J. Lambek - Böcker

This is a textbook on the history, philosophy and foundations of mathematics. One of its aims is to present some interesting mathematics, not normally taught in other courses, in a historical and philosophical setting. The book is intended mainly for undergraduate mathematics students, but also for students in the sciences, humanities and education with a strong interest in mathematics. It proceeds in historical order from about 1800 BC to 1800 AD and then presents some selected topics of foundational interest from the 19th and 20th centuries. Among other material in the first part, the authors discuss the renaissance method for solving cubic and quartic equations and give rigorous elementary proofs that certain geometrical problems posed by the ancient Greeks (e.g. the problem of trisecting an arbitrary angle) cannot be solved by ruler and compass constructions. Among the topics in the second part, they sketch a proof of Godel's incompleteness theorem and discuss some of its implications, and also present the elements of category theory. The authors' approach to most of these matters is new.

In this book the authors reconcile two different viewpoints of the foundations of mathematics, namely mathematical logic and category theory. In Part I, they show that typed lambda-calculi, a formulation of higher order logic, and cartesian closed categories are essentially the same. In Part II, it is demonstrated that another formulation of higher order logic (intuitionistic type theories) is closely related to topos theory. Part III is devoted to recursive functions. Numerous applications of the close relationship between traditional logic and the algebraic language of category theory are given. The authors have included an introduction to category theory and develop the necessary logic as required, making the book essentially self-contained. Detailed historical references are provided throughout, and each section concludes with a set of exercises. Thus it is well-suited for graduate courses and research in mathematics and logic. Researchers in theoretical computer science, artificial intelligence and mathematical linguistics will also find this an accessible introduction to a subject of increasing application to these disciplines.