This book is a fascinating and thought-provoking exposition of the development of the human ability to think mathematically. Of course we do not know (and probably cannot ever know) precisely how we humans developed our mathematical ability, and Devlin acknowledges this fact. Nevertheless, the book lays out a quite plausible sequence of events which could have led to the acquisition of the ability to think mathematically. Keith Devlin is a fine writer (as evidenced by the fact that the Joint Policy Board for Mathematics awarded him the 2001 JPBM Communications Award for his many contributions to public understanding of mathematics), and this latest book continues his string of well-written books.
Before one can address the evolutionary origins of the capacity for mathematical thought, one must of course define what is meant by "mathematics" and "mathematical thought." Unfortunately, most adults never get much beyond arithmetic and algebra in their mathematical education and are therefore left with a false impression of what mathematics is and what mathematicians do. If the typical adult opens an advanced mathematics book, he or she is likely to be turned off by the flood of symbols and may well believe that mathematics is nothing more than the symbols they see on the page. As Devlin says in the book, "Modern mathematics books are awash with symbols, but mathematical notation no more is mathematics than musical notation is music." Many people think that mathematics is somehow the study of and manipulation of numbers (a colleague of mine from another department often jokingly asks me whether we mathematicians have discovered any new numbers lately). Of course, as most people reading this review already know, there is a difference between arithmetic/numerical ability and mathematical ability. As Devlin says, "Mathematics is not about numbers, but about life. It is about the world in which we live. It is about ideas. And far from being dull and sterile, as it is so often portrayed, it is full of creativity." The book goes on to paint a pretty good picture of what mathematicians think of as mathematics. For this reason alone, I would recommend this book to those who are curious about what it is that mathematicians do. Hint: We don't spend much time discovering new numbers.
From the book: "One of my aims in this book is to convince you of just how remarkable and powerful — and uniquely human — language and mathematics are. ... Along the way, I shall examine the questions of what exactly is mathematics, what exactly is language, and how they arose. I shall also consider a third, distinctly human faculty: our ability to formulate — and follow — complicated plans, worked out in advance, incorporating various alternatives to be followed, depending on how things turn out at the time. ... This last human ability ... is, in essence, the source of the other two (mathematics and language). Arguably, therefore, it is the most important of all. ... The principal claim of this book is ... that the feature of our brains that enables us to use language is the same feature that makes it possible for us to do mathematics." If this claim is true (and the book makes a strong case), then since nearly all individuals have the demonstrated capacity to use language, it follows that these same individuals have the capacity to do mathematics.
After getting the reader up to speed about what he means by mathematics and mathematical thought, Devlin describes four levels of abstract thought: level 1 abstraction involves thinking about objects that are perceptually accessible in the immediate environment, level 2 abstraction involves thinking about familiar objects which are not perceptually accessible, level 3 abstraction involves thinking about real objects which have not previously been encountered or imaginary variations of real objects (for example, a unicorn is a variation of a horse which none of us has actually seen but which we can nonetheless think about), and level 4 abstraction involves thinking about objects which are entirely abstract and is where mathematical thought takes place. He points out that while many species of animals are capable of level 1 abstraction and a few are capable of level 2 abstraction, only humans seem to be capable of level 3 and level 4 abstraction. He argues rather convincingly that the critical development which gave the human brain the capacity for language and mathematical thought was one of increased abstraction rather than one of increased complexity.
Briefly, the road to mathematical ability (in Devlin's view) may have gone something like the following. First, brain size steadily increased over a period of about 3.5 million years. The driving force for this development was the richer view of the world, the greater range of responses to stimuli, and the more effective means of communication (a developing protolanguage) which were made possible by the larger brain. Second, the structure of the brain changed to give it the capacity for symbolic thought, language, the ability to formulate complex plans, etc. It is not obvious that all of these capacities (his list is even longer) are related, but, again, he gives a very plausible argument that a single change in the brain could account for all of them. He also gives evidence from the archaeological record that all of these capacities arose at essentially the same time.
So, how does the development of a capacity for language give rise automatically to a capacity for mathematical thought? To help us understand this part of his argument, Devlin first tells us the primary use of language. "[Sociolinguists and psycholinguists] have found that, on average, roughly two-thirds of all conversations are taken up with social matters — who is doing what with whom and whether it's a good thing or not, problems within relationships and how to handle them, and problems and activities at work, school, or in the family. In short, gossip." After a reasonable argument that gossip (and the sense of group membership imparted by gossip) is what gave language an evolutionary foothold, Devlin points out that each of us acquires and maintains a large amount of information about other individuals (names, backgrounds, interests, relationships, etc.). Furthermore, we are able to routinely access this information and use it to understand or predict their behavior, pass judgement, etc. Importantly, we do all of this without effort. We do not stay up late trying to memorize facts about those we interact with, we simply file the information away for ready use. The same is true for those who get caught up in a soap opera. They know vast amounts of information about the various characters in the soap and about how those characters fit together. They do not work hard to memorize this information, it simply gets filed away as they watch. Mathematics is not so different from this. In Devlin's words: "Mathematics studies the properties of, and relationships between, various objects, either real objects in the world (more accurately, idealized versions of those real objects) or else abstract entities that the mathematician creates. ... We have discovered the secret that enables mathematicians to be able to do mathematics: a mathematician is someone for whom mathematics is a soap opera. ... The characters in the mathematical soap opera are not people but mathematical objects — numbers, geometric figures, groups, topological spaces, and so forth. The facts and relationships that are the focus of attention are ... mathematical facts and relationships about mathematical objects." He goes on to say that "mathematicians are not born with an ability that no one else possesses. ... Whatever it is that causes the interest, it is that interest in mathematics that constitutes the main difference between those who can do mathematics and whose who claim to find it impossible."
I have tried to share some of what Keith Devlin has written about so well in this enlightening book, but I urge interested readers to read the book for themselves. This book will likely be well received by a wide audience, and it would certainly find a comfortable home in any library.
Carl D. Mueller (firstname.lastname@example.org) is Associate Professor of Mathematics at Georgia Southwestern State University in Americus, GA.