A brief history of mathematical thought

A brief history of mathematical thought

Luke Heaton

Book - 2017

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Subjects
Mathematics > History.
Math anxiety.
Published
New York, NY : Oxford University Press [2017]
Language
English
Main Author
Luke Heaton (-)
Physical Description
321 pages ; 22 cm
Bibliography
Includes bibliographical references and index.
ISBN
9780190621766
  • Introduction
  • 1. Beginnings
  • 1.1. Language and Purpose
  • 1.2. Human Cognition and the Meaning of Math
  • 1.3. Stone Age Rituals and Autonomous Symbols
  • 1.4. Making Legible Patterns
  • 1.5. The Storage of Facts
  • 1.6. Babylon, Egypt and Greece
  • 1.7. The Logic of Circles
  • 1.8. The Factuality of Math
  • 2. From Greece to Rome
  • 2.1. Early Greek Mathematics
  • 2.2. Pythagorean Science
  • 2.3. Plato and Symmetric Form
  • 2.4. Euclidean Geometry
  • 2.5. The Euclidean Algorithm
  • 2.6. Archimedes
  • 2.7. Alexandria in the Age of Rome
  • 3. Ratio and Proportion
  • 3.1. Measurement and Counting
  • 3.2. Reductio Ad Absurdum
  • 3.3. Eudoxus, Dedekind and the Birth of Analysis
  • 3.4. Recurring Decimals and Dedekind Cuts
  • 3.5. Continued Fractions
  • 3.6. Quadratic Equations and the Golden Ratio
  • 3.7. Structures of Irrationality
  • 3.8. The Fibonacci Sequence
  • 4. The Rise of Algebra
  • 4.1. Zero and the Position System
  • 4.2. Al-Khwarizmi and the Science of Equations
  • 4.3. Algebra and Medieval Europe
  • 4.4. Fermat's Little Theorem
  • 4.5. How to Make a Mathematical Padlock
  • 5. Mechanics and the Calculus
  • 5.1. The Origins of Analysis
  • 5.2. Measuring the World
  • 5.3. The Age of Clocks
  • 5.4. Cartesian Coordinates
  • 5.5. Linear Order and the Number Line
  • 5.6. Isaac Newton
  • 5.7. The Fundamental Theorem of Calculus
  • 5.8. From Algebra to Rates of Change
  • 6. Leonhard Euler and the Bridges of Konigsberg
  • 6.1. Leonhard Euler
  • 6.2. The Bridges of Königsberg
  • 6.3. On Drawing a Network
  • 6.4. The Platonic Solids Revisited
  • 6.5. Poincaré and the Birth of Topology
  • 7. Euclid's Fifth and the Reinvention of Geometry
  • 7.1. Measurement and Direction
  • 7.2. Non-Euclidean Geometry
  • 7.3. The Curvature of Space
  • 7.4. The Unity and Multiplicity of Geometry
  • 7.5. Symmetry and Groups
  • 7.6. The Oddities of Left and Right
  • 7.7. The Möbius Strip
  • 8. Working with the Infinite
  • 8.1. Blaise Pascal and the Infinite in Math
  • 8.2. Reasoning by Recurrence
  • 8.3. The Mathematics of the Infinitely Large
  • 8.4. Cantor's Pairs
  • 8.5. The Diagonal Argument
  • 9. The Structures of Logical Form
  • 9.1. The Formal Logic of AND, OR and NOT
  • 9.2. Classical Logic and the Excluded Middle
  • 9.3. Mechanical Deductions
  • 9.4. Quantifiers and Properties
  • 9.5. Inputs for Predicate Calculus
  • 9.6. Axiomatic Set Theory
  • 10. Alan Turing and the Concept of Computation
  • 10.1. From Mechanical Deductions to Programmable Machines
  • 10.2. Depicting Calculation
  • 10.3. Deterministic Language Games
  • 10.4. Church's Thesis
  • 10.5. Decision Problems
  • 10.6. Figure and Ground
  • 10.7. Semi-Decidable Problems
  • 11. Kurt Godel and the Power of Polynomials
  • 11.1. Matiyasevich's Theorem
  • 11.2. Kurt Gödel
  • 11.3. Searching for Solutions
  • 11.4. The Incompleteness of Arithmetic
  • 11.5. Truth, Proof and Consistency
  • 12. Modelling the World
  • 12.1. Science and the Uses of Models
  • 12.2. Order and Chaos
  • 12.3. Theoretical Biology
  • 12.4. Interactions and Dynamical Systems
  • 12.5. Holism and Emergent Phenomena
  • 13. Lived Experience and the Nature of Facts
  • 13.1. Rules and Reality
  • 13.2. The Objectivity of Math
  • 13.3. Meaning and Purpose
  • Further Reading
  • Acknowledgements
  • Index

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