**Marco
Giunti**

**BEYOND
COMPUTATIONALISM**

In Cottrell, Garrison W., ed. (1996).

Proceedings of the 18th annual conference of the Cognitive Science Society, 71-75. Mahwah NJ: Lawrence Erlbaum Associates Publishers.

**Abstract**

By *computationalism*
in cognitive science I mean the view that cognition essentially
is a matter of the computations that a cognitive system performs
in certain situations. The main thesis I am going to defend is
that computationalism is only consistent with symbolic modeling
or, more generally, with any other type of computational modeling.
In particular, those scientific explanations of cognition which
are based on *(i)* an important class of connectionist
models or *(ii)* nonconnectionist continuous models cannot
be computational, for these models are not the kind of system
which can perform computations in the sense of standard
computation theory. Arguing for this negative conclusion requires
a formal explication of the intuitive notion of computational
system. Thus, if my thesis is correct, we are left with the
following alternative. Either we construe computationalism by
explicitly referring to some nonstandard notion of computation,
or we simply abandon the idea that computationalism be a basic
hypothesis shared by all current research in cognitive science. I
will finally suggest that a different hypothesis, *dynamicism*,
may represent a viable alternative to computationalism. According
to it, cognition essentially is a matter of the state evolutions
that a cognitive system undergoes in certain situations.

**Introduction**

By *computationalism*
in cognitive science I mean the view that cognition essentially
is a matter of the computations that a cognitive system performs
in certain situations. The main goal of this paper is to assess
whether this view may represent a basic hypothesis shared by the
three current approaches to cognition: the symbolic (or classic)
approach, connectionism, and nonconnectionist dynamics.

If we look at the models actually used in cognitive science, we see that a different type of model corresponds to each approach. The symbolic approach (Newell and Simon, 1972; Newell, 1980; Pylyshyn, 1984; Johnson Laird, 1988) employs symbolic processors as models. As a first approximation, we may take a symbolic processor to be any device that operates effective transformations of appropriately defined symbol structures. The connectionist approach (Rumelhart and McClelland, 1986), on the other hand, employs connectionist networks, while nonconnectionist dynamicists use other kinds of continuous systems specified by differential (or difference) equations. Nonconnectionist researchers favoring a dynamical perspective are active in many fields. For examples see Port and van Gelder (1995).

The main thesis I am
going to defend is that computationalism is only consistent with
symbolic modeling or, more generally, with any other type of
computational modeling. In particular, those scientific
explanations of cognition which are based on *(i)* an
important class of connectionist models or *(ii)*
nonconnectionist continuous models cannot be computational, for
these models are not the kind of system which can perform
computations in the sense of standard computation theory.

The thesis that
computationalism is only consistent with computational modeling
is empty unless one gives a sufficiently precise characterization
of what a *computational model* of a cognitive system is. By
this term, I mean any computational system that describes (or, at
least, is intended to describe) some cognitive aspect of the
cognitive system. Intuitively, by the term *computational
system* I refer to any device of the kind studied by standard
computation theory. Thus, for example, Turing machines, register
machines, and finite state automata are three different types of
computational systems. By contrast, socalled analog computers
are not computational systems. I will propose later a formal
explication of this intuitive notion of a computational system.

Thus, if my thesis is
correct, we are left with the following alternative. Either we
construe computationalism by explicitly referring to some
nonstandard notion of computation, or we simply abandon the idea
that computationalism be a basic hypothesis shared by *all*
current research in cognitive science. In the last section of
this paper, I will also suggest that a different hypothesis, *dynamicism*,
may represent a viable alternative to computationalism. According
to it, cognition essentially is a matter of the state evolutions
that a cognitive system undergoes in certain situations.

**The Argument**

The main thesis of this
paper is that computationalism is only consistent with symbolic
modeling or, more generally, with any other type of computational
modeling. The argument I am going to propose is based on two
premises. The first one affirms that all models currently
employed in cognitive science are *mathematical dynamical
systems*. The second premise, on the other hand, affirms that
a computation (in the sense of standard computation theory) can
only be performed by that special type of mathematical dynamical
system which I have called a *computational system*. Having
established these two premises, I will then show that *(a)*
an important class of connectionist models, and *(b)*
nonconnectionist continuous models are not computational systems.
Hence, these models cannot perform computations in the standard
sense. But then, if our scientific explanations of cognition are
based on these models, we cannot maintain that cognition is,
essentially, a matter of the computations performed by the
cognitive system which these models are intended to describe. On
the other hand, *(c)* all symbolic models are computational
systems. Therefore, computationalism is only consistent with
symbolic modeling or, more generally, with any other approach
which employs computational systems as models of cognition.

**The First Premise**

The first premise of my
argument is that all models currently employed in cognitive
science are mathematical dynamical systems. A *mathematical *dynamical
system is an abstract mathematical structure that can be used to
describe the change of a real system as an evolution through a
series of states. If the evolution of the real system is
deterministic, that is, if the state at any future time is
determined by the state at the present time, then the abstract
mathematical structure consists of three elements. The first
element is a set *T* that represents time. *T* may be
either the reals, the rationals, the integers, or the nonnegative
portions of these structures. Depending on the choice of *T*,
then, time is represented as continuous, dense, or discrete. The
second element is a nonempty set *M* that represents all
possible states through which the system can evolve; *M* is
called the *state space* of the system. The third element is
a set of functions *{g*^{t}*}* that
tells us the state of the system at any instant *t* provided
that we know the initial state; each function in *{g*^{t}*}*
is called a *state transition* of the system. For example,
if the initial state is *x **Î** M*, the state at time *t* is
given by *g*^{t}*(x)*, the state at time
*u > t* is given by *g*^{u}*(x)*,
*etc*. The functions in the set *{g*^{t}*}*
must only satisfy two conditions. First, the function *g*^{0}
must take each state to itself and, second, the composition of
any two functions *g*^{t} and *g*^{w}
must be equal to the function *g*^{t+w}.

An important subclass of
the mathematical dynamical systems is that of all systems with
discrete time. Any such system is called a cascade. More
precisely, a mathematical dynamical system *<T, M, {g*^{t}*}>*
is a *cascade* just in case *T* is equal to the
nonnegative integers (or to the integers).

As mentioned, the models
currently employed in cognitive science can basically be
classified into three different types: *(1)* symbolic
processors, *(2)* neural networks, and *(3)* other
continuous systems specified by differential (or difference)
equations. That a system specified by differential or difference
equations is a mathematical dynamical system is obvious, for this
concept is expressly designed to describe this class of systems
in abstract terms. That a neural network is a mathematical
dynamical system is also not difficult to show. A complete state
of the system can in fact be identified with the activation
levels of all the units in the network, and the set of state
transitions, on the other hand, is determined by the differential
(or difference) equations that specify how each unit is updated.
To show that all symbolic processors are mathematical dynamical
systems is a bit more complicated. The argumentative strategy I
prefer considers first a special class of symbolic processors (such
as Turing machines, or monogenic production systems, *etc*.)
and it then shows that the systems of this special type are
mathematical dynamical systems with discrete time, *i.e.*,
cascades. Given the strong similarities between different types
of symbolic processors, it is then not difficult to see how the
argument given for one type could be modified to fit any other
type (Giunti, 1992, 1996). We may thus conclude that all models
currently employed in cognitive science are mathematical
dynamical systems.

**The Second Premise**

The second premise of my
argument affirms that a computation (in the sense of standard
computation theory) can only be performed by a *computational
system*. Intuitively, by this term I refer to any device of
the kind studied by standard computation theory (*e.g.*,
Turing machines, register machines, cellular automata, *etc*.)
I call any computation performed by any such device a *standard
computation*. According to this terminology, then, my second
premise affirms that *a standard computation can only be
performed by a computational system*. It is thus clear that I
in fact take this premise to be true by definition.

Someone might object
that, given my definitions, my second premise is not only true,
but also trivial. According to my imaginary critic, the important
question is not whether a standard computation can be performed
by a noncomputational system but, rather, whether standard
computational methods are sufficient to accurately describe the
behavior of *all* models employed in cognitive science (be
these models computational or not). I will give an answer to this
kind of objection later. Before I can proceed with my argument,
however, I need to give a formal explication of the intuitive
concept of a computational system.

**A Formal Definition
of a Computational System**

To this extent, let us
first of all consider the mechanisms studied by standard
computation theory and ask *(i)* what type of system they
are, and *(ii)* what specific feature distinguishes these
mechanisms from other systems of the same type.

As mentioned, standard
computation theory studies many different kinds of abstract
systems. A basic property that is shared by all these mechanisms
is that they are *mathematical dynamical systems with discrete
time*, that is *cascades*. However, standard computation
theory does not study all cascades. The specific feature that
distinguishes computational systems from other mathematical
dynamical systems with discrete time is that a computational
system *can always be described in an effective way*.
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 mechanism. However, since we want to
arrive at a formal definition of a computational system, we
cannot limit ourselves to this intuitive characterization. Rather,
we must try to put it in a precise form.

Since I have informally
characterized a computational system as a cascade that can be
effectively described, let us ask first what a *description*
of a cascade is. If we take a structuralist viewpoint, this
question has a precise answer. A description (or a representation)
of a cascade consists of a second cascade *isomorphic* to it
where, by definition, a cascade *MDS*_{1}*
= <T, M*_{1}*, {h*^{t}*}>*
is isomorphic to a given cascade *MDS = <T, M, {g*^{t}*}>*
just in case there is a bijection *f: M **®** M*_{1} such that,
for any *t **Î** T* and any *x **Î** M*, *f(g*^{t}*(x))
= h*^{t}*(f(x))*.

In the second place, let
us ask what an *effective* description of a cascade is.
Since I have identified a description of a cascade *MDS = <T,
M, {g*^{t}*}>* with a second cascade *MDS*_{1}*
= <T, M*_{1}*, {h*^{t}*}>*
isomorphic to *MDS*, an effective description of *MDS*
will be an *effective cascade* *MDS*_{1}
isomorphic to *MDS*. The problem thus reduces to an analysis
of the concept of an effective cascade. Now, it is natural to
analyze this concept in terms of two conditions: *(a)* there
is an effective procedure for recognizing the states of the
system or, in other words, the state space *M*_{1}
is a *decidable* set; *(b)* each state transition
function *h*^{t} is effective or *computable*.
As it is well known, these two conditions can be made precise in
several ways which turn out to be equivalent. The one I prefer is
by means of the concept of Turing computability. If we choose
this approach, we will then require that an effective cascade
satisfy: *(a')* the state space *M*_{1}
is a subset of the set *P(A)* of all finite strings built
out of some finite alphabet *A*, and there is a Turing
machine that decides whether an arbitrary finite string is member
of *M*_{1}; *(b')* for any state
transition function *h*^{t}, there is a
Turing machine that computes *h*^{t}.

Finally, we are in the position to formally define a computational system. The following definition expresses in a precise way the informal characterization of a computational system as a cascade that can be effectively described.

**DEFINITION** (computational
system)

*MDS* is a
computational system iff *MDS = <T, M, {g*^{t}*}>*
is a cascade, and there is a second cascade *MDS*_{1}*
= <T, M*_{1}*, {h*^{t}*}>*
such that *MDS*_{1} is isomorphic to *MDS*
and

- if
*P(A)*is the set of all finite strings built out of some finite alphabet*A*,*M*_{1}*Í**P(A)*and there is a Turing machine that decides whether an arbitrary finite string is member of*M*_{1}; - for any
*t**Î**T*, there is a Turing machine that computes*h*^{t}.

It is tedious but not
difficult to show that all systems that have been actually
studied by standard computation theory (Turing machines, register
machines, monogenic production systems, cellular automata, *etc*.)
satisfy the definition (Giunti, 1992, 1996).

**Two Sufficient
Conditions for a System not to Be Computational**

The definition of a
computational system allows us to deduce two sufficient
conditions for a mathematical dynamical system not to be
computational. Namely, a mathematical dynamical system *MDS =
<T, M, {g*^{t}*}>* is not
computational if it is continuous in either time or state space
or, more precisely, if either *(i)* its time set *T* is
the set of the (nonnegative) real numbers, or *(ii)* its
state space *M* is not denumerable.

An immediate consequence
of condition *(ii)* is that *any finite neural network
whose units have continuous activation levels is not a
computational system*. Also note that *the same conclusion
holds for any continuous system specified by differential (or
difference) equations*. Since all these systems are continuous
(in time or state space), none of them is computational.

**Summing up the
Argument**

We have thus seen that *(I)*
all models currently employed in cognitive science are
mathematical dynamical systems; *(II)* a standard
computation can only be performed by a computational system; *(III)*
any finite neural network whose units have continuous activation
levels or, more generally, any continuous system specified by
differential (or difference) equations is not a computational
system. Hence, all connectionist models in this class and all
nonconnectionist continuous models cannot perform standard
computations. But then, if our scientific explanations of
cognition are based on these models, we cannot maintain that
cognition is, essentially, a matter of the standard computations
performed by the cognitive system which these models are intended
to describe. On the other hand, it is obvious that *(IV)*
all symbolic models are computational systems. Therefore,
computationalism is only consistent with symbolic modeling or,
more generally, with any other approach which employs
computational systems as models of cognition.

A word of caution is needed here. Somebody might object to this conclusion in the following way. It is well known that the behavior of virtually all continuous systems considered by physics can be simulated, to an arbitrary degree of precision, by a computational system, even though these systems are not computational systems themselves (Kreisel, 1974). Why should the continuous systems considered in cognitive science be different in this respect? As long as the behavior of a continuous model of a cognitive system can be simulated (to an arbitrary degree of precision) by a computational system, there is nothing, in the model, which is beyond the reach of standard computational methods. Therefore, it is false that computationalism is only consistent with computational modeling.

This objection is
confused because it blurs the distinction between the standard
computations *performed* by a system, and the *simulation*
of its behavior by means of standard computations performed by a
different system. In the first place, this distinction is
essential for the formulation of the computational hypothesis
itself. If computationalism is intended as a very general
hypothesis that indicates the appropriate style of explanation of
cognitive phenomena (namely, a computational style), it is
crucial to affirm that cognition depends on the standard
computations *performed* by the cognitive system we are
studying, for it is precisely by understanding the particular
nature of these computations that we can produce a detailed
explanation of cognition. But then, in formulating the
computational hypothesis, we are in fact implicitly assuming that
the cognitive system *is* a computational system, we are not
just claiming that its behavior can be simulated by a
computational system. In the second place, I have argued that any
continuous model is not a computational system, and thus it *cannot
perform* standard computations. But then, if our scientific
explanations of cognition are based on continuous models, we
cannot maintain that cognition is, essentially, a matter of the
standard computations performed by the cognitive system which
these models are intended to describe. Therefore,
computationalism is indeed inconsistent with continuous modeling.

**Concluding Remarks**

My argument shows that,
unless we construe computationalism by explicitly referring to
some nonstandard notion of computation, we cannot maintain that
computationalism is a basic hypothesis shared by *all*
current research in cognitive science. In view of this fact,
however, we should consider at least two further questions. First,
what kind of nonstandard notion of computation would be needed
for an adequate generalization of the computational hypothesis?
And, second, is there some other hypothesis that might play this
unifying role as well?

As regards the first
question, I will limit myself to just one preliminary remark, for
a critical discussion is beyond the scope of this paper. Even
within these limits, however, it seems quite reasonable to
maintain that a generalized version of the computational
hypothesis should be based on a theory of computation that *(i)*
applies to continuous systems and standard computational systems
as well; *(ii)* in the special case of standard
computational systems, this more general theory reduces to the
standard one, and thus *(iii)* all the standard
computability results should turn out to be special cases of the
more general theory. I leave it up for further discussion whether
these conditions are indeed well chosen, or whether they are in
fact satisfied by some theories which intend to generalize
various aspects of standard computation theory (Blum, Shub, and
Smale, 1989; Friedman, 1971; Shepherdson, 1975, 1985, 1988;
Montague, 1962).

As for the second
question, we have seen that all models currently employed in
cognitive science are mathematical dynamical systems. Furthermore,
in general, a mathematical dynamical system changes its behavior
according to the particular state evolution that the system
undergoes. But then, if our aim is to model cognition by means of
appropriate mathematical dynamical systems, we may very well
claim that *cognition is, essentially, a matter of the
particular state evolutions that a cognitive system undergoes in
certain situations*. I call this hypothesis *dynamicism*.
For two, quite different, articulations and defenses of
dynamicism see van Gelder and Port (1995) and Giunti (1995, 1996).

It is thus clear that dynamicism, unlike (standard) computationalism, is consistent with symbolic, connectionist, and nonconnectionist continuous modeling as well. Therefore, all research on cognition might end up sharing this new hypothesis, independently of the type of model employed. The question remains, however, whether this possibility will really obtain. I believe that the answer to this question depends on whether the explicit assumption of a dynamical perspective can sharply enhance our understanding of cognition. This issue, however, will ultimately be settled by detailed empirical investigation, not by abstract argument.

On the other hand, it is
also quite obvious that the dynamical hypothesis, as stated above,
only gives us an extremely general methodological indication.
Essentially, it only tells us that cognition can be explained by
focusing on the class of the dynamical models of a cognitive
system, where a dynamical model is *any* mathematical
dynamical system that describes some cognitive aspect of the
cognitive system. Now, a standard objection against this version
of dynamicism is that this methodological indication is so
general as to be virtually empty. Unfortunately, a detailed
rebuttal to this charge goes beyond the scope of this paper.
Therefore, I must limit myself to briefly outline the three
defenses that have been adopted by the proponents of the
dynamical approach.

The first line of
defense points out that dynamicism, just like computationalism,
has in fact two aspects. The first one is the specification of a
particular class of models (dynamical *vs.* computational
models), while the second is the proposal of a conceptual
framework (dynamical systems theory *vs.* computation theory)
that should be used in the study of these models. Therefore, if
we also consider this second aspect, we see that the mathematical
tools of dynamical systems theory provide dynamicism with a rich
methodological content, which clearly distinguishes this approach
from the computational one (Giunti 1995, 1996; van Gelder and
Port 1995; van Gelder 1995).

Second, some proponents of the dynamical approach (van Gelder and Port 1995; van Gelder 1995) have in fact restricted the class of models allowed by the dynamical hypothesis. According to their proposal, dynamical models include most connectionist models and all nonconnectionist continuous models, but they exclude computational models. Thus, under this interpretation of dynamicism, it is no longer true that the dynamical hypothesis is consistent with symbolic modeling. These authors, however, do not take this to be a drawback, for they maintain that all symbolic models give a grossly distorted picture of real cognition.

Finally, my line of
defense (Giunti 1995, 1996) also restricts the class of the
dynamical models, but in a different way. The heart of my
proposal lies in the distinction between two different kinds of
dynamical models: simulation models and Galilean ones. This
distinction is an attempt to set apart two, quite different,
modeling practices. *Simulation models* are mathematical
dynamical systems which, to a certain extent, are able to
reproduce available data about certain tasks or domains. Besides
this empirical adequacy (which sometimes is itself quite weak) it
is very difficult, if not impossible, to find an interpretation
which assigns a feature (aspect, property) of the real system to
each component of the model. By contrast, *Galilean models*
are built in such a way that no component of the model is
arbitrary. Rather, each component must correspond to a magnitude
of the real system. Galilean modeling is in principle consistent
with symbolic, connectionist, and nonconnectionist continuous
modeling as well. What I have been arguing for is that we should
take the *ideal* of Galilean modeling more seriously for, if
we are successful, we are going to build a better science of
cognition.

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