A History of Mathematics From Mesopotamia to Modernity

by Hodgkin, Luke

• ISBN13: 9780198529378
• ISBN10: 0198529376
• Format: Hardcover
• Copyright: 2005-08-11
• Publisher: Oxford University Press

Summary

iA History of Mathematics: From Mesopotamia to Modernity/i covers the evolution of mathematics through time and across the major Eastern and Western civilizations. It begins in Babylon, then describes the trials and tribulations of the Greek mathematicians. The important, and often neglected,influence of both Chinese and Islamic mathematics is covered in detail, placing the description of early Western mathematics in a global context. The book concludes with modern mathematics, covering recent developments such as the advent of the computer, chaos theory, topology, mathematical physics,and the solution of Fermat's Last Theorem.Containing more than 100 illustrations and figures, this text, aimed at advanced undergraduates and postgraduates, addresses the methods and challenges associated with studying the history of mathematics. The reader is introduced to the leading figures in the history of mathematics (includingArchimedes, Ptolemy, Qin Jiushao, al-Kashi, al-Khwarizmi, Galileo, Newton, Leibniz, Helmholtz, Hilbert, Alan Turing, and Andrew Wiles) and their fields. An extensive bibliography with cross-references to key texts will provide invaluable resource to students and exercises (with solutions) willstretch the more advanced reader.

Author Biography

Luke Hodgkin has taught mathematics and its history at the universities of Warwick and Algiers, and most recently at King's College, London. He studied mathematics at Balliol College and St. John's College, Oxford, and is a former member of the Institute for Advanced Study, Princeton. He is now a freelance writer and teacher.

Table of Contents

List of figures	xi
Picture Credits	xiv
Introduction	1	(1)
Why this book?	1	(1)
On texts, and on history	2	(3)
Examples	5	(1)
Historicism and `presentism'	6	(2)
Revolutions, paradigms, and all that	8	(2)
External versus internal	10	(2)
Eurocentrism	12	(2)
Babylonian mathematics	14	(19)
On beginnings	14	(3)
Sources and selections	17	(3)
Discussion of the example	20	(1)
The importance of number-writing	21	(3)
Abstraction and uselessness	24	(3)
What went before	27	(3)
Some conclusions	30	(3)
Appendix A. Solution of the quadratic problem	30	(1)
Solutions to exercises	31	(2)
Greeks and `origins'	33	(24)
Plato and the Meno	33	(2)
Literature	35	(1)
An example	36	(3)
The problem of material	39	(3)
The Greek miracle	42	(2)
Two revolutions?	44	(1)
Drowning in the sea of Non-identity	45	(2)
On modernization and reconstruction	47	(2)
On ratios	49	(8)
Appendix A. From the Meno	51	(1)
Appendix B. On pentagons, golden sections, and irrationals	52	(2)
Solutions to exercises	54	(3)
Greeks, practical and theoretical	57	(21)
Introduction, and an example	57	(3)
Archimedes	60	(3)
Heron or Hero	63	(3)
Astronomy, and Ptolemy in particular	66	(3)
On the uncultured Romans	69	(2)
Hypatia	71	(7)
Appendix A. From Heron's Metrics	73	(2)
Appendix B. From Ptolemy's Almagest	75	(1)
Solutions to exercises	76	(2)
Chinese mathematics	78	(23)
Introduction	78	(2)
Sources	80	(1)
An instant history of early China	80	(2)
The Nine Chapters	82	(3)
Counting rods---who needs them?	85	(3)
Matrices	88	(2)
The Song dynasty and Qin Jiushao	90	(5)
On `transfers'---when, and how?	95	(3)
The later period	98	(3)
Solutions to exercises	99	(2)
Islam, neglect and discovery	101	(32)
Introduction	101	(2)
On access to the literature	103	(3)
Two texts	106	(2)
The golden age	108	(2)
Algebra---the origins	110	(5)
Algebra---the next steps	115	(2)
Al-Samaw'al and al-Kashi	117	(6)
The uses of religion	123	(10)
Appendix A. From al-Khwarizmi's algebra	125	(2)
Appendix B. Thabit ibn Qurra	127	(1)
Appendix C. From al-Kashi, The Calculator's Key, book 4, chapter 7	128	(2)
Solutions to exercises	130	(3)
Understanding the `scientific revolution'	133	(28)
Introduction	133	(1)
Literature	134	(1)
Scholastics and scholasticism	135	(3)
Oresme and series	138	(2)
The calculating tradition	140	(3)
Tartaglia and his friends	143	(3)
On authority	146	(3)
Descartes	149	(2)
Infinities	151	(2)
Galileo	153	(8)
Appendix A	155	(1)
Appendix B	156	(1)
Appendix C	157	(1)
Appendix D	158	(1)
Solutions to exercises	159	(2)
The calculus	161	(28)
Introduction	161	(2)
Literature	163	(2)
The priority dispute	165	(2)
The Kerala connection	167	(2)
Newton, an unknown work	169	(3)
Leibniz, a confusing publication	172	(4)
The Principia and its problems	176	(2)
The arrival of the calculus	178	(2)
The calculus in practice	180	(2)
Afterword	182	(7)
Appendix A. Newton	183	(2)
Appendix B. Leibniz	185	(1)
Appendix C. From the Principia	186	(1)
Solutions to exercises	187	(2)
Geometries and space	189	(24)
Introduction	189	(5)
First problem: the postulate	194	(3)
Space and infinity	197	(2)
Spherical geometry	199	(2)
The new geometries	201	(2)
The `time-lag' question	203	(2)
What revolution?	205	(8)
Appendix A. Euclid's proposition I.16	207	(2)
Appendix B. The formulae of spherical and hyperbolic trigonometry	209	(1)
Appendix C. From Helmholtz's 1876 paper	210	(1)
Solutions to exercises	210	(3)
Modernity and its anxieties	213	(22)
Introduction	213	(1)
Literature	214	(1)
New objects in mathematics	214	(3)
Crisis---what crisis?	217	(4)
Hilbert	221	(2)
Topology	223	(5)
Outsiders	228	(7)
Appendix A. The cut definition	231	(1)
Appendix B. Intuitionism	231	(1)
Appendix C. Hilbert's programme	232	(1)
Solutions to exercises	232	(3)
A chaotic end?	235	(25)
Introduction	235	(1)
Literature	236	(2)
The Second World War	238	(2)
Abstraction and `Bourbaki'	240	(3)
The computer	243	(3)
Chaos: the less you know, the more you get	246	(3)
From topology to categories	249	(2)
Physics	251	(3)
Fermat's Last Theorem	254	(6)
Appendix A. From Bourbaki, `Algebra', Introduction	256	(1)
Appendix B. Turing on computable numbers	256	(2)
Solutions to exercises	258	(2)
Conclusion	260	(3)
Bibliography	263	(8)
Index	271