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

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

Published $\text {1968}$

Subject Matter

  • History of Mathematics


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

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