X Marks the Spot: The Lost Inheritance of Mathematics

by Richard Garfinkle, David Garfinkle

Overview

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.

Product Details

ISBN-13:	9780367187040
Publisher:	CRC Press
Publication date:	02/05/2021
Series:	AK Peters/CRC Recreational Mathematics Series
Pages:	482
Product dimensions:	7.00(w) x 10.00(h) x (d)

About the Author

David Garfinkle was born in 1958 and wanted to be a physicist ever since his first year of high school. He got a bachelor's degree from Princeton University in 1980 and a PhD from The University of Chicago in 1985. Since 1991 he has been a physics professor at Oakland University in Michigan.

David is the author of over 100 articles in physics journals. His main areas of research are black holes, spacetime singularities, and gravitational radiation. He performs computer simulations of gravitational collapse to resolve questions about black holes and singularities.

David was named a Fellow of the American Physical Society (APS) with the citation reading "for his numerous contributions to a wide variety of topics in relativity and semiclassical gravity."

David and Richard have written Three Steps to the Universe (U. of Chicago Press, 2008) a book on black holes and dark matter.

For the better part of his early life, Richard Garfinkle thought he wanted to be a mathematician. He went so far as to spend four years studying math at the University of Chicago before discovering that he really wanted to be a writer. He has had several science fiction and fantasy novels published. His first Celestial Matters (Tor 1996) won the Compton Crook award for best first novel. Richard and David have written Three Steps to the Universe (U. of Chicago Press, 2008) a book on black holes and dark matter. For his day job, he programs computers. Richard lives in Chicago with his wife and children.

Table of Contents

List of Figures ix

Preface xvii

Authors xix

1 Why This Book? 1

Part I The Roots of Mathematics

2 Sticks and Stones 9

The Unnumbered World 10

Stones: Counting 11

Sticks: Measuring 17

Multiplying the Stick 23

Tricks with Sticks 27

Rate of Change 32

Units Multiply Like Rabbits 34

The volume? 35

Infinity 37

The Numbered World 47

3 Abstraction, Mistrust, and Laziness 49

Abstraction 49

Mistrust 56

Premises 57

Methods of Reasoning in Proofs 59

Kinds of Proof 62

Elegance 63

Laziness 64

Putting the Three Together 67

4 Algebra, Geometry, Analysis: The Mathematical Mindsets 69

Algebra 72

Here it is: Value Abstraction. Let's bring him out again, x the unknown, Variable Extraordinaire! 75

Geometry 92

Sticks Are Lines 95

When Are Shapes the Same? 102

Shapes as Tools 104

Analysis 115

Building Machines 119

Function Behavior and Unknown Functions 122

Logic 123

Logic in Math 125

Logic and Meaning 128

Digging up the Roots 128

Part II Theory in Practice

5 Analytic Geometry 133

Planes and Space, Numbers by Numbers by Numbers 135

Function is Shape 144

Parametric Graphing 153

The Fault Is Not in Our Stars 158

Curves and Surfaces 162

Coordinate Systems 169

More Dimensions than You Can Shake a Stick at 176

6 Calculus: Motion and Size 183

The Derivative 185

Limits 187

Derivative Redux (Reduced, That Is) 191

When Can We Differentiate? 200

Function Approximation and Taylor Series 202

Vectors 207

Integral Calculus 210

The Fundamental Theorem of Calculus 217

Logarithm and Exponential 224

The Absence and Presence of Calculus 227

7 The Language of Motion 229

Linear Digression 231

8 Sound, Notes, and Harmonics 237

9 Probability and Statistics 247

Cards and Randomness 262

Combinatorics: Counting Without Counting 263

Counting Cards 264

Probability Distributions: Stick the Chances 266

Stochastic Processes 270

Statistics 272

10 Other Geometries: Not So Straight, These Sticks 279

The Universe Is Bent 281

Shortest Distance Between Two Points 288

Metric Spaces 291

Nature Spiky in Coast and Leaf 297

Why Are They Called Fractals? 307

Iteration and Chaos 311

11 Algebra and the Rise of Abstraction 315

Construction and Its Limits 326

Imaginary and Complex Numbers 333

Algebraic Structures: Abstraction Spreads Wide Its Arms 338

Morphisms: Preservatives Added and Multiplied 343

Algebraic View of Geometry: Slouching Toward Topology 345

Category Theory 348

Part III Toolkit of the Theoretical Universe

12 The Smith and the Knight 353

How the User Sees the Tool 354

The Maker's View of the Tool 356

Beauty in the Use of Tools 358

Beauty in the Making of Tools: Artists and Artisans 359

Mathematics as Toolkit 360

Using the Toolkit 362

Expanding the Toolkit 363

13 Building the Theoretical Universe 365

Fluids 365

Electricity and Magnetism 371

14 Computers 379

Logic Embodied 381

Binary Arithmetic Embodied 386

Computer Programming 389

Modular Programming, Procedures, and Functions 400

Speed, Memory, Bandwidth, Price, Size, and Efficiency 407

Human Thought and Computer "Thought" 408

Computer Use in Math and Science 411

15 The Theoretical Universe of Modern Physics: Toolkit Included 415

Relativity 415

General Relativity 427

Quantum Mechanics 432

Quantum Field Theory 441

16 Math Education and Math in Education 445

Welcome to Math Problem World 449

How to Teach It 450

Math in Education 451

Reclaiming the Lost 453

A Stone's Throw 454

Index 455

Editorial Reviews

"Few mathematics books succeed as well as this one in creating a feeling that the author is engaged in discussion with readers. The Garfinkles' focus on "the human heritage of mathematical thinking" offers enrichment, going beyond mere theoretical abstraction. The authors start from basic counting and arithmetic; continue through algebra and geometry; and lead readers to statistics, calculus, topology, and more. This approach is refreshing and should provide readers with insights into how various aspects of mathematics developed. A broad spectrum of expressions and equations appear throughout the text, but the emphasis is not on symbolic manipulation and algorithms. Rather, readers are encouraged to think deeply about the richness that today's mathematics has inherited from the past. The final chapter on education calls for teachers to broaden their traditional classroom approach by incorporating this sense of mathematical heritage. The authors argue persuasively that students retain more from meaningful learning experiences than from temporary memorization, benefit more from a lesson built around interrelated concepts than from a piecemeal presentation of scattered ideas, and succeed by building on past learning rather than mimicking an ever-growing series of unmotivated procedures and tricks. This text demonstrates the value of exploring what originally motivated mathematical ideas, in preference to blindly performing processes."
—Choice

"Deftly organized into three major sections (The Roots of Mathematics; Theory in Practice; Toolkit of the Theoretical Universe), X Marks the Spot: The Lost Inheritance of Mathematics is enhanced for academia and the reader with figures, an informative introduction (Why This Book?), and a seven page Index. An ideal textbook, X Marks the Spot: The Lost Inheritance of Mathematics is very highly recommended for college and university library Mathematics collections and supplemental studies curriculums."
—Midwest Book Review

"The Garfinkles’ book is written in a light-hearted manner without sacrificing mathematical rigor. The authors begin by taking us on a journey of discovering the roots of mathematics and then delving into what is described as its three primary processes, which they playfully dub as abstraction, mistrust, and laziness.
[. . .] X Marks the Spot is a book for anyone who teaches mathematics at any level. This book will be helpful for any course, particularly in developing the introduction portion of a course. The professor or teacher who is looking for practical examples to help answer the eternal question of “When is this stuff ever used?” will find a multitude of examples to help answer such questions. And those interested in the history and philosophy of mathematics would find this a useful text and a welcome addition to their library. The book is well documented and contains over 200 drawings and illustrations within its over 450 pages."
—MAA Reviews

From the Publisher