A brief history of mathematical thought / Luke Heaton.

Format:

Book

Author/Creator:

Heaton, Luke, author.

Language:

English

Subjects (All):

Mathematics--History.

Mathematics.

Math anxiety.

History.

Genre:

History.

Physical Description:

321 pages ; 22 cm

Place of Publication:

New York, NY : Oxford University Press, [2017]

Summary:

Emblazoned on many advertisements for the wildly popular game of Sudoku are the reassuring words, "no mathematical knowledge required." Anxiety about math plagues many of us, and school memories can still summon intense loathing. In A Brief History of Mathematical Thought, Luke Heaton shows that much of what many think-and fear-about mathematics is misplaced, and to overcome our insecurities we need to understand its history. To help, he offers a lively guide into and through the world of mathematics and mathematicians, one in which patterns and arguments are traced through logic in a language grounded in concrete experience. Heaton reveals how Greek and Roman mathematicians like Pythagoras, Euclid, and Archimedes helped shaped the early logic of mathematics; how the Fibonacci sequence, the rise of algebra, and the invention of calculus are connected; how clocks, coordinates, and logical padlocks work mathematically; and how, in the twentieth century, Alan Turing's revolutionary work on the concept of computation laid the groundwork for the modern world. A Brief History of Mathematical Thought situates mathematics as part of, and essential to, lived experience. Understanding it requires not abstract thought or numbing memorization but an historical imagination and a view to its origins. -- Provided by publisher.

Contents:

1 Beginnings

1.1 Language and Purpose 13

1.2 Human Cognition and the Meaning of Math 16

1.3 Stone Age Rituals and Autonomous Symbols 20

1.4 Making Legible Patterns 24

1.5 The Storage of Facts 28

1.6 Babylon, Egypt and Greece 31

1.7 The Logic of Circles 35

1.8 The Factuality of Math 37

2 From Greece to Rome

2.1 Early Greek Mathematics 42

2.2 Pythagorean Science 44

2.3 Plato and Symmetric Form 48

2.4 Euclidean Geometry 53

2.5 The Euclidean Algorithm 56

2.6 Archimedes 58

2.7 Alexandria in the Age of Rome 61

3 Ratio and Proportion

3.1 Measurement and Counting 64

3.2 Reductio Ad Absurdum 67

3.3 Eudoxus, Dedekind and the Birth of Analysis 70

3.4 Recurring Decimals and Dedekind Cuts 77

3.5 Continued Fractions 81

3.6 Quadratic Equations and the Golden Ratio 84

3.7 Structures of Irrationality 87

3.8 The Fibonacci Sequence 90

4 The Rise of Algebra

4.1 Zero and the Position System 98

4.2 Al-Khwarizmi and the Science of Equations 104

4.3 Algebra and Medieval Europe 107

4.4 Fermat's Little Theorem 112

4.5 How to Make a Mathematical Padlock 114

5 Mechanics and the Calculus

5.1 The Origins of Analysis 119

5.2 Measuring the World 124

5.3 The Age of Clocks 127

5.4 Cartesian Coordinates 129

5.5 Linear Order and the Number Line 134

5.6 Isaac Newton 137

5.7 The Fundamental Theorem of Calculus 141

5.8 From Algebra to Rates of Change 147

6 Leonhard Euler and the Bridges of Konigsberg

6.1 Leonhard Euler 152

6.2 The Bridges of Königsberg 154

6.3 On Drawing a Network 156

6.4 The Platonic Solids Revisited 158

6.5 Poincaré and the Birth of Topology 161

7 Euclid's Fifth and the Reinvention of Geometry

7.1 Measurement and Direction 167

7.2 Non-Euclidean Geometry 173

7.3 The Curvature of Space 175

7.4 The Unity and Multiplicity of Geometry 180

7.5 Symmetry and Groups 186

7.6 The Oddities of Left and Right 193

7.7 The Möbius Strip 196

8 Working with the Infinite

8.1 Blaise Pascal and the Infinite in Math 199

8.2 Reasoning by Recurrence 202

8.3 The Mathematics of the Infinitely Large 205

8.4 Cantor's Pairs 206

8.5 The Diagonal Argument 211

9 The Structures of Logical Form

9.1 The Formal Logic of AND, OR and NOT 215

9.2 Classical Logic and the Excluded Middle 219

9.3 Mechanical Deductions 221

9.4 Quantifiers and Properties 222

9.5 Inputs for Predicate Calculus 225

9.6 Axiomatic Set Theory 227

10 Alan Turing and the Concept of Computation

10.1 From Mechanical Deductions to Programmable Machines 232

10.2 Depicting Calculation 236

10.3 Deterministic Language Games 238

10.4 Church's Thesis 242

10.5 Decision Problems 243

10.6 Figure and Ground 247

10.7 Semi-Decidable Problems 249

11 Kurt Godel and the Power of Polynomials

11.1 Matiyasevich's Theorem 253

11.2 Kurt Gödel 258

11.3 Searching for Solutions 260

11.4 The Incompleteness of Arithmetic 261

11.5 Truth, Proof and Consistency 266

12 Modelling the World

12.1 Science and the Uses of Models 273

12.2 Order and Chaos 277

12.3 Theoretical Biology 280

12.4 Interactions and Dynamical Systems 284

12.5 Holism and Emergent Phenomena 288

13 Lived Experience and the Nature of Facts

13.1 Rules and Reality 292

13.2 The Objectivity of Math 298

13.3 Meaning and Purpose 304.

Notes:

Includes bibliographical references and index.

ISBN:

9780190621766

0190621761

OCLC:

942885049