X Marks the Spot

✍ Scribed by Richard Garfinkle, David Garfinkle

• Publisher: CRC Press
• Year: 2021
• Tongue: English
• Leaves: 483
• Series: AK Peters/CRC Recreational Mathematics Series
• Edition: 1
• Category: Library

✦ Synopsis

X Marks the Spot is written from the point of view of the users of mathematics. Since the beginning, mathematical concepts and techniques (such as arithmetic and geometry) were created as tools with a particular purpose like counting sheep and measuring land areas.

Understanding those purposes leads to a greater understanding of why mathematics developed as it did. Later mathematical concepts came from a process of abstracting and generalizing earlier mathematics. This process of abstraction is very powerful, but often comes at the price of intuition and understanding. This book strives to give a guided tour of the development of various branches of mathematics (and what they’re used for) that will give the reader this intuitive understanding.

Features

• Treats mathematical techniques as tools, and areas of mathematics as the result of abstracting and generalizing earlier mathematical tools
• Written in a relaxed conversational and occasionally humorous style making it easy to follow even when discussing esoterica.
• Unravels how mathematicians think, demystifying math and connecting it to the ways non-mathematicians think and connecting math to people’s lives
• Discusses how math education can be improved in order to prevent future generations from being turned off by math.

✦ Table of Contents

Cover
Half Title
Title Page
Copyright Page
Table of Contents
List of Figures
Preface
Authors
Chapter 1 Why This Book?
Part I The Roots of Mathematics
Chapter 2 Sticks and Stones
The Unnumbered World
Stones: Counting
Sticks: Measuring
Multiplying the Stick
Tricks with Sticks
Rate of Change
Units Multiply Like Rabbits
The volume?
Infinity
The Numbered World
Chapter 3 Abstraction, Mistrust, and Laziness
Abstraction
Mistrust
Premises
Methods of Reasoning in Proofs
Kinds of Proof
Elegance
Laziness
Putting the Three Together
Chapter 4 Algebra, Geometry, Analysis: The Mathematical Mindsets
Algebra
Here it is: Value Abstraction. Let’s bring him out again, x the unknown, Variable Extraordinaire!
Geometry
Sticks Are Lines
When Are Shapes the Same?
Shapes as Tools
Analysis
Building Machines
Function Behavior and Unknown Functions
Logic
Logic in Math
Logic and Meaning
Digging up the Roots
Part II Theory in Practice
Chapter 5 Analytic Geometry
Planes and Space, Numbers by Numbers by Numbers
Function is Shape
Parametric Graphing
The Fault Is Not in Our Stars
Curves and Surfaces
Coordinate Systems
More Dimensions than You Can Shake a Stick at
Chapter 6 Calculus: Motion and Size
The Derivative
Limits
Derivative Redux (Reduced, That Is)
When Can We Differentiate?
Function Approximation and Taylor Series
Vectors
Integral Calculus
The Fundamental Theorem of Calculus
Logarithm and Exponential
The Absence and Presence of Calculus
Chapter 7 The Language of Motion
Linear Digression
Chapter 8 Sound, Notes, and Harmonics
Chapter 9 Probability and Statistics
Cards and Randomness
Combinatorics: Counting Without Counting
Counting Cards
Probability Distributions: Stick the Chances
Stochastic Processes
Statistics
Chapter 10 Other Geometries: Not So Straight, These Sticks
The Universe Is Bent
Shortest Distance Between Two Points
Metric Spaces
Nature Spiky in Coast and Leaf
Why Are They Called Fractals?
Iteration and Chaos
Chapter 11 Algebra and the Rise of Abstraction
Construction and Its Limits
Imaginary and Complex Numbers
Algebraic Structures: Abstraction Spreads Wide Its Arms
Morphisms: Preservatives Added and Multiplied
Algebraic View of Geometry: Slouching Toward Topology
Category Theory
Part III Toolkit of the Theoretical Universe
Chapter 12 The Smith and the Knight
How the User Sees the Tool
The Maker’s View of the Tool
Beauty in the Use of Tools
Beauty in the Making of Tools: Artists and Artisans
Mathematics as Toolkit
Using the Toolkit
Expanding the Toolkit
Chapter 13 Building the Theoretical Universe
Fluids
Electricity and Magnetism
Chapter 14 Computers
Logic Embodied
Binary Arithmetic Embodied
Computer Programming
Modular Programming, Procedures, and Functions
Speed, Memory, Bandwidth, Price, Size, and Efficiency
Human Thought and Computer “Thought”
Computer Use in Math and Science
Chapter 15 The Theoretical Universe of Modern Physics: Toolkit Included
Relativity
General Relativity
Quantum Mechanics
Quantum Field Theory
Chapter 16 Math Education and Math in Education
Welcome to Math Problem World
How to Teach It
Math in Education
Reclaiming the Lost
A Stone’s Throw
Index