The Logic, Philosophy, and History of the Lambda-Calculus: Theory and Applications

by Levis Zerpa

Overview

This is the first book focused on the logico-philosophical aspects of the lambda-calculus since the inception of the field in 1932 in the pioneering work of Alonzo Church. The book starts a new field called “lambda-philosophy”: a branch of logic-based analytic philosophy in the tradition of Frege and Russell, directly constructed from the lambda-calculus. Another innovation of the book is a new graphical and intuitive logico-mathematical notation for the lambda-calculus called “the container notation”.

The book covers in detail some episodes of the history of the subject, including three comparative studies of the lambda-calculus with Viète's algebra, Descartes' analytic geometry, and Wittgenstein’s Tractatus. A didactic approach to the logico-mathematical aspects of the lambda-calculus, partially based on cognitive science, provides the technical basis for the analysis. In this way, the book provides a systematic and coherent treatment of diverse logico-philosophical aspects and applications of the lambda-calculus as part of the platform offered by lambda-philosophy.

The book includes the following: a detailed treatment of the ambiguities of the concept of function (under the traditional or Euler’s notation); an elucidation of the notion of transformative philosophical analysis; an account of Church’s methodology which shows that the lambda-calculus is an adequate solution to the problem of a philosophical analysis of functions as rules of computation; a didactic treatment of the formal aspects of the lambda-calculus through the container notation; and diverse arithmetical and logical examples of the container notation. The book questions a dogma of algorithmic thinking by arguing that the lambda-calculus is more intuitive and natural than Turing machines.

The Logic, Philosophy, and History of the Lambda-Calculus is essential reading for all scholars and researchers of the history of analytic philosophy and especially those focussing on logic-based analytic philosophy.

Product Details

ISBN-13:	9783031728501
Publisher:	Springer Nature Switzerland
Publication date:	01/07/2025
Series:	History of Analytic Philosophy
Edition description:	2025
Pages:	255
Product dimensions:	5.83(w) x 8.27(h) x 0.00(d)

About the Author

Levis Zerpa is Researcher-Professor at Yachay Tech University, Ecuador.

Table of Contents

Part I Introduction.- 1 General Introduction and Motivation.- Part II: The Logic and Philosophy of the Lambda-Calculus: Lambda-Philosophy.- 2 Lambda-Philosophy: The General Program.- 3 The λ-Calculus as a Solution to a Problem of Philosophical Analysis.- 4 The λ-Calculus as a Formal System of Symbolic Logic and the Container Notation.- Part III: Two Applications: Arithmetic and Propositional Logic.- 5 The Container Notation in the λ-Calculus (1): Arithmetic.- 6 The Container Notation in the λ-Calculus (2): Propositional Logic.- Part IV: The Philosophy of Computer Science and Computability Theory in Lambda-Philosophy.- 7 Questioning a Dogma of Algorithmic Thinking in Our Time.- Part V: Ontology in Lambda-Philosophy.- 8 The Criterion of Ontological Commitment as a Logical Tester in Ontology.- 9 Church’s Ontological Development.- 10 Compiler Output as a Programming Tester in Ontology.- Part VI: The History of The Lambda-Calculus: Three Comparative Studies.- 11 Comparative Historical Studies (0): Introduction.- 12 Comparative Historical Studies (1): Viète’s Analytic Art in λ-Philosophy.- 13 Comparative Historical Studies (2): A Brief Analysis of Descartes’ Correlation-Hypothesis and Church’s Thesis.- 14 Comparative Historical Studies (3): Wittgenstein Anticipation of Church Numerals.