Logic, Mathematics, and Computer Science

Despite its expansive title, this book is primarily about set theory. Overall the book is slanted to logic and to methods of proof. It also includes a final chapter of unrelated applications.

Strictly speaking the development assumes no knowledge of anything mathematical, but it would be much easier to follow if you have had some prior exposure to syllogisms, proof methods, and informal set theory. In fact, without this prior exposure you might be able to follow the book but would have no idea why its content is important.

The first third of the book deals with propositional logic and first-order logic. The treatment is fairly conventional, although it does include some treatment of multi-valued logic and fuzzy logic and considers several alternative axiom systems.

Most of the book deals with set theory. The approach is mostly axiomatic (rather than naive), based on the Zermelo–Fraenkel axioms, but switching viewpoints often to logic. There’s also a construction of the number system up through the rationals. There’s quite a lot on well-ordering and induction and a chapter on the axiom of choice.

The last chapter is applications: an outline of game theory, Gale & Shapley’s match-making algorithms, and Arrow’s impossibility theorem about voting systems. None of these have anything in particular to do with the rest of the book, but as before the exposition is based on reasoning and first principles rather than mathematical knowledge.

Although logic and reasoning might be a unifying theme of the book, I think it still doesn’t hang together well. It is further harmed by a lot of gratuitous photographs of purported applications; for example, on p. 304 there are two photos of early rockets, which are supposed to illustrate the nuclear arms race in the context of game theory (although neither rocket is carrying a nuclear warhead).

Bottom line: a difficult book to like, because it lacks focus.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.