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

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## R.L. Wilder:

## Contents

## R.L. Wilder: *Evolution of Mathematical Concepts: An Elementary Study*

Published $\text {1968}$.

## 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

- 1.1 The Notion of Culture

**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

- 2.2.1 Sumerian-Babylonian and Mayan Numerals: Place Value; Zero Symbol
- 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

- 2.1 Inception of Counting

**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

- 3.3.1 Number and Geometric Magnitude
- 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?

- 4.1 The Real Numbers

**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

- 6.1 Mathematics and Its Relation to the Other Sciences

*Bibliography*

*Index*