The ambitious title of this book suggests a potentially wide scope. The back cover claims that the book intends to “demonstrate the practical computing applications behind seemingly abstract ideas.” With such a long list of topics thus identified, one might expect a large book to cover them all. Instead, the book is surprisingly small, with 275 pages spread across 16 chapters. This hints at a possible superficiality of coverage, which is unfortunately confirmed on further examination.

O’Regan tries to inventory all common mathematical tools and theories used in computing, as well as some historical data from ancient to modern times. Topics include ancient era mathematics; the basics of set theory and propositional logic; a cursory view of software engineering; formal methods for the specification and validation of programs, with examples such as the Z specification language; fundamentals of number theory; cryptography; an overview of coding theory and error codes; programming language grammars and parsing; computability and decidability; the fundamentals of probabilities and statistics; basic matrix theory and operations; complex numbers and quaternions; basic calculus; and graph theory.

While the variety of topics does encompass the large spectrum of mathematical topics involved in computer science, the individual chapters often fail to bridge the gap from the mathematical discipline to its practical applications. This leaves the reader wondering where and how the mathematical apparatus briefly presented fuses with computation. While a few applied examples are presented, they are not well explained, due probably to the thinness of the chapters. Indeed, some chapters lack any computational details whatsoever, or offer trivial examples that do not really reflect the complexity of the mathematical concept or its applicability to computation. The chapters on set and function theory, matrices, complex numbers, quaternions, and calculus do not contain any computational applications, not even a discussion of how these topics intersect with computation. Instead, they contain a complex set of descriptions of axioms, basic formulas, and a few theorems, which one might expect in short mathematics monographs, but not in a book claiming to demonstrate the computational intricacies enabled by these tools. On the other hand, the chapters on formal methods, language theory, parsing, and the Z specification language seem well balanced and better written.

For these reasons, it is doubtful that computer science students would benefit significantly from this book, as their curricula typically include applied computational topics preceded by the mathematical information required to formalize them. For general readers, the book offers a wide but shallow view of mathematical subjects involved in computing. Mostly, the book amounts to a summary of the theoretical formulation of the mathematical concepts, with few examples that truly link them to computation. Because of the axiomatic presentation of each mathematical concept, general readers without prior exposure to fundamental mathematics may find the book challenging, while readers acquainted with formal mathematics may be frustrated by the simplicity (or absence) of computational examples. As such, the optimal audience for the book is difficult to define.

In conclusion, this book has an enticing title, but does not live up to the goals claimed by the author. If measured against those claims, it is disappointing overall. Perhaps some college freshmen interested both in mathematics and computer science could use it for a list of mathematical concepts linked to algorithmic and applied computer science, but they will certainly need to go beyond the content of the book if some of the topics catch their interest. For that purpose, there is a short bibliography.