Cognitive Systems and the Scientific Explanation of Cognition

A cognitive system is any real system that has some cognitive property. Therefore, cognitive systems are a special type of K-systems (see chapter 3, section 3). Note that this definition includes both natural systems such as humans and other animals, and artificial devices such as robots, implementations of AI (artificial intelligence) programs, some implementations of neural networks, etc. Focusing on what all cognitive systems have in common, we can state a very general but nonetheless interesting thesis: All cognitive systems are dynamical systems. Section 2 explains what this thesis means and why it is (relatively) uncontroversial. It will become clear that this thesis is a basic methodological assumption that underlies practically all current research in cognitive science. The goal of section 3 is to contrast two styles of scientific explanation of cognition: computational and dynamical. Computational explanations are characterized by the use of concepts drawn from computation theory, while dynamical explanations employ the conceptual apparatus of dynamical systems theory. Further, I will suggest that all scientific explanations of cognition might end up sharing the same dynamical style, for dynamical systems theory may well turn out to be useful in the study of all types of models currently employed in cognitive science. In particular, a dynamical viewpoint might even benefit those scientific explanations of cognition which are based on symbolic models. Computational explanations of cognition, by contrast, can only be based on symbolic models or, more generally, on any other type of computational model. In particular, those scientific explanations of cognition which are based on an important class of connectionist models cannot be computational, for this class of models falls beyond the scope of computation theory. Arguing for this negative conclusion requires the formal explication of the concept of a computational system that I gave in chapter 1 (see definition 3). Finally, section 4 explores the possibility that scientific explanations of cognition might be based on Galilean models of cognitive systems (see chapter 3, section 5). Most cognitive scientists have not yet considered this possibility. The goals of this section are to contrast this proposal with the current modeling practice in cognitive science, to make clear its potential benefits, and to indicate possible ways to implement it.

Galilean Models and Explanations

Given a natural kind K (for example, mechanical, chemical, biological, cognitive, etc.), I say that a real system is a K -system if and only if it has some K-property. The main goal of this chapter is to analyze a particular type of scientific explanation, which I call a Galilean explanation. This analysis is based on a more general view of scientific explanation, according to which scientific explanations are solutions of problems of a special type. This type of problem essentially consists of two parts: first, considering a certain K-system and, second, setting the goal of scientifically explaining some K -property of this system. A scientific explanation of a K -property of a K -system is an explanation obtained by studying a model of the K -system, and the type of scientific explanation we construct in general depends on the type of model that we are going to study. Galilean explanations are a particular type of scientific explanations, for they are based on the study of models of a special type. I call a model of this special type a Galilean model of a K -system. The next four sections of this chapter explain, exactly, what I mean by a Galilean model of a K-system. A Galilean explanation of a K -property of a K -system is an explanation obtained by studying a Galilean model of the K -system. But if we need to study a Galilean model of the K -system, we must first specify or describe the model in such a way that we are able to use the model for constructing the explanation we seek. Therefore, the problem of constructing a Galilean explanation of a K -property of a K -system presupposes the problem of specifying a Galilean model of the K -system. we will see in section 6 that this second problem is equivalent to that of producing a correct Galilean framework of the K -system. Thus, if our main goal is to produce a Galilean explanation of a particular K -property P of a fixed K -system KRS, we must first construct a correct Galilean framework of KRS. Furthermore, this correct framework must also be P-explanatory, in the sense that it must specify a Galilean model of KRS whose study will then allow us to produce an explanation of the particular K -property P we want to explain.

Generalized Computational Systems

The definition of a computational system that I proposed in chapter 1 (definition 3) employs the concept of Turing computability. In this chapter, however, I will show that this concept is not absolute, but instead depends on the relational structure of the support on which Turing machines operate. Ordinary Turing machines operate on a linear tape divided into a countably infinite number of adjacent squares. But one can also think of Turing machines that operate on different supports. For example, we can let a Turing machine work on an infinite checkerboard or, more generally, on some n-dimensional infinite array. I call an arbitrary support on which a Turing machine can operate a pattern field. Depending on the pattern field F we choose, we in fact obtain different concepts of computability. At the end of this chapter (section 6), I will thus propose a new definition of a computational system (a computational system on pattern field F) that takes into account the relativity of the concept of Turing computability. If F is a doubly infinite tape, however, computational systems on F reduce to computational systems. Turing (1965) presented his machines as an idealization of a human being that transforms symbols by means of a specified set of rules. Turing based his analysis on four hypotheses: 1. The capacity to recognize, transform, and memorize symbols and rules is finite. It thus follows that any transformation of a complex symbol must always be reduced to a series of simpler transformations. These operations on elementary symbols are of three types: recognizing a symbol, replacing a symbol, and shifting the attention to a symbol that is contiguous to the symbol which has been considered earlier. 2. The series of elementary operations that are in fact executed is determined by three factors: first, the subject’s mental state at a given time; second, the symbol which the subject considers at that time; third, a rule chosen from a finite number of alternatives.

Mathematical Dynamical Systems and Computational Systems

The main thesis of this chapter is that a dynamical viewpoint allows us to better understand some important foundational issues of computation theory. Effective procedures are traditionally studied from two different but complementary points of view. The first approach is concerned with individuating those numeric functions that are effectively calculable. This approach reached its systematization with the theory of the recursive functions (Gödel, Church Kleene).This theory is not directly concerned with computing devices or computations. Rather, the effective calculability of a recursive function is guaranteed by the algorithmic nature of its definition. In contrast, the second approach focuses on a family of abstract mechanisms, which are then typically used to compute or recognize numeric functions, sets of numbers, or numbers. These devices can be divided into two broad categories: automata or machines (Turing and Post), and systems of rules for symbol manipulation (Post). The mechanisms that have been studied include: a. Automata or Machines 1. gate-nets and McCulloch-Pitts nets 2. finite automata (Mealy and Moore machines) 3. push-down automata 4. stack automata 5. Turing machines 6. register machines 7. wang machines 8. cellular automata b. Systems of Rules 9. monogenic production systems in general 10. monogenic Post canonical systems 11. monogenic Post normal systems 12. tag systems. I call any device studied by computation theory a computational system. Computation theory is traditionally interested in studying the relations between each type of computational system and the others, and in establishing what class of numeric functions each type can compute. Accordingly one proves two kinds of theorem: (1) that systems of a given type emulate systems of another type (examples: Turing machines emulate register machines and cellular automata; cellular automata emulate Turing machines, etc.), and (2) that a certain type of system is complete relative to the class of the (partial) recursive functions or, in other words, that this type of system can compute all and only the (partial) recursive functions (examples of complete systems: Turing machines, register machines, cellular automata, tag systems, etc.). All different types of computational systems have much in common. Nevertheless, it is not at all clear exactly which properties these mechanisms share.