A flourishing new genre in the trade book market might be called "academics rushing into public print with abstruse speculations and an occasional joke." Many of these books, if not about cosmology or molecular biology, focus on the human brain. The question often asked is something like "Why are we as smart as we seem to be (or think we are)?" and the answer is usually a put-down: "Because of an evolutionary accident."
The standard scientific explanation for all our human talents and complex social organization is that we have a "big brain," with billions of individual neurons and trillions of interconnections. Given the big brain and a few environmental pressures and opportunities, all of our great accomplishments and big ideas (especially those about ourselves) were almost bound to become manifest.
I can't imagine anyone doubting that human abilities are unrelated to the kind of brain we have, so it might seem that there isn't too much to argue about. Nevertheless, all these books are characterized by an earnest intention to persuade, though it is difficult to determine whether the entreaty is addressed to lay readers or academic colleagues.
Keith Devlin, a British mathematician working in California, has already explored mathematics as a subject in a score of books and other media, both academic and popular. In The Math Gene, he wants to account for how mathematics came on the scene in the first place, and why, like language, it seems to be an exclusively human characteristic.
He begins by disowning his title. There is no such thing as a math gene, he says, but it is a common metaphor for an innate facility. (It is?) His explanation for mathematics is that the potential is wired into our brains, and he has some interesting and provocative ideas about why it actually emerged. But his exposition is patchy because it looks in two directions at the same time. He addresses colleagues who share the same knowledge base and can answer him back, but also readers whose understanding of the technical issues must be assumed to be minimal, and who probably couldn't care less about the actual conclusions that Devlin reaches. So Devlin has to wrap his hypotheses in a cocoon of stories (and that misleading title), blunting the sharp edges of argument and transforming academic slog into a breathtaking tale of excitement and discovery.
Surprisingly, Devlin doesn't have much time for arithmetic and numbers, though this doesn't stop him from devoting his opening chapters to the topics. Infants, for example, can discriminate between two and three objects or sounds soon after birth, and by the age of two can make some simple calculations. Many other animals can do the same things. These abilities are probably not mathematical in any intellectual sense, but more like estimation, or a sensitivity to quantities that are not counted. Devlin has a reason for this difference, but can't bring himself to discuss it before other digressions.
The first is a discussion of individuals who lost aspects of their ability to understand numbers or perform mathematical operations after severe injury to their brain. There is a certain morbid fascination about all this, but it surely doesn't come as a surprise that damage to a crucial part of the brain is likely to affect intellectual functioning, just as jamming a screwdriver into the motherboard of your computer will probably interfere with its speed and accuracy.
The neurological exposition is followed by an abrupt excursion into group theory, which essentially concerns the basic relationships of mathematics without the numbers. If group theory doesn't mean anything to you, don't worry. Devlin says it doesn't matter, except that it will give an idea of the kind of thing he is talking about when he refers to higher mathematics.
Devlin defines mathematics as the science of patterns. One could argue that mathematics isn't always a science (certainly not the way most people do it), and that it is arbitrary, or at least metaphorical, to refer to mathematical relationships as patterns, when they are unlike most other kinds of patterns that we encounter. However, the definition enables Devlin to make his first significant point: that our mathematical ability comes from having a brain that is basically a pattern-creating and pattern-recognizing organ.
This is why he thinks higher mathematics is probably easier and more natural than arithmetic, because counting, calculating and other activities with numbers are linear, and unsuited to the kind of brain we have. He even suggests that it might be a mistake to begin children's mathematical education with numbers, rather than plunging them directly into more patterned mathematical activities, such as group theory (an idea reminiscent of the educational turmoil of the 1950s and 1960s when children were introduced to the so-called New Math precisely through group theory and other topics normally taught in higher grades).
Devlin is now ready to explain why mathematics is possible. The technical term he uses is "mental representation," although "reflection" and "imagination" would probably serve just as well. He argues that the human brain is able to achieve four levels of representation (at least when thinking about mathematics) while other creatures must be satisfied with just one or two.
The first level of representation is the here and now. All you can think about, if you can think at all, is the situation you are currently in. The second level of representation is when you can think about something not present at the moment, but that you know because you have encountered it before. The third level is when you can think about something that is new, but put together from elements of your past experience. Chimps may be able to do this, Devlin thinks, but he is dubious.
Only humans, however, are able to attain the fourth level, which is representing (reflecting upon) abstractions that were never part of concrete experience. He calls this "off-line thinking." It enables us, he says, to live in the "wide-open spaces" of symbolic thought, seeing real and imaginary events as patterns rather than as sequences of rules.
What gives us that unique ability for abstract thought, apart from our large brain? Devlin argues that it is only language that enables us to pursue and understand mathematical patterns and other aspects of off-line thought. Given language, mathematics is inevitable.
But language itself would never exist without off-line thinking, continues Devlin. In fact, language and off-line thinking boil down to the same thing. You can't have one without the other, and they developed together¨not as a direct result of evolution, but as a by-product of our big brain.
What brought about the big brain, and therefore language and mathematics? Evolutionary chance. Some people, by chance, were born with slightly bigger brains. Others, no doubt, were born with slightly bigger feet. Those with bigger brains begot chains of bigger-brained offspring, at least one of which survived to beget everyone in the world today. Those with bigger feet, fortunately, did not survive.
And why did off-line thinking and language develop? It is here that Devlin is most original, or outrageous. When our ancestors left the trees (Devlin blames meteorites) and took to an organized terrestrial life, they needed to keep track of personal relationships. Language as we know it, with its complex reflective syntax and story patterns, developed as a way to talk about what everyone was doing. Gossip, in other words (Devlin's word).
Our attention to patterns and relationships is the reason, in the long run, we got numbers and mathematics. Numbers, as Devlin's sub-title asserts, are like gossip. Mathematicians do their job by exploring the relationships they find in the topics they study.
Why should many people find mathematics inaccessible? Here Devlin is relentless in asserting that "math is hard." Math involves tedious exercises and its value is inadequately explained, so many students aren't motivated to made the effort. It's like running a marathon, he says. You have to want to succeed.
It's an interesting tale, like all speculations about evolution, even if untestable and devoid of any practical utility. ˛
Frank Smith lives in Victoria, BC. His latest publication is The Book of Learning and Forgetting (Teachers College Press).