From the Calculus to Set Theory, 1630-1910: An Introductory History

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I. Grattan-Guinness, H. J. M. Bos
Princeton University Press, 2000 - 306 pàgines
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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.

 

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Continguts

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
Bibliography
256
Name Index
283
Subject Index
291
Copyright

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Quant a l’autor (2000)

I. Grattan-Guinness is Professor of the History of Mathematics and Logic at Middlesex University. Founder of the journal History and Philosophy of Logic and past President of the British Society for the History of Mathematics, he has authored or edited numerous books, including The Norton History of Mathematics, Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, and Convolutions in French Mathematics, 1800-1840, and The Search for Mathematical Roots, 1870-1940.

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