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.
Author(s): I. Grattan-Guinness
Publisher: PUP
Year: 1980
Language: English
Pages: 315
Front Matter......Page _003.djvu
Contents......Page _004.djvu
Preface to the Princeton Edition......Page _007.djvu
0.1. Possible uses of history in mathematical education......Page 002.djvu
0.2. The chapters and their authors......Page 003.djvu
0.3. The book and its readers......Page 006.djvu
0.4. References and bibliography......Page 008.djvu
0.5. Mathematical notations......Page 009.djvu
1.1. Introduction......Page 010.djvu
1.2. Mathematicians and their society......Page 012.djvu
1.3. Geometrical curves and associated problems......Page 013.djvu
1.4. Algebra and geometry......Page 015.djvu
1.5. Descartes's method of determining the normal, and Hudde's rule......Page 016.djvu
1.6. Roberval's method of tangents......Page 020.djvu
1.7. Fermat's method of maxima and minima......Page 023.djvu
1.8. Fermat's method of tangents......Page 026.djvu
1.9. The method of exhaustion......Page 031.djvu
1.10. Cavalieri's method of indivisibles......Page 032.djvu
1.11. Wallis's method of arithmetic integration......Page 037.djvu
1.12. Other methods of integration......Page 042.djvu
1.13. Concluding remarks......Page 046.djvu
2.1. Introduction and biographical summary......Page 049.djvu
2.2. Newton's fluxional calculus......Page 054.djvu
2.3. The principal ideas in Leibniz's discovery......Page 060.djvu
2.4. Leibniz's creation of the calculus......Page 066.djvu
2.5. I'Hopital's textbook version of the differential calculus......Page 070.djvu
2.6. Johann Bernoulli's lectures on integration......Page 073.djvu
2.7. Euler's shaping of analysis......Page 075.djvu
2.8. Two famous problems: the catenary and the brachistochrone......Page 079.djvu
2.9. Rational mechanics......Page 084.djvu
2.10. What was left unsolved: the foundational questions......Page 086.djvu
2.11. Berkeley's fundamental critique of the calculus......Page 088.djvu
2.12. Limits and other attempts to solve the foundational questions......Page 090.djvu
2.13. In conclusion......Page 092.djvu
3.1. Mathematical analysis and its relationship to algebra......Page 094.djvu
3.2. Educational stimuli and national comparisons......Page 095.djvu
3.3. The vibrating string problem......Page 098.djvu
3.4. Late-18th-century views on the foundations of the calculus ......Page 100.djvu
3.5. The impact of Fourier series on mathematical analysis......Page 104.djvu
3.6. Cauchy's analysis: limits, infinitesimals and continuity......Page 109.djvu
3.7. On Cauchy's differential calculus......Page 111.djvu
3.8. Cauchy's analysis: convergence of series......Page 116.djvu
3.9. The general convergence problem of Fourier series......Page 122.djvu
3.10. Some advances in the study of series of functions......Page 127.djvu
3.11. The impact of Riemann and Weierstrass......Page 131.djvu
3.12. The importance of the property of uniformity......Page 133.djvu
3.13. The post-Dirichletian theory of functions......Page 138.djvu
3.14. Refinements to proof-methods and to the differential calculus......Page 141.djvu
3.15. Unification and demarcation as twin aids to progress......Page 145.djvu
4.1. Introduction......Page 149.djvu
4.2. Fourier analysis and arbitrary functions......Page 150.djvu
4.3. Responses to Fourier, 1821-1854......Page 153.djvu
4.4. Defects of the Riemann integral......Page 159.djvu
4.5. Towards a measure-theoretic formulation of the integral......Page 164.djvu
4.6. What is the measure of a countable set?......Page 172.djvu
4.7. Conclusion......Page 180.djvu
5.1. Introduction......Page 181.djvu
5.2. The trigonometric background: irrational numbers and derived sets......Page 182.djvu
5.3. Non-denumerability of the real numbers, and the problem of dimension......Page 185.djvu
5.4. First trouble with Kronecker......Page 188.djvu
5.5. Descriptive theory of point sets......Page 189.djvu
5.6. The Grundlagen: transfinite ordinal numbers, their definitions and laws......Page 192.djvu
5.7. The continuum hypothesis and the topology of the real line......Page 197.djvu
5.8. Cantor's mental breakdown and non-mathematical interests......Page 199.djvu
5.9. Cantor's method of diagonalisation and the concept of coverings......Page 203.djvu
5.10. The Beitrage: transfinite alephs and simply ordered sets......Page 206.djvu
5.11. Simply ordered sets and the continuum......Page 210.djvu
5.12. Well-ordered sets and ordinal numbers......Page 213.djvu
5.13. Cantor's formalism and his rejection of infinitesimals......Page 216.djvu
5.14. Conclusion......Page 219.djvu
6.1. Introduction ......Page 220.djvu
6.2. Dedekind on continuity and the existence of limits......Page 222.djvu
6.3. Dedekind and Frege on natural numbers......Page 226.djvu
6.4. Logical foundations of mathematics......Page 231.djvu
6.5. Direct consistency proofs......Page 234.djvu
6.6. Russell's antinomy......Page 237.djvu
6.7.The foundations of Principia mathematica......Page 240.djvu
6.8. Axiomatic set theory......Page 245.djvu
6.9. The axiom of choice......Page 250.djvu
6.10. Some concluding remarks......Page 255.djvu
Bibliography......Page 256.djvu
Name index......Page 283.djvu
Subject index......Page 291.djvu
