From the Calculus to Set Theory 1630-1910,9780691070827

From the Calculus to Set Theory 1630-1910

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ISBN13: 9780691070827
ISBN10: 0691070822
Format: Paperback
Pub. Date: 2000-11-20
Publisher(s): Princeton Univ Pr
List Price: $85.60

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Summary

From the Calculus to Set Theorytraces the development of the calculus from the early seventeenth century through its expansion into mathematical analysis to the developments in set theory and the foundations of mathematics in the early twentieth century. It chronicles the work of mathematicians from Descartes and Newton to Russell and Hilbert and many, many others while emphasizing foundational questions and underlining the continuity of developments in higher mathematics. The other contributors to this volume are H. J. M. Bos, R. Bunn, J. W. Dauben, T. W. Hawkins, and K. Moslash;ller-Pedersen.

Table of Contents

Preface to the Princeton Edition viii
Introductions and explanations
I. Grattan-Guinness
Possible uses of history in mathematical education
2(1)
The chapters and their authors
3(4)
The book and its readers
7(1)
References and bibliography
8(1)
Mathematical notations
9(1)
Techniques of the calculus, 1630-1660
Kirsti Moller Pedersen
Introduction
10(2)
Mathematicians and their society
12(1)
Geometrical curves and associated problems
13(2)
Algebra and geometry
15(1)
Descartes's method of determining the normal, and Hudde's rule
16(4)
Roberval's method of tangents
20(3)
Fermat's method of maxima and minima
23(3)
Fermat's method of tangents
26(5)
The method of exhaustion
31(1)
Cavalieri's method of indivisibles
32(5)
Wallis's method of arithmetic integration
37(5)
Other methods of integration
42(5)
Concluding remarks
47(2)
Newton, Leibniz and the Leibnizian tradition
H. J. M. Bos
Introduction and biographical summary
49(5)
Newton's fluxional calculus
54(6)
The principal ideas in Leibniz's discovery
60(6)
Leibniz's creation of the calculus
66(4)
l'Hopital's textbook version of the differential calculus
70(3)
Johann Bernoulli's lectures on integration
73(2)
Euler's shaping of analysis
75(4)
Two famous problems: the catenary and the brachistochrone
79(5)
Rational mechanics
84(2)
What was left unsolved: the foundational questions
86(2)
Berkeley's fundamental critique of the calculus
88(2)
Limits and other attempts to solve the foundational questions
90(2)
In conclusion
92(2)
The emergence of mathematical analysis and its foundational progress, 1780-1880
I. Grattan-Guinness
Mathematical analysis and its relationship to algebra and geometry
94(1)
Educational stimuli and national comparisons
95(3)
The vibrating string problem
98(2)
Late-18th-century views on the foundations of the calculus
100(4)
The impact of Fourier series on mathematical analysis
104(5)
Cauchy's analysis: limits, infinitesimals and continuity
109(2)
On Cauchy's differential calculus
111(5)
Cauchy's analysis: convergence of series
116(6)
The general convergence problem of Fourier series
122(5)
Some advances in the study of series of functions
127(4)
The impact of Riemann and Weierstrass
131(2)
The importance of the property of uniformity
133(5)
The post-Dirichletian theory of functions
138(3)
Refinements to proof-methods and to the differential calculus
141(4)
Unification and demarcation as twin aids to progress
145(4)
The origins of modern theories of integration
Thomas Hawkins
Introduction
149(1)
Fourier analysis and arbitrary functions
150(3)
Responses to Fourier, 1821-1854
153(6)
Defects of the Riemann integral
159(5)
Towards a measure-theoretic formulation of the integral
164(8)
What is the measure of a countable set?
172(8)
Conclusion
180(1)
The development of Cantorian set theory
Joseph W. Dauben
Introduction
181(1)
The trigonometric background: irrational numbers and derived sets
182(3)
Non-denumerability of the real numbers, and the problem of dimension
185(3)
First trouble with Kronecker
188(1)
Descriptive theory of point sets
189(3)
The Grundlagen: transfinite ordinal numbers, their definitions and laws
192(5)
The continuum hypothesis and the topology of the real line
197(2)
Cantor's mental breakdown and non-mathematical interests
199(4)
Cantor's method of diagonalisation and the concept of coverings
203(3)
The Beitrage: transfinite alephs and simply ordered sets
206(4)
Simply ordered sets and the continuum
210(2)
Well-ordered sets and ordinal numbers
212(4)
Cantor's formalism and his rejection of infinitesimals
216(3)
Conclusion
219(1)
Developments in the foundations of mathematics, 1870-1910
R. Bunn
Introduction
220(2)
Dedekind on continuity and the existence of limits
222(4)
Dedekind and Frege on natural numbers
226(5)
Logical foundations of mathematics
231(3)
Direct consistency proofs
234(3)
Russell's antinomy
237(3)
The foundations of Principia mathematica
240(5)
Axiomatic set theory
245(5)
The axiom of choice
250(5)
Some concluding remarks
255(1)
Bibliography 256(27)
Name index 283(8)
Subject index 291

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