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Index
Cover Page
Title Page
Copyright Page
Contents
Introduction
Chapter 1: Ancient Western Mathematics
Ancient Mathematical Sources
Mathematics in Ancient Mesopotamia
The Numeral System and Arithmetic Operations
Geometric and Algebraic Problems
Mathematical Astronomy
Mathematics in Ancient Egypt
The Numeral System and Arithmetic Operations
Geometry
Assessment of Egyptian Mathematics
Greek Mathematics
The Development of Pure Mathematics
The Pre-Euclidean Period
The Elements
The Three Classical Problems
Geometry in the 3rd Century BCE
Archimedes
Apollonius
Applied Geometry
Later Trends in Geometry and Arithmetic
Greek Trigonometry and Mensuration
Number Theory
Survival and Influence of Greek Mathematics
Mathematics in the Islamic World (8th–15th Century)
Origins
Mathematics in the 9th Century
Mathematics in the 10th Century
Omar Khayyam
Islamic Mathematics to the 15th Century
Chapter 2: European Mathematics Since the Middle Ages
European Mathematics During the Middle Ages and Renaissance
The Transmission of Greek and Arabic Learning
The Universities
The Renaissance
Mathematics in the 17th and 18th Centuries
The 17th Century
Institutional Background
Numerical Calculation
Analytic Geometry
The Calculus
The 18th Century
Institutional Background
Analysis and Mechanics
History of Analysis
Other Developments
Theory of Equations
Foundations of Geometry
Mathematics in the 19th and 20th Centuries
Projective Geometry
Making the Calculus Rigorous
Fourier Series
Elliptic Functions
The Theory of Numbers
The Theory of Equations
Gauss
Non-Euclidean Geometry
Riemann
Riemann’s Influence
Differential Equations
Linear Algebra
The Foundations of Geometry
The Foundations of Mathematics
Cantor
Mathematical Physics
Algebraic Topology
Developments in Pure Mathematics
Mathematical Physics and the Theory of Groups
Chapter 3: South and East Asian Mathematics
Ancient Traces
Vedic Number Words and Geometry
The Post-Vedic Context
Indian Numerals and the Decimal Place-Value System
The “Classical” Period
The Role of Astronomy and Astrology
Classical Mathematical Literature
The Changing Structure of Mathematical Knowledge
Mahavira and Bhaskara II
Teachers and Learners
The School of Madhava in Kerala
Exchanges with Islamic and Western Mathematics
Mathematics in China
The Textual Sources
The Great Early Period, 1st–7th Centuries
The Nine Chapters
The Commentary of Liu Hui
The “Ten Classics”
Scholarly Revival, 11th–13th Centuries
Theory of Root Extraction and Equations
The Method of the Celestial Unknown
Chinese Remainder Theorem
Fall into Oblivion, 14th–16th Centuries
Mathematics in Japan
The Introduction of Chinese Books
The Elaboration of Chinese Methods
Chapter 4: The Foundations of Mathematics
Ancient Greece to the Enlightenment
Arithmetic or Geometry
Being Versus Becoming
Universals
The Axiomatic Method
Number Systems
The Reexamination of Infinity
Calculus Reopens Foundational Questions
Non-Euclidean Geometries
Elliptic and Hyperbolic Geometries
Riemannian Geometry
Cantor
The Quest for Rigour
Formal Foundations
Set Theoretic Beginnings
Foundational Logic
Impredicative Constructions
Nonconstructive Arguments
Intuitionistic Logic
Other Logics
Formalism
Gödel
Recursive Definitions
Computers and Proof
Category Theory
Abstraction in Mathematics
Isomorphic Structures
Topos Theory
Intuitionistic Type Theories
Internal Language
Gödel and Category Theory
The Search for a Distinguished Model
Boolean Local Topoi
One Distinguished Model or Many Models
Chapter 5: The Philosophy of Mathematics
Mathematical Platonism
Traditional Platonism
Nontraditional Versions
Mathematical Anti-Platonism
Realistic Anti-Platonism
Nominalism
Logicism, Intuitionism, and Formalism
Mathematical Platonism: For and Against
The Fregean Argument for Platonism
The Epistemological Argument Against Platonism
Ongoing Impasse
Glossary
Bibliography
Index
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