From the Calculus to Set Theory, 1630-1910: An Introductory HistoryI. Grattan-Guinness From the Calculus to Set Theory traces 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. Møller-Pedersen. |
Contents
Introductions and Explanations | 1 |
01 Possible uses of history in mathematical education | 2 |
02 The chapters and their authors | 3 |
03 The book and its readers 1 | 7 |
04 References and bibliography | 8 |
05 Mathematical notations | 9 |
Techniques of the Calculus 1630l660 | 10 |
12 Mathematicians and their society | 12 |
310 Some advances in the study of series of functions | 127 |
311 The impact of Riemann and Weierstrass | 131 |
312 The importance of the property of uniformity | 133 |
313 The postDirichletian theory of functions | 138 |
314 Refinements to proofmethods and to the differential calculus | 141 |
315 Unification and demarcation as twin aids to progress | 145 |
The Origins of Modern Theories of Integration | 149 |
42 Fourier analysis and arbitrary functions | 150 |
13 Geometrical curves and associated problems | 13 |
14 Algebra and geometry | 15 |
15 Descartess method of determining the normal and Huddes rule | 16 |
16 Robervals method of tangents | 20 |
17 Fermats method of maxima and minima | 23 |
18 Fermats method of tangents | 26 |
19 The method of exhaustion | 31 |
110 Cavalieris method of indivisibles | 32 |
111 Walliss method of arithmetic integration | 37 |
112 Other methods of integration | 42 |
113 Concluding remarks | 47 |
Newton Leibniz and the Leibnizian Tradition | 49 |
22 Newtons fluxional calculus | 54 |
23 The principal ideas in Leibnizs discovery | 60 |
24 Leibnizs creation of the calculus | 66 |
25 IHopitaIs textbook version of the differential calculus | 70 |
26 Johann Bernoullis lectures on integration | 73 |
27 Eulers shaping of analysis | 75 |
the catenary and the brachistochrone | 79 |
29 Rational mechanics | 84 |
the foundational questions | 86 |
211 Berkeleys fundamental critique of the calculus | 88 |
212 Limits and other attempts to solve the foundational questions | 90 |
213 In conclusion | 92 |
The Emergence of Mathematical Analysis and its Foundational Progress 1780l880 | 94 |
32 Educational stimuli and national comparisons | 95 |
33 The vibrating string problem | 98 |
34 Late18thcentury views on the foundations of the calculus | 100 |
35 The impact of Fourier series on mathematical analysis | 104 |
limits infinitesimals and continuity | 109 |
37 On Cauchys differential calculus | 111 |
convergence of series | 116 |
39 The general convergence problem of Fourier series | 122 |
43 Responses to Fourier 18211854 | 153 |
44 Defects of the Riemann integral | 159 |
45 Towards a measuretheoretic formulation of the integral | 164 |
46 What is the measure of a countable set ? | 172 |
47 Conclusion | 180 |
The Development of Cantorian Set Theory | 181 |
irrational numbers and derived sets | 182 |
53 Nondenumer ability of the real numbers and the problem of dimension | 185 |
54 First trouble with Kronecker | 188 |
55 Descriptive theory of point sets | 189 |
transfinite ordinal numbers their definitions and laws | 192 |
57 The continuum hypothesis and the topology of the real line | 197 |
58 Cantors mental breakdown and nonmathematical interests | 199 |
59 Cantors method of diagonalisation and the concept of coverings | 203 |
transfinite alephs and simply ordered sets | 206 |
511 Simply ordered sets and the continuum | 210 |
512 Wellordered sets and ordinal numbers | 213 |
513 Cantors formalism and his rejection of infinitesimals | 216 |
514 Conclusion | 219 |
Developments in the Foundations of Mathematics 18701910 | 220 |
62 Dedekind on continuity and the existence of limits | 222 |
63 Dedekind and Frege on natural numbers | 226 |
64 Logical foundations of mathematics | 231 |
65 Direct consistency proofs | 234 |
66 Russells antinomy | 237 |
67 The foundations of Principia mathematica | 240 |
68 Axiomatic set theory | 245 |
69 The axiom of choice | 250 |
610 Some concluding remarks | 255 |
| 256 | |
Name Index | 283 |
Subject Index | 291 |
Other editions - View all
From the Calculus to Set Theory 1630-1910: An Introductory History I. Grattan-Guinness Limited preview - 2020 |
From the Calculus to Set Theory, 1630-1910: An Introductory History I. Grattan-Guinness Limited preview - 2000 |
Common terms and phrases
aleph algebraic antinomies arithmetic axiom of choice Beiträge Bernoulli Borel Cantor cardinal numbers Cauchy Cauchy's century chapter concept considered continuous continuum continuum hypothesis convergence corresponding countable set curve Dedekind defined definition dense set denumerable derived set determined differential calculus Dirichlet discontinuous elements equal equation Euler example existence Fermat figure finite number fluxions formulated foundations Fourier series Frege function f(x geometrical Grattan-Guinness idea infinitely small infinitesimal infinity integral interval introduced Jakob Bernoulli Johann Bernoulli Kronecker Leibniz limit logic mathe mathematical analysis mathematicians maxima and minima measure method natural numbers Newton notation one-one ordinal outer content Papers Peano point sets principles problem proof proved quadrature quantities rational numbers real numbers relation Riemann Russell sequence set theory symbols tangent textbooks theorem tion transfinite numbers trigonometric series uniform convergence variable Weierstrass Zermelo zero