Labyrinth of Thought
Subtitle: A History of Set Theory and Its Role in Modern Mathematics (Science Networks Historical Studies, Vol. 23) (Science Networks. Historical Studies)
Contributors: Erwin Hiebert (Editor); Eberhard Knobloch (Editor); Erhard Scholz (Editor)
Edition: 1st ed. 1999. 2nd printing edition
Publisher: Birkhäuser Basel
Published: November 23, 1999
Language: eng
Format: Hardcover
Number of pages: 464
ISBN-13: 9783764357498
Subjects: History of ideas, intellectual history; History of science; Philosophy of mathematics; Set Theory; 20th century; c 1800 to c 1900; Science/Mathematics; General; Science; Mathematics; History; Mathematics / History; Mathematics / Set Theory; Medical : General; Science : General
First sentence: It is a common characteristic of the various attempts to integrate the totality of mathematics into a coherent whole - whether we think of Plato, of Descartes, or of Leibniz, of arithmetization, or of the logicists of the nineteenth-century - that they have all been made in connection with a philosophical system, more or less wide in scope; always starting from a priori views concerning the relations of mathematics with the twofold universe of the external world and the world of thought.1
Description: <i>Labyrinth of Through</i> discusses set theory's emergence and development, and the set-theoretical approach to mathematics during 1850-1840.  Rather than focusing on Georg Cantor, it analyzes his work and transfinite set theory's emergence within the broader context of modern mathematics' rise.  Questions addressed include: Why and how did mathematicians begin paying close attention to the notion of a set?  What role did the notion of set play in the emergence of modern mathematics?  How did set theory turn into an autonomous branch of mathematics, and how did our present conception of the theory become widely accepted? (from back cover copy)
Table of contents: title: . Introduction, xi; title: 1: Aims and Scope, xiii; title: 2: General Historical Remarks, xvii; title: . Part One: The Emergence of Sets within Mathematics, 1; title: . I Institutional and Intellectural Contexts in German Mathematics, 1800-1870, 3; title: 1: Mathematics at the Reformed German Universities, 4; title: 2: Traditional and 'Modern' Foundational Viewpoints, 10; title: 3: The Issue of the Infinite, 18; title: 4: The G&#246;ttingen Group, 1855-1859, 24; title: 5: The Berlin School, 1855-1870, 32; title: . II A New Fundamental Notion: Reimann's Manifolds, 39; title: 1: The Historical Context: Grossenlehre, Gauss, and Herbart, 41; title: 2: Logical Prerequisites, 47; title: 3:The Mathematical Context of Riemann's Innovation, 53; title: 4: Riemann's General Definition, 62; title: 5: Manifolds, Arithmetic, and Topology, 67; title: 6 Riemann's Influence on the Development of Set Theory, 70; title: <i>Appendix</i>: Riemann and Dedekind, 77; title: . III Dedekind and the Set-Theoretical Approach to Algebra, 81; title: 1: The Algebraic Origins of Dedekind's Set Theory, 1856-58, 82; title: 2: A New Fundamental Notion for Algebra: Fields, 90; title: 3: The Emergence of Algebraic Number Theory, 94; title: 4: Ideals and Methodology, 99; title: 5: Dedekind's Infinitism, 107; title: 6: The Diffusion of Dedekind's Views, 111; title: . IV The Real Number System, 117; title: 1: 'Construction' vs. Axiomatization, 119; title: 2: The Definitions of the Real Numbers, 124; title: 3: The Influence of Riemann: Continuity in Arithmetic and Geometry, 135; title: 4: Elements of the Topology of <b>R</b>, 137; title: . V Origins of the Theory of Point-Sets, 145; title: 1: Dirichlet and Riemann: Transformations in the Theory of Real Functions, 147; title: 2: Lipschitz and Hankel on Nowhere Dense Sets and Integration, 154; title: 3: Cantor on Sets of the First Species, 157; title: 4: Nowhere Dense Sets of the Second Species, 161; title: 5: Crystallization of the Notion of Content, 165; title: . Part Two: Entering the Labyrinth - Toward Abstract Set Theory, 169; title: . VI The Notion of Cardinality and the Continuum Hypothesis, 171; title: 1: The relations and Corespondence Between Cantor and Dedekind, 172; title: 2: Non-denumerability of <b>R</b>, 176; title: 3: Cantor's Exposition and the 'Berlin Circumstances', 183; title: 4: Equipollence of Continua <b>R</b> and <b>R</b><sup><i>n</i></sup>, 187; title: 5: Cantor's Difficulties, 197; title: 6: Derived Sets and Cardinalities, 202; title: 7: Cantor's Definition of the Continuum, 208; title: 8: Further Efforts on the Continuum Hypothesis, 210; title: . VII Sets and Maps as A Foundation for Mathematics, 215; title: 1: Origins of Dedekind's Program for the Foundations of Arithmetic, 218; title: 2: Theory of Sets, Mappings, and Chains, 224; title: 3: Through the Natural Numbers to Pure Mathematics, 232; title: 4: Dedekind and the Cantor-Bernstein Theorem, 239; title: 5: Dedekind's Theorem of Infinity, and Epistemology, 241; title: 6: Reception of Dedekind's Ideas, 248; title: . VIII The Transfinit Ordinals and Cantor's Mature Theory, 257; title: 1: "Free Mathematics", 259; title: 2: Cantor's Notion of Set in the Early 1880s, 263; title: 3: The Transfinite (Ordinal) Numbers, 267; title: 4: Ordered Sets, 274; title: 5: The Reception in Early 1880s, 282; title: 6: Cantor's Theorem, 286; title: 7: The <i>Beitrage zur Begr&#252;ndung der transfiniten Mengenlehre</i>, 288; title: 8: Cantor and the Paradoxes, 290; title: . Part Three: In Search of an Axiom System, 297; title: . IX Diffusion, Crisis, and Bifurcation: 1890 to 1914, 299; title: 1: Spreading Set Theory, 300; title: 2: The Complex Emergence of the Paradoxes, 306; title: 3: The Axiom of Choice and the Early Foundational Debate, 311; title: 4: The Early Work of Zermelo, 317; title: 5: Russell's Theory of Types, 325; title: 6: Other Developments in Set Theory, 333; title: . X Logic and Type Theory in the Interwar Period, 337; title: 1: An atmosphere of Insecurity: Weyl, Brouwer, Hilbert, 338; title: 2: Diverging Conceptions of Logic, 345; title: 3: The Road to the Simple Theory of Types, 348; title: 4: Type Theory at its Zenith, 353; title: 5: A Radical Proposal: Weyl and Skolem on First-Order Logic, 357; title: . XI Consolidation of Axiomatic Set Theory, 365; title: 1: The Contributions of Fraenkel, 336; title: 2: Toward the Modern Axiom System: von Neumann and Zermelo, 370; title: 3: The System von Neumann-Bernays-G&#246;del, 378; title: 4: G&#246;del's Relative Consistency Results, 382; title: 5: First-Order Axiomatic Set Theory, 386; title: 6: A Glance Ahead: Mathematics and Foundations after World War II, 388; title: . Bibliographical References, 393; title: . Index of Illustrations, 422; title: . Name Index, 423; title: . Subject Index, 430