Conceptual Spaces: The Geometry of Thought (A Bradford Book) by Peter Gardenfors (PDF)

Description

Within cognitive science, two approaches currently dominate the problem of modeling representations. The symbolic approach views cognition as computation involving symbolic manipulation. Connectionism, a special case of associationism, models associations using artificial neuron networks. Peter Gärdenfors offers his theory of conceptual representations as a bridge between the symbolic and connectionist approaches.Symbolic representation is particularly weak at modeling concept learning, which is paramount for understanding many cognitive phenomena. Concept learning is closely tied to the notion of similarity, which is also poorly served by the symbolic approach. Gärdenfors’s theory of conceptual spaces presents a framework for representing information on the conceptual level. A conceptual space is built up from geometrical structures based on a number of quality dimensions. The main applications of the theory are on the constructive side of cognitive science: as a constructive model the theory can be applied to the development of artificial systems capable of solving cognitive tasks. Gärdenfors also shows how conceptual spaces can serve as an explanatory framework for a number of empirical theories, in particular those concerning concept formation, induction, and semantics. His aim is to present a coherent research program that can be used as a basis for more detailed investigations.

User’s Reviews

Editorial Reviews: Review This is a fearless book that casts a wide net around key issues in cognitive science. It offers the kind of coherent, unified view that the field badly needs.―Steven Sloman, Associate Professor, Cognitive and Linguistic Sciences, Brown University About the Author Peter Gärdenfors is Professor of Cognitive Science at Lund University, Sweden. He is the author of Conceptual Spaces: The Geometry of Thought (MIT Press) and other books.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐If one is to design a machine that can formulate concepts and engage in such things as inductive inference and its corollary scientific discovery, then one must be able to quantify the notion of a concept in such a way that it can be implemented into the cognitive structure of the machine. One must be able to distinguish one concept from another, be able to tell when one concept is similar to another, and understand in detail how concepts are related across domains. It would not be enough to have qualitative notions of these distinctions or similarities, since they must be able to be formatted in such a way, either via coding, language, or electronically, so as to be used by the machine. This book gives an interesting approach to the problem of concept classification, but it does so only from a qualitative point of view. It is a good start in this regard, and readers will gain a lot of insight into the problems that it addresses. It does not however give any advice on how to implement its ideas into a real thinking machine. Mathematical concepts are brought in order to talk more meaningfully about spaces of concepts, but they are really restricted to metric spaces and not general enough to deal with the plethora of concepts that could present themselves in typical environments. The book should be considered more as a work in philosophy, so those interested in this field might enjoy the book more than those who were expecting a book more geared towards artificial intelligence and computer science. Those readers interested in automated theorem proving or automated mathematical discovery might find the discussion on geometric categorization models of interest, and will find an interesting application of Voronoi tessellations, namely that of accounting for the varying sizes of concepts in a categorization.By far the most interesting chapter in the book is chapter 6, wherein the author gives a highly original discussion of inductive inference. The ability of human cognition to generalize from a limited number of observations is viewed (correctly) by the author as very impressive, but he is careful to note that inductive inference cannot be done free of side constraints. Quoting the philosopher J.S. Peirce and his evolutionary explanation of why induction is so effective, the author uses his theory of conceptual spaces to develop a theory of constraints for inductive inferences. The main notion in this theory is that of “projectability”, which attempts to delineate the properties and concepts that are may be used in inductive inference. The author wants to arrive at a computational model of induction, and he offers interesting proposals for doing so, even if they lack immediate empirical justification. Central to the problem of induction the author argues is how observations are to be represented. This has been neglected in the history of philosophy he says, and so he then proceeds to outline his ideas on how to represent observations, distinguishing three levels, namely the `symbolic’, the `conceptual’, and the `subconceptual.’ At the symbolic level, observations are represented by describing them in a specified language. At the conceptual level, observations are characterized relative to a conceptual space. At this level induction is viewed as concept formation. At the subconceptual level observations are characterized by inputs from sensory receptors. Induction is then viewed as the attaining of connections between various inputs. The author views the processing taking place in artificial neural networks as an example of modeling at the subconceptual level. The problem of induction is more complicated than is typically presented in the literature, the author argues. Inductive inference will look different depending on which approach to observations is taken. In his elaborations on the processes of induction, one of the key issues that arises is the how discovery takes place across different domains. The process of conceptualizing across different domains takes place, as expected, at the subconceptual and conceptual levels. The symbolic level is delegated to formulating laws.

⭐Profound piece of work. I am not a cognitive scientist, and this book is a bit technical, but it is still within reach of the motivated lay person.Gardenfors puts forward a a model to explain cognition that he calls “conceptual spaces.” These conceptual spaces are at a level of abstraction in between the symbolic (used by AI types) and connectionist (Neural Nets). But what makes his conceptual spaces interesting and plausible is the position he takes that in this conceptual space, most reasoning is done by evaluating the analog of a distance between two aspects of a perception. Or, we find things to be similar if they are “geometrically” (measurably) closer on some limited number of dimensional scales.This is easy to follow for things like colors, but he doesn’t stop there. He goes on to describe how this explains a wide variety of perceptions, as well as how we form and reform categories and concepts, and shows how this informs semantics and the process of induction.My only criticism is that some of the illustratios would have been more powerful in color.

⭐I like the book content-wise, but the binding arrived shattered. I’m going to try to fix it because I don’t want the hassle of mailing it back. This was supposed to be a new book, so of course I’m disappointed.

⭐What are conceptual spaces? What does the concept of conceptual spaces mean? Good questions! Basically, they are a framework for representations of reality, defined in terms of relations between categories of reality. In the words of Peter Gärdenfors (p. 5):”The epistemological role of the theory of conceptual spaces….is to serve as a tool in modeling various relations among our experiences, that is: what we perceive, remember or imagine”.When I purchased the book and read through it, I instantly appreciated its abstract emphasis, but also wanted to explore how or to what extent its quality aspects might be more quantified, and then applied to actual practical investigations. I consumed more than two months exploring these, especially in terms of Voronoi Tesselations, as well as other aspects. The ultimate aim was to see if they might be applied in a practical form to something like the JFK assassination (say in comparing claimed (by Dartmouth computeer scientist, Hany Farid) authentic images…like the Oswald backyard photos…with fake ones and possibly deducing authenticity….or the hand of fakery).In general any given conceptual space approach will deliberately separate the relational concepts into domains. The choice of domain by category will depend on the issue or topic for which the modeling is done. Each domain is configured then into quality dimensions from which the causes, agents and outcomes can be better assessed. Okay, but how might one use that to approach analysis of aspects of a historical event?Well, in the most generic portrayal of interacting agents in the primary historical event of 11/22/63, the three conceptual space quality dimensions (appended to each of 3 orthogonal axes) might designate: P = Political (political inputs, including fear of proceeding or attaining a full investigation, Sc = scientific or what the scientific inputs, e.g. ballistic analysis show, and H = Historical or factors which affect the information quality and content via action that are purely historical. H can also dimensionalize where (potentially) disinformation has arisen, or has contaminated previous work, perceptions or conclusions.Fine as it goes, but how to extrapolate some of the more basic approaches (e.g. covered in Chapters 3, 4) into concrete formats to see where they lead? Key in here Gärdenfors inclusion of perception. How so in the case of the historical event under study?Shadows, angles, perspective lines, displacements from image normality (i.e. tilt of a person’s standing angle beyond what a vertical plumbline through his center of gravity would allow), all of these can betray image manipulation and deception. The trick then is to be able to ferret them out and expose what was attempted by the image technician or manipulator. Many techniques have been used, especially to do with study of detailed images from the JFK assassination (Z-film frames, backyard photos, etc.), some even relying only on computer digital technology alone.Basically, the most rudimentary but quantitative conceptual analysis must then be contingent on specific geometry and recognition of basic properties of spaces. The two spaces used almost interchangeably – are called Euclidean and Cartesian. (Euclidean space can be generalized to an n-dimensional Cartesian space, R^n.)In this light, Chapter Three was particularly useful to me (especially Gärdenfors’ mapping template on p. 94), to quantify and show how photos of the time could easily have been manipulated, as I did in Ch. 8 of my book:’The JFK Assassination: The Final Analysis’.Second, a higher level of quantitative approach was needed by the time I got to how pixels might have been manipulated in images from photo to photo, frame to frame. Gärdenfors’ Chapter Four was ideally suited for this and specifically the ‘prototypes’ -e.g. ith coordinate of such, as discussed on pages 124, 125.At some point, however, in order to show finer details – for example of how Gärdenfors’ equation 4.2 would apply to pixel displacements that might be quanitified in fractions of a mm, I had also to incorporate fractional calculus. (See also my book review of ‘The Fractional Calculus’) But this proved very useful and I was able to actually quantify a number of significant manipulated displacements, which I retained in terms of ‘delta p_i’. (I reserved this extensive analysis for my Appendix).Gärdenfors’ treatment of vertical and horizontal “expansion rates” (pp. 144-45) also proved to be quite useful in showing at least one photo of LHO was obviously fudged, in that his standing angle would – if actually valid- have forced him to fall over to one side! (Stevens’ power law, given on p. 144, also factored into this).My point in all this is to try to show that Gärdenfors’ superb text can be read (and used) at different levels. It can be read, first, as a remarkable introduction to a “geometry of thought” whereby one assigns quality dimensions which “can be used to assign properties to objects and specifiy relations between them”. (p. 6)But the material (especially in Chapters Three and Four) can also form the foundation for a more rigorous, quantitative analysis when applied to particular investigations in which perspective and perception might play a role. I also really applauded his discussion (in Chapter One, e.g. pp. 20-21) of how a Euclidean space can be generalized to an n-dimensional Cartesian space. I wonder, for example, how many readers are fully aware of all the uses of the ‘Minkowski metrics’ he shows on page 20.What I’d basically say is this: the full scope and use of the material of this wonderful text is limited only by the imagination and determination of the user. For me, it not only opened doors of insight, but enabled definite breakthroughs to accomplishing objectives that had hitherto been stymied.The book is not for everyone, but for those with inquiring minds and imaginations, it is a welcome treat! And perhaps even an indispensable tool!

⭐Gärdenfors’ Anliegen ist es, die Kluft zwischen symbolischen und neuronalen Ansätzen in der Kognitionswissenschaft zu schließen, und es dürfte kaum überraschen, daß er dabei neben der Erklärung auch konstruktive Anwendungen im Blick hat. Seinen Vorschlag eines »conceptual space« führt er mit einleuchtenden Argumenten ein: demnach ist es hilfreich, sich Konzepte und Eigenschaften als mehrdimensionale Vektorräume vorzustellen, mit deren Hilfe sich Prototypen und Ähnlichkeiten ebenso zwanglos erklären lassen wie nichtmonotones Schließen – alles Bereiche, die in der symbolischen KI zu Kopfzerbrechen führen.Angesichts der gegenwärtigen Debatte um das »semantische Web«, in der Konzepte aus der symbolischen KI wie first-order logic, inference engine oder frame wie selbstverständlich gehandhabt werden, kommt das Buch auch gerade zur rechten Zeit – als Denkanstoß nämlich, daß das Web auf diese Art zwar strukturierter, aber nicht unbedingt automatisch intelligenter werden dürfte. [Wohlgemerkt: Es ist kein Buch über das semantische Web, sondern kognitionswissenschaftliche Grundlagenforschung.]In meiner Sammlung kognitionswissenschaftlicher Klassiker steht das Buch »Conceptual Spaces« jedenfalls ziemlich genau zwischen Lakoff’s »Women, Fire, and Dangerous Things« und »Understanding Intelligence« von Pfeifer/Scheier.

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