A Brief History of Mathematical Thought

Luke Heaton



Advertisements for the wildly popular game of Sudoku often feature the reassuring words, "no mathematical knowledge required." In fact, the only skill Sudoku does require is the use of mathematical logic. For many people, anxiety about math is so entrenched, and grade school memories so haunting, that these disclaimers - though misleading - are necessary to avoid intimidating potential buyers.

In A Brief History of Mathematical Thought, Luke Heaton provides a compulsively readable history that situates mathematics within the human experience and, in the process, makes it more accessible. Mastering math begins with understanding its history. Heaton's book therefore offers a lively guide into and through the world of numbers and equations-one in which patterns and arguments are traced through logic in the language of 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 does not require the application of various rules or numbing memorization, but rather a historical imagination and a view to its origins. Moving from the origin of numbers, into calculus, and through infinity, Heaton sheds light on the language of math and its significance to human life.

Table of Contents

Introduction

1 Beginnings
1.1 Language and Purpose
1.2 Human Cognition and the Meaning of Maths
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 Maths

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 1
5.7 The Fundamental Theorem of Calculus
5.8 From Algebra to Rates of Change

6 Leonhard Euler and the Bridges of Königsberg
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 Maths
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 Gödel 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 Maths
13.3 Meaning and Purpose

Further Reading
Acknowledgements
Index



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