- Hardcover: 328 pages
- Publisher: Diane Pub Co (March 1, 2000)
- Language: English
- ISBN-10: 0756765927
- ISBN-13: 978-0756765927
- Product Dimensions: 6.2 x 1.2 x 9.2 inches
- Shipping Weight: 1.5 pounds
- Average Customer Review: 40 customer reviews
- Amazon Best Sellers Rank: #16,544,007 in Books (See Top 100 in Books)
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Math Gene: How Mathematical Thinking Evolved and Why Numbers Are Like Gossip
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About the Author
Keith Devlin is the Dean of the School of Science at St. Mary's College, Moraga, California, and a Senior Researcher at the Center for the Study of Language and Information at Stanford University.
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The main theses of this book are:
1) Mathematical thought is a type of linguistic thought.
2) The math gene is the genetic endowment for Universal Grammar.
3) Universal Grammar specifies the "fundamental language tree" - i.e., the basic X-bar tree consisting of an XP, which consists of a SPEC and an X-bar, which consists of an X which optionally combines with another XP.
4) During the first hundred of thousands, if not millions of years of evolution of the genus "homo", brain size grew rapidly. Given the cost of growing a bigger brain, it had to be very useful. Growing the brain was useful because it allowed homo to recognize more types (bananas, monkeys, palm trees, etc...)
5) Then, sometime around 200,000 thousand years ago full-blown language appears. How? With the emergence of "off-line thinking": the brain can self-trigger patterns of activity that used to have to be caused by external stimuli.
6) off-line thinking requires more than types; it requires representations of structural relations between them (e.g., cause-effect).
7) types + structural relations = syntax
8) being able to talk about relations between people is the same as being able to talk about mathematical relations
There are several problems with these theses. First, the author rarely considers alternatives. Take for example the argument that goes: a) the increase in brain size had to be useful to make up for its cost; b) being able to recognize types is useful; c) therefore, the brain grew because brain growth allowed homo to recognize more types.
This could be. The problem is that there are many other alternatives. It could be that brain growth allowed homo to have a larger working memory, or to store more mental images of individual objects and events, or to inhibit immediate desires in the interest of delayed gratification, etc...To show that the brain grew to represent more types, Devlin would have to consider some alternatives and show that the alternatives are wrong. However, far too little of that happens. Moreover, there are other sources of evidence that could support Devlin's thesis. For example, if evolution endowed us with the capacity to recognize multiple types, then type recognition should be innate - it should be available to young human babies. Much infant research suggests that indeed human infants do distinguish basic types - e.g., animate-inanimate. However, this is a far cry from the whole inventory of types Devlin seems to have in mind. More importantly, Devlin doesn't present other types of evidence that could support or disconfirm his hypothesis.
Unfortunately, this is typical of most other arguments in the book. Devlin presents arguments that are consistent with his theses, but rarely considers alternatives. Moreover, the evidence he considers is usually deplorably thin.
The other strain of problems comes from loose and usually erroneous analyses of mental representations. For example, Devlin characterizes syntax as the fundamental language tree, a characterization that finds much support in linguistic theory. But then later on syntax is said to be the same as representations of relations between types (or of the structure of the world). The fundamental language tree does not represent any contentful relations between objects or types. Syntax is pure form, not meaning. So the relation between the fundamental language tree and representations of the structure of the world escapes me. More generally, I don't see why Devlin dedicated a whole chapter to the fundamental language tree (which by itself is quite good) because he never goes back to that idea. Be that as it may, there is another problem with the thesis. Syntax is not a representation of structural relations of the physical or of the social kind. For example, consider these sentences: "Joe convinced Bob," "Joe kissed Bob," "Joe purchased a log" and "Joe burned a log". These sentences have the same syntactic structure: NP VP NP. However, they are about completely different types of relations. The first is about mental causation, the second is about contact, the third is about transfer, and the fourth is about physical causation. Clearly then, syntactic structure is not the same as representation of relations between objects and people.
This problem recurs in many the arguments of the second half of the book. For example, if off-line thinking evolved the way Devlin says it evolved (i.e., it is an internally generated simulation of on-line thinking), then the capacity to think about the past or the future does not necessarily follow from off-line thinking. On its own, a simulation of on-line thinking (thinking about what is here now), is no more a representation of the past or future than on-line thinking itself. To think about the past or the future, one needs to represent the structure of time. If no such structure is available in on-line thinking, then it cannot appear in a simulation of on-line thinking. The general problem is that, other than being able to run without an external cause, the simulation cannot have properties that are not in what is being simulated.
The latter point also holds for the book's final thesis - i.e., that our capacity for math grew out of our capacity to gossip - to talk about relations between people. Devlin's argument for this seems to be that all relations are equal. So if i can talk about relations between people then, necessarily, I can also talk about relations between geometric transformations or between numbers. But this is not so, at least, not patently. Love, argue with, hide from, embarrass, and flirt with are all social relations. Each of them has a particular content, as seen by the fact that each of them entails particular things (e.g., if A embarrassed B, then it is plausible that B's ego dropped temporarily but if A flirted with B, it is likely that A's ego enjoyed a temporary boost). Likewise, mathematical relations like SUCCESSOR, IDENTICAL, or SIMILAR have their particular content. How the content of any of these relations can be derived from social relations is quite unclear. To argue for his thesis that gossip is the origin of math, Devlin owes us an explanation of how one gets from the content of social relations such as embarass and flirt with, to relations like successor and identical. However, no such explanation is to be found. Therefore, as far as I can tell, there are no reasons to believe Devlin's final thesis. Rather, there are some pretty good ones to doubt it - i.e., that you cannot get representations of mathematical relations out of representations of social relations.
What Devlin really needs is evidence for abstract symbols that only capture the most basic logical properties of relations (say whether they are symmetrical or transitive). Here syntactic categories and the fundamental language tree could be part of the answer. But unfortunately Devlin does not make this connection.
I would recommend this book for anyone who is interested in understanding the evolutionary and cognitive origins of mathematical thinking and ability.
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