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Conceptual Spaces: The Geometry of Thought; Peter Gärdenfors; MIT Press, Cambridge, MA, 2000
Acta Psychologica 106 (2001) 333±336
www.elsevier.com/locate/actpsy
Book Review Conceptual Spaces: The Geometry of Thought; Peter Gardenfors; MIT Press, Cambridge, MA, 2000 Peter Gardenfors is the author of the now classic monograph entitled ``Knowledge in ¯ux: Modeling the dynamics of epistemic states'' which appeared in 1988. Hence I was very curious about the new book of this distinguished author. Below I will try to summarize and discuss what I see as the basic tenets of the new book, mainly from a psychologist's point of view. As Gardenfors indicates in the preface, this book deals with issues related to at least ®ve dierent disciplines: philosophy, computer science, psychology, neuroscience and linguistics. Given my own background, and in view of what I expect to be the main interests of readers of Acta Psychologica, it would seem prudent to emphasize those aspects of the book which might have a bearing on psychology. The book is devoted to the way in which representations should be modeled (nowhere in the book is the necessity of introducing representations questioned; for an insightful critique of this assumption, cf., e.g., Shannon, 1993). Alongside the mainstream symbolic and connectionistic models of representation, Gardenfors discerns an additional class of so-called conceptual models. A conceptual model of representation consists of quality dimensions spanning up a topological space. With respect to connectionistic models a conceptual model is considered to be more coarse-grained, whereas it is more ®ne-grained with respect to symbolic models. A topological space comes equipped with structure about neighborhoods, and in Chapter 1 Gardenfors explains betweenness and equidistance relations which, together with the introduction of some metric, enable the natural de®nition of similarity relations which turn out to be of prime importance in the remainder of the book. All this is explained very clearly and does not require any special expertise of the reader. Chapter 1 ends with methods to identify quality dimensions (multidimensional scaling) and ideas about the origin of quality dimensions (e.g., nativism). Before proceeding, I have to mention some additional distinctions made by G ardenfors, namely, between representations within a constructive context and those within an explanatory context. Representations within a constructive context need not have any psychological validity, but serve scienti®c and constructive (e.g., robotics) purposes. On the other hand, representations within an explanatory context aim to describe psychological structure, and hence are called phenomenal. The use of multidimensional scaling and deliberations about the origin of quality dimensions should be located at this phenomenal level. Another distinction at the phenomenal level concerns integral versus separable dimensions. In contrast to 0001-6918/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 1 - 6 9 1 8 ( 0 0 ) 0 0 0 6 7 - 6
334
Book Review
separable dimensions, a set of dimensions is integral if an object cannot be assigned a value in one dimension without giving it a value in the other ones (for instance, hue and brightness are integral). Like most of the book, I will concentrate on the phenomenal level (which becomes especially important in the chapter on semantics). The distinction between integral and separable dimensions is used in Chapters 3 and 4 in the de®nitions of property and concept. Chapter 2 discusses at some length the symbolic, connectionistic (subconceptual), and conceptual approaches to representation. The author is at his best in describing the limitations of symbolic representations, including the frame problem and the problem concerning the origin of new predicates. The latter problem is related to the learning paradox in cognitive developmental psychology (cf. Molenaar, 1986). At the subconceptual level, some problems of connectionistic representations are brie¯y considered (large number of training trials; questions about the relationships with conceptual and symbolic levels). The section on conceptual models stresses, among other things, the importance of similarity relations and the ease with which these can be handled within geometrical representations. Comparisons with similar theories of Gallistel (1990), Harnad (1990), Mandler (1992) and Rademacher (1996) are made. Chapters 1 and 2 comprise the general introduction of conceptual models of representation. A conceptual model is a geometrical model, and I expected that the author would pay due attention to one of the hallmarks of any geometrical approach, namely the speci®cation of invariants under coordinate transformations. Gardenfors clearly indicates that quality dimensions at the phenomenal level have to be identi®ed in painstaking empirical research. Hence the choice of a particular set of quality dimensions could be made more robust by determining which aspects remain invariant under alternative choices (similar remarks can be made with respect to the choice of a metric). Unfortunately, however, this fundamental topic is not addressed in the present book. Chapter 3 is a beautiful chapter on properties, de®ned as regions in conceptual spaces spanned up by sets of integral dimensions, while naturalness is de®ned as the convexity of such regions. G ardenfors discusses many interesting implications of these geometrical de®nitions, including relationships with the prototype theory of categorization. A powerful technique to create convex regions (Voronoi tesselation) is presented in detail. G ardenfors shows by means of a clear illustration that convexity is not invariant under coordinate transformations and/or change of metric. Given the importance of convexity in the de®nition of natural properties, and because the choice of a particular set of dimensions and metric is to some extent arbitrary, one would like to have available some analogue of convexity which is invariant under coordinate transformations and/or changes of metric. This is a particular instance of the general point which I made in the previous paragraph. To wit, Gardenfors partly addresses this point in the section discussing the relativism of conceptual spaces. In that section, however, invariants are not discussed, but the relativism concerned is moderated by an appeal to evolutionary theory. The next chapter about concepts proceeds in a similar way. A concept is de®ned as a weighted combination of regions in a conceptual space spanned up by a set of separable, usually oblique, dimensions, while naturalness is again de®ned as con-
Book Review
335
vexity of the regions concerned. The weights in each combination are called saliences and are understood to be context-dependent. Gardenfors introduces an extremely interesting concept combination rule to cover cases like: ``a tall Chihuahua'', which of course is not the same as ``a tall dog''. Concept learning is explained by means of generalization from a ®nite number of exemplars, using again the Voronoi tesselation technique. Whether this would also explain stage transitions involving complete qualitative restructuring of conceptual spaces, such as has been envisioned in Piagetian theory, would seem to be an open question. The chapter ends with a detailed presentation of two experiments on categorization, using shell shapes drawn by a graphics program. Such graphics programs are important tools in theoretical morphology, where also relevant dynamical theories of form evolution and ontogenesis have been proposed (the latter dynamical theories might provide clues for alternative concept learning processes; cf. McGhee, 1998, for a recent summary). In Chapter 5 G ardenfors presents a detailed defense of cognitive semantics, i.e., the theory that the meaning of language expressions consists of a mapping to conceptual structures in the head of an individual, in contrast to the more standard theory that semantics is truth-functional and consists of a mapping of language to the world or a set of possible worlds. A number of interesting implications of cognitive semantics are discussed, like the primacy of semantics, more speci®cally the primacy of image-schemas, to syntax (which reminded me of the now abandoned case grammar of Fillmore). Another implication is that meanings in the heads of dierent individuals may not be the same, which leads to the postulation of a process of socio-cognitive mutual coordination of meanings to approximate common sense. This chapter also contains technical, but very interesting, sections on lexical semantics and the analysis of metaphors, which provide good illustrations of the strengths of cognitive semantics. Chapter 6 separately discusses induction at the symbolic level, the conceptual level, and the subconceptual (arti®cial neural network) level. I will only comment on induction at the conceptual level, but before doing so I would like to cite the closing sentence of the section on induction at the symbolic level: ``In brief, the symbolic approach to induction sustains no creative inductions, no genuinely new knowledge, and no conceptual discoveries.'' This statement nicely captures the source of the learning paradox to which I referred earlier. I was, therefore, very curious to know whether the learning paradox could be solved at the conceptual level. Gardenfors argues at some length that induction at the conceptual level involves a shift in the underlying conceptual space, and refers in this context to the Piagetian theory of cognitive development. However, he does not present a clear causal description of the way in which such a shift could take place. For one thing, I fear that even major changes in the saliences underlying concepts (see above) are insucient to explain genuinely new knowledge such as a child's transition to conservation. A more plausible causal model of major shifts in conceptual spaces is provided by catastrophe theory (see van der Maas & Molenaar, 1992, for an application to Piagetian stage transitions). I have the impression that perhaps Gardenfors himself has considered this possibility, given the number of references to the works of Thom (1972) and Zeeman (1977) (the founders of catastrophe theory) throughout the book. As a
336
Book Review
matter of fact, the same general dierential geometrical theory of catastrophes also can be applied at the subconceptual level of arti®cial neural networks (cf. Raijmakers, van Koten, & Molenaar, 1996). The book closes with chapters on computational aspects and a closing chapter. I recommend the book to readers of Acta Psychologica for its clear and principled introduction and overview of geometrical models of information processing, with expert expositions of especially the manifold relationships with theories and topics in cognitive science in general. References Gallistel, C. R. (1990). The organization of learning. Cambridge, MA: MIT Press. Harnad, S. (1990). The symbol grounding problem. Physica D, 42, 335±346. McGhee, G. R. (1998). Theoretical morphology. New York: Columbia University Press. Mandler, J. (1992). How to build a baby: Conceptual primitives. Psychological Review, 99, 587±604. Molenaar, P. C. M. (1986). On the impossibility of acquiring more powerful structures: A neglected alternative. Human Development, 29, 245±251. Rademacher, F. J. (1996). Cognition in systems. Cybernetics and Systems, 27, 1±41. Raijmakers, M. E. J., van Koten, S., & Molenaar, P. C. M. (1996). On the validity of simulating stagewise development by means of PDP networks: Application of catastrophe analysis. Cognitive Science, 20, 101±136. Shannon, B. (1993). The representational and the presentational: An essay on cognition and the study of the mind. New York: Harvester Wheatsheaf. Thom, R. (1972). Stabilite structurelle et morphogenese. New York: Benjamin. van der Maas, H. L. J., & Molenaar, P. C. M. (1992). Stagewise cognitive development: An application of catastrophe theory. Psychological Review, 99, 395±417. Zeeman, C. (1977). Catastrophe theory: Selected papers 1972±1977. Redwood City, CA: Addison-Wesley.
Peter C.M. Molenaar Department of Psychology University of Amsterdam 1018 WB Amsterdam Netherlands
www.elsevier.com/locate/actpsy
Book Review Conceptual Spaces: The Geometry of Thought; Peter Gardenfors; MIT Press, Cambridge, MA, 2000 Peter Gardenfors is the author of the now classic monograph entitled ``Knowledge in ¯ux: Modeling the dynamics of epistemic states'' which appeared in 1988. Hence I was very curious about the new book of this distinguished author. Below I will try to summarize and discuss what I see as the basic tenets of the new book, mainly from a psychologist's point of view. As Gardenfors indicates in the preface, this book deals with issues related to at least ®ve dierent disciplines: philosophy, computer science, psychology, neuroscience and linguistics. Given my own background, and in view of what I expect to be the main interests of readers of Acta Psychologica, it would seem prudent to emphasize those aspects of the book which might have a bearing on psychology. The book is devoted to the way in which representations should be modeled (nowhere in the book is the necessity of introducing representations questioned; for an insightful critique of this assumption, cf., e.g., Shannon, 1993). Alongside the mainstream symbolic and connectionistic models of representation, Gardenfors discerns an additional class of so-called conceptual models. A conceptual model of representation consists of quality dimensions spanning up a topological space. With respect to connectionistic models a conceptual model is considered to be more coarse-grained, whereas it is more ®ne-grained with respect to symbolic models. A topological space comes equipped with structure about neighborhoods, and in Chapter 1 Gardenfors explains betweenness and equidistance relations which, together with the introduction of some metric, enable the natural de®nition of similarity relations which turn out to be of prime importance in the remainder of the book. All this is explained very clearly and does not require any special expertise of the reader. Chapter 1 ends with methods to identify quality dimensions (multidimensional scaling) and ideas about the origin of quality dimensions (e.g., nativism). Before proceeding, I have to mention some additional distinctions made by G ardenfors, namely, between representations within a constructive context and those within an explanatory context. Representations within a constructive context need not have any psychological validity, but serve scienti®c and constructive (e.g., robotics) purposes. On the other hand, representations within an explanatory context aim to describe psychological structure, and hence are called phenomenal. The use of multidimensional scaling and deliberations about the origin of quality dimensions should be located at this phenomenal level. Another distinction at the phenomenal level concerns integral versus separable dimensions. In contrast to 0001-6918/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 1 - 6 9 1 8 ( 0 0 ) 0 0 0 6 7 - 6
334
Book Review
separable dimensions, a set of dimensions is integral if an object cannot be assigned a value in one dimension without giving it a value in the other ones (for instance, hue and brightness are integral). Like most of the book, I will concentrate on the phenomenal level (which becomes especially important in the chapter on semantics). The distinction between integral and separable dimensions is used in Chapters 3 and 4 in the de®nitions of property and concept. Chapter 2 discusses at some length the symbolic, connectionistic (subconceptual), and conceptual approaches to representation. The author is at his best in describing the limitations of symbolic representations, including the frame problem and the problem concerning the origin of new predicates. The latter problem is related to the learning paradox in cognitive developmental psychology (cf. Molenaar, 1986). At the subconceptual level, some problems of connectionistic representations are brie¯y considered (large number of training trials; questions about the relationships with conceptual and symbolic levels). The section on conceptual models stresses, among other things, the importance of similarity relations and the ease with which these can be handled within geometrical representations. Comparisons with similar theories of Gallistel (1990), Harnad (1990), Mandler (1992) and Rademacher (1996) are made. Chapters 1 and 2 comprise the general introduction of conceptual models of representation. A conceptual model is a geometrical model, and I expected that the author would pay due attention to one of the hallmarks of any geometrical approach, namely the speci®cation of invariants under coordinate transformations. Gardenfors clearly indicates that quality dimensions at the phenomenal level have to be identi®ed in painstaking empirical research. Hence the choice of a particular set of quality dimensions could be made more robust by determining which aspects remain invariant under alternative choices (similar remarks can be made with respect to the choice of a metric). Unfortunately, however, this fundamental topic is not addressed in the present book. Chapter 3 is a beautiful chapter on properties, de®ned as regions in conceptual spaces spanned up by sets of integral dimensions, while naturalness is de®ned as the convexity of such regions. G ardenfors discusses many interesting implications of these geometrical de®nitions, including relationships with the prototype theory of categorization. A powerful technique to create convex regions (Voronoi tesselation) is presented in detail. G ardenfors shows by means of a clear illustration that convexity is not invariant under coordinate transformations and/or change of metric. Given the importance of convexity in the de®nition of natural properties, and because the choice of a particular set of dimensions and metric is to some extent arbitrary, one would like to have available some analogue of convexity which is invariant under coordinate transformations and/or changes of metric. This is a particular instance of the general point which I made in the previous paragraph. To wit, Gardenfors partly addresses this point in the section discussing the relativism of conceptual spaces. In that section, however, invariants are not discussed, but the relativism concerned is moderated by an appeal to evolutionary theory. The next chapter about concepts proceeds in a similar way. A concept is de®ned as a weighted combination of regions in a conceptual space spanned up by a set of separable, usually oblique, dimensions, while naturalness is again de®ned as con-
Book Review
335
vexity of the regions concerned. The weights in each combination are called saliences and are understood to be context-dependent. Gardenfors introduces an extremely interesting concept combination rule to cover cases like: ``a tall Chihuahua'', which of course is not the same as ``a tall dog''. Concept learning is explained by means of generalization from a ®nite number of exemplars, using again the Voronoi tesselation technique. Whether this would also explain stage transitions involving complete qualitative restructuring of conceptual spaces, such as has been envisioned in Piagetian theory, would seem to be an open question. The chapter ends with a detailed presentation of two experiments on categorization, using shell shapes drawn by a graphics program. Such graphics programs are important tools in theoretical morphology, where also relevant dynamical theories of form evolution and ontogenesis have been proposed (the latter dynamical theories might provide clues for alternative concept learning processes; cf. McGhee, 1998, for a recent summary). In Chapter 5 G ardenfors presents a detailed defense of cognitive semantics, i.e., the theory that the meaning of language expressions consists of a mapping to conceptual structures in the head of an individual, in contrast to the more standard theory that semantics is truth-functional and consists of a mapping of language to the world or a set of possible worlds. A number of interesting implications of cognitive semantics are discussed, like the primacy of semantics, more speci®cally the primacy of image-schemas, to syntax (which reminded me of the now abandoned case grammar of Fillmore). Another implication is that meanings in the heads of dierent individuals may not be the same, which leads to the postulation of a process of socio-cognitive mutual coordination of meanings to approximate common sense. This chapter also contains technical, but very interesting, sections on lexical semantics and the analysis of metaphors, which provide good illustrations of the strengths of cognitive semantics. Chapter 6 separately discusses induction at the symbolic level, the conceptual level, and the subconceptual (arti®cial neural network) level. I will only comment on induction at the conceptual level, but before doing so I would like to cite the closing sentence of the section on induction at the symbolic level: ``In brief, the symbolic approach to induction sustains no creative inductions, no genuinely new knowledge, and no conceptual discoveries.'' This statement nicely captures the source of the learning paradox to which I referred earlier. I was, therefore, very curious to know whether the learning paradox could be solved at the conceptual level. Gardenfors argues at some length that induction at the conceptual level involves a shift in the underlying conceptual space, and refers in this context to the Piagetian theory of cognitive development. However, he does not present a clear causal description of the way in which such a shift could take place. For one thing, I fear that even major changes in the saliences underlying concepts (see above) are insucient to explain genuinely new knowledge such as a child's transition to conservation. A more plausible causal model of major shifts in conceptual spaces is provided by catastrophe theory (see van der Maas & Molenaar, 1992, for an application to Piagetian stage transitions). I have the impression that perhaps Gardenfors himself has considered this possibility, given the number of references to the works of Thom (1972) and Zeeman (1977) (the founders of catastrophe theory) throughout the book. As a
336
Book Review
matter of fact, the same general dierential geometrical theory of catastrophes also can be applied at the subconceptual level of arti®cial neural networks (cf. Raijmakers, van Koten, & Molenaar, 1996). The book closes with chapters on computational aspects and a closing chapter. I recommend the book to readers of Acta Psychologica for its clear and principled introduction and overview of geometrical models of information processing, with expert expositions of especially the manifold relationships with theories and topics in cognitive science in general. References Gallistel, C. R. (1990). The organization of learning. Cambridge, MA: MIT Press. Harnad, S. (1990). The symbol grounding problem. Physica D, 42, 335±346. McGhee, G. R. (1998). Theoretical morphology. New York: Columbia University Press. Mandler, J. (1992). How to build a baby: Conceptual primitives. Psychological Review, 99, 587±604. Molenaar, P. C. M. (1986). On the impossibility of acquiring more powerful structures: A neglected alternative. Human Development, 29, 245±251. Rademacher, F. J. (1996). Cognition in systems. Cybernetics and Systems, 27, 1±41. Raijmakers, M. E. J., van Koten, S., & Molenaar, P. C. M. (1996). On the validity of simulating stagewise development by means of PDP networks: Application of catastrophe analysis. Cognitive Science, 20, 101±136. Shannon, B. (1993). The representational and the presentational: An essay on cognition and the study of the mind. New York: Harvester Wheatsheaf. Thom, R. (1972). Stabilite structurelle et morphogenese. New York: Benjamin. van der Maas, H. L. J., & Molenaar, P. C. M. (1992). Stagewise cognitive development: An application of catastrophe theory. Psychological Review, 99, 395±417. Zeeman, C. (1977). Catastrophe theory: Selected papers 1972±1977. Redwood City, CA: Addison-Wesley.
Peter C.M. Molenaar Department of Psychology University of Amsterdam 1018 WB Amsterdam Netherlands