network of relatively simple
processing elements, where the global behavior is determined by the connections between the
processing elements and element parameters. The original inspiration for the technique was from
examination of bioelectrical networks in the brain formed by neurons and their synapses. In a neural
network model, simple nodes (or "neurons", or "units") are connected together to form a network of
nodes — hence the term "neural network".
Figure 16 (left) A neural network, an interconnected group of nodes, akin to the vast network of
neurons in the human brain. Figure 17 (right) Neurosynaptic pathways, illustrated through
Connectionists are interested in modeling cognitive processes through neural
networks, which are designed to incorporate a variety of parameters and constraints,
but their sophistication also depends on the mathematical and informational interests
of their engineers. Thus not all artificial neural networks are meant to model
cognitive processes, since ANNs are now general computing tools to facilitate the
solution of problems in any and all areas one might think of, from engineering
design to management, and automated navigation. This section presents an overview
of a number of connectionist models, with the aforementioned considerations in
mind. The purpose is to assess the potential of connectionism as a model of choice
for cognitive science, and its ‘situation’ in view of computational and dynamical
Prima facie, what are connectionist models, namely ANNs, used for
generally, in cognitive science? Glymour (Glymour 1997) sketches four mainstream
types of connectionist research to link neural network models and the study of
cognition: (i) systems simulation, which endeavors to “describe as completely as
possible the nerve connections of very simple animals and simulates them on a
computer” (id., p. 3), (ii) functional analysis, which focuses on the physiological
properties of single neurons, by framing them into information processing
explanations, constituting a significant part of neurobiological research (also referred
to as single neuron electrophysiology, e.g. Stein, Wallace, and Stanford, 1998), (iii)
serial implementation, which purports to establish serial, and sometimes discrete
processes onto parallel and distributed processes, and (iv) abnormal cognition
simulation, which attempts to simulate the evidence gathered from abnormal
neuropsychology in neural networks through the characterization of similar features,
such as the graceful degradation of artificial networks being quite similar to brain
Now, artificial neural networks are much simpler and smaller than actual
neural subsystems (as can be seen in figures 16 and 17 above, for example), but
connectionists are interested in bridging functional features of such models with
actual cognitive features and performances, most prominent of which probably is
learning, on smaller scales. Formal neural units operate similarly to biological
neurons, typically in layers (at least three layers are necessary to exhibit any
interesting kind of calculation, namely input, hidden, and output layers ), by the
summation of weighted synaptic inputs, which may or may not be sufficient to
activate a given unit, depending on a given threshold value (usually determined by a
sigmoid function). The process is repeated on a massive scale, for all interconnected
units. Thus, artificial networks are mathematically designed over ‘transfer’
functions, representing the activation relation between biological neurons. Such
calculations span from very simple (algebraically outputting 0s and 1s) to rather
complex (if they are to be representative of biological processes, or computationally
useful in any way). Sigmoid and tanh (hyperbolic tangent) functions are usually
employed as transfer functions since they introduce nonlinearity in the calculations
of a network, while restricting the domain and codomain’s values to a range of [0,1]
for the sigmoid function, or [-1,1] for the hyperbolic tangent. A derived advantage of
such functions is that their derivatives are simple, and as such allow easier errorcorrection calculations for neural networks. Such calculations are usually set on
random initial values (the state of a system which has no information), then ‘trained’
by feeding input values that are to be matched to output values through the gradual
modification of synaptic weights, in order to obtain a network that can be said to
have ‘learnt’ useful associative patterns, thus constituting some kind of
representations, or memory, of relevant data.
The information processing usually occurs in both hidden and output layers, as the input layer
typically only serves the purpose of feeding the information to the rest of the network.
Neural networks exhibit many heuristic features of biological neural
mechanisms, as well as their ‘higher-level’ cognitive counterparts: (i) learning,
through exposure to an environment by means of sensory inputs, (ii) auto-organized
representations, which result from the learning process on repeated exposition to
diverse sources of inputs, (iii) fault-tolerance, as representations are redundantly
formed as prototypical informations, which are massively distributed, local damage
to the network does not impair the network’s overall performance, (iv) flexibility and
scalability, as noisy and partial inputs are handled efficiently through such models,
and they can handle different problems related to a similar inputs-outputs
environment, and (v) real-time processing, as the implementation of an ANN can be
made to operate on bounded real-valued (continuous) data. As mentioned above,
different types of networks can achieve different types of tasks, with a
computational might proportional to the degree of sophistication involved in its
mathematical and architectural design. We can sketch the following taxonomy as a
coarse characterization of the various types of neural networks
(such a
classification is by no means exhaustive of the ever expanding field of parallel and
distributed architectures in computer science):
Feed-forward networks
Feed-forward models are ANNs with inputs to outputs activation, the information
flowing only in one way. Thus, the outputs from all neurons go to following but not
preceding layers, so there are no feedback loops. While such networks can be useful
for simple calculation tasks, they do not qualify as dynamical in the preferred way
described throughout this paper.
Single-layer perceptron
Frank Rosenblatt’s (1958) first attempt at modeling parallel and
distributed, neural-like information processing, the very simple single-layer
perceptron is built as a unique layer of output neurons, to which input values are
All references for such models: Elman 1998, Elman, Bates, Johnson, Karmiloff-Smith, Parisi, and
Plunkett 1997, Gurney 1997, Haykin 1998, Rumelhart 1989, Rumelhart and McClelland 1986,
Smolensky 1989, Stein, Wallace, and Stanford 1998, Sun 1998.
directly fed, through a set of weights. The output values are simply either
‘activated’, or ‘deactivated’, which are given through a rudimentary learning
algorithm named the delta rule. Such networks can only solve linearly separable
Multi-layer perceptron
A network usually possessing at least three layers, input, hidden, and
output, where all the units from one layer are interconnected with each unit in the
subsequent layer. Learning is formalized through the back-propagation algorithm,
which compares output values with the expected values to calculate an errorfunction. This error calculation is in turn used to adjust the weights of the
connections in order to minimize the error of further network computations. This
weight adjustment algorithm is known as gradient descent calculation.
Feedback (recurrent) networks
Recurrent networks are designed to include bi-directional data flow, whereby a
function of the output signal of a system is passed (fed back) to the input. This is
done in order to control the dynamic behavior of the network. Such networks
obviously fit well into our discussion on the complementarity of computation and
Simple recurrent network (SRN)
Designed like feed-forward multi-layer networks, such models also
include ‘context’ units in the input layer. Such units are used to maintain a copy of
the previous values of the hidden units, allowing complex computations involving
sequence prediction, for instance.
Fully recurrent network
A non-layered network where every unit is connected to everyone
else. Some subset of the network’s units also receives external inputs, whereas
another subset performs the opposite task of outputting values outside of the
Hopfield network/Boltzmann machine
Such recurrent networks have symmetrical connections, and exhibit
dynamical properties quite useful for complex calculations. A Boltzmann machine
has the additional feature of involving noisy variables, making it a stochastic
Another example of recurrent networks, cascade-correlation is a
constructive learning algorithm. It starts as a minimal network, consisting only of an
input and an output layer. Minimizing the overall error of the network through
backpropagation, it adds (‘recruits’) at each computational step new hidden units to
the hidden layer, until the network has assimilated its training input vectors. This
allows cascade-correlation networks to learn much faster.
Integrated networks
Committee of machines (CoM)
Tricks of design can be greatly beneficial to connectionism’s
computing endeavors, and integrating many networks together is one such clever
idea. The idea is to have multiple networks sharing the same architecture, but
different initial random weights and input training values, ‘vote’ together on a given
problem. While it doesn’t translate into faster processing, it has the advantage of
greater output stability over its many calculations.
Time-based networks
Whereas recurrent networks offered a first dynamic outlook to computational
processes by including feedback and continuous, simultaneous interactivity,
networks that integrate the timing and latency of processes have an even greater
edge in the race to account for cognitive processes. Such networks can thus be said
to have essential dynamic features, on par with the type of continuous dynamical
models championed by the proponents of the DHCS.
Spiking neural networks (SNN)
Spiking networks propose to model the intrinsic timing of neural
spikes, and spike trains, properties essential to the dynamics of biological neural
networks. Thus are considered the latency of inputs, the all-or-nothing type of event
that is neural activation, and the processes are achieved continuously. Such a model
would probably do well in confronting Piccinini’s arguments in chapter III, section
Adaptive time-delay neural network (ATDNN)
A type of multi-layer, feed-forward or recurrent network, the
ATNN’s architecture has a set of neurons which can ‘store’ their energy level, and
are connected to other neurons. In a standard network, each neuron can be connected
to any number of neurons in the next layer, but they can only have a single
connection to any given neuron. The difference with an ATNN is the use of delayed
weights. That is, a given neuron in a preceding layer can be connected to a neuron in
further layer many times with different weights. With each weight is associated a
time delay, which acts as a memory. Thus the ATNN offers a significant gain in
memory, but also the possibility of changing the delay values during training,
another significant gain in flexibility.
Hybrid networks
Autonomous robotics control, using Continuous-time recurrent neural
networks (CTRNNs) and genetic algorithms (GAs)
Want to make your artificial neural network even more ‘biologically
inspired’? No problem, combine it with yet another type of biocomputing model, a
genetic algorithm. Genetic algorithms are a subset of evolutionary algorithms, which
find solutions to optimization problems through heuristics inspired by biological
phenomena such as inheritance, mutation, natural selection, and recombination. By
combining a CTRNN, which is similar to the time-based networks mentioned above
(with the additional advantage of being capable, in principle, of approximating any
dynamical system), with a genetic algorithm, one can design very sophisticated
computational models to achieve an artificial sensorimotor control system. The
genetic algorithm’s input values are the neural networks’ parameters, and fitness is
measured through the comparative adequacy of outputted motor behaviors.
While the aforementioned list of connectionist models is by no means as
nearly sophisticated as are biological neural mechanisms, they do show how much
progress can be, and has been, achieved in modeling cognitive processes, or for
other purposes, such as developing computing applications and tools. Are
connectionist models computational? They are, in fact, essentially so, and were
always meant to model cognition as informational processes. Some may be
dynamical, in van Gelder et al’s view of dynamics, appealing to the continuous,
time-based coevolution of variables. Ultimately, connectionist models, as far as they
can be used in explanatory endeavors in the study of cognition, are but one type of
models of intelligent processes and behaviors. While we were concerned with the
comparative advantages and limitations of the computational and dynamical
frameworks at large, viz. on all levels of description relevant to cognitive science,
connectionism constitutes but one such level of cognitively related phenomena.
Nevertheless, this examination has only served the purpose of summarily showing
how cognition can be appropriately studied through both computational and
dynamical concepts and methods. The following sections offer a more
comprehensive summary of just how exactly connectionism is both computational
and dynamical, and moreover, constitutes a good candidate as a model supporting a
theory of cognition.
Connectionism as a computational framework. Connectionism is a
mathematical model that departs from symbolic computation, but is nevertheless
built on the latter. The symbolic approach of the GOFAI, mostly concerned with a
literal understanding of Turing’s thesis on a formal manipulation of symbols being
functionally implemented by mechanical means, was a preliminary approach to a
theory of cognition, emphasizing a strong resemblance between symbolic processors
and the linguistic and logico-mathematical competences of natural cognitive agents .
But the machine metaphor was too narrow and restrictive, and a more biologically
inspired model of cognition was needed. Connectionism, with its massively parallel
and distributed architecture of informational processes, emphasizes on an integrated
account of cognition spanning from neurological, psychological, and linguistic
considerations on how mental processes should be modeled, while keeping essential
features of its computational origin. Most importantly, connectionism is still
a functionalist account of cognition,
an information processing model,
based on representations, albeit not strictly symbolic ones,
a model that realizes a formal representation of neural processes,
implemented on a computational architecture,
a good candidate to accommodate higher level symbolic processing as
originally conceived by symbolicists.
Thus, connectionism is a computational theory, because (i) its concepts are still
integral to a functional and informational stance on cognition, while enjoying a
considerable gain in explanatory power in the field of cognitive science (we could
say that connectionism is a conservative extension of a symbolic theory of cognition,
with a much larger scope), and (ii) its models, viz. artificial neural networks, are
computational ones, i.e. they perform calculations by means of an idealized formal
model implemented on a computational machine (a digital computer usually). Those
calculations are vectorial transformations, which are made in the language of linear
algebra. Indeed, connectionist computations usually involve algorithms defining
functions of linear and nonlinear algebra, and dynamical and statistical analyses are
generally used to observe relevant qualitative and quantitative features of such
algorithms, or enhance their computational performance.
Noteworthy is the absence of sensorimotor processes, agency and perception from such an
approach to cognition, which were relegated to cybernetics and early robotics. Likewise, a theory of
animal cognition was but awkwardly possible under such a framework, since formal and semantic
properties of cognition could hardly be attributed to nonhuman animals.
53 I include logical and semantic properties as linguistic ones, for the sake of simplicity.
Connectionism as a subset of dynamical systems. Any connectionist theory
of cognition that pretends to a nontrivial degree of explanatory power must draw
upon dynamical systems theory and dynamical modeling to better describe
informational processes, and the intricate relationships between cognition, body,
context, and environment. While there can be dynamical models of cognition that
deal with nonconnectionist issues, such as behavioral and psychophysiological
phenomena, the opposite is hardly relevant anymore to model cognition, as seen in
the abovementioned feed-forward models. This is due to the intrinsically dynamic
nature of cognitive processes, including the relevant ones for connectionism, namely
the neurobiological processes involved in natural cognition. Neural information
processing occurs in feedback loops, through the reinforcement of synaptic
connections (by means of electrochemical catalysts), or the impoverishment of such
connections (by means of inhibitory electrochemical reactions), and such
connections may in turn be globally cooperative or antagonistic in the activation of
yet other neural processes. Since the architecture of any nontrivial artificial neural
network requires the use of formal concepts and tools drawn from dynamics and
statistical analysis, we can support without further ambiguities the claim that
connectionism is indeed dynamical in nature.
That connectionist models are dynamical is hardly surprising then, and
computationalists have as such never denied it, as seen in chapter III. Thus was the
issue only a matter of emphasis, as dynamicists would have some connectionist
models (the nontrivial, relevant ones in cognitive science, incidentally) belong
exclusively to their view. We have seen that such a claim is untenable, if we
correctly distinguish between the narrow definition of symbolic computation, and
the larger definition of computability. The fact is that connectionism is a
computational theory of cognitive processes, albeit one that is not concerned only
with symbols, and incorporates formal elements drawn from many areas of pure and
Feed-forward models were a first good approximation of just how neural processes might be
designed, but are nowadays relegated to purely instrumental purposes, such as the design of inductive
algorithms for task-specific computational problems.
applies mathematics, both qualitative and quantitative. It so happens that as real
cognitive processes are dynamical, i.e. they happen in time, are complex systems
whose variables fluctuate interdependently, and can not be reduced to simple linear
equations. Any pretender to an accurate account of cognition must thus explain (and
consequently be able to model) the dynamical nature of mental phenomena.
Connectionism as a theory of cognition, and neural networks as models of
cognitive processes. I had intended to present connectionism as but one example of a
theory in cognitive science that is both computational and dynamical for very
specific reasons, namely the popularity and dominance of connectionist accounts of
specific cognitive processes on one hand, but mostly because connectionism was
claimed on both sides of the controversy concerning which account of cognition is
more accurate, between the CHCS and the DHCS. A fair and impartial account of
the benefits of adopting a connectionist theory of cognition must delimitate its
scope: connectionism is concerned with, and only with, cognitive phenomena
occurring at the level of neurological events, and even then, at the one postulated to
be relevant to just the kind of phenomena of interest to cognitive science, namely the
informational level involved in perception, memory, language use, deliberation, et
caetera. So on the one hand, connectionism may take into account neurobiological
processes of a lower level, such as electrochemical dynamics, and indirect
neurobiological events such as synaptic reinforcement through interactions with glial
cells for example, but such lower or indirect levels of biological activity are not the
focus of such a theory. On the other hand, connectionism is scarcely exhaustive in
the modeling of all psychological phenomena, and does not pretend to be able to
explain linguistic, sensorimotor, or social cognition solely through the workings of
neural processes.
That connectionism does not offer a unified and complete account of
cognition is again hardly surprising, as such an account would be quite suspicious
from a mechanistic outlook anyhow, and more so, what counts as cognitive is rather
inclusive, or permissive. Cognitive science is for this very reason an
interdisciplinary endeavor, and connectionism is but one theory among many,
phenomena. Being focused on specific aspects of such complex phenomena is but a
necessity while threading mechanistic explanations.
- Conclusion - Strange bedfellows? Computational and dynamical models in
cognitive science
“11h15, restate my assumptions: 1. Mathematics is the language of nature. 2. Everything around us
can be represented and understood through numbers. 3. If you graph these numbers, patterns
emerge. Therefore: There are patterns everywhere in nature.” - Maximillian Cohen, in the movie Pi
(Darren Aronofsky, 1998)
“Hold on. You have to slow down. You're losing it. You have to take a breath. Listen to yourself.
You're connecting a computer bug I had with a computer bug you might have had and some religious
hogwash. You want to find the number 216 in the world, you will be able to find it everywhere. 216
steps from a mere street corner to your front door. 216 seconds you spend riding on the elevator.
When your mind becomes obsessed with anything, you will filter everything else out and find that
thing everywhere […] But, Max, as soon as you discard scientific rigor you are no longer a
mathematician. You become a numerologist.” - Sol Robeson, in the movie Pi (Darren Aronofsky,
Cleaning up: dealing with some conceptual vagaries in cognitive science
Early on in the course of my argumentation, I have exposed fine distinctions
pertaining to the definitions of concepts like cognition, computation, and that of
representation, concepts that find themselves equivocal in the literature of cognitive
science. I have also argued that part of the conflict between the proponents of both
computational and dynamical models of cognition originated from such conceptual
Quining (some) concepts. In an entertaining take on the problem of qualia in
philosophy and cognitive science, Daniel Dennett (Dennett 1988) advocated a
radical position by pretending to simply eliminate such a problem, or in his own
sarcastic vernacular, by ‘quining’ it, that is, by refusing to deal with a seemingly
important issue on grounds of its not being a real problem in the first place: "quine,
v. To deny resolutely the existence or importance of something real or significant".
While I do not pretend to do the same about concepts relevant to cognitive science, I
do want to adopt a rigorous and strict, positivist-like view on the aforementioned
conceptual vagaries: we simply can not allow ourselves the extravagance of
polysemic references in science, however useful that might turn out to be for
analogical reasoning and scientific revolutions. We thus need necessary and
sufficient criteria for concepts like cognition, computation, and representation, an
issue which has only been covered superficially in the present paper. Many of the
arguments proposed by supporters of the CHCS and the DHCS are based on
conveniently vague and questionable definitions of what such concepts stand for.
For mechanists, there simply isn’t a question of whether cognition extends outside
the brain, or the body, since most of them are strictly interested in design and
functional issues of the internal structures underlying intelligent behavior. Not that it
need be so, but the point is that you can’t accuse someone of neglecting
embeddedness or social cognition on the basis of their choice of a level of enquiry
that focuses on internal processes. Granted, it may turn out (and is a liable
hypothesis) that social and environmental factors have direct consequences in the
shaping of cognitive processes, but the study of neural pathways or syntactical
performance does not entail that one has to include such top-down considerations in
every aspect of their studies. Better start rigorously at the bottom and work your way
up, while nevertheless be wary of external factors that might turn out to be essential
in the understanding of your domain of enquiry. Dynamicists, on the other hand,
seem to often toss away considerations about the underlying mechanisms of
cognition in favor of behavioral and systematic descriptions of what counts as
cognitive to them, but neither can they be guilty of being concerned with such
factors. What both sides are guilty of, if it is the case, is being parsimonious in their
conception of what counts as cognitive, and to what extent one should be concerned
with it. This intransigence towards different intensions of cognition doesn’t indeed
facilitate the debate. Cognitive science doesn’t restrict itself to a theory of behavior,
a theory of brain, or a theory of mental faculties, it aims to integrate such endeavors.
Don Quixote’s take on computation. Likewise, computation has been
mistreated from the very beginning of the discussion, since the dynamicists’ take on
the CHCS was obstinately directed towards one very narrow characterization of
computability, namely symbolic computation, or Turing’s formal account of
computable systems. The formal and empirical components of both frameworks
aren’t equivalent, and some theoretical exemplars of such frameworks (such as
digital computers and Watt’s centrifugal governor) translate poorly as exemplars of
biological cognition. By focusing strictly on symbolicism, and conflating
computation and symbolic processes, supporters of the DHCS did little in
discrediting computational endeavors in cognitive science, struggling with straw
men. Not only did most of the dynamicists’ arsenal of arguments not hold the road in
convincing us of potential shortcomings in computability theory with regards to the
study of cognition, it also marks an embarrassing anachronism in that such an issue
was already being debated decades before, in the discussion on the relative
advantages of connectionism versus symbolicism. More so, connectionists always
explicitly endorsed a commitment to the use of dynamics in modeling the cognitive
processes of concern. So much for a revolutionary stance. On the issue of
representations, we have seen that while such a problem did indeed pervade the
arguments supporting both stances towards cognition, it was necessary to posit them
in mechanistic explanations. By simply adopting a minimalist concept of
representation, we can then only concern ourselves on matters of representational
format, which seems to indeed play a significant role into the framing of accurate
explanations of cognitive phenomena.
Everything you’ve always wanted to know about maths but never dared to
ask. Another important issue concerns the mathematics involved in both conceptual
repertoires. Time and again, we have seen that cognitivists clash on formal issues
supporting their respective position. Some of those arguments are essential to the
debate on the relative advantages of such and such model of cognition, but find
themselves squarely regressing back to foundational issues in theoretical
mathematics. Continuity and discreteness, integers and reals, algorithms and
computable functions, isomorphism and effective representations, and the
formalisation of time are many such grounds of dispute. It comes down to a petitio
principii on the possibility of adequately representing cognitive phenomena through
a given mathematical model, and connectionism has been shown to develop in
heuristic avenues, to mention only one such model.
What about the symbolic view? So much has been said against an essentially
symbolic approach to cognition, but is there still a place for it in cognitive science?
Indeed, since no one would reasonably doubt the fundamentally symbolic nature of
some aspects of cognition, namely the acquisition and use of language, high-level
processes such as deliberation and logical reasoning, parts of mnemonic and
perceptual processes, social cognition, et caetera. Regardless of one’s semiotic take
on the role of icons, indices, and symbols, deeply representation-laden topics in
cognitive science are as important as concerns about psychophysics and neurological
studies. Since today’s cognitive science has a broader conception of what constitutes
cognitive phenomena, it should be considered as a conservative extension of older
definitions of what is involved in the realm of the mental.
Of evidence and use
If the recipe works, why change it? Dynamicism, flawed logical reasoning,
and the burden of proof. van Gelder et al profess the openness of the dynamical
hypothesis, viz. that only future research and evidence in cognitive will prove the
righteousness of adopting a dynamic stance towards cognition. On the other hand,
dynamicicts accuse computational models of failing to meet the standards of
cognitive science, two rather dubiously prejudiced takes on the same ground of
argumentation. As Glymour most acerbically states it:
In almost all of [cognitive science’s] work, an essential assumption is that
cognition depends on computable biological processes. And here is where the
radical character of van Gelder's thesis begins to come home: van Gelder's
thesis is that the thousands of papers on the computational biology of nervous
systems relevant to cognition are scientific junk, pursued under some
fundamental metaphysical error. […] make no mistake about the boldness (or,
to be less generous, the crankiness) of his claim. What should replace a
century's scientific investigation of cognitive physiology and its computational
aspects is ‘dynamical systems’. (Glymour 1997, pp. 4-5)
Likewise, on the issue of cognitive systems possibly belonging to the class of
uncomputable dynamical systems, the burden of proof should be on the dynamicists,
for the successes so far gathered in computation-based cognitive science don’t seem
to stress any cognitivist’s possible angst towards such an issue. In fact, since most
radical dynamicists rely on the promotion of dynamics as a ‘larger’ framework,
encompassing even computable systems, and then taking credit in that cognitive
processes might be uncomputable but nevertheless still always dynamical processes.
Such a line of thinking has more to do with rhetoric than a sound, reasonable
position. Promoting the DH as an open empirical investigation project is reasonable,
as is bearing in mind some pragmatical concerns about the evolution of empirical
enquiry in cognitive science anyway. But doing so, far from invalidating the
computational stance towards cognition, actually bestows additional merit to
computability theory, having progressed that far, and through controversies spanning
more than half a century already. A more lenient pragmatical position would be to
say, as Bechtel does, that it has proven to be useful to adopt both views towards
cognitive processes, and that their complementarity promises even more
sophisticated means of explanation in the study of cognition.
(Not so) strange bedfellows?
The best of both worlds. After behaviorism’s downfall around the middle of
the twentieth century, mentalistic explanations, of which the CHCS is an offspring,
gained popularity because they offered more and more accurate and useful
depictions of cognitive phenomena. Obviously, the paradigmatic shift was to
eventually reintegrate what it had tossed away, by means of a renewed interest in
matters of cognitive performances in given contexts and environments. Dynamics
offer one such opportunity of explaining cognition in a systematic, embodied and
embedded way. The mechanists that are computationalists posit actual functional
features and components in cognitive systems, bearing a strong realism in their type
of explanation, while the more nomologically inclined dynamicists have a skeptical
outlook to their humean empiricist take on causal explanations. It would seem that
this whole debate might revolve around the philosophical problem of causal
relations, with optimistic realists clashing with noncommittal empiricists. Indeed, the
whole debate might just be formulated as following: is it preferable to be
noncommittal towards causal explanations such as the ones involved in mechanistic
explanations, by simply sticking to a correlational stance, for a wide array of
Or maybe we should shed some dennettian light on the debate, by drawing
on an analogy derived from Dennett’s (Dennett 1987) take on explanations in the
philosophy of mind and cognitive science. Dennett’s classic argument consisted of a
harmonious division in three ‘stances’ towards cognitive phenomena, an intentional
stance, a design stance, and a physical stance. Now, the intentional stance appeals to
mentalistic explanations, ones involving the intentional vernacular of propositional
attitudes, such as the attribution of beliefs, desires, and intentions. Unlikely to be
dismissed on pragmatical grounds, such a level of explanation of cognition is
nevertheless not the preferred means of scientific enquiry in cognitive science. The
design stance, on the other hand, posits causal relations and functional roles much in
the same way that the aforementioned mechanistic type of explanation involved in
the CHCS does, and it was indeed the motivation of Dennett to segregate it in view
of the role of computational explanations. The third stance, that of physicalism, is
interested in the properties of the substrate of cognition, and appeals to the covering
laws of physics and the regularities observed in the life sciences. Obviously, the
design and physical stances translate well into our ‘competing’ schemes that are
computational and dynamical explanations. But just as Dennett, some twenty years
ago, advocated the complementarity of all such stances both on conceptual and
pragmatical grounds, so do I for computability and dynamics, in the wake of
Bechtel. All such distinctions, from Dennett to Bechtel, were meant to emphasize
the various contributions championed in the name of cognition, and their
complementarity is also consequent of just how far reaching the concept of cognition
can be, as discussed above. Cognitive science is said to be interdisciplinary precisely
for such reasons, bearing on many levels of description, and drawing on concepts
and methodologies from all concerned areas of research.
- Finale -
To summarize our concluding position regarding the comparative advantages
and limitations of the computational and dynamical stances in the study of cognition,
we can say firstly that much of the dispute between proponents of computationalism
and dynamicism is based on conceptual confusions of the nature of what counts as
cognitive, on the nature and role of computation, and also on the nature, role, and
format of representations. Secondly, that the complementarity of computational and
dynamical models of cognition has been established by virtue of the type of
explanation involved through such characterization, and as such, no exclusive claim
on cognition can be made by their respective proponents. Such explanations have
been segregated as mechanistic and nomological types of analysis, and find their
place in the study of cognition by means of integration. Thirdly, the aforementioned
integration has a theoretical exemplar in the form of the applied mathematical and
informational theory that is connectionism. While we do not claim that such a
framework is an exclusive means to model cognition, our endeavors were to show
the possibility of drawing upon all available formal and empirical tools and evidence
to frame cognitive phenomena in a heuristic, informative manner. Fourthly, the
functional account of connectionism can be reciprocally constrained and informed
by and with psychophysical evidence, as behavior and internal processing must be
integrated into a coherent framework, should there be any claim of an
interdisciplinary account of cognition. Similarly, mechanistic and nomological
concepts and methods should be employed in concert even where behavior and
psychophysical enquiries are concerned, as computational and dynamical tools of
analysis are not restricted to the study of internal processes.
- Appendices -
- Appendix I - Definitions of computation
The class of computable functions is equivalent to the class of functions
defined by the following models:
- recursive functions
Class of functions from natural numbers to natural numbers.
Axioms and operators:
(i) The constant function 0 is primitive recursive;
(ii) The successor function S, which takes one argument and returns the succeeding
number as given by the Peano postulates, is primitive recursive;
(iii) The projection functions Pin, which take n arguments and return their ith
argument, are primitive recursive;
(iv) Composition: Given f, a k-ary primitive recursive function, and k l-ary primitive
recursive functions g0,...,gk-1, the composition of f with g0,...,gk-1, i.e. the function
h(x0,...,xl-1) = f(g0(x0,...,xl-1),...,gk-1(x 0,...,xl-1)), is primitive recursive;
(v) Primitive recursion: Given f a k-ary primitive recursive function and g a (k+2)ary primitive recursive function, the (k+1)-ary function defined as the primitive
recursion of f and g, i.e. the function h where h(0,x0,...,xk-1) = f(x0,...,xk-1) and
h(S(n),x0,...,xk-1) = g(h(n,x0,...,xk-1),n,x0,...,x k-1), is primitive recursive;
(vi) Extension to partial functions: f is many-to-one, or functional: if x f y and x f z,
then y = z. i.e., many input values can be related to one output value, but one input
value cannot be related to many output values. The function f need not be total, or
entire (for all x in X, there exists a y in Y such that x f y (x is f-related to y), i.e. for
each input value, there is at least one output value in Y);
References for computable functions equivalents and algorithms: J. L. Hein (1996), and R. G.
Taylor (1998).
(vii) Unbounded search operator: If f(x,z1,z2,...,zn) is a partial function on the natural
numbers with n+1 arguments x, z1,...,zn, then the function μx f is the partial function
with arguments z1,...,zn that returns the least x such that f(0,z1,z2,...,zn), f(1,z1,z2,...,zn),
..., f(x,z1,z2,...,zn) are all defined and f(x,z1,z2,...,zn) = 0, if such an x exists; if no such x
exists, then μx f is not defined for the particular arguments z1,...,zn.
- lambda calculus
A formal system designed to investigate the definition and applications of
functions, as well as the concept of recursion. The lambda calculus consists of a
single transformation rule (variable substitution) and a single function definition
Axioms and operators:
(i) Composed of a countably infinite set of identifiers, for example, {a, b, c, ..., x, y,
z, x1, x2, ...};
(ii) The set of all lambda expressions can be described by the following context-free
1) <expr> ::= <identifier>
2) <expr> ::= (λ <identifier> . <expr>)
3) <expr> ::= (<expr> <expr>)
The first two rules generate functions, while the third describes the application of a
function to an argument. Usually the brackets for lambda abstraction (rule 2) and
function application (rule 3) are omitted if there is no ambiguity under the
assumptions that (1) function application is left-associative, and (2) a lambda binds
to the entire expression following it. For example, the expression ((λ x. (x x)) (λ y.
y)) can be simply written as (λ x. x x) λ y.y;
(iii) Lambda expressions such as λ x. (x y) do not define a function because the
occurrence of the variable y is free, i.e., it is not bound by any λ in the expression.
The binding of occurrences of variables is (with induction upon the structure of the
lambda expression) defined by the following rules:
1) In an expression of the form V where V is a variable this V is the single free
2) In an expression of the form λ V. E the free occurrences are the free
occurrences in E except those of V. In this case the occurrences of V in E are
said to be bound by the λ before V.
3) In an expression of the form (E E' ) the free occurrences are the free
occurrences in E and E' ;
(iv) Over the set of lambda expressions an equivalence relation (here denoted as ==)
is defined that captures the intuition that two expressions denote the same function.
This equivalence relation is defined by the so-called alpha-conversion rule (v) and
the beta-reduction rule (vi);
(v) alpha-conversion rule (expresses the idea that the names of the bound variables
are unimportant): if V and W are variables, E is a lambda expression, and E[V/W]
means the expression E with every free occurrence of V in E replaced with W, then λ
V. E == λ W. E[V/W];
(vi) beta-reduction rule (expresses the idea of function application): ((λ V. E ) E' ) ==
E [V/E' ] if all free occurrences in E' remain free in E [V/E' ]. The relation == is then
defined as the smallest equivalence relation that satisfies these two rules;
(vii) Eta-conversion rule (expresses the idea of extensionality): two functions are the
same iff they give the same result for all arguments. Eta-conversion converts
between λ x . f x and f, whenever x does not appear free in f.
The class of computable functions is also definable as algorithms calculable
- Markov algorithms
A string rewriting system that uses grammar-like rules to operate on strings
of symbols.
Vocabulary and operations:
(elements of the Markov algorithm)
(i) A vocabulary composed of symbols/strings of symbols, and
(ii) grammatical rules;
(iii) Check the rules in order from top to bottom to see whether any of the strings to
the left of the arrow can be found in the symbol string;
(iv) If none are found, stop executing the algorithm;
(v) If one or more is found, replace the leftmost matching text in the Symbol string
with the text to the right of the arrow in the first corresponding Rule;
(vi) Return to step (iii) of operations and iterate.
- register machines
An abstract machine used to study decision problems. Also called counter
machines, Minsky machines, or program machines.
Vocabulary and operations:
(i) A register machine consists of a finite set of registers r1 ... rn, each of which can
hold a non-negative integer, and
(ii) a finite list of instructions I1 ... Im. Each instruction can only be either:
INC (j, k) — increment the value of rj by 1, then jump to instruction Ik;
DEC (j, k, z) — check if the value of rj is zero. If so, jump to instruction Iz;
otherwise, decrement rj by 1 and jump to Ik;
HALT — halts the computation.
- Post systems
A deterministic finite automaton with a queue. There is no separate input
(mechanical description)
(i) At the start of the computation, the input string x is loaded on the queue. The
input string is followed by a special symbol Zo. At the start of the computation, the
contents of the queue are xZo. The first symbol of x is at the front of the queue and
Zo is at the end of the queue;
(ii) A transition of a Post machine depends on the symbol at the front of the queue
and on the state. Each transition will delete the symbol at the front of the queue. A
transition has two components: the next state and the string to be added at the end of
the queue;
(iii) This string can be the empty string.
- Turing machines
An abstract machine introduced by Turing to give a mathematically precise
definition of an algorithm.
(mechanical description)
(i) A Turing machine consists of:
1) A tape which is divided into cells, one next to the other. Each cell contains a
symbol from some finite alphabet. The alphabet contains a special blank
symbol (here written as '0') and one or more other symbols. The tape is
assumed to be arbitrarily extendible to the left and to the right, i.e., the
Turing machine is always supplied with as much tape as it needs for its
computation. Cells that have not been written to before are assumed to be
filled with the blank symbol;
2) A head that can read and write symbols on the tape and move left and right;
3) A state register that stores the state of the Turing machine. The number of
different states is always finite and there is one special start state with which
the state register is initialized;
4) An action table (or transition function) that tells the machine what symbol to
write, how to move the head ('L' for one step left, and 'R' for one step right)
and what its new state will be, given the symbol it has just read on the tape
and the state it is currently in. If there is no entry in the table for the current
combination of symbol and state then the machine will halt.
(ii) Note that every part of the machine is finite, but it is the potentially unlimited
amount of tape that gives it an unbounded amount of storage space.
(formal definition)
(iii) A (one-tape) Turing machine is a 7-tuple M = (Q,Γ,Σ,s,b,F,δ), where
Q is a finite set of states ;
Γ is a finite set of the tape alphabet ;
Σ is a finite set of the input alphabet (Σ ⊆ Γ) ;
s ∈ Q is the initial state ;
b is the blank symbol (b ∈ Γ \ Σ) ;
F ⊆ Q is the set of final or accepting states ;
(for a one-tape Turing machine) δ : Q X Γ → Q X Γ X {L, R} is a partial
function called the transition function, where L is left shift, R is right shift, or
(for a k-tape Turing machine) δ : Q X Γk → Q X (Γ X {L, R, S})k is a partial
function called the transition function, where L is left shift, R is right shift, S
is no shift.
- Appendix II - Computational and dynamical models of low-level cognitive
What follows are formal representations of, respectively, an inverse
kinematics problem from a computational perspective (related to figures 8, 9, 10,
and 11 in chapter II), and a MDS (mathematical dynamical system) of the ‘A-not-B
error’ task (figures 13 and 15 in chapter II).56
Partial code for the inverse kinematics problem: (a) gradient by
measurement, (b) gradient by calculation, (c) alternative (faster) gradient following,
(d) defining a target through a vector field. One can think of a simple neural
network that would implement such algorithms rather easily.
function Calc_Distance(angle_A, angle_B)
work out the tip position for joint A = angle_A and joint B = angle_B
return distance from calculated tip position to target
end function
dist = Calc_Distance(a, b)
while (dist > 0.1)
gradient_a = Calc_Distance(a+1, b) - Calc_Distance(a-1, b)
gradient_b = Calc_Distance(a, b+1) - Calc_Distance(a, b-1)
a -= gradient_a
b -= gradient_b
dist = Calc_Distance(a, b)
for each joint
if 3D: axis = axis of rotation for this joint
if 2D:
axis = (0, 0, 1)
ToTip = tip - joint_centre
ToTarget = target - tip
movement_vector = crossproduct(ToTip, axis)
gradient = dotproduct(movement_vector, ToTarget)
end loop
dist = Calc_Distance(a, b)
are,, and Thelen, Schöner, Scheier and Smith
old_gradient_a = 0
old_gradient_b = 0
while (dist > 0.1)
gradient_a = Calc_Distance(a+1, b) - Calc_Distance(a-1, b)
gradient_b = Calc_Distance(a, b+1) - Calc_Distance(a, b-1)
have we gone past it?
if sign(old_gradient_a) != sign(gradient_a) then
a -= speeda * old_gradient_a / (gradient_a-old_gradient_a)
speeda = 0
speeda += ga
if sign(old_gradient_b) != sign(gradient_b) then
b -= speeda * old_gradient_b / (gradient_b-old_gradient_b)
speedb = 0
speedb += gb
a -= speed_a
b -= speed_b
dist = Calc_Distance(a, b)
constant POINT = 1
constant PLANE = 2
constant RING = 3
structure TARGET
integer Target_Type
vector centre
vector axis
number size
end structure
function to_target(TARGET T, vector Tip_Position)
if T.Target_Type = POINT
v = T.centre - Tip_Position
return v
end if
if T.Target_Type = PLANE
p = T.centre - Tip_Position
v = T.axis * dotproduct(p, T.axis)
return v
end if
if T.Target_Type = VECTOR_FIELD
return vector at this Tip_Position
end if
end function
Algebraic characterizations of the MDS for the ‘A-not-B error’ task (see
figures 13 and 15): equation (i) is the dynamic field of the ‘A-not-B error’ task when
inputs are added together, (ii) time scale parameter, (iii) interactions within the
dynamic field (cooperation), (iv) interaction kernel of cooperation function, (v)
threshold function of cooperation function, (vi) isolated cooperation function, (vii)
motor field evolution function, including time scale, cooperativity, inertia, and
sensory inputs, (viii) overall field dynamics (precedent function coupled with
Gaussian noise), (ix) input sources function for motor planning field dynamics, (x)
task input specification function for ‘A-not-B error’ task, (xi) specific input function
during time interval T, (xii) memory field dynamics (another bias of the dynamic
field for the ‘A-not-B error’ task).
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