Mathematics as a Cultural System

1st Edition - January 1, 1981
Author: Raymond L. Wilder
Editor: Mario Bunge
Language: English
eBook ISBN:
9 7 8 - 1 - 4 8 3 1 - 0 0 6 1 - 6

Mathematics as a Cultural System discusses the relationship between mathematics and culture. The book is comprised of eight chapters discussing topics that support the concept of… Read more

Purchase options

LIMITED OFFER

Save 50% on book bundles

Immediately download your ebook while waiting for your print delivery. No promo code is needed.

Institutional subscription on ScienceDirect

Request a sales quote

Mathematics as a Cultural System discusses the relationship between mathematics and culture. The book is comprised of eight chapters discussing topics that support the concept of mathematics as a cultural system. Chapter I deals with the nature of culture and cultural systems, while Chapter 2 provides examples of cultural patterns observable in the evolution of mechanics. Chapter III treats historical episodes as a laboratory for the illustration of patterns and forces that have been operative in cultural change. Chapter IV covers hereditary stress, and Chapter V discusses consolidation as a force and process. Chapter VI talks about the singularities in the evolution of mechanics, while Chapter 7 deals with the laws governing the evolution of mathematics. Chapter VIII tackles the role and future of mathematics. The book will be of great interest to readers who are curious about how mathematics relates to culture.

Chapter I. The Nature of Culture and Cultural Systems

1. Evolution of a Cultural Artifact

2. The Things That Make up a Culture

3. Culture as a Collection of Elements in a Communications Network

4. Mathematics as a Cultural System

5. Cultural and Conceptual Evolution

Chapter II. Examples of Cultural Patterns Observable in the Evolution of Mathematics

1. Multiples

2. "Clustering of Genius"

3. The "Before His Time" Phenomenon

4. The Operation of Cultural Lag in Mathematics

5. Patterns of Thought. Mathematical Reality and Mathematical Existence

6. Evolution of Greater Abstraction

7. Forced Origins of New Concepts

8. Selection in Mathematics

9. The Effect of the Occurrence of Paradox, or the Discovery of Inconsistency

10. The Relativity of Mathematical Rigor

11. Growth Patterns of Fields of Mathematics

12. A Problem

Chapter III. Historical Episodes; A Laboratory for the Study of Cultural Change

1. The Great Diffusions

2. Symbolic Achievements

3. Pressure from the Environment; Environmental Stress

4. Motivation for Multiple Invention; Exceptions to the Rule

5. The Great Consolidations

6. Leaps in Abstraction

7. Great Generalizations

Chapter IV. Potential of a Theory or Field; Hereditary Stress

1. Hereditary Stress

2. Components of Hereditary Stress

(i) Capacity

(ii) Significance

(iii) Challenge

(iv) Conceptual Stress

(v) Status

(vi) Paradox and/or Inconsistency

3. General Remarks

Chapter V. Consolidation: Force and Process

Part I. General Theory

Ia. Consolidation as a Social or Cultural Phenomenon

Ib. Effects of Diffusion

Part II. The Consolidation Process in Mathematics

Part IIa. Examples

IIb. Cultural Lag and Cultural Resistance in the Consolidation Process

IIIc. Analysis

Part III. Concluding Remarks

Chapter VI. The Exceptional Individual; Singularities in the Evolution of Mathematics

1. General Remarks. Mendel, Bolzano, Desargues

2. Historical Background of Desargues' Work

2a. Girard Desargues and "PG17"

3. Why Was PG17 not Developed into a Field?

3a. The Mathematical Environment of the 17th Century

3b. The Internal Nature of PG17

4. Avenues of Possible Survival

5. The Success of Projective Geometry in the 19th Century

6. General Characteristics of the "before-His-Time" Phenomenon

6a. The Premat as a Loner

6b. Tendency of the Premat to Create a Vocabulary That Repels Possible Readers

6c. The Capacity and Significance of the New Concepts Embodied in the Prematurity not Recognized

6d. The Culture not Ready to Incorporate and Extend the New Concepts Embodied in the Prematurity

6e. Lack of Personal Status of the Premat in the Scientific Community

6f. Insufficient Diffusion of the New Ideas Presented by the Prematurity

6g. An Unusual Combination of Interests on the Part of the Premat

7. Comment

Chapter VII. "Laws" Governing the Evolution of Mathematics

1. Law Governing Multiple Discovery

la. Law Governing First Proof of a Theorem

2. Law Re. Acceptance of a New Concept

3. Law Re. Evolution of New Concepts

4. Law Re. the Status of Creator of a New Concept

5. Law Re. Continued Importance of a Concept

6. Law Re. The Solution of an Important Problem

7. Law Re. The Occurrence of Consolidation

7a. Law of Consolidation

8. Law Re. Interpretation of "Unreal" Concepts

9. Law Re. The Cultural Intuition

10. Law Re. Diffusion

11. Law Re. Environmental Stresses

12. Law Re. Great Advances or Breakthroughs

13. Law Re. Inadequacies of a Conceptual Structure

14. Law Re. Revolutions in Mathematics

15. Law Re. Mathematical Rigor

16. Law Re. Evolution of a Mathematical System

17. Law Re. The Individual and Mathematics

18. Law Re. Mathematics Becoming "Worked out"

19. Law Re. Beginnings

20. Law Re. Ultimate Foundation of Mathematics

21. Law Re. Hidden Assumptions

22. Law Re. Emergence of Periods of Great Activity

23. Law Re. Absolutes in Mathematics

Chapter VIII. Mathematics in the 20th Century; Role and Future

1. The Place of Mathematics in 20th-Century Culture

2. Future "Dark Ages?"

3. The Role of Mathematics in the 20th Century

4. The Uses of Mathematics in the Natural and Social Sciences

5. Relevance to Historiography

Appendix: Footnote for the Aspiring Mathematician

Bibliography

Index