From the Calculus to Set Theory 1630-1910

by Grattan-Guinness, Ivor; Bos, Henk J. M.

• ISBN13: 9780691070827
• ISBN10: 0691070822
• Format: Paperback
• Copyright: 2000-11-20
• Publisher: Princeton Univ Pr

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