Mathematics is the investigation of an artificial world: a universe populated by abstract entities and rigid rules governing those entities. Mathematicians devoted to the study and advancement of pure mathematics have an extremely well-developed respect for these rules, for the integrity of this artificial world depends on them. In order to preserve the integrity of their artificial world, their collective work must be rigorous, never allowing for sloppy handling of the rules or allowing intuitive leaps to be left unproven.
However, many of the tools and techniques developed by mathematicians for their artificial world happen to be extremely useful for understanding the real world in which we live and work, and therein lies a problem. In applying mathematical rules to the study of real-world phenomena, we often take a far more pragmatic approach than any mathematician would feel comfortable with.
The tension between pure mathematicians and those who apply math to real-world problems is not unlike the tension between linguists and those who use language in everyday life. All human languages have rules (though none as rigid as in mathematics!), and linguists are the guardians of those rules, but the vast majority of human beings play fast and loose with the rules as they use language to describe and understand the world around them. Whether or not this “sloppy” adherence to rules is good depends on which camp you are in. To the purist, it is offensive; to the pragmatist, it is convenient.
I like to tell my students that mathematics is very much like a language. The more you understand mathematics, the larger “vocabulary” you will possess to describe principles and phenomena you encounter in the world around you. Proficiency in mathematics also empowers you to grasp relationships between different things, which is a powerful tool in learning new concepts.
This book is not written for (or by!) mathematicians. Rather, it is written for people wishing to make sense of industrial process measurement and control. This chapter of the book is devoted to a very pragmatic coverage of certain mathematical concepts, for the express purpose of applying these concepts to real-world systems.
Mathematicians, cover your eyes for the rest of this chapter!