More direction needed: Representing Direction in Language and Space, edited by E. van der Zee, and J. Slack, Oxford University Press, 2003. $99.00 (hbk)/ $39.95 (pabk) (282 pages) ISBN 0 19 926018 4

View Large Image | Download PowerPoint SlideThis is a book of separately authored chapters derived from a workshop where linguists, psychologists, neuroscientists and computers scientists interested in spatial language and the mental representation of space came together to discuss their findings and theories. For researchers who want to know what is being done in this area, it is a very useful book. However, like most such assemblages, it lacks overall cohesion, and there is considerable redundancy. The editors open with an overview chapter. There are several chapters on axes, directions and vectors, two chapters on spatial prepositions in English, and a third that tries to anchor the treatment of English spatial prepositions in neuroscientific findings. Then there is a chapter on spatial prepositions in Finnish, one on how spatial features determine reference axis categorization, and one on memory for locations relative to the principal axes of objects.The book cannot be recommended for students, because it contains considerable confusing material about vectors, axes, half lines, directions, and so on. For example, in the opening paragraph of the chapter on ‘reconciling axis and vector representations’, we read ‘reference frames are dependent on an axial system, whereas spatial templates can be best characterized as a vector representation.’ The student who never had linear algebra would not suspect that axes and vectors are a conceptual package. If you want to make vectors (that is, ordered n-tuples, or strings of n numbers) refer to sensible locations in the world or to points in the pure (insensible) space of a geometer, you need axes. That was the essence of Descartes' and Fermat's insight. Moreover, there is little point in defining axes if you do not use them to relate a vector space to a plane or volume or to locations in the sensible world (that is, to a physical space).In the introductory chapter, we read ‘There is no a priori reason why the primitives encoding direction must be spatial entities such as axes or vectors. The currency in which direction can be formally or cognitively encoded might as well be algebraic or propositional.’ The student would not know from this that vector is an algebraic, not a geometric concept. Modern textbooks of geometry often begin with a ‘purely geometric’ treatment, using the proof methods of Euclid and his successors over the 2000 years between his work and the 17th century. These purely geometric methods make no use of axes and vectors. Then, most modern textbooks go on to develop the subject algebraically, building on the (relatively recent) inventions of Fermat and Descartes, which made it possible to do geometry algebraically. That's when the axes and vectors come in.Later in the introductory chapter, we read ‘[other authors'] axial systems can capture direction but not distance. To incorporate distance would require basic metrical concepts to be added to the framework.’ But direction is derivative of angle, and angle is a metric concept. Formally, distance and angle come together as a package (we have the dot product to thank for both). There is no geometry in which angle is defined and distance is not.Confusion about what vectors are runs through half or more of the chapters. They are treated as point pairs (as in the tail and head of an arrow), straight lines (that is, infinite sets of points), axes (ditto), and, above all as arrows, the overly literal interpretation of which seems to be the source of much of the confusion, because arrows have a tail and a head connected by a straight line. Vectors in a two-dimensional vector space are ordered pairs of numbers. They may be interpreted geometrically either as point vectors (coordinates), in which case vector is synonymous with (isomorphic to) point, or as instructions to move, in which case a vector is an operator that maps the whole plane onto itself.Near the end of her chapter, Barbara Landau writes: ‘In the literature, there appears to be considerable variability in what people mean by reference system, frame of reference, coordinate system, and vector representation. Significant progress awaits clear, precise and consistent use of these terms.’ I agree. It might also be noted that clear, precise and consistent use of these terms has been the norm in college level courses on linear algebra and algebraic geometry for generations. That might be a good lead to follow. This is one area where cognitive scientists do not need to reinvent the wheel.