Book:R.L. Wilder/Evolution of Mathematical Concepts

Subject Matter

 * History of Mathematics

Contents

 * Preface


 * Introduction
 * 1 Conceptions of the Nature of Mathematics
 * 2 Mathematics in the Schools
 * 3 Humanistic Aspects of Mathematics
 * 4 Modern 'Reforms' in Mathematical Education


 * 1 Preliminary Notions
 * 1.1 The Notion of Culture
 * 1.1.1 A Culture as an Organic Whole
 * 1.1.2 Relations between Cultures and Peoples
 * 1.1.3 Contrast between the 'Lives' of a Culture and of a People
 * 1.2 Processes of Culture Change and Growth
 * 1.3 Mathematics as a Culture
 * 1.4 Systems of Number Notation


 * 2 Early Evolution of Number
 * 2.1 Inception of Counting
 * 2.1.1 Environmental Stress, Physical and Cultural
 * 2.1.2 Primitive Counting
 * 2.1.2a Distinction between 'Numeral' and 'Number'
 * 2.1.2b Distinction between 'Cardinal' and 'Ordinal'
 * 2.1.2c 'Two-Counting'
 * 2.1.2d Tallying: One-to-One Correspondence
 * 2.1.2e Number Categories: Adjectival Forms
 * 2.2 Written Numeral Systems
 * 2.2.1 Sumerian-Babylonian and Mayan Numerals: Place Value; Zero Symbol
 * 2.2.1a The Bases 10 and 60
 * 2.2.1b Place Value in the Babylonian and Mayan Numeral System
 * 2.2.1c Zero Symbols
 * 2.2.1d Sexagesimal Fractions
 * 2.2.2 Cipherization
 * 2.2.2a The Ionian Numerals
 * 2.2.3 Fusion of Place Value and Cipherization
 * 2.2.4 Decimal Fractions
 * 2.3 Evolution of the Conceptual Aspect of Number
 * 2.3.1 Number Mysticism; Numerology
 * 2.3.2 A Number Science
 * 2.3.3 Status of the Number Concept and Its Symbolization at the End of the Babylonian Ascendancy
 * 2.3.4 The 'Pythagoran' School
 * 2.4 Interlude


 * 3 Evolution of Geometry
 * 3.1 The Position of Geometry in Mathematics
 * 3.2 Pre-Greek 'Geometry'
 * 3.3 Why Did Geometry Become Part of Mathematics?
 * 3.3.1 Number and Geometric Magnitude
 * 3.3.1a Geometric Number Theory
 * 3.3.2 Number Theory in Euclid; Number and Magnitude
 * 3.3.3 Concept of Form in Number and Geometry
 * 3.4 Later Developments in Geometry
 * 3.4.1 Non-Euclidean Geometry
 * 3.4.2 Analytic Geometry
 * 3.5 Effects of the Diffusion of Geometric Modes throughout Mathematics
 * 3.5.1 Axiomatic Method; Introduction of Logic
 * 3.5.2 Revolution in Mathematical Thought
 * 3.5.3 Effects on Analysis
 * 3.5.4 Labels and Modes of Thought


 * 4 The Real Numbers. Conquest of the Infinite
 * 4.1 The Real Numbers
 * 4.1.1 The Irrational Numbers and Infinity
 * 4.1.2 The Infinite Decimal Symbol for a Real Number
 * 4.1.3 The Real Number as a 'Magnitude'
 * 4.1.4 The Real Numbers Based on the Natural Numbers
 * 4.2 The Class of Real Numbers
 * 4.2.1 The Cantor Diagonal Method
 * 4.3 Transfinite Numbers; Cardinal Numbers
 * 4.3.1 Extension of 'Counting Numbers' to the Infinite
 * 4.3.2 Transfinite Ordinal Numbers
 * 4.4 What is Number?


 * 5 The Process of Evolution
 * 5.1 The Pre-Greek Elements
 * 5.2 The Greek Era
 * 5.3 The Post-Green and European Developments
 * 5.3.1 Non-Euclidean Geometry
 * 5.3.2 Introduction of the Infinite
 * 5.4 The Forces of Mathematical Evolution
 * 5.4.1 Commentary and Definitions
 * 5.4.2 The Individual Level
 * 5.4 Stages in the Evolution of Number


 * 6 Evolutionary Aspects of Modern Mathematics
 * 6.1 Mathematics and Its Relation to the Other Sciences
 * 6.1.1 Relation to Physics
 * 6.1.2 Tendencies towards Greater Abstraction in Science
 * 6.1.3 Relation to Other Sciences in General
 * 6.1.4 Specialization
 * 6.1.5 Pure and Applied Mathematics
 * 6.2 The 'Foundations' of Mathematics
 * 6.2.1 The Mathematical Subculture
 * 6.2.2 The Emergence of Contradictions
 * 6.2.3 Mathematical Logic and Set Theory
 * 6.3 Mathematical Existence
 * 6.4 'Laws' Governing the Evolution of Mathematical Concepts
 * 6.4.1 Discussion
 * 6.4.2 Conclusion


 * Bibliography


 * Index