Book Review: The Number Sense: How the Mind Creates Mathematics by Stanislas Dehaene

Numbers seem so basic. The vast majority (but by no means all) of people learn to count at a very early age, and have no problems doing it.

But behind this seeming simplicity lurks a world of complication. It is that world that Stanislas Dehaene explores in The Number Sense .

In the preface, Dehaene states that The Number Sense is aimed at answering a question first posed by Warren McCulloch:

"What is a number, that a man may know it, and a man, that he may know a number?"

Contents of The Number Sense

The Number Sense has 10 chapters, organized into four parts. Part One is entitled "Our Numerical Heritage" and has three chapters:

Chapter 1: Talented and gifted animals covers the surprising amount of number sense that many animals have; this is most prominent in the other primates, but also exists in birds, rats and other animals. Many of them are capable of distinguishing number of objects, if the number is very small. That is, they can tell one object from two, and two from three, but the ability to tell 10 from 20 is rarer. Some primates have a sense of addition.

Chapter 2: Babies who count gives information about how babies, even very young babies, have a sense of number, and evidence surprise at different types of numerical presentations. A certain portion of number sense is present even in newborns, and this develops over time.

Chapter 3: The adult number line gets into how adults represent numbers, and why we have more problems with some parts of this than others.

Part Two of The Number Sense is entitled "Beyond Approximation" and it too has three chapters:

Chapter 4: The Language of Numbers discusses the history of representation of numbers, from the use of body parts to early visual representations, to Roman numerals and on to our current system. It also discusses variations in our current system, and their implications. For example, Chinese uses fewer words than English or other European languages to represent all numbers, and these have fewer syllables as well. In addition, Chinese people are able to remember longer number series. (For instance, in Chinese, there is no separate word for "eleven" or "twelve" or "twenty", and Chinese number words are all one short syllable.

Chapter 5: Small heads for big calculations discusses how children develop algorithms for doing arithmetic, how these go wrong, and why calculators should be used in classrooms from a very early age.

Chapter 6: Geniuses and prodigies is all about (you guessed it) geniuses and prodigies, and how they are and aren't different from the rest of us.

The third part of The Number Sense is "Of neurons and numbers", and it also has three chapters.

Chapter 7: Losing number sense discusses things that can go wrong with number sense when the brain is injured. There is incredible diversity in what skills can be lost while others are retained, and this chapter presents summaries of case histories of people with these various difficulties.

Chapter 8: The computing brain is about what is known about the actual parts of the brain that do arithmetic.

Chapter 9: What is a number? Is more philosophical, and covers the debate among Platonists, formalists and intuitionists. Very briefly, Platonists believe that mathematics is "out there" in the universe, and that we only discover, rather than invent it. Formalists believe mathematics is a game, played by a set of formal rules. Intuitionists believe that humans have intuitions about numbers, and this allows math to develop. Dehaene is very much an intuitionist.

Part four of The Number Sense is Dehaene's updating of the book. It has one chapter entitled: The Number Sense, fifteen years later.

Best parts of The Number Sense by Stanislas Dehaene

For anyone (such as me) who likes thinking about numbers, this is an absolutely fascinating book. Dehaene writes clearly and well, and is good at summarizing vast amounts of information. He argues his case well, but makes it clear when he is stating his opinion as opposed to stating a fact.

I learned a lot reading this book, and it stimulated my thoughts on these matters.

I especially liked chapters 1 through 5 and chapter 9.

Worst parts of The Number Sense by Stanislas Dehaene

No book is without flaws. There are several in this book, wonderful though it is. The biggest problem, for me, was that the final chapter is tacked on at the end, rather than incorporated into the text. This is particularly problematic for Chapter 8, which relied on what are now very obsolete methods of brain imaging. Another problem is that there are some minor errors that are not quite typos, but aren't major. For instance, Dehaene misrepresents what statistical significance is.

Summary of The Number Sense by Stanislas Dehaene If you like math or cognitive psychology, you will like this book.