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Reality Rules, The Fundamentals (Wiley Professional: Reality Rules: Picturing the World in Mathematics) (Volume 1)

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Mathematical models of the world are the toolset that we use to build everything, from the simplest machine to the largest skyscraper. For example, when a large and tall building is being planned, the foundations are laid based on the models for the type of soil, width of the building, the weight of the materials and the height. It is all constructed in model form before the first shovel lifts the first piece of dirt. The fact that it is a rare occasion when a building or anything else spontaneously comes crashing down demonstrates how sound our mathematical models are.
In this book, Casti presents some of the more recent and abstract models of the world, these models are not nearly as precise. These are not basic equations, but generally ideas regarding how models can be constructed. The chapters are:

*) The ways of model making: natural systems and formal mathematical representations - this chapter introduces the idea of a system being modeled by a set of states and transitions between those states in response to events.
*) Catastrophes, dynamics and life: the singularities of ecological and natural resource systems - this chapter covers the models for systems that reach cusps of dramatic change, such as populations of living creatures where the model contains cusps. The most interesting graph in this chapter is on page 131. It is a graph of Scottish witchcraft trials from 1600 to 1700 plotted so the graph is linear.
*) Pattern and the emergence of living forms: cellular automata and discrete dynamics - in this chapter the models of cellular and finite automata are presented and used to describe some types of biological activity.
*) Order in chaos- the mathematics of chaos theory are presented and used to describe some situations such as population cycles, free markets and the spread of disease.

As the contents of the chapters demonstrate, this is not a book on simple modeling; the topics require a degree of mathematical sophistication. However, if your interest is in these models of the world, then this is an excellent resource. There is a small set of exercises at the end of each section and larger sets of discussion questions and problems at the ends of the chapters.