|Mathematics and Sex|
|publisher||Allen & Unwin|
|summary||A very nice introduction to the modelling of inter-personal behavior|
The way one studies patterns mathematically is by building models for the behavior being modeled. This is why most of this book is about mathematical models for interpersonal behavior. Well, that together with some amusing anecdotes that make the book a fun read even if you know the literature very well. Still, before I go any further with this review I want to remind everyone that the key question to ask oneself when reading any book that does mathematical modeling of any topic is always the same: are the models built realistic?. Mathematicians can't answer this question: only research by scientists (i.e., experience) can. Einstein probably put it best when he said:
"As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality."
While we do study models for their applicability and their eventual predictive use by and for science, mathematicians can and do also study them for their intrinsic mathematics beauty, and some of the models Cresswell discusses in this book are certainly very pretty (in the mathematical sense of beauty--because the solutions are elegant, though the pun is intended.)
As an example of what this whole subject is like let me tell you about a long-studied model of interpersonal behavior that the author discusses in Chapter 3, a chapter titled "Road Testing the Bed"--I kid you not.
"You have to choose your life mate. The rules we adopt for this model are that you will be presented 100 choices one after another, you may date them, sleep with them, whatever. But, at the end, you must say yea or nay and if you say nay, you will never see them again."
What strategy should you adopt? Well, if you wait to the end, the odds are only 1/100 that the last person is the optimal choice; ditto if you choose the first person. The modeler then asks: what strategy should you adopt for optimum results? A little bit of mathematics involving infinite series gives the answer. You can prove mathematically that the best strategy is to look at (approximately) the first 36.787944117144235 people (rounding it to, say, 37 people) and then you should choose the first person from that point on that is 'better' then the previous 37 people. This increases the odds of your finding the best match from 1% to about 37%- roughly a 37 times improvement. (In the pre-politically correct literature this model was called "The Sultan's Dowry Problem," or "The Secretary Problem"; now, alas, it is usually called simply an example of an "Optimal Stopping Problem." )
Is this a good model for how we behave? Is this a strategy that one can realistically adopt? Certainly, 100 possibilities seems like a lot of choices to have if one is not the current day equivalent of a sultan -- a movie star or an athlete. But the model is intriguing, if not totally realistic and applicable.
Models that spring from modification of the rules of the Sultan problem have always been one of my favorites in this area. This makes Chapter 3 my favorite chapter: it is chock full of goodies with lots of interesting variations of the original problem, and thus even more interesting models. Some may be far more applicable. For example, if you get to play the cad and can keep potential mates 'stockpiled,' then, by stockpiling seven potential mates, there's a strategy that you can use to increase the odds of finding the best one to 96% or so! Or, in another variation of the model, whose solution she refers to as the "twelve bonk rule," there's a result that says that if you simply want to ensure that your choice is better than 90% of the other choices available, simply 'sample' the first 12 possibilities and pick the first person who is better after the first 12. This strategy gives you a 77% possibility of success.
I obviously can't go over all the models she builds, the interesting results she cites, or the interesting observations she makes in a review so let me simply give you some of the high points of the remaining chapters:
Chapter 1 is entitled "Love, sweeeet love" and mostly consists of showing you various differential equations that can model love's attraction and repulsion i.e. variations on standard "prey-predator models." For example, she mentions the following model of attraction:
This model gives rise to a standard and very simple first order differential equation. She then talks about more sophisticated versions of this model including one by Rinaldi that tries to model a famous love poem by Petrarch. (Personally, I think these models are only useful for learning differential equations but don't shed much light on the problem.)"The more Romeo loves Juliet, the More Juliet wants to run away ... Romeo gets discouraged and backs off, Juliet finds him strangely attractive. Romeo tends to echo her..."
Chapter 2 is called "Marriage and the Happily Ever After" and describes models for behavior in a relationship, including an analysis of how absurd the folk tale is that more sex occurs in the first year of marriage then in all subsequent years combined. Probably the most interesting work she talks about in this chapter are the models by Guttman et al. intended to analyze conversations between lovers to determine if the relationship is on the rocks. In this case the models they build are known to be highly accurate in predicting problems in the relationship.
Chapter 4 is entitled "Dating Services -- are you really being served?" and it has a fascinating analysis of the perils of questionnaires that try to match too many variables (i.e. why those questionnaires don't help that much). As she points out, this is called the "curse of dimensionality" in the literature. The problem is that if you are trying to determine whether two points are very close in n-dimensional space where n is large, you are unlikely to get a whole lot of difference between points and so closeness doesn't really matter much.
Chapter 5 is called "Pairing Up," and shows how Game Theory can (should?) enter into the problem of "choice" preferences. This chapter is a very nice gateway into models that are studied in the greatest depth in economics; there is an incredibly interesting literature on these issues. One should start with Arrow's paradox on voting (that most logical axiom systems for building choice models are actually inconsistent and can't simultaneously be satisfied) and then work up to real problems with how congressional seats are allocated in the United States. Wikipedia has good articles to start with on these models.
Chapter 6 is called "Action Reaction Attraction" and is about ways to model people's attractiveness. This means things like symmetry as a cross cultural model for beauty, and waist-to-hip ratio for females as a cross-cultural model for male choice. Whether these models are correct is an extremely active area of research in anthropology and evolutionary psychology. The jury seems to still be out, but the evidence for their truth is certainly growing.
Chapter 7 is called "Pick a Sex, Any Sex" and is a tantalizing hint of what the mathematics of evolution is all about. In particular this chapter includes a nice discussion of how sex itself can evolve. (It seems paradoxical that the question of how sex itself can evolve is not yet resolved. After all, in a naive "selfish gene" approach to evolution, it would seem seem that asexual methods of reproduction win hands down. But, as usual, the issues are more complex then naive models would predict. For example, who would have thought that parasites might be the reason sex arose? Again, for more details on the science behind the models the author discusses, you can start with a useful Wikipedia article. Ridley's popular science book called the Red Queen (or anything by Maynard Smith) is where to go next.
Chapter 8 is titled "How Ovaries Count and Balls Add Up," and is about models for feedback levels of hormone concentration and circadian rhythms and didn't particular interest me.
Finally, Chapter 9 is called "Orgasm" and I'm not going to summarize it, since that would be telling.
To sum up, is this book perfect? No. I think more mathematically literate people would like appendices which give some indication of the deeper math behind what she discusses. For example, the math that shows why the answer I gave above to the Sultan's choice problem really is approximately 36.787944117144235 - or more correctly n/e, where e is the base of natural logarithms and n is the number of choices one has to go through, is well within the reach of any 2nd year calculus student. The differential equations she introduces in other chapters can be understood by anyone with a good engineering or math background. The game theory and even a proof of Arrow's theorem should be accessible to any literate person etc. As is, though, anyone with even some knowledge of or interest in mathematics will find this book great fun.
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