Living by numbers and counting on testicles

The Mathematical Brain
February 18, 2000

Brian Butterworth begins by reminding us of our utter dependence on numbers. He paraphrases the front page of his daily newspaper: prices, dates, ages, telephone numbers, page numbers, bar codes - numbers everywhere, underlying all that we do and think. His book regales us with nuggets such as how the men of the Yupno of Papua New Guinea use their right testicle to stand for the number 32; why it is easier to count to large numbers in Chinese; the story of Signora Gaddi, whose stroke left her abilities at language, memory and reasoning intact but made her unable to deal with numbers greater than four; and how Brazilian street kids calculate costs when selling coconuts. His achievement is to weave such intriguing facts, observations and theories, from disciplines as diverse as archaeology and neuroscience, into a powerful argument that the brain contains a "number module". This, according to Butterworth, is a part of the brain that contains specialised circuits for identifying "numerosities": it automatically recognises the number of objects in small collections without needing to count them.

The existence of such a module is claimed at the outset, and the book aims to persuade the reader that this claim is convincing. Butterworth begins by establishing the fact that everybody counts and that the various counting systems used throughout human history are simply different means to build on the same biologically based mathematical skills. Then we meet newborn babies who lack any linguistic skills but who seem particularly interested and able at identifying numerosities. Such babies enable Butterworth to argue that we are indeed "born to count". Experiments with chimpanzees and birds are described to make the case that animals have latent capacities to do simple numerical tasks. It remains unclear, however, whether such animals are like humans and able to identify absolute numerosities or whether they rely on relative estimates of quantity.

When Butterworth deals with "numbers in the brain", we meet a cast of characters who have suffered various forms of brain damage leaving some of their mental faculties intact and demolishing others. The author's key task is to establish double dissociations of number skills with those of language, memory and reasoning. To establish that number skills rely on neural circuits independent from those of language, for instance, Butterworth required not only someone who had lost their number skills but maintained their language abilities, as had Gaddi, but also the converse - someone who had lost their language abilities but remained good at numbers. He found the latter in a certain Mr Bell, who was suffering from a degenerative brain disease, which left him able to mutter no more than a few stereotyped phrases including the curious "millionaire bub". Yet he remained quite competent at arithmetic.

Midway through the book, Butterworth attempts to locate the number module within the brain, providing substantial evidence that the inferior lobule inside the left parietal lobe is its home. Why there? Butterworth explains that the fingers and hands are represented there, and the fingers play a critical role in the development of counting and arithmetic skills. But he displays quite appropriate caution as to the value of brain-imaging techniques when hunting for such modules, comparing them to Galileo's first use of telescopes.

Butterworth goes on to examine why some individuals are bad at mathematics and others good. Cases of disorders that appear to have inhibited the development of the number module, producing dyscalculia (the mathematical equivalent of dyslexia), are discussed, while Butterworth argues that zeal and hard work are the ingredients of mathematical success, there being no biological-based "gift" even behind the accomplishments of Ramanujan. How then, he asks, should schools teach mathematics? Butterworth stresses the need for discovery learning, fun and understanding, and expresses appropriate disdain for the current British government's Numeracy Task Force's emphasis on drill and practice. Finally, he asks why certain mathematical ideas seem particularly hard - negative numbers, probabilities, fractions - and argues that mistakes are made with these because such problems are distant from the idea of numerosities that the number module has been "designed" by evolution to address.

This is a splendid book, outlining an important and rigorous scientific argument in an entertaining and absorbing fashion. It goes far beyond many recent books on the brain by getting down to the nitty-gritty of locating specific neural circuits. It also shows a great degree of caution, always aware that more evidence is needed and frequently suggesting the type of experiments required. Throughout, there are mathematical teasers. My rather disastrous attempts to solve these frequently involved pencil and paper. This is significant because the most compelling argument in this book is not about the mathematical brain at all. In fact from the evidence Butterworth provides - he identifies no more than the numerosities of collections of up to four members - the mathematical prowess of the unaided brain appears remarkably limited. The human number module seems little different from that of apes and birds. But when this number module is attached to a counting system and extended into the material world of human culture, the rather innocuous set of neural circuits has powerful consequences for human thought. As a neuroscientist, Butterworth appreciates that studying the brain in isolation from the body and material world will tell us very little about human intelligence and thought.

Steven Mithen is reader in early prehistory, University of Reading.

The Mathematical Brain

Author - Brian Butterworth
ISBN - 0 333 735 7
Publisher - Macmillan
Price - £20.00
Pages - 446

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