Governments worry over their citizens' mathematical abilities - or lack of them. Education secretary David Blunkett introduced the National Numeracy Strategy in 1999 to raise standards in primary schools. Before him, Tory ideologue Sir Keith Joseph wrote that "few subjects are as important to the future of the nation as mathematics". Why? The reason for this emphasis is that, as Napoleon figured out, "the advancement and perfection of mathematics are ultimately connected with the prosperity of the state". If we fall behind our economic competitors -and we have fallen a long way behind in the average mathematics performance of schoolchildren -this could diminish our own prosperity as a country and as individuals.
The basis of mathematical abilities is a matter of urgent concern to present-day policy-led funding bodies. But, of course, relevant research is not new. We can look to Francis Galton in the late 19th century and Piaget, who famously studied the development of numerical concepts in children of four and five years of age. More recently, psychologists have focused on how children learn the counting sequence, how they cope with place-value numerals, how they master the four mathematical operations, and so on. With the invention of functional brain imaging, it has been possible to identify the brain systems most active when carrying out mathematical tasks.
There have been two recent surveys of the bases of mathematical cognition by neuroscientists specialising in this field: Stanislas Dehaene's The Number Sense and my own The Mathematical Brain . Dehaene argues that the distinctive aspect of mathematical thinking is a sense of quantity, a kind of number line in the head onto which numerical expressions are mapped. To support this view, he has excellent evidence from brain imaging, computational modelling and infant studies. I, on the other hand, argue that we are born with a sense of a set and its numerosity that is fundamental to building advanced mathematical concepts. My position, too, has evidence in its favour.
Keith Devlin is a mathematician, not a neuroscientist. What can he bring to this debate? We would expect the mathematical virtues: careful definition, rigorous logic with unsupported postulates clearly labelled as such. From Devlin, author of many excellent popular books on mathematics, we also expect an accessible and entertaining argument. Devlin's claim is that mathematics is not special. Rather, mathematical thinking depends on the evolution of language.
For Devlin, language is fundamental in two ways. First, our capacity for syntax -putting elements together in different ways -provides the combinatorial power that we need for maths. This is not a new idea. Noam Chomsky proposed it some years ago, and it has been followed up in important papers by Paul Bloom, though one would not have known this from Devlin's book. Second, fully syntactic language, it is claimed, allows us to "think off-line", another prerequisite for maths. "Protolanguage" -a simple system that allows the combination of just two terms without syntax, such as "mammoth come", is regarded as insufficient to think off-line, sufficient only to contemplate the here and now. Devlin attributes capacity for proto-language to Homo erectus . But it is unclear to me why H. erectus or other hominids capable of forming a simple proposition should not be able to take propositional attitude: hope (mammoth come), want (mammoth come) and so on. Children go through a stage that looks very like protolanguage as described here, producing just one and two-word utterances. Can they not think off-line?
Michael Morwood and his colleagues have found that H. erectus colonised an island in Indonesia. What makes this remarkable is that the island has never been connected to mainland Asia, so these protolanguage speakers must have sailed 30km across open sea to reach it. Is it possible to build a seaworthy craft and plan to sail to an invisible destination without being able to think off-line?
Devlin makes the bold, and unsupported (insupportable?), proposal that "language did not evolve primarily as a communicative device" although "many authorities claim exactly the opposite". It evolved precisely for off-line thinking. However, later in the book, the primary purpose of language seems to be gossip, and its evolutionary role is to create esprit de corps in prehistoric hunting groups. Devlin also argues that it is necessary to be able to think about abstract entities, though these are never defined, and symbols.
His treatment of symbols is, I am tempted to say, symbolic of this book's many confusions. First, symbols are held to arise by association - a word sound with an object ("cat" with a cat). Pavlov's bell is not simply associated with, but symbolises food for the dog. Then, Devlin uses Peirce's account of symbols, in which X becomes a symbol for Y by "agreed convention". Finally, he asks whether vervet monkeys, which have distinct cries for different situations (such as the presence of leopard), use symbols. The association account entails answering yes, but the Peirce account entails no, because vervets do not learn these cries, are not agreed conventions, but innate responses.
Devlin concludes that "the language gene and the maths gene are one and the same". Even if his arguments are sketchy and confused, what about his conclusion? First, The Maths Gene has nothing about genetics. Actually, there is little evidence available. We do know, however, that newborn infants are sensitive to the number of things they see long before they have acquired any language.
We also know, from the work of Neil O'Connor and Beate Hermelin, that it is possible for a child to become a superb calculator without being able to learn any type of language or proto-language. And there are many cases of people with good language skills who have profound difficulties with simple numerical tasks such as comparing two numbers.
Finally, there is a body of evidence going back more than 60 years, and confirmed by recent brain-imaging studies, that the parts of the brain involved in mathematical tasks differ from those involved in linguistics. Language involves Broca's area in the left frontal lobe and Wernicke's area in the left temporal lobe (Devlin seems not to know about the latter), while the critical brain area for number is in the left parietal lobe (also not mentioned), with some role for the right parietal on very simple tasks, and for the prefrontal cortex when chains of reasoning are needed. It seems unlikely that a single gene or set of genes is responsible for building these very separate neural structures.
If, as Devlin avers, we all have the relevant gene (whatever it is), why are we not all good mathematicians? Despite appearing in the title of the book, just three and half pages are devoted to this problem, which is a serious one whether you are a policy-maker or a citizen evaluating policy-makers' statistical claims. Close to the end is Devlin's answer:
"The key to be able to do mathematics is wanting to."
His evidence: two anecdotes. The first shows that cashiers in a Paris department store, who (this being the 19th century) had many years on the tills, outperformed mathematical prodigies, showing anyone can do it with practice. The second shows that illiterate street traders in northeast Brazil can develop calculation strategies that are quite different from what they would have learnt in school. This is interesting evidence, which both Dehaene and I also cite, but it is is scarcely a proof.
So that is Devlin's explanation for why "most people don't use it". And we end up with nothing on "it", the gene, for maths or anything else. Why everyone has a maths gene is a mixture of largely unsupported speculation about the origins of language, combined with a few facts and ideas well known from many other popular accounts. Nice title, shame about the book.
Brian Butterworth is professor of cognitive neuropsychology, University College London.
The Maths Gene: Why Everyone Has It, But Most People Don't Use It
Author - Keith Devlin
ISBN - 0 297 64571 4
Publisher - Weidenfeld and Nicolson
Price - £9.99
Pages - 320