Numbers, language and the human mind (review)

This book is sure to become a classic in its field. As declared on the dust jacket by luminaries such as Terrence Deacon (evolutionary anthropology), Karen Wynn (developmental psychology), and Ray Jackendoff (linguistics), it provides an impressive synthesis of findings and leading ideas across multiple fields, including the philosophy of mathematics, as well as those just mentioned. That it does all this in clear and accessible language will make the book an invaluable reference work for virtually anyone interested in 'exploring the distinctive way in which numerical cognition is intertwined with the human language faculty' (8). Just as importantly, Wiese advances her own account of how key insights can be stitched together into a coherent story that has explanatory force at different levels—cognitive, linguistic, developmental, and evolutionary. The book has an introduction, eight substantive chapters, and five appendices containing technical information on mathematical modeling of number systems.

What are numbers really? This question is the subject of the two opening chapters: 'Numbers and objects' (9–46) and 'What does it mean to be a number?' (47–74). These run through some of the leading ideas from the philosophy of mathematics about the nature of number and what is involved in making a number assignment (saying, for example, 'there are three pens on my desk'). W reviews different views on the nature of numbers by Gottlob Frege, Bertrand Russell, and Richard Dedekind, all of which rely on set-theoretical concepts in one way or another. The key ideas are that cardinality ('how many') is a property of a set of items, and that among the essential properties of cardinal numbers are that they be distinct (well distinguished) from one another and form a strict progression (i.e. they are in a successor relationship). In addition to indicating cardinality (three runners), W distinguishes two other kinds of number assignment: ordinal (the third runner) and nominal/labeling (runner #3).

Ch. 1 also informally introduces the representational theory of measurement (RTM), which W says can provide a unified account of all forms of number assignment. The key element is the notion of dependent linking, that is, the systematic mapping from an 'empirical relational structure' of some kind onto the numerical-reference relationship structure. The psychological plausibility of this theory is perhaps open to question, but W keeps it in the background for most of the book. This is a good thing in the sense that explicit reference to the RTM would interfere with an otherwise accessible exposition, but it also means that certain of W's theoretical or ontological assumptions are kept out of view. Simply put, W seems to accept a realistic view of numbers, with the implication that number words are conceptual tools for working with preexisting realities. Alternative intuitionalist, constructivist, or cognitivist views of numerical cognition (cf. Shapiro 2000, Lakoff & Núñez 2000) do not feature in this book.

The book really takes off with Ch. 3, 'Can words be numbers?' (75–100), the longest chapter.

The first part of the argument is that numbers begin ontogenetically as counting words, and that counting words do not REFER to numbers, but are better viewed as numerical tools that can be used as numbers. The English language of course has an infinite progression of counting words generated by morphosyntactic rules, but as Wpoints out, in some cultures, such as the Oksapmin of Papua New Guinea, the same function is served by a system involving sequential touching of body parts. The crucial thing is to have a fixed sequence of clearly differentiated items—be they words, fingers or other body parts, or some other kind of 'props'—that can be paired oneto one with the items being enumerated.

Counting words are initially acquired by young children as a series of meaningless words in an unbreakable string or chain, like the elements in a nursery rhyme or the letters of the alphabet. It is only afterwards that children learn to pair the counting words with objects 'in a stylised repetitive...