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To a mathematician, the answer is that all three describe algebraic structures. To a linguist, however, all three are examples of "ambiguous" words, that is, words with more than one meaning.

Human beings are quick to determine the meanings of such ambiguous words from their context if the list had been "ring, clip, and pin", or "ring, vault, mat, and bars", our mental image of the word "ring" would change.

Two supposed examples are the apocryphal translation of these two common expressions: "The spirit is willing, but the flesh is weak," and "Out of sight, out of mind," into "The vodka is good, but the meat is rotten," and "Invisible Idiot" respectively. Both of these translations were the invention of human sense of humor, although urban legend attributes these to computer error; the persistence of these legends attests to the important role that ambiguity plays in language.

My own one-sentence oversimplification of the strategy is this: words linked by "and" are likely to have related meanings. To drive home this point, I challenge the reader to create a natural-sounding sentence that includes the phrase "parsnips and honor".

Faced with two dissimilar words that are linked by "and" — such as "parsnips and honor" or "Geometry and Meaning" — we have to work hard to seek a context that includes both. If you prefer, you can skip to the end of this digression. I would place myself in an outer ring of this cult.

Her work — particularly that found in the first book — has changed the way art instructors everywhere teach their classes, albeit in the same way that Calculus Reform has changed the approach of all mathematics instructors. That is, either artists incorporate the Betty Edwards approach, or they make a conscious, defendable choice not to use it.

It had always seemed to me that drawing, compared with other kinds of learning, was easy: after all, everything you need to know in order to draw is right there in front of the eyes.

Just look at it, see it, and draw it. What is the problem? Then one day, out of sheer frustration, I announced to my class, "Right. Today we are going to draw upside down. The students, I believe, thought I had gone "round the bend. Her conclusion from this experiment, and the thesis of her first book, is that teaching students to draw really means teaching students to see the world around them. Her second book, appropriately enough, turns this conclusion upside down, and proposes the notion that in order to "see" the answers to perplexing riddles in our own world, it helps to understand the "vocabulary" of drawing.

Drawings whose main feature is a simple horizontal line — many landscapes, for example — tend to be seen as "tranquil", and the lines in an "angry" painting have very different characteristics than those in a "joyful" one.

In both cases, the visual display allows the viewer to ask questions of the display, and with any luck to answer those questions. In both cases, this visual display is personal in the sense that it expects that a person will be its audience. Widdows differs from both Edwards and Tufte in that his work is not about visual displays although there are some of these in the book. The "Geometry" is not the paper-and-pencil geometry of similar triangles or ruler-and-compass constructions.

Instead, it is the geometry of weighted graphs, linear algebra on high-dimensional vector spaces, lattices, and logic. Enter Widdows. The three main audiences for this book are linguists, mathematicians, and computer scientists particularly those interested in artificial intelligence or information engineering.

Widdows has used the material from this book in a graduate course in language and informatics, and has exercises that are available "upon request". In spite of this, Widdows hopes that this book is a gentle introduction to the field for those with a "general scientific interest", and true to this goal, he carefully distinguishes sections that go into great detail for the benefit of a select group, reminding the rest of his readers of the main points and urging them not to get bogged down in details.

His introduction states "my dearest and for a mathematical book, most ambitious desire has been that Geometry and Meaning should be enjoyable. That said, anyone who reads reviews on MAA Online is likely to find this book "enjoyable"; I might read snippets of this book to my linear algebra students, and I would recommended it to my mathematical colleagues. So what is in this book? Chapter 1 "Geometry, Numbers, and Sets" is an introduction to just what its title describes.

In fact, I cruised through the first chapter wondering if it seemed so clear because I am a mathematician, or because of his writing. I got to test this question in Chapter 2 "The Graph Model: Networks of Concepts" , which introduced linguistic terminology like thesauri, corpora a body of texts , kinds, and tokens. Also in Chapter 2 we get the first examples of linguistic graphs, where nouns for example are the vertices, linked by an edge if they are linked by an "and" in the corpus.

It is here also that we get a hint of how to use the graph structure to identify ambiguous words like "ring". These metrics give us a computational way of determining that "Rings, ideals, and fields are all algebraic structures," is a more informative statement than "Rings, ideals, and fields are all mathematical structures," Chapter 4 "Measuring Similarity and Distance" begins by using distance notions to help identify "prototypes" of a class of objects a prototypical algebraic structure might be a group, ring, or field, rather than a monoid.

The topic then turns to symmetry, in particular noticing that "and" is not a transitive way of identifying the meanings of words, and using several notions such as curvature of a graph to identify ambiguous words and keep their various synonyms separate. It is hard to imagine that the reader who does not know the material of Chapter 1 can get his mind all the way around all of these concepts; to his credit, Widdows continues to give both technical and intuitive descriptions of the mathematical techniques he uses.

The linguistic accomplishments that these techniques produce, however, are clear and aimed to impress especially to a novice like me. Widdows supposes that we want to find a place to clean our business suit, and so we do a search of "suit NOT lawsuit". For such searches, the method of Vector Negation which relies on orthogonal projections allows search engines to remove documents with words related to B such as "plaintiff" , and are therefore more efficient than the traditional Boolean methods at calling up only the most relevant documents.

The book ends as every good book should with a chapter or two that sums everything up and points toward future research. Perhaps I have made it clear that I am a stranger to mathematical linguistics. On the other hand, I have the advantage of being able to judge how accessible this material might be to a new audience, and I can say quite honestly that this book opened up a whole new world to me.

I can now speak confidently of latent semantic analysis and corpus data. When I next teach Linear Algebra, I have a whole new set of examples close at hand — examples that are probably more immediate to my student than electrical circuit diagrams.

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