The Math Gene: How Mathematical Thinking Evolved and Why Numbers Are Like Gossip [Englisch] [Taschenbuch]

This is an intriguing question. After all, it's a fairly new part of human behavior - having been around maybe 10,000 years - that we all can do, at least a bit, and the rest of the animal kingdom can't, at least as far as we know.

Devlin's the first mathematician I know of who's looked deeply into this subject using recent research in the area; he's done a great job fitting the available data to a theory that starts to answer the question, how it is we can do mathematics?

First, though, you have to understand what mathematics really is. Devlin's definition is the "science of patterns" and he explains clearly and convincingly why it's the right one.

His premise, roughly, is that however we acquired language, and he stays mostly on the sidelines of the heated debates about that, mathematical ability came along for the ride. His reasoning is that "off-line reasoning" is an essentially equivalent to language, as you can't have one without the other, and that this plus some other abilities, such as a number sense and spatial reasoning, give us the ability to do mathematics.

He then explains why so many of us find the subject difficult. A simplified version is that we use language mainly to talk about interpersonal relationships. In a word, gossip. Note he's not claiming this to have been the purpose for it's development, just that it's what we mostly do with it now. And we're very good at gossiping. In fact, it's so easy we consider it to be a form of relaxation. To Devlin, you need to have the same kind of relationship with mathematical objects in order to be able to work with them.

The book's greatest strength, to my mind, is its gathering of results in cognitive psychology into a coherently developed thesis regarding the origins of mathematical ability. It's a worthy contribution to the discussion, even if the theory proposed is completely wrong, as it may well be. Devlin's open and clear about it being highly speculative.

I do have quibbles, but they're just that. Its major weakness, if the book can be said to have any, is that it doesn't make much by the way of predictions based on his theory, which would make it far more convincing. But this is a terrific starting point for other work.

Then there's a fairly long discussion of mathematics from the inside--are mathematician's brains different? What is it mathematicians do?--including a moderately detailed description of the basics of mathematical groups. I think Devlin does this to provide non-mathematicians with a sense of what mathematics is about, to make the rest of the book more plausible. This section is well-written and fluent, but I found myself getting a little impatient for the meat of his argument, which comes in the last half of the book. I suspect any reader with a good mathematics background would react the same way.

The next piece of the argument is to demonstrate that language is unlikely to have developed solely as a result of evolutionary pressure towards communication. This is a subtle point I haven't seen made before, but Devlin (who acknowledges his debts to other workers in this area) makes the case quite convincing. In summary: apart from extremely simple messages like "Danger!" and "Mammoth here" you can't communicate what you don't have a mental representation of. The evolution of communication can't have driven representation; it must have always lagged a half-step behind. So mental representation must have evolved first. I am not doing this argument justice here, but Devlin buttresses it well.

The inference is that language is simply a natural but lucky result of our ability to represent the world in our minds. Devlin's key point, however, is that since mathematics is essentially the ability to construct and work with increasingly abstract representations, the same mental changes that gave rise to language have also given rise to mathematics. His conclusion is that we all have the ability to do mathematics: there is no "math gene" except in the same way there is a "language gene": it's universal.

As a side note, not critical to his main argument, he points out that the most likely reason for the growth of representational ability in human brains was to foster understanding of other humans in the group; to encourage a sense of group-ness. For a creature that was more effective in group actions (e.g. hunting) there would have been a strong evolutionary advantage to having an emotional investment in the success of the group. Hence much of the early use of this ability would have been to represent others in the group; when language was added, it would have enabled people to talk about each other. In Devlin's words, "Having arisen as a side-effect of off-line thinking, language was immediately hijacked to facilitate gossip." (Off-line thinking is used to mean representational thinking that doesn't result in or from actions in the immediate environment.)

Two particular items in the book are worth mentioning. One is a followup to some famous experiments done by child psychologist Piaget in the 1930's. Piaget thought he'd demonstrated that children don't acquire a fully-developed number sense till around six years old. More recent work has demonstrated that children are much smarter than Piaget realized: there was a subtle and fascinating methodological flaw in Piaget's experiment. The second item is a little test of logical reasoning, presented with four cards on a table. Even mathematicians, who will probably get the test right, may be surprised at the coda to the test, which forms one of the few methods of direct verification of Devlin's claim that everyone can do mathematics.

The case is well-argued, but one problem with theories like these is that there *are* so few ways of finding out if they're true. "The Math Gene" is reminiscent of Julian Jaynes' "The Origin of Consciousness in the Breakdown of the Bicameral Mind" in this way; a fascinating argument that we may never be able to test. However, it's thought-provoking and plausible, and left me, at least, convinced of its likely truth.

I thoroughly enjoyed this book, and have recommended it to friends and colleages alike. I would also recommend another one of Devlin's books, The Language of Mathematics, for a glimpse into the diverse and beautiful world of math any person could understand and appreciate.