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Mathematics as a Cultural System
- 1st Edition - May 20, 2014
- Author: Raymond L. Wilder
- Editor: Mario Bunge
- Language: English
- Paperback ISBN:9 7 8 - 1 - 4 8 3 1 - 0 6 9 5 - 3
- 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
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Request a sales quoteMathematics 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
- No. of pages: 194
- Language: English
- Edition: 1
- Published: May 20, 2014
- Imprint: Pergamon
- Paperback ISBN: 9781483106953
- eBook ISBN: 9781483100616
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