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This book is a different approach to teaching the foundations of mathematical analysis and of computation. The main idea is to delay the use of "formal definitions", which are definitions that nobody can understand without working with them. The approach of this book is to employ the history of mathematics to first develop fundamental concepts of mathematical analysis and the theory of computation and to only introduce formal definitions after the concepts are understood by the students.The historical order clarifies what analysis is really about and also why the theory of computation came about. The book provides students with a broader background involving for instance glimpses of cardinal arithmetic, predicate logic background, as well as the importance of a sound theory of the infinitesimal (which is in essence the foundations of mathematics and computation).There is a wealth of exercises and numerous graphical illustrations which give an experienced instructor lots of possibilities to select a stimulating course with a broader background. Even for just browsing by general readers, this book presents stories, insights and mathematical theories, covering a window of ancient times to the present.The book is self explanatory and self sufficient, so any staff member in the departments of mathematics or computer science can teach this course.This book will give the students the right techniques and skills to work with mathematical analysis and the theory of computation and to go on further to study more advanced courses on the subject.

In 1922, Curry started reading Principia Mathematica and was intrigued by the complications of its substitution rule. As a result of trying to analyze substitution, Curry conceived the combinators in 1926. This collection is dedicated to Jonathan Seldin's 80th anniversary. Seldin is the penultimate PhD student of Curry and the guardian of Curry's paradigm.The search at the beginning of the 20th century for powerful systems that combine computations and deductions (functions and logic) and that are able to formalise mathematics has led to the birth of the mighty ¿-calculus of Church, Combinatory Logic of Curry and Category Theory of Eilenberg and Mac Lane, all of which are well represented in this collection. The struggle for internalising as much as possible while keeping the system consistent is clear in the evolution of the ¿-calculus and combinatory logic and can be felt again in the articles in this volume. Similarly, the struggle for elegant theories that minimise the number of basic concepts while remaining as close as possible to the language's structure is clear. Generalising concepts, connecting areas that may seem far apart and applying useful techniques from one area to the other is also represented well in this volume where for example notions like coherence, confluence, commuting diagrams, are extended between ¿-calculus, rewriting systems and category theory, and where embedding relations are given to allow a lot of disciplines from logic to mathematics to computer science to meet.