Nearly every teaching mathematician has heard some form of the question “What is math good for?” This book starts with the work of George Boole and his “Laws of Thought” that are quite abstract and takes the implementation of those ideas by Alan Turing and Claude Shannon to the development and structure of the modern computer. Through the treatment, which includes some basic digital logic design and probabilistic computations, the reader is taken on a journey from the development of some abstract mathematical ideas through a nearly ubiquitous application of those ideas within the modern world with so many embedded digital computers.
While most of the treatment is understandable, readers who are unfamiliar with Turing machines will likely struggle a bit when reading the section that covers them. While simple in structure, Turing machines are an abstraction that is best understood by seeing some examples, which is not done well here. The sections on digital devices such as logic gates and flip-flops may also prove challenging.
I enjoyed the discussion of Claude Shannon. In the history of the computer and development of the internet and World Wide Web, his ideas and contributions are too often overlooked. He is one of my heroes and I believe that everyone that reads this book will come to the same conclusion. If you read this book and hear the question, “What good is algebra?” you will have a ready and irrefutable answer, “Boolean algebra is the basis for describing and designing the circuits of computers.”
Charles Ashbacher splits his time between consulting with industry in projects involving math and computers, teaching college classes and co-editing The Journal of Recreational Mathematics. In his spare time, he reads about these things and helps his daughter in her lawn care business.
1 What You Need to Know to Read This Book 1
Notes and References 5
2 Introduction 6
Notes and References 14
3 George Boole and Claude Shannon: Two Mini-Biographies 17
4 Boolean Algebra 43
5 Logical Switching Circuits 67
6 Boole, Shannon, and Probability 88
7 Some Combinatorial Logic Examples 114
8 Sequential-State Digital Circuits 139
9 Turing Machines 161
10 Beyond Boole and Shannon 176
For the Future: The Anti-Amphibological Machine 210
Fundamental Electric Circuit Concepts 219