Hodgkin L.H. A History of Mathematics: From Mesopotamia to Modernity

Oxford University Press, 2005. — 294 p. — ISBN 978-0-19-852937-8, 978-0-19-967676-7.A History of Mathematics: From Mesopotamia to Modernity 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 (including Archimedes, Ptolemy, Qin Jiushao, al-Kashi, al-Khwasizmi, 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) will stretch the more advanced reader.Why this book?
On texts, and on history
Examples
Historicism and ‘presentism’
Revolutions, paradigms, and all that
External versus internal
Eurocentrism
Babylonian mathematics
On beginnings
Sources and selections
Discussion of the example
The importance of number-writing
Abstraction and uselessness
What went before
Some conclusions
Appendix
Solution of the quadratic problem
Solutions to exercises
Greeks and ‘origins’
Plato and the Meno
Literature
An example
The problem of material
The Greek miracle
Two revolutions?
Drowning in the sea of Non-identity
On modernization and reconstruction
On ratios
Appendixes
From the Meno
On pentagons, golden sections, and irrationals
Solutions to exercises
Greeks, practical and theoretical
Introduction, and an example
Archimedes
Heron or Hero
Astronomy, and Ptolemy in particular
On the uncultured Romans
Hypatia
Appendixes.
From Heron’s Metrics
From Ptolemy’s Almagest
Solutions to exercises
Chinese mathematicsSources
An instant history of early China
The Nine Chapters
Counting rods—who needs them?
Matrices
The Song dynasty and Qin Jiushao
On ‘transfers’—when, and how?
The later period
Solutions to exercises
5. Islam, neglect and discoveryOn access to the literature
Two texts
The golden age
Algebra—the origins
Algebra—the next steps
Al-Samaw’al and al-Kāshī
The uses of religion
Appendixes.
From al-Khwārizmī’s algebra
Thābit ibn Qurra
From al-Kāshī, The Calculator’s Key, book 4, chapter 7
Solutions to exercises
Understanding the ‘scientific revolution’Literature
Scholastics and scholasticism
Oresme and series
The calculating tradition
Tartaglia and his friends
On authority
Descartes
Infinities
Galileo
Appendixes
Solutions to exercises
The calculusLiterature
The priority dispute
The Kerala connection
Newton, an unknown work
Leibniz, a confusing publication
The Principia and its problems
The arrival of the calculus
The calculus in practice
Afterword
Appendixes
Newton
Leibniz
From the Principia
Solutions to exercises
Geometries and spaceFirst problem: the postulate
Space and infinity
Spherical geometry
The new geometries
The ‘time-lag’ question
What revolution?
Appendixes
Euclid’s proposition I.16
The formulae of spherical and hyperbolic trigonometry
From Helmholtz’s 1876 paper
Solutions to exercises
Modernity and its anxietiesLiterature
New objects in mathematics
Crisis—what crisis?
Hilbert
Topology
Outsiders
Appendixes
The cut definition
Intuitionism
Hilbert’s programme
Solutions to exercises
1A chaotic end?Literature
The Second World War
Abstraction and ‘Bourbaki’
The computer
Chaos: the less you know, the more you get
From topology to categories
Physics
Fermat’s Last Theorem
Appendixes
From Bourbaki, ‘Algebra’, Introduction
Turing on computable numbers
Solutions to exercises