### Geometry and Meaning

C. J. "Keith" van Rijsbergen
2006 Computational Linguistics
Let no man enter here who is ignorant of geometry. -Plato Geometry and Meaning is an interesting book about a relationship between geometry and logic defined on certain types of abstract spaces and how that intimate relationship might be exploited when applied in computational linguistics. It is also about an approach to information retrieval, because the analysis of natural language, especially negation, is applied to problems in IR, and indeed illustrated throughout the book by simple
more » ... using search engines. It is refreshing to see IR issues tackled from a different point of view than the standard vector space (Salton, 1968) . It is an enjoyable read, as intended by the author, and succeeds as a sort of tourist guide to the subject in hand. The early part of the book concentrates on the introduction of a number of elementary concepts from mathematics: graph theory, linear algebra (especially vector spaces), lattice theory, and logic. These concepts are well motivated and illustrated with good examples, mostly of a classificatory or taxonomic kind. One of the major goals of the book is to argue that non-classical logic, in the form of a quantum logic, is a candidate for analyzing language and its underlying logic, with a promise that such an approach could lead to improved search engines. The argument for this is aided by copious references to early philosophers, scientists, and mathematicians, creating the impression that when Aristotle, Descartes, Boole, and Grassmann were laying the foundations for taxonomy, analytical geometry, logic, and vector spaces, they had a more flexible and broader view of these subjects than is current. This is especially true of logic. Thus the historical approach taken to introducing quantum logic (chapter 7) is to show that this particular kind of logic and its interpretation in vector space were inherent in some of the ideas of these earlier thinkers. Widdows claims that Aristotle was never respected for his mathematics and that Grassmann's Ausdehnungslehre was largely ignored and left in obscurity. Whether Aristotle was never admired for his mathematics I am unable to judge, but certainly Alfred North Whitehead (1925) was not complimentary when he said: The popularity of Aristotelian Logic retarded the advance of physical science throughout the Middle Ages. If only the schoolmen had measured instead of classifying, how much they might have learnt! (page 41)