Many years ago I read somewhere how curious it is that people say, for example, "three white sheep" but never "white three sheep". "Numbers are a particular kind of adjective," I read, but there the analysis ended. I remember reflecting that the "particularity" probably lay in the fact that, while "white" modifies each sheep, the word "three" modifies the *set* of sheep.

In the last chapter of our present book, I was finally glad to read these, and other thoughts towards an analysis of the above-described phenomenon. Here "three", along with other "referential number words", is, in fact, not thought of as an "adjective" but as a "quantifier", like "few" and "many". Moreover, "number words" that are not cardinals, but ordinals ("first", "second"...) are compared, grammatically, to "superlatives" ("best", "worst"...), while "number words" that are "nominal" (meaning, the number is used like a name, such as "the #21 bus", and might or might not have anything to do with the actual number used) are (predictably) compared grammatically to proper names. Other interesting phenomena are also brought out in this chapter, as well as throughout the rest of this book. For example, the author explores the question to what extent an ordinal number *sounds like* its corresponding cardinal number. Of course, this could depend on the language, and in fact, even in English, "first" sounds nothing like "one", nor does "second" sound anything like "two". In certain other languages, there is even more discrepancy, not only with the smaller numbers. Moreover, there is discrepancy between some "non-referential counting words", and their corresponding "quantifiers". Indeed, there is much here that is interesting — and interesting in a mathematical kind of way.

However, the author's main point in writing this book is to make "the case for a linguistic foundation of systematic numerical cognition" (p. 4) — that is, to show that (from the book jacket), "language as a human faculty plays a crucial role in the emergence of systematic numerical thinking". And while I don't doubt this premise, I found it difficult to relate it to the many interesting passages, and vice versa. Moreover, it seems to me that it's pretty clear, to just about anybody who has lived for any reasonable length of time, that numbers and words are intimately and intricately related, and that cognition of any one depends on cognition of the other. And of course, to do *anything* systematic (in particular, to write anything down) — be it numbers or politics or art — we need language. In other words, although the book has some fascinating things to say about "numbers, language, and the human mind", I feel that, all told, its message isn't anything terribly surprising.

Years ago I reviewed (for The American Mathematical Monthly) a book by Keith Devlin called *The Math Gene* . The point of Devlin's book is, in his words, to show that "the ability to do mathematics is based on our facility for language". In other words, his book seems to have been written for the same pupose as our present book. Yet Wiese seems unaware of Devlin's work and — I checked the index, besides reading every word of this book — does not ever refer to it. I found this curious, but not surprising, given that the two authors work in different fields, and in general belong to different "clubs".

Perhaps more important in comparing these two books: In my review of Devlin's book I wrote, "Correlation is not proof. Nor are interesting ideas. Nor 'worlds'." Perhaps mathematicians are too hard to please or convince. But — I said it for Devlin's book and I'll say it for Weise's — "In the end, this, for me, is just another interesting and impressive book."

Many of her arguments are based on the concept of "dependent linking". A "dependent linking" is defined to be a mapping between relational structures. For example, the relation "greater than", among cardinal numbers, is associated with the "empirical relation" "has more elements than", among sets. (In math language, it's an isomorphism between the categories of cardinals and sets.) Her argument is that, since dependent linking is a linguistic thing, and since we need dependent linking in order to systematically study numbers, we therefore need language in order to systematically study numbers. To me, however, that argument seems to involve too many variables. For instance, *is* dependent linking a linguistic thing? Also, the mathematical mind often works intuitively and "skips steps", thus perhaps not needing dependent linking, or not as obviously as the author believes.

Here, on p. 112, are some of her main arguments: "(1) the emergence of language as a symbolic system provided the basis for the use of numbers as tools, by giving us a cognitive pattern for dependent linking, (2) the linguistic routine of conventional association (associations between linguistic signs and their referents) supports the use of counting words as conventional tools in this linking, and (3) the recursive rules underlying linguistic constructions give us a handle on infinity and can be put to use in our number domain through the medium of counting words... "

Indeed, commendably, there are many "internal summaries" throughout the book. In fact, there is much repetition. I'm an author and teacher who likes to summarize, and to repeat, and in general to explain. So I was interested to find that too much summary and explanation can be confusing. "Haven't I seen this before?" I'd think, and "Why is this in *here*?" When I teach, I try to *announce* when what I'm doing is review, to spend the appropriate amount of time reviewing (not too much), and in general to *place* the reviewing in a comfortable context. I hope that I'm successful in this, and that I don't make students feel the way I felt while reading certain passages in this book.

It is, again, commendable that the author takes great pains to be clear. (And many of her metaphors relating her subject matter to everyday life are extremely well-taken.) However, at least for me, she was often *not* clear, or at least I had trouble, again, seeing the grand scheme. Perhaps, for that, a second, and third, reading is necessary. What I got out of the book was indeed great headway towards (in her words, p. 8) "exploring the distinctive way in which numerical cognition is intertwined with the human language faculty." (Often I couldn't tear myself away — for instance, when she writes about babies and animals "counting", or the effects of various mental disabilities on mathematical ability. And I'd liked the section on "number forms" (beginning p. 224); these are mental pictures which some individuals have of the numbers. I enjoyed asking myself whether I have a number form, and I realize that my number form is like a website; I visualize a "home page", which includes the numbers up to approximately 100. I can then "click" forward, or backward, whenever I want to, even to the transfinites (and I can "zoom in" to very small "fractions", or infinitesimals" (when I choose to believe in them...). )

However, again, I didn't get much beyond this exploration. The book jacket says, "She characterizes number sequences as powerful and highly flexible mental tools that are unique to humans and shows that it is language that enables us to go beyond the perception of numerosity and to develop such mental tools: language as a mental faculty not only lies at the heart of what makes us human, it is also the capacity that lays the ground for our concept of number." However, I didn't get much broad insight into the human condition, nor was my pride and joy in being human enhanced. Are there implications for teachers, students, parents? Are there "life lessons" to be learned, by individuals or by society? Indeed, does this book give us anything truly profound?

Marion D. Cohen has just published a poetry book, available from Plain View Press (http://www.plainviewpress.net), about the experience of mathematics. The title of the book is “Crossing the Equal Sign”. She would love to receive emails at: [email protected]