- Author:
- Yves Nievergelt
This text for the first or second year undergraduate in mathematics, logic, computer science, or social sciences, introduces the reader to logic, proofs, sets, and number theory. It also serves as an excellent independent study reference and resource for instructors. Adapted from Foundations of Logic and Mathematics: Applications to Science and Cryptography 2002 Birkhuser, this second edition provides a modern introduction to the foundations of logic, mathematics, and computers science, developing the theory that demonstrates construction of all mathematics and theoretical computer science from logic and set theory. The focuses is on foundations, with specific statements of all the associated axioms and rules of logic and set theory, and provides complete details and derivations of formal proofs. Copious references to literature that document historical development is also provided.Answers are found to many questions that usually remain unanswered: Why is the truth table for logical implication so unintuitive? Why are there no recipes to design proofs? Where do these numerous mathematical rules come from? What issues in logic, mathematics, and computer science still remain unresolved? And the perennial question: In what ways are we going to use this material? Additionally, the selection of topics presented reflects many major accomplishments from the twentieth century and includes applications in game theory and Nash's equilibrium, Gale and Shapley's match making algorithms, Arrow's Impossibility Theorem in voting, to name a few. From the reviews of the first edition:"...All the results are proved in full detail from first principles...remarkably, the arithmetic laws on the rational numbers are proved, step after step, starting from the very definitions!...This is a valuable reference text and a useful companion for anybody wondering how basic mathematical concepts can be rigorously developed within set theory."MATHEMATICAL REVIEWS"Rigorous and modern in its theoretical aspect, attractive as a detective novel in its applied aspects, this paper book deserves the attention of both beginners and advanced students in mathematics, logic and computer sciences as well as in social sciences."Zentralblatt MATH
Index Terms
- Logic, Mathematics, and Computer Science: Modern Foundations with Practical Applications
Recommendations
Martin D. Schweitzer
Probably my first observation of this book was that, like many Springer books, it's hardcore. I do not mean that the material is particularly difficult or intractable, but that the book does not pander to the reader. There is no breezy humor, nor are there long explanations. Rather, it gets straight down to the business of the topics. The topics, as the first part of the name suggests, are logic and mathematics. While there are some computer science topics, I feel a more accurate title may have been Logic and mathematics for computer science . This is very much a book that teaches the mathematics that underlies a lot of computer science (also often referred to as discrete mathematics). After some brief introductory material, the book jumps straight into propositional logic. In under 100 pages, it covers a lot of material. While this book is probably suitable for self-guided study, and probably can be understood by someone with only high school mathematics, it would take a lot of discipline and focus to work through it. On the other hand, it is quite suitable for a first- or second-year university course where the hand of the lecturer can guide the student through the material. There are fairly comprehensive sets of problems at the end of each chapter, and solutions are given to many of the odd numbered problems. The same observation about the level of focus required applies to the rest of the book. Predicate logic and first-order logic occupy about the first third of the book. The next part introduces set theory, which also runs into mathematical functions. Once again, the material is solid and comprehensive for the level at which it is aimed. This material also covers roughly one-third of the book. The third part of the book (note that the distinctions between the three different parts of the book are mine, not imposed by the book) builds on set theory and introduces the integers and the rationals. Chapter 4, "Mathematical Induction," lays the groundwork for the integers and rationals. Since I have a computer science background, it is hard to assess what it would be like for someone without this background to read the later chapters and assimilate them, but I suspect it would be rather challenging. The material is dense and unforgiving, but thorough. Those who prefer definitions, axioms, and theorems to prose will find the approach welcoming. The penultimate chapters delve more deeply into set theory (chapter 5: "Well-Formed Sets: Proof by Transfinite Induction with Already Well-Ordered Sets" and chapter 6: "The Axiom of Choice: Proofs by Transfinite Induction"). Finally, the last chapter is perhaps the "chattiest" of the chapters and looks at the applications of earlier work. Chapter 7, "Applications: Nobel-Prize Winning Applications of Sets, Functions, and Relations," looks at some practical applications of the work introduced in the earlier sections. In summary, the book is a very thorough treatment of the topics mentioned above. It is quite suitable as a textbook for an undergraduate course in the logic and mathematics that underpin a computer science course. The exercises are comprehensive. I would probably not recommend this book to someone without a solid mathematical background as an introduction to the topics for self-guided learning. Online Computing Reviews Service
Access critical reviews of Computing literature here
Become a reviewer for Computing Reviews.