Mathematics for the Environment

669 Pages·2011·3.651 MB·English

by  Walter, Martin

MATHEMATICS
FOR THE
ENVIRONMENT

MATHEMATICS
FOR THE
ENVIRONMENT
MARTIN WALTER
University of Colorado
Boulder, Colorado, USA
Chapman & Hall/CRC
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Dedication
I dedicate this book to the memory of Judson “Sandy” Sanderson, to Baby
Lorax, and to the Joy of my life.

Contents
List of Tables xiii
List of Figures xv
Why Did I Write This Book? xvii
Reading, Learning and/or Teaching from This Book xxi
Acknowledgments xxv
I Mathematics Is Connected to Everything 1
1 Earth’s Climate and Some Basic Principles 3
1.1 One of the Greatest Crimes of the 20th Century . . . . . . . 3
1.2 Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Edison’s Algorithm: Listening to Nature’s Feedback . . . . . 5
1.4 Fuzzy Logic, Filters, the Bigger Picture Principle . . . . . . 8
1.5 Consequences of the Crime: Suburbia’s Topology . . . . . . 12
1.6 A Toxic Consequence of the Crime . . . . . . . . . . . . . . . 14
1.7 Hubbert’s Peak and the End of Cheap Oil . . . . . . . . . . 16
1.8 Resource Wars: Oil and Water . . . . . . . . . . . . . . . . . 20
1.9 The CO Greenhouse Law of Svante Arrhenius . . . . . . . . 21
2
2 Economic Instability: Ongoing Causes 31
2.1 Necessary Conditions for Economic Success . . . . . . . . . . 32
2.2 The Mathematical Structure of Ponzi Schemes . . . . . . . . 34
2.3 Dishonest Assessment of Risk . . . . . . . . . . . . . . . . . 37
2.4 One Reason Why Usury Should Again Be Illegal . . . . . . . 42
3 What Is Mathematics? More Basics 45
3.1 The Definition of Mathematics Used in This Book . . . . . . 45
3.2 The Logic of Nature and the Logic of Civilization . . . . . . 47
3.3 Box-Flow Models . . . . . . . . . . . . . . . . . . . . . . . . 51
3.4 Cycles and Scales in Nature and Mathematics . . . . . . . . 55
3.5 The Art of Estimating . . . . . . . . . . . . . . . . . . . . . . 67
v
vi Contents
4 We All Soak in a Synthetic Chemical Soup 73
4.1 Thomas Latimer’s Unfortunate Experience . . . . . . . . . . 73
4.2 What’s in the Synthetic Chemical Soup? . . . . . . . . . . . 75
4.3 Synthetic Flows and Assumptions . . . . . . . . . . . . . . . 79
4.4 The Flow of Information about Synthetic Flows . . . . . . . 82
4.5 You Cannot Do Just One Thing: Two Examples . . . . . . . 90
5 Mathematics: Food, Soil, Water, Air, Free Speech 99
5.1 The “Hour Glass” Industrial Agriculture Machine . . . . . . 99
5.2 Industrial Agriculture Logic vs. the Logic of Life . . . . . . . 103
5.3 Fast Foods, Few Foods, and Fossil Fuels . . . . . . . . . . . . 108
5.4 Genetic Engineering: One Mathematical Perspective . . . . . 113
5.5 Toxic Sludge Is Good for You! . . . . . . . . . . . . . . . . . 125
5.6 Media Concentration . . . . . . . . . . . . . . . . . . . . . . 131
5.7 Oceans: Rising Acidity and Disappearing Life . . . . . . . . 131
5.8 Stocks, Flows and Distributions of Food . . . . . . . . . . . . 138
5.9 My Definition of Food . . . . . . . . . . . . . . . . . . . . . . 143
5.10 Choices: Central vs. Diverse Decision Making . . . . . . . . 143
5.11 Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6 Mathematics and Energy 151
6.1 How Much Solar Energy Is There? . . . . . . . . . . . . . . . 151
6.2 Solar Energy Is There, Do We Know How to Get It? . . . . 154
6.3 Four Falsehoods . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.4 Nuclear Power: Is It Too Cheap to Meter? . . . . . . . . . . 161
6.5 Net Primary Productivity and EcologicalFootprints . . . . . 165
6.6 NPP, Soil, Biofuels, and the Super Grid . . . . . . . . . . . . 166
7 The Brower–Cousteau Model of the Earth 173
7.1 How Heavily Do We Weigh upon the Earth? . . . . . . . . . 173
7.2 Mining and Damming: Massive Rearrangements . . . . . . . 174
7.3 Fish, Forests, Deserts, and Soil: Revisited . . . . . . . . . . . 176
7.4 The Brower–CousteauEarth Model . . . . . . . . . . . . . . 180
8 Fuzzy Logic, Sharp Logic, Frames, and Bigger Pictures 187
8.1 Sharp (Aristotelian) Logic: A Standard Syllogism . . . . . . 187
8.2 Measuring Truth Values: Fuzzy/Measured Logic . . . . . . . 188
8.3 Definitions, Assumptions and the Frame of Debate . . . . . . 193
8.4 Humans in Denial–Nature Cannot be Fooled–Gravity Exists 195
8.5 The Bigger Picture Principle . . . . . . . . . . . . . . . . . . 198
9 The Dunbar Number 203
9.1 The Sustainability Hypothesis: Is It True? . . . . . . . . . . 203
9.2 The Dunbar Number . . . . . . . . . . . . . . . . . . . . . . 204
Contents vii
9.3 Public Relations, Political Power, and the Organization of So-
ciety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
9.4 Political Uses of Fear . . . . . . . . . . . . . . . . . . . . . . 214
9.5 Confronting Fear (and Apathy): Organizing Your Community
for Self-Preservationand Sustainability . . . . . . . . . . . . 217
II Math and Nature: The Nature of Math 223
10 One Pattern Viewed via Geometry and Numbers: Mathese 225
10.1 The Square Numbers of Pythagoras . . . . . . . . . . . . . . 225
10.2 The Language of Mathematics: Mathese . . . . . . . . . . . 228
10.3 AGeneralExpressioninMathese: AFormulaforOddNumbers 228
10.4 An Important Word in Mathese: Σ . . . . . . . . . . . . . . 229
10.5 Sentences in Mathese: Equations with Σ and a Dummy Vari-
able . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
10.6 Induction,Deduction,MathematicalResearch,andMathemat-
ical Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
10.7 What Is a Mathematical Proof? . . . . . . . . . . . . . . . . 232
10.8 What Is a Deductive System? . . . . . . . . . . . . . . . . . 234
10.9 Originalidad es volver al Origen . . . . . . . . . . . . . . . . 234
11 Axioms and Atoms 237
11.1 Molecules and Atoms; the Atomic Number and the Atomic
Mass Number of an Atom . . . . . . . . . . . . . . . . . . . . 237
11.2 Scaling and Our First Two Axioms for Numbers . . . . . . . 239
11.3 Our First Axiom for Numbers . . . . . . . . . . . . . . . . . 240
11.4 Number 1: Its Definition, Properties, Uniqueness . . . . . . . 241
11.5 The Definition of Multiplicative Inverse . . . . . . . . . . . . 242
11.6 Our Second Axiom for Numbers . . . . . . . . . . . . . . . . 243
11.7 If ... , Then ... . Our First Proofs . . . . . . . . . . . . . . 244
11.8 Return to the Problem: How Many Protons in One Gram of
Protons? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
11.9 What Is a Mole? Scaling Up from the Atomic to the Human
Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
12 Five More Axioms for Numbers 255
12.1 Associativity, Identity, and Inverses for + . . . . . . . . . . . 256
12.2 Commutativity of + and ∗ . . . . . . . . . . . . . . . . . . . 257
12.3 Distributivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
13 What Patterns Can Be Deduced in Our Deductive System? 261
13.1 Playing the Mathematics Game . . . . . . . . . . . . . . . . 261
13.2 Rules for Playing the Mathematics Game . . . . . . . . . . . 262
13.3 The Usual Rules for Fractions Are Partof Our Deductive Sys-
tem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264