Conceptual Spaces: The Geometry of Thought (Bradford Books) [Hardcover]

Your choice of qualitative measures deeply affects how you understand the world. 'Spose reality is an infinitely dimensional, then we have lots of choices for axes. We simplify and correlate by using all that coordinate transformation and axis projection stuff from 3D graphics! Heck Gardenfors even uses Delauney Triangulation (or polyhedralization).

Criterion P, page 71

A natural property is a convex region of a domain in a conceptual space.

Criterion C, page 105

A natural concept is represented as a set of regions in a number of domains together with an assignments of salience weights to the domains and information about how the regions in the different domains are correlated.

Concept Combination, page 122

The combination CD of two concepts C and D is determined by letting the regions for the domains of C, confined by D replace the values of the corresponding regions for D. (contrast class p. 119), for example the "stone lions" outside the NYC library.

Six Tenets of Cognitive Semantics, page 160

i) Meaning is a conceptual structure in a cognitive system (not truth conditions in possible worlds)
ii) Conceptual Structure are embodied (meaning is not independent of perception or of bodily experience).
iii) Semantic elements are constructed from geometrical or topological structures (not symbols that can be composed according to some system of rules).
iv) Cognitive models are primarily image-schematic (not propositional). Image-schemas are transformed by metaphoric and metonymic operations (which are treated as exceptional features on the traditional views).
v) Semantics is primary to syntax and partly determines it (syntax cannot be described independently of semantics).
vi) Concepts show prototype effects (instead of showing the Aristotelian paradigm based on necessary and sufficient conditions).

Process of Abstraction, page 191 - Start with a collection of things. Identify and quantify individual objects. The determine the clusters. Step three: abstract the clusters into dimensions. Simple!

I especially liked the notion that a metaphor is taking the spatial relationship of a cluster of concepts in one domain and using them in a new domain to help understand the new domain.

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.

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.