From the Calculus to Set Theory 1630-1910

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.

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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From the Calculus to Set Theory 1630-1910

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