From the Calculus to Set Theory 1630-1910

Name: From the Calculus to Set Theory 1630-1910 | An Introductory History
Brand: Princeton University Press
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An Introductory History

Ivor Grattan-Guinness(Herausgeber*in)

Princeton University Press

1. Auflage

Erschienen am 6. Oktober 2020

320 Seiten

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978-0-691-21966-0 (ISBN)

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Inhalt

Cover Page
Half-title Page
Title Page
Copyright Page
Contents
Preface to the Princeton Edition
0. Introductions and explanations
0.1. Possible uses of history in mathematical education
0.2. The chapters and their authors
0.3. The book and its readers
0.4. References and bibliography
0.5. Mathematical notations
1. Techniques of the calculus, 1630-1660
1.1. Introduction
1.2. Mathematicians and their society
1.3. Geometrical curves and associated problems
1.4. Algebra and geometry
1.5. Descartes's method of determining the normal, and Hudde's rule
1.6. Roberval's method of tangents
1.7. Fermat's method of maxima and minima
1.8. Fermat's method of tangents
1.9. The method of exhaustion
1.10. Cavalieri's method of indivisibles
1.11. Wallis's method of arithmetic integration
1.12. Other methods of integration
1.13. Concluding remarks
2. Newton, Leibniz and the Leibnizian tradition
2.1. Introduction and biographical summary
2.2. Newton's fluxional calculus
2.3. The principal ideas in Leibniz's discovery
2.4. Leibniz's creation of the calculus
2.5. l'Hopital's textbook version of the differential calculus
2.6. Johann Bernoulli's lectures on integration
2.7. Euler's shaping of analysis
2.8. Two famous problems : the catenary and the brachistochrone
2.9. Rational mechanics
2.10. What was left unsolved : the foundational questions
2.11. Berkeley's fundamental critique of the calculus
2.12. Limits and other attempts to solve the foundational questions
2.13. In conclusion
3. The emergence of mathematical analysis and its foundational progress, 1780-1880
3.1. Mathematical analysis and its relationship to algebra and geometry
3.2. Educational stimuli and national comparisons
3.3. The vibrating string problem
3.4. Late-18th-century views on the foundations of the calculus
3.5. The impact of Fourier series on mathematical analysis
3.6. Cauchy's analysis : limits, infinitesimals and continuity
3.7. On Cauchy's differential calculus
3.8. Cauchy's analysis : convergence of series
3.9. The general convergence problem of Fourier series
3.10. Some advances in the study of series of functions
3.11. The impact of Riemann and Weierstrass
3.12. The importance of the property of uniformity
3.13. The post-Dirichletian theory of functions
3.14. Refinements to proof-methods and to the differential calculus
3.15. Unification and demarcation as twin aids to progress
4. The origins of modern theories of integration
4.1. Introduction
4.2. Fourier analysis and arbitrary functions
4.3. Responses to Fourier, 1821-1854
4.4. Defects of the Riemann integral
4.5. Towards a measure-theoretic formulation of the integral
4.6. What is the measure of a countable set ?
4.7. Conclusion
5. The development of Cantorian set theory
5.1. Introduction
5.2. The trigonometric background : irrational numbers and derived sets
5.3. Non-denumerability of the real numbers, and the problem of dimension
5.4. First trouble with Kronecker
5.5. Descriptive theory of point sets
5.6. The Grundlagen : transfinite ordinal numbers, their definitions and laws
5.7. The continuum hypothesis and the topology of the real line
5.8. Cantor's mental breakdown and non-mathematical interests
5.9. Cantor's method of diagonalisation and the concept of coverings
5.10. The Beitrdge : transfinite alephs and simply ordered sets
5.11. Simply ordered sets and the continuum
5.12. Well-ordered sets and ordinal numbers
5.13. Cantor's formalism and his rejection of infinitesimals
5.14. Conclusion
6. Developments in the foundations of mathematics, 1870-1910
6.1. Introduction
6.2. Dedekind on continuity and the existence of limits
6.3. Dedekind and Frege on natural numbers
6.4. Logical foundations of mathematics
6.5. Direct consistency proofs
6.6. Russell's antinomy
6.7. The foundations of Principia mathematica
6.8. Axiomatic set theory
6.9. The axiom of choice
6.10. Some concluding remarks
Bibliography
Name index
Subject index

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