Is mathematics all around us? Why, if you want to see it, it is; if you do not, you may also pass by and think of anything else.
John A. Adam chose to espy mathematics in many aspects of urban life. His new book X and the City consists of 25 chapters, each ending with the words "the city", for example, Getting into the city, Eating in the city, Sex and the city, Nighttime in the city - all 25 of them.
John A. Adam - professor of mathematics at Old Dominion University - is a coauthor of an admirable book Guesstimation. Some of the questions he asks - and answers - while strolling in the city, could have come right from the latter: How many fast food restaurants may one expect in a city, how many barber shops? How many leaves are on that lonely tree you stroll by? How many tomatoes are eaten in the city every year? These have to do with figuring out estimates under rough, simplified assumptions. Posing and analyzing such questions is an entertaining activity in itself but also helps develop a skill frequently employed when it comes to making on-the-spot-decisions.
The art of making simplifying assumptions is the foundation of mathematical modeling that is demonstrated through out the book. Some of the models are simple (an average tree is of such-and-such height, there are so many branches on a tree per unit length, so many leaves on every branch, so the number of the leaves on a tree is about ...) Other models lead to differential equations (say, the phenomenon of air pollution in the city is modeled by the diffusion equation - a partial differential equation of second order. (Most of the models in the book are described by ordinary differential equations, though.) The author goes to a considerable length to explain the ways of deriving those equations and the approaches to solving them. It is also possible to read the book by skipping the equations, but just checking the assumptions and the ensuing conclusions - an activity that would give the reader the sense of what the mathematical modeling is about. As the author put it (p. 108):
At this point the reader may throw up his hands in disgust and say - "These mathematicians! Nothing is ever realistic - when do such conditions ever occur?" He has my sympathies, but this is the way modeling is usually done: take the simplest nontrivial situation and see what the implications are for the real-world problem, and modify, tweak, and improve as necessary ... trial and error are important in constructing models.
The book unequivocally helps dispel the notion that mathematics (represented in the book via mathematical modeling) is about plugging values into formulas and punching calculator keys. Solving problems takes a lot of insight and thinking and - surprise! - even some experimentation.
Some of the chapters in the book deal with probabilities. What is the chance that a couple on a city shopping spree will miss each other for a lunch? How long one may expect a line of post office customers be? What is the behavior of gaps in the traffic flow?
John Adam discerns mathematics in a multitude of situations. Should you decelerate or pass a slow moving car in front? What is the dynamics of the population growth or of the real estate as a function of the location in the city? Should you drive a car during the peak times or use public transportation?
He masterfully blends into the narrative some classical problems of popular mathematics. How much weight will lose watermelons being delivered to the city if on the way there they lose 1% of their water contents? According to legend, what shape had Carthage - one of the first cities built on the Southern shores of the Mediterranean? What is the average speed of the car that moved at 40 mph in one direction and 60 mph on the way back? How fast does one's shadow grow when one moves away from a street lamp?
Years ago another math professor - Underwood Dudley of DePauw University - published a paper Is Mathematics Necessary? He answered that question in the last sentence: "Is mathematics necessary? No. But it is sufficient." (See also my column Necessary And Sufficient.) John Adam's book teaches us a lesson of what mathematics is sufficient for. A person can live his or her life without ever needing mathematics beyond its most rudimentary aspects. However, as John Adam shows convincingly, mathematics can make one's life more colorful and entertaining. Mathematics makes us notice and ponder many engaging things that most of the populace nonchalantly stroll by. I have no doubt that they would regret it if they only knew what they are missing. According to John Adam, they miss a lot.