Space and Spaces Author(s): Helen Couclelis and ... - Martin Raubal

Space and Spaces Author(s): Helen Couclelis and Nathan Gale Source: Geografiska Annaler. Series B, Human Geography, Vol. 68, No. 1 (1986), pp. 1-12 Published by: Blackwell Publishing on behalf of the Swedish Society for Anthropology and Geography Stable URL: http://www.jstor.org/stable/490912 Accessed: 01/04/2009 18:36 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=black. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact [email protected].

Swedish Society for Anthropology and Geography and Blackwell Publishing are collaborating with JSTOR to digitize, preserve and extend access to Geografiska Annaler. Series B, Human Geography.

http://www.jstor.org

SPACE AND SPACES BY HELEN COUCLELIS and NATHAN GALE*

Lespace m'a toujours rendu silencieux. (Space has always reduced me to silence). - Jules Valles, L'Enfant, p. 238. ABSTRACT. Alternative conceptions of space abound in geography, and drawing a line between them is not always easy. A distinction that seems particularly difficult to make is the one between perceptual and cognitive space. Although it is usually recognized that these are different concepts, the lack of any hard distinctions between them continues to be the cause of some confusion in behavioral studies. In this paper we show that a formal distinction between perceptual and cognitive space, as well as some other concepts of space currently used in geography, can be drawn within the framework of algebraic structures of the group family. Indeed, it appears that different concepts of space can be associated with particular members of that family. The paper ends with a speculative discussion of the possible wider significance of these results, in particular with respect to the structure of the formal languages most appropriate for the analysis of each of these concepts of space, and consequently, of the empirical phenomena viewed in their context. Resume. La g6ographie comprend une grande variete de conceptions diff6rentes de l'espace qui ne sont pas toujours faciles a distinguer l'une de l'autre. Par exemple, une distinction particulierement difficile a faire est celle entre espace perqu et espace mental. Bien qu'il soit souvent reconnu qu'il s'agit la de deux concepts bien distincts, l'absence de criteres concrets de diff6renciation est la cause d'un certain degr6 de confusion dans la g6ographie du comportement. Nous montrons ici qu'il est possible d'e6tablir une distinction formelle entre espace perqu et espace mental, ainsi qu'entre certains autres concepts spatiaux utilis6s en geographie, dans le cadre des structures alg6briques de la famille du groupe. En effet, il semble que les differents concepts spaciaux peuvent etre directement associ6s avec les divers membres de cette famille. L'article aboutit a une discussion sp6culative de la signification probable plus generale de ces r6sultats, tout particulierement en ce qui concerne la structure des languages math6matiques mieux adapt6s a la description du contenu de chacun de ces concepts spatiaux. Ceci aurait des consequences directes pour lanalyse des ph6nomenes empiriques conceptualis6s dans ces divers espaces.

Introduction On the subject of space it would be safer to remain silent. After all, it has been said thet even a fool, if he keeps his peace, is esteemed wise (Proverbs 17:28). Space is a difficult to talk of sensibly as it

* Dr. Helen Couclelis and Dr. Nathan Gale, Department of Geography, University of California, Santa Barbara, Ca 93106, U.S.A. Partial support for this research was provided by NSF Grant SES81-10253. GEOGRAFISKA ANNALER - 68 B (1986) * 1

is perplexing to think about; nonetheless, it is not easy to ignore. Most geographers are likely to agree that the concept of space in one of the fundamentals of geographic research. A brief perusal of the literature, however, is all that is needed to realize that there is not one concept of space, but a multitude. Indeed, the last two decades in particular have seen a proliferation of spatial concepts used by geographers (e.g Harvey, 1969, Ch. 14; Entrikin, 1977; Sack, 1980). This, in itself, need be neither good nor bad; it is probably both. It is good in the sense that the variety of concepts of space now deemed to be significant to geography has the potential of enriching the discipline. It is bad in the sense that the pool of familiar constructs is being expanded by the inclusion of notions which in many cases are not well defined or generally intelligible - and yet often they are assumed to be so. More specifically, with the increased interest in individual spatial cognition and behavior, and thus, in "subjective" space, there has been, and continues to be, considerable confusion regarding the notions of perceptual and cognitive space. Geographers, psychologists, and (alas) philosophers have used these terms in many different contexts, with a wide variety of intended meanings. Without necessarily implying causality in either direction, it is apparent that the muddle in terminology reflects the degree of conceptual confusion in this area. Part of the reason why the concept of space appears so elusive seems to lie in the plethora of contrasting and even contradictory qualities it is associated with. It is not surprising that something that can be described as being subjective or objective, psychological or mathematical, relative or absolute, empirical or formal, Euclidean or nonEuclidean , metric or non-metric, proves so difficult to pin down. Geographers and others have sometimes used selected pairs of such dichotomous attributes (e.g., abstract - concrete, public - private,

primitive

- sophisticated;

see Sack,

1981; Couclelis, 1982) to define conceptual axes along which to structure arguments concerning particular conceptions of space. However, the contradictions inherent in any more general dis1

COUCLELIS/GALE

cussion of the notion are so numerous that they seem unlikely to be overcome within any framework defined at the level of the conflicting attributes themselves. In this paper, we suggest that the algebraic theory of groups can provide a view of the issue that is general enough to transcend much of the current confusion. To illustrate the potential value of such a framework as a tool for discriminating between related though different conceptions of space, we shall focus on one of the most subtle distinctions needed to be made in behavioural geography, that between perceptual and cognitive space. Thus, in

the first part, we discuss some of the attempts made by psychologists, behavioral geographers., and others to draw a line between spatial perception and spatial cognition on the basis of experimental evidence, and the great conceptual difficulties encountered in this endeavor. We then go on to outline a hierarchical framework of general spatial categories based on the definition of algebraic structures of the group family, and show how this can be associated with different members of the group family. Thus, we argue, there exists at least cal, sensorimotor, perceptual and cognitive. Indeed, it appears that different concepts of space can be associated with different numbers of the group family. Thus, we argue, there exists at least one formal framework within which a hard distinction can be drawn between perceptual and cognitive space; moreover, this distinction is of the same nature as that which can be made between certain other pairs of space concepts, so that it becomes possible to discuss spatial notions that seem very different qualitatively within the same general context. The paper ends with a speculative discussion of the possible wider significance of these results, in particular with respect to the structure of the formal languages most appropriate for the analysis of each of these spaces.

usage may make perception a useful term to have in the title of a review article, a point which has been made several times in the literature is that it needs to be defined more precisely if it is to be effectively employed as a working construct in the advancement of geographic research (e.g. Hart and Moore, 1973; Moore and Golledge, 1976; Golledge, 1981). It is unlikely that universally agreed upon definitions of perception and cognition will ever be found, but some attempts to find useful distinctions have been made. In what might be called the general view of experimental psychology, the standard criteria for perception have to do with stimulus dependency and immediacy (Epstein, 1967). Downs and Stea (1973, p. 14) present a definition that distinguishes perception and cognition along these lines as follows: . .. we reserve the term perception for the process that occurs because of the presence of an object, and that results in the immediate apprehension of that object by one or more of the senses. Temporally, it is closely connected with events in the immediate surroundings and is (in general) linked with immediate behavior. . . . Cognition need not

be linked with immediate behavior and therefore need not to be directly related to anything occuring in the proximate environment. Consequently, it may be connected with what has passed (or is past) or what is going to happen in the future. The authors qualify this definition with the remark that it is a "distinction of convenience" that "falls short of establishing a clear dichotomy" (Downs and Stea, 1973, pp. 13-14). Nevertheless, this type of definition may well lead to a scale-dependent distinction (of convenience) in the spatial context. Accordingly, perceptual space may be considered to be basically that which can be seen or sensed at one place and one time, while cognitive space includes the larger-scale space beyond Perceptual space and cognitive space the sensory horizon about which information must In a recent review, Saarinen and Sell (1980, p. 525) be mentally organized, stored, and recalled. The begin by stating that the term "environmental per- emphasis on scale, however, although intuitively ception" is "the term most frequently used by geo- appealing to geographic thought, may give rise to graphers when referring to the broad interdiscipli- some confusion if it is interpreted strictly spatially nary behavior-environment-design nexus." In- to mean that cognitive space only "begins," as it deed, many geographers appear to be using "per- were, where perceptual spaces leaves off. Clearly, ception" in a very broad sense to refer to virtually one can think about, or cognize, space within the all mental processes and outcomes, including sen- sensory field as well as beyond it. There is obvioussations, memories, images, evaluations, preferen- ly a spatial limit to sense perception, but there is ces and attitudes, and so on. While such a loose no spatial boundary between perceptual and cog2

GEOGRAFISKA ANNALER *68 B (1986) ? I

SPACE AND SPACES

nitive space; rather, the latter seems somehow to include the former (see Downs, 1981, pp. 99-100). The difference between the two cannot be measured quantitatively, for example in feet to the wall or in miles to the horizon, for it is fundamentally a qualitative difference. As a philosopher who dealt extensively with concepts of space, Cassirer (1953, pp. 67-68) expressed this difference in terms of the distinction between the acquaintance and knowledge of space. Acquaintance means only presentation; knowledge includes and presupposes representation. The representation of an object is quite a different act from the mere handling of the object. The latter demands nothing but a definite series of actions, of bodily movements coordinated with each other or following each other. It is a matter of habit acquired by constantly repeated unvarying performance of certain acts. But the representation of space and spatial relations means much more. To represent a thing it is not enough to be able to manipulate it in the right way for practical uses. We must have a general conception of the object, and regard it from different angles in order to find its relations to other objects. We must locate it and determine its position in a general system. An example taken from the Army-Air Force battery of aptitude tests (Guilford, Fruchter, and Zimmerman, 1952) may help to illustrate the concept of representation: All surfaces of a 3-inch cube are painted red and it is then cut into twenty-seven 1-inch cubes. Questions: 1. How many cubes have three faces painted red? 2. How many cubes have one face painted red? 3. How many cubes have no faces painted red? 4. How many cubes have two faces painted red? Given the physical presence of such an object, the questions could be answered with the aid of handling and sense perception. The 3-inch cube could be dismantled and the number of painted faces on each of the 1-inch cubes visually perceived. On the other hand, the questions could be answered without the actual viewing of a proximate object. In this case, it is assumed that some kind of cognitive spatial representation, or visual imagery, is employed (Anderson, 1980). By "seeing," as it were, in the "minds eye," each of GEOGRAFISKA ANNALER *68 B (1986) * I

the smaller cubes can be regarded from different angles and their relation to the more general system of the larger cube can be determined. Although this simple example may not be the best, it does raise a number of important issues. Not only does it help to make a distinction between perception and cognition, it also indicates to some degree the relationship between the two. Lee (1973) has pointed out that while the distinction between percepts and concepts is well taken, it is very difficult to find an instance of perception free of all concepts, or an instance of conceptual knowledge free from all percepts. Recall, for instance, the failure of the logical positivist attempts to build a pure language of sense perception (Nagel, 1961, p. 121). As Downs (1981, p. 99) has stated, the attempt to make an "overly fine distinction" is "heroic, but ultimately futile." Indeed, returning to our simple example, it seems close to impossible to conceive of a red face on a cube if one has never perceived the color red. On the other hand, how can a cube be perceived as such, without at least some proto-conceptualization of what "cubeness" is? Viewed from this angle, what belongs to perception, and what to cognition, appears to boil down to a question of definition. On the other hand, two major developmental psychologists, Werner and Piaget, have tended to view the distinction as an empirical one. Wernergposition is that perception is basically a subsystem of cognition (Werner, 1948;Wapner and Werner, 1957). In this view there are three major means by which cognition of the human world is constructed: these are sensorimotor acts, perceptions, and contemplative operations. As developmental progress occurs, the less differentiated sensorimotor and perceptual means become increasingly subordinate to the higher cognitive means of contemplation. It is questionable, therefore, whether what might be called pure perception, independent of any contemplative judgement, ever occurs in normal adult behavior. Thus, Werner emphasizes the constructive nature of perception as a means to and a subsystem of cognition, as well as a function of cognitive processes. Piaget (1969; 1971) would reserve the term perception for an immediate collection of sensations, where sensations refer merely to elements and not to structuring. He identifies two aspects of knowing; one is figurative and immediate, the other is operative and mediated. Perception, in his view, is one form of figurative knowing, while cognition 3

COUCLELIS/GALE

is fundamentally operative. Knowledge, in general, arises not from sensation nor perception alone, but from the entire action, of which perception constitutes the function of signalization. An object is only really known by acting on it and transforming it. This can be done in two ways. One involves changing its positions, movements, or characteristics, and is essentially physical. The other involves enriching the object with characteristics or new relationships having to do with systems of classification, numerical order, measure, and so on, and is known as logico-mathematical (Piaget, 1971, p. 67). With regard to space, perceptual space consists of momentary figurations, while cognitive, or "notional" space implies (cognitive) action and introduces a system of operatory transformations. Piaget and his co-workers spent years studying relations between notions of space and corresponding perceptions. His work is voluminous, and rather than attempt a comprehensive summary, only a few examples will be briefly discussed. It should be remembered, however, that his conclusions and theoretic statements were based on many such studies. The objective of the work outlined here was to explore the relations of the notion of projective space and the perception of projective sizes (Piaget, 1971, pp. 74-75). In the average child, representation of perspective generally appears spontaneously in drawing about the age of nine or ten. When presented with common objects in different positions and instructed to choose among two or three drawings the one which most closely corresponds to the chosen perspective, correct responses were obtained only after the age of seven or eight. In another experiment, children were shown a cardboard model of three mountains differring in size, location, color, and the object at their summit. When asked to identify pictures representing the mountains from the four principle viewpoints, it was found that the children had great difficulty in freeing themselves from their own egocentric perspective. The problem is not solved until the age of nine or ten. Thus, Piaget suggests that the notion of perspective space begins about the age of seven or eight and reaches a point of equilibrium around nine or ten. On the other hand, young children do much better at estimating projective size. Children were asked to compare the apparent (not real) size of a stick measuring about 4 inches, placed a yard away from the subject, and a stick of variable length 4

about 13 feet away. Once the children understood what was being asked (which took considerable explanation and demonstration), they gave perceptual estimates far better than those of older children, and even better than those of most adults. It appears, therefore, that the notion of projective space only arises at the level at which the perception of projective sizes deteriorates. Cognition, then, is not a generalization or abstraction of perceptions. If this were so, the notion of projective space would be formed when projective perception is best. Hence, Piaget concludes that the notion of space is "infinitely richer" than the corresponding percept because of the inclusion of a coordination of viewpoints and a complex mechanism of transformational operations (Piaget, 1971, pp. 68-69). This brief overview of work involving the distinction between perceptual and cognitive space only confirms the impression of confusion surrounding generalized notions of space. On the one hand, the apparent impossibility to draw a line between perception and cognition is seen by many as an indication that the distinction may be merely definitional. On the other hand, Piaget's work seems to suggest that perceptual and cognitive space are "real" things formed at particular stages of the development of a growing child, not unlike the appearance of facial hair or breasts. From the analytic perspective, problems of space have been traditionally approached through the medium of geometry, and more recently, also topology. In behavioral geography, the study of cognitive geometry is emerging as a promising new direction (Tobler, 1976; Golledge and Hubert, 1982). Still, the difficulties and controversies involved in this type of work - the issues of Euclidean versus non-Euclidean, metric versus nonmetric, two or three-dimensional versus multidimensional, measurable versus non-measurable, and so on - indicate that the formal languages of conventional geometry and even topology may be too constrained (not general enough) to tackle some of the most fundamental aspects of the question. For this reason, we propose to go back to the most general mathematical definition of space available, such as that given in Alexandroff (1961, p. 9), who writes: The concept of topological space is only one link in the chain of abstract space constructions which forms an indispensable part of all modern geometric thought. All these GEOGRAFISKA ANNALER *68 B (1986) * 1

SPACE AND SPACES

constructions are based on a common conception of space which amounts to considering one or more systems of objects points, lines, etc. - together with systems of axioms describing the relations between these objects. Moreover, this idea of a space depends only on these relations and not on the nature of the respective objects. We can then look for a formal framework that meets the simple requirements of this definition (a set or undefined objects, and a system of relations between these objects) without adding any other conditions that could result in a loss of generality. As will be shown in the following section, the concept of group meets just these requirements, since all the members of its family are defined on the basis of a set of operands (abstract objects) and a particularly interesting class of relations (more specifically, functions) called binary operations. By gradually endowing the abstract operands with larger and larger sets of empirical properties, and reinterpreting the operations accordingly, one can address some of the basic properties of a series of qualitatively different, familiar kinds of space. Throughout this procedure, the notions of operands and operators provide the fundamental invariants that identify the different constructs in that sequence as instances of the same basic concept, space. The concept of group and concepts of space The theory of groups can provide an illuminating framework for the analysis of problems in which the notions of operations and transformations are particularly important. At first, group theory was confined to the domains of algebra and combinatorics where it had been developed by the mathematicians Galois and Cauchy in the beginning of the nineteenth century. The generality and fertility of this form of abstraction was then further established in the latter half of that century, primarily through the work in geometry of Lie and Klein. Their developments of the theory made it an important part of the general reorientation of thought which followed the discoveries of nonEuclidean geometries. Helmholtz (1868), Poincare (1913), Eddington (1939), and Cassirer (1944) were probably the first to speculate about possible connections between the mathematical and psychological problems of space in terms of a grouptheoretic framework; and later, Piaget (1954) inGEOGRAFISKA ANNALER *68 B (1986) ?1

troduced some group concepts into his theories of child development. Thus far, however, it appears that the notion of group has not found its way into the geographic literature. In a broad and abstract definition, group theory has been described as ... a branch of mathematics in which one does something to something and then compares the result with the result obtained from doing the same thing to something else, or something else to the same thing. (Newman 1956, p. 1534) Thus, in the most general sense, the theory of groups has to do with operations performed on a set of operands ("doing something to something"), and with the results of such operations, whereby the original operands are transformed into something else. Furthermore, group theory is concerned with the invariants, or structural constants, of sets of operations or transformations; it involves the quest for the general, the permanent, the unchanged. But the mathematical systems known as groups are themselves only one member of a family of algebraic structures [S,*] involving a set S of operands and one binary operation, *. Formally, a binary operation * on a set S is a function *:SxS-> S, that is, a rule which assigns to every ordered pair SxS in S a unique element in S (Preparata and Yeh, 1974, p. 17). Familiar examples of such rules are of course the operations +, -, x,-, of ordinary algebra performed on the set of reals, R. The family of algebraic structures we wish to consider involves the following five axioms (Larson, 1970, p. 97): G1 Closure law For all a, b E S, a*b E S G2 Associative law For all a,b,c E S, (a*b)*c=a*(b*c) G3 Existence of an identity element For all a E S, there exists an element e E S such that a*e=e*a=a G4 Existence of inverses For every a E S, there exists an element b E S such that a*b=b*a=e G5 Commutative law For all a, b E S, a*b=b*a. The algebraic structure conforming to all five axioms is called an abelian group. The set of integers (I) form an abelian group under the operation of 5

COUCLELIS/GALE

addition (+), as is easily verified by checking through the axioms G1-G5: (1) a+b = c (another integer); (2) (a+b)+c=a+(b+c); (3) there exists an identity element, 0: a+0 =O+a=a; (4) for every integer there exists an inverse: a+(-a) = (-a) +a=0; (5) for all integers, a +b=b+a. Three other important algebraic structures are obtained by dropping one or more of these axioms: - axioms G1-G4 (but not G5) define the best known member of the family, the group, - axioms G1-G3 (but not G4 and G5) define a monoid, - axioms G1 and G2 (but not G3-G5) define a semigroup.

2.

3.

4.

any two vectors in the space must be another element (vector) of that space. This is, in fact, a fundamental property of vectors. Associative law: For all a, b, cE S, we must have (a-b)-c = a(b-c) This holds for vectors, obviously. To give a concrete illustration, "go left and straight ahead, then go right" is generally equivalent to "go left, then go straight ahead and to the right." Existence of an identity element: This is of course the null vector, but in more concrete terms it may be thought of as the stayas-you-are "move." Existence of inverses: Again this is obvious in the case of vectors. More concretely, the inverse of a move is another move that can "undo" the first one and bring you back exactly where you started. Commutative law: The commutativity of vectors is exemplified in the parallelogram of forces. Physical interpretations of this axiom also abound.

What we have here, therefore, is something like a nested hierarchy of algebraic structures, with each structure in the sequence resulting from the preceding one through the deletion (or, going in the opposite direction, through the addition) of a further axiom. With these preliminaries out of the way, we must now return to the theme of the paper and explore how these algebraic structures might relate to the notion of space - and spaces. Before we can go any further, however, it is necessary to show that the five axioms have meaning when translated into spatial terms. Consider, as a first approximation, a set S consisting of all possible vectors linking any two places or points in a space. "Places" and "points" are to be understood here in the most general possible sense, and the vectors themselves may indicate any kind of directed association between two points or places, for example: going from A to B, moving something from A to B, drawing a line from A to B, associating A to B in one's mind, whatever - we will be more specific later. Consider in addition a binary relation of composition, , (to be interpreted loosely as "and" or "then"), imposed on the set S. We have now defined the framework [S, ?] within which we may start searching for the existence of particular algebraic structures. Let us now consider the five axioms, one at a time.

5.

1. Closure law (also known as the group property): For all a, b, E S, we must have a-b E S. In other words, the result of the composition of

Towards a hierarchy of spaces In a classic popular exposition of the theory of groups, Eddington (1956) introduces the notion of

6

It is thus quite straightforward to show that, in principle, the framework provided by the family of algebraic structures introduced earlier is fully interpretable with respect to space. The question now becomes, what space is it that we are talking about? The discussion in the first part of this paper has focussed on the distinction (or lack thereof) between cognitive and perceptual space; but physical space, abstract geometrical space, symbolic space, sensorimotor space, and so on, are also current notions in geography and other fields of study. Are these all different manifestations of the same fundamental entity? Or are they mere synonyms for what is basically the same thing? Are they all different things? How do they relate to one another? Are any differences between them purely or mainly definitional? Are the same methods of analysis suitable for all? These seem to be largely philosophical questions, yet mathematical analysis along the lines of algebraic structures can go some way towards providing answers to them. Before we can proceed in that direction, however, a brief digression is needed in order to introduce another important formal concept.

GEOGRAFISKA ANNALER *68 B (1986) * 1

SPACE AND SPACES

u

a/

/

B=ObOaU

Table 1. A hierarchy of concepts of space Concepts of space

Atoms

Operands

/Sl

Symbolic space Cognitive space Perceptual space Sensorimotor space Physical space Pure Euclidean space

spaces places vantage points positions locations points

connection association travel move displacement vector

[S6] [S5] [S4] [S3] [S-] [S,]

OcObOaU

Figure. Hierarchical

filtering of sets by selective

operators.

selective operators, as used in modern spectral theory (Roman, 1975, Vol. 2). A selective operator may be thought of as a sieve, or filter, that sorts the entities corresponding to some particular description out of a universe of entities, U. If Oa is the selective operator that selects out of U whatever answers to the description of A, then OaU is a representation for the set of entities A. Now, A itself may comprise several other kinds of entities, among which those answering to the description of B may be of particular interest. In this case, if Oh is the operator that selects the B's, OhA =Oh (O,U) is a way of representing B as a function of A and U. This procedure can of course be iterated for as many steps as necessary or possible, so that if we have a hierarchy of entities A, B, C, D, . .. such that DcCcBcAc

U, we may represent

these as: OaU=A, ObOaU=B, OcObOaU=C, OdOcObOaU=D, and so on (see figure). Thus, while A and D may be as qualitatively different as forests are from squirrels' hairs, it is possible to represent them consistently within the same formal framework. This same principle may be used to bring the various concepts of space together, whether or not substantial qualitative differences exist between them. Let us therefore set up a tentative hierarchy of concepts of space according to the relative "richness," or experiential content, of these concepts as intuitively apprehended: some more solid resons for the particular ordering, based on the definition of the basic terms, will emerge later on. To help clarify these concepts, we will suggest names for two of the most characteristic features of each space: the "atoms" (the points or places GEOGRAFISKA ANNALER *68 B (1986) * 1

as introduced earlier), and the links (vectors) between such atoms constituting our set of operands, S (see Table 1). Let us focus first on the class of all six sets of operands, [S]. Starting from the bottom of the hierarchy, and moving upwards, the elements of each successive [Sn] may be defined as the elements of [Sn-l] augmented by a new set of concepts. Thus, the notion of displacement in physical space is the notion of mathematical vector augmented by the concepts of physical mass and time; a move in sensorimotor space is a physical displacement augmented by the notions of (sensory) inputs, motility, and time irreversibility which characterize organisms; a travel in perceptual space is a move associated with particular sets of perceptual structures - mainly views, but also sonic and other sensory "images," linked to particular vantage points; an association of places in cognitive space is like a travel in that it involves generalized images relating to these places, but these are enriched and modified by cognitive factors such as beliefs, knowledge and memory; finally, a connection in symbolic space involves everything to be found in an association in cognitive space, but here images are invested with meaning. Meaning, as we understand it here, is related to the cognitive component in the same way that the sense, intention, or connotation of a linguistic expression relates to its reference, extension, or denotation (Frege, 1982/1971) - or, more simply, in the way the idea of a place called "home" relates to any actual instance of a home in a person's mind. There is a permanence implied here, a universality, and a relative independence from spatiotemporal and psychological specifics, that makes symbolic space a largely shared experience for human groups, in contrast with cognitive space which is understood to be much more strongly individualistic. There is of course a rich literature in geography about the symbolic meaning of spaces (e.g. Tuan, 1977; 7

COUCLELIS/GALE

Sack, 1980), but within the framework of the present discussion we can do no more than acknowledge one more basic qualitative distinction that seems to exist between concepts of space. What we have just done in this conceptual exercise is to reverse the abstractive procedure embodied in the repeated application of selective operators outlined earlier. Since that procedure is fully reversible, we may accept that in principle, the structure of our set [S] of operand sets [Sn] is that of a hierarchy of entities obtained from each other by means of appropriate selective operators. That is, if U is some most comprehensive notion of spatial association between places, then O6U = S6, 05S6 = S5, 04S5 = S4, and so on. A particularly

interesting aspect of this representation is that it is not necessary to specify what the entities, U, S6, S5, . . . are, intrinsically, in themselves - only the selective operations by means of which these entities may be distinguished from each other. As Eddington (1956) notes, modern science rests on operational definitions of this kind, based on relations rather than essences. Spaces as algebraic structures This interesting possibility notwithstanding, the original challenge of defining a distinction between cognitive and perceptual space still remains to be met. Within the general framework just defined, we are now ready to search for the existence of particular algebraic structures of the group family, [S,-]. We shall do this for each space concept in the sequence, starting with pure Euclidean space. Table 2 summarizes the results. 1. Pure Euclidean space [S,, ] As mentioned earlier, axioms G1-G5 all hold for vector operations in Euclidean space. Thus Euclidean space has the structure of an abelian group.

Table 2. Concepts of space and algebraic structures Axioms Concept of space Symbolic space Cognitive space Perceptual space Sensorimotor space Physical space Pure Euclidean space

8

G, 0 0 1 I I I

G, G3 0 0 0 0 0I 1 1 1 I I I

G4 G5 0 0 0 0 0 0 0 0 0 I I

Structure ? ? semigroup monoid group abelian group

2. Physical space [S2, ] Displacements in (small-scale) physical space obey axioms G1-G4, as can be easily verified. However, G5 (the commutativity property), though often true, does not hold with the force of an axiom, because of the directionality of physical space (e.g., "up, then down" is generally not the same as "down, then up"), because of the presence of physical objects, and so on. Thus physical space has the structure of a group. 3. Sensorimotor space [S3, ] The moves of an organism (or machine) in sensorimotor space have the closure property (G1), the associative property (G2), and there is an identity element (no move), as before. However, because biological time is irreversible, the exact inverse of every move does not exist in the general case: the organism may return to the physical location where it started, but it will no longer be in the same position - it will have aged, become more tired, more hungry, etc., in the process. Because of the directionality of entropy, the same would be true of a machine in that space. Thus G4 does not hold as an axiom, nor does G5, since physical space underlies sensorimotor space. Accordingly, the structure of sensorimotor space appears to be that of a monoid. 4. Perceptual space [S4, ] We have defined perceptual space by means of its atoms, the vantage points, and the basic element, the "travel" from vantage point to vantage point. At any time, a vantage point is defined by a set of "views" associated with it. The notion of view has been introduced in set-theoretic frameworks compatible with the present one by researchers such as Kuipers (1981) and Benedict (1979). Since perceptual space should not be identified with visual space, the term "view" should be understood to encompass not only visual structures, but also more general sensory images. Thus the geometric notion of point is here replaced by the vantage point or set of views associated with any particular position at any time, and a travel is any move that takes a viewer from one vantage point to a different one. The term "travel"is used here in a rather special sense similar to that found in the film industry, where it designates an action resulting in a change of view, whether this involves physical displacement of the viewer or just a change in eye focus. GEOGRAFISKA ANNALER ?68 B (1986) ? 1

SPACE AND SPACES

Defined in this way, perceptual space has much in common with sensorimotor space: the group property (G1) holds, because two consecutive travels will always take a viewer where some other travel would; associativity (G2) holds trivially; there is no inverse (G4), because of the irreversibility of biological time; and commutativity (G5) does not hold as an axiom because of the constraints of the underlying physical space. However, in one important respect perceptual space differs from sensorimotor space: because views tend to change continuously (even by as little as the effect of continuously changing lighting conditions), even the "stay-as-you-are" travel is no guarantee that perceptual identity will be maintained. Thus G3 cannot be accepted as an axiom, and perceptual space appears to have the structure of a semigroup. 5. Cognitive space [S5,'] Here things become somewhat confused. Compared to all the preceding spaces, cognitive space is substantially different because of the presence of memory, or the potential to register what has gone on before; it is also virtually free of the constraints of physical space. As a result, the regular pattern of gradual relaxation of structure noted thus fas is broken at this stage. A brief examination of current notions of cognition will show why this must be so. Obviously, because the study of cognition is a relatively new and rapidly evolving field, it is not possible to speak of the properties of cognitive space with any degree of assurance; however, certain features seem well established, and these will be sufficient for our purposes. Cognitive space has been defined here by means of its atoms, the places, and its basic elements or operands, the (cognitive) associations between places. A place is defined as the set of perceptual structures (views) associated with it at any time, augmented by the cognitive factors (in particular, knowledge, memories and beliefs) interpreting each view in the set. In the node-and-link formalism used in semantic-network representations of knowledge and memory, a place is thus a concept, corresponding to a set of interconnected nodes which represent the views and other facts, beliefs, etc,. defining that place. According to the most popular models of associative memory (Quillian, 1968; Barr and Feigenbaum, 1981, p. 185), associating a pair of places in this case amounts to finding connections between the sets of nodes that represent the corresponding concepts. Once such GEOGRAFISKA ANNALER *68 B (1986) * 1

connections have been found, the two sets of nodes representing the two concepts become temporarily unified. Thus the associations between places, the operands a, b, . . .,CS5 of cognitive space as defined here, are "superconcepts" constructed from the union of two place concepts, with the intersection of the two sets of nodes representing the cognized commonalities between the places. Consequently, the operation a-b of combining two such superconcepts may be seen as a higher-level repetition of the same cognitive process of association whereby commonalities between pairs of superconcepts are identified and new, more general, concepts are formed by their union. This means, however, that the result, c, of the operation a b is no longer a member of the set S5, but of its power set: the group or closure property does not hold here. This conclusion makes intuitive sense, since the possibility to create new concepts by combining existing ones is among the most striking properties of cognition. Accordingly, this latter observation gives support to the hypothesis that the above result does not depend on the correctness of semantic-network theory, or any other theory of cognitive representation, as long as we accept the premise that in cognitive space, conceptual innovation must be possible. As fas as the present discussion is concerned, this means that whatever the structure of cognitive space may turn out to be, it does not seem to be an algebraic structure of the group family. It would be a futile exercise to go on and speculate on the structure of symbolic space within the present framework. It is quite obvious that the algebraic structures of the group family are inadequate for a description of the richer concepts of space in the hierarchy: much more elaborate kinds of structures are probably required. But this is another matter. The main purpose of the discussion in this paper, which was to show that perceptual space can be distinguished formally, if not empirically, from cognitive space, has been achieved. Concluding speculations: space, structure and language That very general result deserves some further comments. First, we must make it clear that what we have not done is to say anything at all about the intrinsic nature of perceptual and cognitive spaces, in themselves. What we have done instead is to show that there exists at least one analytic 9

COUCLELIS/GALE

framework within which these two concepts can be systematically related and contrasted, along with several other current concepts of space. Thus we can say that at least under one set of formal criteria, cognitive space is qualitatively different from perceptual space. Is that difference then definitional or real? It is of course both, just as the difference between any two comparable mathematical structures may be seen as definitional or real, depending on whether one subscribes to an idealist or a platonist philosophy of mathematics. In a sense, the fact that cognitive space involves a synthesis (in pratical terms: coordination) of sensorimotor, perceptual, and cognitive operations, lends support to Piaget's view that the formation of the concept of cognitive space is a "real"developmental event. On the other hand, it may be argued that cognitive space, or perceptual space, only exist because of the definitional act of the analyst, the mental act necessary to bring a suitable collection of concepts in proper relation with each other. Unlike that of chipmunks or rainfall, the reality of spaces - of any description - cannot be observed. But there are also some more practical questions worth asking. As geographers, we feel that the description of phenomena relating to relevant notions of space is both our privilege and our duty. Not all of us choose to do so in the analytic mode, but those who do must eventually face the question: Given specific differences in algebraic structure between concepts of space, can they all be approached with the same set of analytic languages? The issue becomes particularly pertinent when it comes to the description of higher-order space concepts whose algebraic structure differs substantially from that of the pure mathematical spaces which conventional geometries describe. Far from constituting idle theoretical speculation, this question is of direct relevance to the analytic study of human phenomena such as those of spatial decision making and behavior, which implicitly involve the notions of physical, sensorimotor, perceptual, cognitive, and symbolic space. In the case of large-scale physical space, the question of the appropriate spatial language was answered implicitly through the development of space-time geometry. Space-time geometry is a geometry of and for physical space: not only is the notion of time, with its intrinsic ordering relation, undefinable in pure geometry, but the special mathematical status of the "fourth dimension" testifies to the fact that we are not here dealing with any ordinary four-dimensional abstract space. Not 10

incidentally, the existence and distinctiveness of the fourth dimension of physical space is reflected in its group structure, as Eddington (1956) demonstrates. Medium-scale, "locally Euclidean" physical space is still well served by conventional geometric language. By contrast, physical space at the sub-atomic scale (which most likely does not obey the group axioms) is still very much an unsolved riddle (DeWitt, 1983), just as are, conceptually, many of the physical processes observed at that level. Distinct classes of processes imply distinct geometries, as Riemann, Minkowski, Einstein, and others have shown (Nagel, 1961, Ch. 9; Harvey, 1969, Ch. 14). Moving up from the relatively well-explored world of physical space, to the level of sensorimotor space, which has a monoid, rather than a group structure, things become much more uncertain. To our knowledge, no geometry has been specifically developed to describe such a space, although it might be a conjecture worth some further investigation that Thom's (1975) geometrical theory of models does just that.' On the other hand, other (non-geometrical) formalisms exist that may be seen to describe processes in sensorimotor space very adequately. For example, there is a theorem in discrete systems theory stating that every finite state machine has associated with it a monoid, which is isomorphic to every finite monoid (Preparata and Yeh, 1973, p. 162).2 The special interest of the theorem, from our point of view, lies in the fact that a finite state machine can be seen as a model of an organism moving in sensorimotor space; indeed, what we have is none other than the formalization of stimulus-response behavior. Given the earlier discussion, it is no coincidence that a monoid structure underlies finite state machines. The pattern that seems to emerge so far encourges one further speculation: namely, that each concept of space may have a set of most appropriate descriptive languages associated with it such that the basic algebraic structure of each space is preserved in the structure of the corresponding languages (in other words, that the algebraic structures of the space and the appropriate languages be homomorphic). If this is true, perceptual space should be best describable by languages that have the structure of a semi-group, whatever these may be. Cognitive space immediately poses a problem, since no recognizable algebraic structure is associated with it under the present framework. Interestingly, natural language, the language used GEOGRAFISKA ANNALER *68 B (1986) * I

SPACE AND SPACES

for millenia to describe cognition in philosophy, literature, and everyday life, exhibits just that same lack of structure under the rough set of criteria used here.3 The structure of natural language is obviously far more subtle than that of the members of the group family, as several years of work in theoretical linguistics and natural language modeling in artificial intelligence testify. The conjecture here is that whatever the algebraic structure of cognitive space turns out to be, it ought to be homomorphic to that of the best possible models of natural language. But the speculation must now come to a close, lest we become tempted to extend it to the level of symbolic spaces. Symbolic spaces, and whatever may lie beyond these, are better left outside the realm of structures and descriptive languages. Structure is the outcome of a drastic abstractive act, of a set of selective operations which when performed on a universe of concern, yield results which may be analyzed, synthesized, manipulated, and most importantly, rationally communicated. Thus, while searching for the hard languages that will best describe this or the other concept of space, we might be well advised to remember that pure space, physical space, sensorimotor space, perceptual space, cognitive space, and so on, are all structures obtained through an abstractive impoverishment of human space - the source of them all, the structure-less experienced totality. No rational knowledge is possible without the act of destruction of the original object, and if it is rational knowledge we seek, we must go on creating structures out of human space, up to the limits of analytic language. What really matters may well lie beyond, but this, as Wittgenstein thought, may by definition be the unsayable. "Space has always reduced me to silence": since no geographer could afford to say as much, let the poet remind us. Notes 1. Thom's general theory of models, widely known as "catastrophe theory," was developed as a purely mathematical system based on differential topology (a generalized geometry). That framework explicitly involves features that may be interpreted as time irreversibility and environmental inputs to a system, both being defining aspects of sensorimotor space. Whether Thom's spaces have the general algebraic structure of a monoid, rather than that of a group or semigroup, say, is a question we have not investigated. 2. A finite state machine M is a system [S, I. Z. 6, .], where S is a finite non-empty set of states, I is a set of input symbols, Z is a set of output symbols, 6: Sxl--I S is the transition function, and X:SxI-- Z is the output function (Preparata and Yeh 1973, p. 51). Translated into the language of sensorimotor beGEOGRAFISKA ANNALER *68 B (1986) * 1

havior, these definitions mean the following: S is a finite nonempty set of positions in space, I is a set of stimuli, Z is a set of responses, 6 is the function that determines the next state the system will assume given an initial state and a stimulus, and k is the function that determines the response given an initial state and a stimulus. 3. Indeed, seen as a system [C,-], where C is a set of words or phrases denoting concepts, and the operation of composition (juxtaposition), natural language defies all five axioms of the group family: (1) There is no closure, since the composition of two concepts does not always give another concept in the language (e.g. "big *dog" is in C, but "big *milk" is not); (2) Associativity does not hold as an axiom, since, for example, (antique *brass) *lamp # antique *(brass - lamp); (3) There is no identity element, since there does not exist a concept which, when combined with any other concept, leaves it unchanged; (4) There are no inverses (concepts cannot be "undone" by other concepts); and (5) Commutativity obviously does not hold, since word order is usually important.

References Alexandroff, P., 1961: Elementary concepts of topology. New York: Dover Publications. Anderson, J.R., 1980: Cognitive psychology and its implications. San Francisco: W.H. Freeman. Barr, A. and Feigenbaum, E.A., 1981: The handbook of artificial intelligence. Vol. I. Los Altos: HeurisTech Press/Kaufmann. Benedict, M.L., 1979: To take hold of space: isovists and isovist fields. Environment and Planning B 6, 47-65. Cassirer, E., 1944: The concept of group and the theory of perception. Philosophy and Phenomenological Research 5, 135. 1953: An essay on man. Garden City, N.Y.: Doubleday. Couclelis, H. 1982: Philosophy in the construction of geographic reality. In Gould, P. and Olsson, G., editors, A search for common ground. London: Pion, 105-138. DeWitt, B.S., 1983: Quantum gravity. Scientific American 249, 112-129. Downs, R.M. and Stea, D., 1973: Cognitive maps and spatial behavior: process and products. In Downs, R.M. and Stea, D., editors, Image and environment. Chicago: Aldine, 8-26. Eddington, A., 1939: The philosophy of physical science. New York: MacMillan. 1956: The theory of groups. In Newman, J.R., editor, The world of Mathematics. New York: Simon and Schuster, 1558-1573. Entrikin, J.N., 1977: Geography's spatial perspective and the philosophy of Ernst Cassirer. The Canadian Geographer 21, 209-222. Epstein, W., 1967: Varieties of perceptual learning. New York: McGraw-Hill. Frege, G., 1892/1971: Sens et denotation. In Imbert, C., editor, Gottlob Frege: tcrits logiques et philosophiques. Paris: Seuil, 102-126. Golledge, R.G. and Hubert, L.J., 1982: Some comments on nonEucliden mental maps. Environment and Planning A 14, 107-118. Guilford, J.P., Frutcher, B. and Zimmerman, W.S. 1952: Factor analysis of the Army Air Force's battery of experimental aptitude tests. Psychometrica 17, 45-68. Hart, R.A. and Moore, G. T., 1973: The development of spatial cognition: a review. In Downs, R.M. and Stea, D., editors, Image and Environment. Chicago: Aldine, 246-288. Harvey, D., 1969: Explanation in geography. New York: St. Martin's Press.

11

COUCLELIS/GALE Helmholtz, H., 1868: Uber die Tatsachen welche der Geometrie zugrunde liegen. Gottinger gelehrte Nachrichten 193-221, as cited in Jammer, M., 1954: Concepts of space. Cambridge, Mass.: Harvard University Press. Kuipers, B., 1982: The "map in the head" metaphor. Environment and Behavior 14, 202-220. Larsen, M.D., 1970: Fundamental concepts of modern mathematics. Reading, Mass.: Addison-Wesley. Lee, H.N., 1973: Percepts, concepts, and theoretic knowledge. Memphis: Memphis State University Press. Moore, G.T. and Golledge, R.G., 1976: Environmental knowing: concepts and theories. In Moore, G.T. and Golledge, R.G., editors, Environmental knowing. Stroudsburg, Pa.: Dowden, Hutchinson and Ross, 3-24. Nagel, E., 1961: The structure of science: problems in the logic of scientific explanation. London: Routledge and Kegan Paul. Newman, J.R., 1956: Certain important abstractions. In Newman, J.R., editor, The world of mathematics. New York: Simon and Schuster, 1534-1537. Piaget, J., 1954: The construction of reality in the child. New York: Basic Books. 1969: Thepsychology of the child. New York: Basic Books. 1971: Psychology and epistemology. New York: Penguin Books. Poincare, H., 1913: The foundations of science. New York: The Science Press. Preparata, E P and Yeh, R. T., 1974:Introduction to discretestructures. Reading, Mass.: Addison-Wesley.

12

Quillian, M.R., 1968: Semantic memory. In Minskv, M., editor, Semantic information processing. Cambridge, Mass.: MIT Press. Quine, W.VO., 1980: Notes on the theory of reference. In Quine, W.V.O., editor, From a logical point of view. Cambridge, Mass.: Harvard University Press, 130-138. Roman, P, 1975: Some modern mathematics for physicists and other outsiders. New York: Pergamon Press. Saarinen, TE and Sell, J.L., 1980: Environmental perception. Progress in Human Geography 4, 525-548. Sack, R. D., 1980: Conceptions of space in social thought. Minneapolis: University of Minnesota Press. 1981: Conceptions of geographic space. Progress in Human Geography 4, 313-345. Thom, R., 1975: Structuralstability and morphogenesis. Reading, Mass.: W.A. Benjamin. Tobler, W.R. 1976: The geometry of mental maps. In Golledge, R.G. and Rushton, G., editors, Spatial choice and spatial behavior. Columbus, Ohio: Ohio State University Press, 6981. Tuan, Y, 1977: Space and place. Minneapolis: University of Minnesota Press. Wapner,S. and Werner,H., 1957: Perceptual development. Worcester, Mass.: Clark University Press. Werner, H., 1948: Comparative psychology of mental development. New York: International University Press.

GEOGRAFISKA ANNALER *68 B (1986) * I