Conceptual Spaces: The Geometry of Thought [Hardcover]

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.

Drawbacks of the book? The lack of conceptualization when it comes to dynamic concepts (treated very superficially). Also, the theory is deficient when modeling the functional aspects of concepts (a "sin" already recognized by the author).

But considering the pioneering character of this piece of art, these drawbacks are just compelling invitations for further research in the field.

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.