Conflicts Between Generalization, Rigor, and Intuition,9780387228365

Conflicts Between Generalization, Rigor, and Intuition

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ISBN13: 9780387228365
ISBN10: 0387228365
Format: Hardcover
Pub. Date: 2005-06-14
Publisher(s): Springer Verlag
List Price: $237.68

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Summary

This book deals with the development of the terms of analysis in the 18th and 19th centuries, the two main concepts being negative numbers and infinitesimals. Schubring studies often overlooked texts, in particular German and French textbooks, and reveals a much richer history than previously thought while throwing new light on major figures, such as Cauchy.

Table of Contents

Preface v
Illustrations xiv
I. Question and Method 1(14)
1. Methodological Approaches to the History of Science
1(7)
2. Categories for the Analysis of Culturally Shaped Conceptual Developments
8(2)
3. An unusual Pair: Negative Numbers and Infinitely Small Quantities
10(5)
II. Paths Toward Algebraization - Development to the Eighteenth Century. The Number Field 15(136)
1. An Overview of the History of Key Fundamental Concepts
15(17)
1.1. The Concept of Number
16(3)
1.2. The Concept of Variable
19(1)
1.3. The Concept of Function
20(2)
1.4. The Concept of Limit
22(3)
1.5. Continuity
25(2)
1.6. Convergence
27(3)
1.7. The Integral
30(2)
2. The Development of Negative Numbers
32(119)
2.1. Introduction
32(3)
2.2. An Overview of the Early History of Negative Numbers
35(5)
From Antiquity to the Middle Ages
35(4)
European Mathematics in the Middle Ages
39(1)
2.3. The Onset of Early Modem Times. The First "Ruptures" in Cardano's Works
40(5)
2.4. Further Developments in Algebra: From Viète to Descartes
45(4)
2.5. The Controversy Between Arnauld and Prestet
49(12)
A New Type of Textbook
49(1)
Antoine Arnauld
50(2)
Jean Prestet
52(2)
The Controversy
54(3)
The Debate's Effects on Their Textbook Reeditions
57(4)
2.6. An Insertion: Brief Comparison of the Institutions for Mathematical Teaching in France, Germany, and England
61(6)
Universities and Faculties of Arts and Philosophy
61(3)
The Status of Mathematics in Various Systems of National Education
64(2)
New Approaches in the Eighteenth Century
66(1)
2.7. First Foundational Reflections on Generalization
67(6)
2.8. Extension of the Concept Field to 1730/40
73(26)
2.8.1. France
73(15)
2.8.2. Developments in England and Scotland
88(7)
2.8.3. The Beginnings in Germany
95(4)
2.9. The Onset of an Epistemological Rupture
99(15)
2.9.1. Fontenelle: Separation of Quantity from Quality
99(3)
2.9.2. Clairaut: Reinterpreting the Negative as Positive
102(2)
2.9.3. D'Alembert: The Generality of Algebra An Inconvénient
104(10)
2.10. Aspects of the Crisis to 1800
114(35)
2.10.1. Stagnant Waters in the French University Context
114(7)
2.10.2. The Military Schools as Multipliers
121(5)
2.10.3. Violent Reaction in England and Scotland
126(6)
2.10.4. The Concept of Oppositeness in Germany
132(17)
2.11. Looking Back
149(2)
III. Paths toward Algebraization-The Field of Limits: The Development of Infinitely Small Quantities 151(106)
1. Introduction
151(2)
2. From Antiquity to Modem Times
153(4)
Concepts of the Greek Philosophers
154(3)
3. Early Modern Times
157(4)
4. The Founders of Infinitesimal Calculus
161(13)
5. The Law of Continuity: Law of Nature or Mathematical Abstraction?
174(12)
6. The Concept of Infinitely Small Quantities Emerges
186(6)
7. Consolidating the Concept of Infinitely Small Quantities
192(14)
8. The Elaboration of the Concept of Limit
206(32)
8.1. Limits as MacLaurin's Answer to Berkeley
206(3)
8.2. Reception in the Encyclopédie and Its Dissemination
209(4)
8.3. A Muddling of Uses in French University Textbooks
213(7)
8.4. First Explications of the Limit Approach
220(8)
8.5. Expansion of the Limit Approach and Beginnings of Its Algebraization
228(10)
9. Operationalizations of the Concept of Continuity
238(17)
10. A Survey
255(2)
IV. Culmination of Algebraization and Retour du Refoulé 257(52)
1. The Number Field: Additional Approaches Toward Algebraization in Europe and Countercurrents
257(22)
1.1. Euler: The Basis of Mathematics is Numbers, not Quantities
257(3)
1.2. Condillac: Genetic Reconstruction of the Extension of the Number Field
260(5)
1.3. Buée: Application of Algebra as a Language
265(3)
1.4. Fundamentalist Countercurrents
268(11)
Klostermann: Elementary Geometry versus Algebra
276(3)
2. The Limit Field: Dominance of the Analytic Method in France After 1789
279(16)
2.1. Apotheosis of the Analytic Method
279(4)
2.2. Euler's Reception
283(3)
2.3. Algebraic Approaches at the École Polytechnique
286(9)
3. Le Retour du Refoulé: The Renaissance of the Synthetic Method at the École Polytechnique
295(14)
3.1. The Original Conception
295(1)
3.2. Changes in the Structure of Organization
296(2)
3.3. The First Crisis: Pressure from the Corps du Génie and the Artillery
298(4)
3.4. The New Teaching Concept of 1800
302(1)
3.5. Modernization of the Corps du Génie and Extension of the School in Metz
303(3)
3.6. The Crisis at the École Polytechnique in 1810/11
306(3)
V. Le Retour du Refoulé: From the Perspective of Mathematical Concepts 309(118)
1. The Role of Lazare Carnot and His Conceptions
309(56)
1.1. Structures and Personalities
309(3)
1.2. A Short Biography
312(6)
1.3. The Development of Carnot's ideas on Foundations: Analyse and Synthèse
318(7)
1.4. The Change in Carnot's Ideas
325(5)
1.5. Relations Between Mechanics and the Foundations of Mathematics
330(4)
1.6. Infinitesimal Calculus: Carnot's Shift from the Concept of Limit to the Infinitely Small
334(19)
1.6.1. The Memoir for the Berlin Academy
334(12)
1.6.2. The 1797 Version
346(2)
1.6.3. The 1813 Version
348(5)
1.7. Substituting Negative Numbers with Geometric Terms
353(12)
1.7.1. Carnot's Writings on Negative Quantities: An Overview
353(1)
1.7.2. Carnot's Basic Concepts and Achievements
354(1)
1.7.3. The Development of Carnot's Concept of Negative Quantities
355(6)
1.7.4. Later Work: L 'analyse. La Science de la Compensation des Erreurs
361(4)
2. Carnot's Impact: Rejecting the Algebraization Program
365(45)
2.1. Initial Adoption
365(1)
2.2. Limits and Negative Numbers in the Lectures at the École Normale
366(3)
2.3. Dissemination of the Algebraic Conception of Analysis
369(2)
2.4. Analysis Concepts at the École Polytechnique to 1811
371(27)
2.4.1. The Reorganization About 1799
371(1)
2.4.2. Lacroix: Propagator of the Méthode des Limites
372(8)
2.4.3. Further Concept Development: Lacroix, Gamier, and Ampère
380(10)
2.4.4. Prony: An Engineer as a Worker on Foundations
390(4)
2.4.5. Lagrange's Conversion to the Infiniment Pétits?
394(4)
2.5. The Impact of the Return to the Infiniment Petits
398(12)
2.5.1. Impact Inside the École: The "Dualism" Compromise
398(4)
2.5.2. The Impact Outside the École
402(6)
2.5.3. The First Overt Criticism Back at the École: Poinsot in 1815
408(2)
3. Retour to Synthèse for the Negative Numbers
410(17)
3.1. Lacroix Propagates the Absurdité of the Negative
410(10)
3.1.1. ...in Algebra
411(5)
3.1.2. ...in the Application of Algebra to Geometry
416(4)
3.2. The Criticisms of Gergonne and Ampère
420(6)
3.3. A Further Look at England
426(1)
VI. Cauchy's Compromise Concept 427(54)
1. Cauchy: Engineer, Scientist, and Politically Active Catholic
427(4)
2. Conflicting Reception of Cauchy in the History of Mathematics
431(2)
3. Methodological Approaches to Analyzing Cauchy's Work
433(3)
4. Effects of the Context
436(5)
5. The Context of Cauchy's Scientific Context: "I'm Far from Believing Myself Infallible"
441(4)
6. Cauchy's Basic Concepts
445(36)
6.1. The Number Concept
446(4)
6.2. The Variable
450(1)
6.3. The Function Concept
451(1)
6.4. The Limit and the Infiniment Petit
452(5)
6.5. Continuity
457(9)
6.6. Convergence
466(11)
6.7. Introduction of the Definite Integral
477(3)
6.8. Some Final Comments
480(1)
VII. Development of Pure Mathematics in Prussia/Germany 481(86)
1. Summary and Transition: Change of Paradigm
481(2)
2. The Context of Pure Mathematics: The University Model in the Protestant Neohumanist System
483(3)
3. Negative Numbers: Advances in Algebra in Germany
486(48)
3.1. Algebraization and Initial Reactions to Carnot
486(9)
H. D. Wilckens (1800)
486(2)
F.G. Busse (1798, 1801, 1804)
488(5)
J.F. Fries (1810)
493(2)
3.2. The Conceptual-Structural Approach of W.A. Förstemann
495(10)
On Förstemann's Biography
496(2)
His 1817 Work
498(7)
3.3. Reception Between Refutation and Adaptation
505(20)
M. Metternich
505(5)
J.P.W. Stein
510(6)
W. A. Diesterweg and His Students
516(5)
Martin Ohm
521(4)
3.4. The Continued Dominance of the Quantity Concept
525(9)
4. The "Berlin Discussion" of Continuity
534(6)
5. The Advance of Pure Mathematics
540(27)
5.1. Summary of Dirksen's Work
541(17)
5.1.1. Number
542(2)
5.1.2. Series
544(4)
5.1.3. Theory of Functions
548(10)
5.2. Dirichlet's Work on Rigor
558(3)
5.3. The Reception of Pure Mathematics in Textbook Practice
561(6)
VIII. Conflicts Between Confinement to Geometry and Algebraization in France 567(34)
1. Keeping Up the Confinement of Negative Quantities to Geometry
567(7)
2. Last Culmination Points of the Infiniment Petits
574(27)
2.1. Poisson's Universalization of the Infiniment Petits
574(13)
2.2. The Apotheosis of the Dualist Compromise: Duhamel
587(11)
2.2.1. The Principle of Substitution for Infiniment Petits
590(8)
2.3. The End of the Classical Infiniment Petits
598(3)
IX. Summary and Outlook 601(18)
1. Principle of Permanence and Theory of Forms
601(5)
2. On the New Rigor in Analysis
606(3)
3. On the Rise of Modern Infiniment Petits
609(7)
4. Some Closing Remarks
616(3)
Appendix 619(12)
A. The Berlin Contest of 1784 Reassessed
619(1)
B. Carnot's Definitions of Quantités Infiniment Petites
620(4)
Part I
620(1)
Part II
621(1)
Part III
622(1)
Part IV
623(1)
C. Calendar of Cauchy's Traceable Correspondence
624(7)
References 631(40)
Sources
631(1)
Publications
632(39)
Index of Names 671

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