General Introduction and Motivation

The main goal of this book is to explore in detail the logico-philosophical and historical aspects of the lambda-calculus ( \(=\lambda \) -calculus). This first introductory chapter defines the \(\lambda \) -calculus and explains, in an intuitive way, its main ideas and two important motivations for its invention. The chapter also points out the deep connections that the \(\lambda \) -calculus has not only with logic but with other fundamental branches of analytic philosophy in the tradition of Frege, Russell, and Quine. In fact, Frege, Russell, and Wittgenstein anticipated crucial aspects of the \(\lambda \) -calculus. Furthermore, these crucial connections (and anticipations) motivate the creation of a new kind of logic-based analytic philosophy: lambda-philosophy ( \(=\lambda \) -philosophy). Several results of \(\lambda \) -philosophy are explored in detail in many chapters of the book. The \(\lambda \) -calculus can be described as a non-extensional logical theory of functions as rules (of computation or correspondence) and a programming language. Besides intuitive explanations of basic concepts like Church numerals and functional abstraction, this chapter and the whole book are motivated by certain rebellious questions like this one: being computability theory a branch of both symbolic logic and software-oriented computer science, then, why so many researchers and teachers focus in a hardware-oriented model (as Turing-Post machines)?